CEJM 3(3) 2005 342–397
Diffusion times and stability exponents for nearly integrable analytic systems Pierre Lochak∗, Jean-Pierre Marco† Analyse alg´ebrique, Universit´e Paris VI, UMR 7586, 4 place Jussieu, 75252 Paris, Cedex 05, France
Received 20 December 2004; accepted 30 March 2005 n = {x ∈ Rn | kxk Abstract: For a positive integer n and R > 0, we set BR ∞ < R}. Given R > 1 and n ≥ 4 we construct a sequence of analytic perturbations (Hj ) of the completely 2 n , with unstable orbits for which integrable Hamiltonian h(r) = 12 r12 + · · · 12 rn−1 + rn on Tn × BR we can estimate the time of drift in the action space. These functions Hj are analytic on a n , and setting ε := kh − H k fixed complex neighborhood V of Tn × BR j j C 0 (V ) the time of drift of
these orbits is smaller than exp(c(1/εj )1/2(n−3) ) for a fixed constant c > 0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n − 2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg’s conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift. c Central European Science Journals. All rights reserved.
Keywords: Perturbations, normal forms, small divisors, Arnol’d diffusion MSC (2000): 37J40, 37B, 37D
1
Introduction and main results
The present work is devoted to the optimality of stability exponents for analytic quasiconvex near-integrable Hamiltonian systems, which amounts to the search for an example of an unstable orbit with the highest possible speed of drift. ∗ †
E-mail:
[email protected] E-mail:
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We begin with a short reminder on stability over exponentially long times (as pioneered by N.N.Nekhorhoshev) in the analytic and Gevrey categories and the optimaity problem for the stability exponents. We then state our main instability results in the framework of discrete as well as continuous systems.
1.1 The general problem 1.1.1 Let T = R/Z and An = T ∗ Tn = Tn × Rn for a positive integer n. In this paper we deal with Hamiltonian systems close to an integrable one on the annulus An , of the form H(θ, r) = h(r) + εf (θ, r), which gives rise to the following vector field ˙ θi = ∂ri h(r) + ε∂ri f (θ, r), XH r˙i = −ε∂θi f (θ, r), i = 1, . . . , n.
The canonical coordinates (θ, r) ∈ Tn × Rn are angle-action coordinates for the integrable part h. When ε = 0, the actions ri are first integrals of the system and the motion takes place on the corresponding invariant tori Tn × {r}, all the solutions being quasiperiodic. For a generic real-analytic function h and for any real-analytic perturbation f , Nekhoroshev theorem [29] asserts that all solutions remain stable in action over exponentially long time intervals: there exist positive numbers a and b, depending only on h, such that for each small enough ε > 0 any initial condition (θ0 , r0 ) gives rise to a solution (θ(t), r(t)) which is defined at least for |t| ≤ exp( const ( 1ε )a ) and satisfies kr(t) − r(0)k ≤ const εb in that range. When n = 2 and h is nondegenerate (or simply isoenergetically nondegenerate), the KAM Theorem yields more than Nekhoroshev theorem, since on each energy level the trajectories are confined on or between invariant tori. For n ≥ 3 however, KAM tori do not a priori prevent the projection in action space of a solution from drifting arbitrarily far from its initial location; in this case Nekhoroshev theorem becomes fully relevant. A main question now is to determine how large the stability exponents a and b can be taken in general. This is especially relevant for the first one: the larger a, the longer the time of stability guaranteed by the theorem. As for b, its value controls the closeness of the action variables ot their initial values. 1.1.2 The generic condition imposed by Nekhoroshev on the unperturbed Hamiltonian h is a transversality property called steepness. Here we will confine attention to quasiconvex functions h, which is a simple particular class of steep functions. Recall that a function h is quasiconvex when it has no critical points on its domain, and when there exists m > 0 such that, at any point r of that domain, the inequality D 2 h(r)(v, v) ≥ m kvk2 holds for all vectors v orthogonal to ∇h(r).
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As noticed by the Italian school ([5], [18], [4]), assuming the (quasi) convexity of the unperturbed Hamiltonian h yields refined results from the point of view of finite time stability. The introduction of simultaneous (diophantine) approximation, in conjunction with these remarks were the main ingredients in [22] which was designed to determine the best possible stability exponents a and b. As a result of this paper and minor subsequent improvements, one finds that if h is assumed to be quasiconvex, Nekhoroshev result holds with the exponents. 1 a=b= 2n as proved independently in [24, 25] and [30]; the latter paper actually takes up Nekhoroshev’s original strategy and improves it to reach the above mentioned values. Note that the prediction of these values was part of the problem and comes from heuristic ideas of B.V.Chirikov, as formalized in [22] (see further references in that paper). Moreover, again as predicted by B.V.Chirikov [14] and proved in [22] one can state local results in action space, near resonant surfaces. If m ∈ {1, . . . , n − 1}, a set of m independent linear relations with integer coefficients to be satisfied by the ∂ri h(r) determines a resonant surface of multiplicity m in the action space. Given any ̺ > 0, for the trajectories starting at a distance less than ̺ ε1/2 of such a surface one can take the larger exponents 1 . a=b= 2(n − m) This may be rather surprising at first sight as it shows that resonance enhances the stability of the nearby trajectories, at least over exponentially long times, whereas it is usually thought of as a cause of instability. 1.1.3 The optimality question for the exponents amounts to looking for systems which are arbitrarily close to integrable, admit unstable orbits, i.e. orbits experiencing a drift in action independent of the size of the perturbation (we will let aside the role of the second exponent b), and such that one can prove an asymptotic upper bound for the time of drift which is close as possible to the lower bound exp( const ( 1ε )a ) provided by the stability results. In Arnold’s famous note [1], an example of a three-degree-of-freedom system was proposed in view of exploring the complement of KAM tori in phase space, and instability was obtained from heteroclinic connections between whiskered tori. It is by no means obvious that the diffusion time one obtains in Arnold’s example (or immediate higher-dimensional generalizations) is comparable with the predictions of the stability theory. Again, the first heuristic arguments in this direction are to be found in [14], see also [22]. The first rigorous results on this problem were proved by U. Bessi. Making use of Arnold’s model and a four degrees of freedom variant, he obtained in ([6, 7]) an answer for the optimality of the exponents in the cases n = 3, 4. He succeded in constructing orbits drifting in a time exp( const ( 1ε )1/2 ) for n = 3, and exp( const ( 1ε )1/4 ) for n = 4. These
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orbits pass close enough to a double resonance, thus the exponents cannot be improved for such trajectories; this shows that the exponent 1/2(n−2) for doubly-resonant surfaces is optimal when n = 3 or 4. It seems however difficult to generalize these ideas to higher dimensional systems, essentially due to the lack of a satisfactory higher dimensional analog of the continued fraction theory. 1.1.4 Following new ideas of Herman the framework was enlarged in [28] so as to be able to deal with Gevrey perturbations of integrable systems. Recall that given two real numbers α ≥ 1 and L > 0, and a positive radius R, a C ∞ function ϕ on K = Tn × B ∞ (0, R) ⊂ An is said to be Gevrey-(α, L) on K when kϕkα,L
X L|k|α
∂ k ϕ 0 <∞ := C (K) k!α k∈N2n
(1.1)
with the usual notation for multi-indices of derivation: |k| = k1 +· · ·+k2n , k! = k1 ! · · · k2n !, 2n ∂ k = ∂xk11 · · · ∂xk2n . We denote by Gα,L (K) the Banach algebra formed by such functions. Real-analytic functions are recovered in the special case when α = 1, the number L then indicates the size of a complex domain of analytic extension. For h ∈ Gα,L (B R ) which is quasi-convex and without critical point, it is proved in [28] that Nekhoroshev theorem holds with the exponents a=
1 2(n − m)α
,
b=
1 2(n − m)
,
for orbits passing close enough to resonant surfaces of multiplicity m. In the same paper examples of unstable systems are constructed when the Gevrey exponent α is > 1. In this case the Gevrey class Gα is effectively larger than the space of real analytic functions, as it contains compactly supported functions which gives a lot of flexibility in the construction of examples. The main result of [28] goes as follows: let n ≥ 3 and α > 1, and set 1 a∗ = . 2(n − 2)α Given L > 0 and R > 1, there exist a sequence of functions (fj )j≥0 converging to 0 in the space Gα,L (Tn × B R ) and an increasing sequence of integers (τj )j≥0 such that, for each j ≥ 0, the Hamiltonian system generated by 1 2 Hj (θ, r) = (r12 + · · · + rn−1 ) + rn + fj (θ, r) 2 admits a solution (θ(t), r(t)) defined at least for t ∈ [0, τj ] and for which r1 (0) = 0 and r1 (τj ) = 1. Moreover, there exist positive constants C1 < C2 such that the time of drift τj and the norm εj = kfj kα,L are related by 1 a∗ C2 1 a∗ exp C ≤ τ ≤ exp C , 1 j 2 ε2j εj ε2j εj
C1
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for j ≥ 0. Moreover, one proves that our solution passes through doubly-resonant domains, so the corresponding stability exponent is a = 1/2(n − 2)α. Therefore our result proves the optimality of that local exponent for the Gevrey classes of exponent α > 1.
1.2 Main results of the paper In this paper we are concerned with the same optimality problem in the analytic category, for which we have to introduce new ideas to construct examples. Let d∞ denote the product distance (supnorm) in Cn . We adopt the following notation for complex domains: for ρ > 0, we write Vρ (Tn ) (or simply Vρ ) for the closed neighborhood of width ρ of the real torus Tn in Cn /Zn , that is Vρ = {z ∈ Cn | d∞ (z, Tn ) ≤ ρ}, and for a domain D in Rn , we set Wρ (D) = {z ∈ Cn /ζ n | d∞ (z, D) ≤ ρ}. We write Uρ (D) = Vρ × Wρ (D). We endow the spaces of bounded analytic functions on these domains with its usual C 0 norm. We first state our instability result in the framework of exact symplectic diffeomorphisms. Given a point z ∈ An , we denote by ri (z) the component of rank i of its action variable r. Given a Hamiltonian H, we denote by ΦH the corresponding time-one map (provided it exists). Theorem A (Instability example in the discrete case). Let n be an integer ≥ 3, and set 1 . a∗d = 2(n − 2) Let h(r) = 12 (r12 +· · ·+rn2 ). Then there exist ρ > 0 and a sequence (Ψj )j≥0 of real-analytic exact symplectic diffeomorphisms of An , with analytic continuation to Uρ = Uρ (Rn ), verifying
εj := Ψj − Φh 0 → 0 when j → ∞, C (Uρ )
such that each Ψj admits a wandering point z (j) . Moreover there exists a sequence (κj )j≥0 of positive integers and a constant C > 0 satisfying κj ≤
1 a∗ exp C ε2j εj C
such that κ
r2 (Ψj j (z (j) )) − r2 (z (j) ) ≥ 1 for j ≥ 0. The constant C depends only on n and R. The main part of this paper is devoted to the proof of Theorem A. We then easily deduce the following result from the analytic suspension technique of [21].
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Theorem B (Instability example in the continuous case). Let n ≥ 4 and set a∗c =
1 2(n − 3)
.
Let R > 1. Then there exist ρ > 0, a sequence of analytic functions (fj )j≥0 with analytic continuation to the domain Uρ = Uρ (BR ), and an increasing sequence of integers (τj )j≥0 such that, for each j ≥ 0, the Hamiltonian system generated by 1 2 Hj (θ, r) = (r12 + · · · + rn−1 ) + rn + fj (θ, r) 2 admits a solution (θ(t), r(t)) defined at least for t ∈ [0, τj ] and for which r2 (0) = 0 and r2 (τj ) = 1. Moreover, there exists a positive constant C which depends only on n and R, such that the time of drift τj and the norm εj = kfj kC 0 (Uρ ) are related by τj ≤
1 ∗ ac exp C , 2 εj εj C
j ≥ 0.
As in the Gevrey case the orbits we construct pass very close to double resonant surfaces, so the optimal value for the exponent a∗c would be 1/2(n − 2). We could not reach this value due to some technical difficulties in the construction of our analytic example, but this result is almost optimal, and becomes more and more so when the number of degrees of freedom tends to infinity, which was our original goal. Nevertheless we think that an improved construction could yield the correct exponent, as well as unstable orbits which are close to simple resonances. But these methods will contain technical refinements which can obscure the underlying ideas, so we see the present work as a first significant step in the direction of optimality in the analytic category, as well as a basis for further work.
1.3 Description of the method and general comments The proof splits into two main parts. The first one (Sections 2 and 3) gathers the dynamical constructions: roughly speaking the whole method relies on an embedding of a two-dimensional normally hyperbolic annulus with homoclinic connections into a higher dimensional near-integrable system, which enables us to obtain the drifting orbits by means of a semi-local analysis combining the dynamics near the annulus with heteroclinic excursions. In order to perform an accurate enough analysis of the local dynamics in the neighborhood of the annulus it is necessary to conjugate our system to a direct product. This kind of result, close in spirit to Sternberg’s conjugacy theorem, is in large part classical but we could not find in the literature a version that would suit our needs. The second part of the proof (Section 4) is devoted to the extension of Sternberg’s theorem to our normally hyperbolic and symplectic framework.
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For the convenience of the reader we describe our constructions a little bit more in the present section. We take the opportunity to point out the similarities between our method and that introduced by J. Bourgain and V. Kaloshin in [8, 9], which nevertheless yields qualitatively very different results. We conclude the section with a discussion of the respective scopes of the two approaches. 1.3.1 The dynamical constructions The construction of our examples is reminiscent of that of [28], with which it presents some similarities: the major part of the work consists in producing discrete systems with wandering points (first theorem), and we then recover the continuous setting (second theorem) thanks to an analytic suspension process. Also, we “embed” here low dimensional diffeomorphisms with controlled dynamics essentially the existence of wandering points with estimates on the speed of drift into high-dimensional near integrable ones, and deduce the existence of instability in these systems from that of the wandering points in the low dimensional ones. But here, due to analytic rigidity, we are led to vary the geometry of our diffeomorphisms. While in [28] our low dimensional systems were particular standard maps on the two-dimensional annulus A, here we have to make use of suitable discrete systems defined on the annulus A2 . The construction of these diffeomorphisms on A2 is indeed the main part of the present work. The main point is that in [28] the “embeddings” of the standard map were made possible by the existence of compactly supported functions in the Gevrey category. Here, we can only obtain approximate embeddings and it is necessary to introduce perturbative techniques in order to keep control of the orbits of the wandering points. This is precisely the reason why we first construct intermediate systems on the annulus 2 A , in the the same way as in [27], into which we are able to embed approximate standard maps defined on A. These systems on A2 are analytic perturbations of the time-one map of the (hyperbolic) Hamiltonian K(θ, r) = 21 (r12 + r22 ) + cos 2πθ1 on A2 , i.e. the product of a pendulum and an oscillator. These perturbations still admit the annulus A = (0, 0) × A (that is the product of the hyperbolic point of the pendulum with the second factor) as a normally hyperbolic invariant manifold. In [27] we proved that A admits a homoclinic two-dimensional annulus, and (even if we will not make use of such an elaborate construction) it is possible to prove the existence of a family of two-dimensional annuli which are invariant under the q th -iterates of the system, for each q large enough. The perturbation is chosen in such a way that the induced dynamics on these invariant annuli uniformly approximate suitable standard maps with wandering points, which we see this way as approximately embedded in our system. Indeed, it will not be necessary to perform such a refined dynamical analysis. Our construction may also be viewed as a discrete version of Arnold’s example in which every quantity is (almost) explicitly computable. The invariant annulus A is foliated by invari-
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ant circles, and the effect of the perturbation is to create heteroclinic connections between their invariant manifolds. Our strategy will be to use Easton’s windowing technique to detect and localize the drifting points located in the neighborhood of these heteroclinic intersections. One could check a posteriori that the drifting points coincide with those of the embedded standard map. Let us describe more precisely our method. One main remark is that the wandering points have to stay most of the time in a small neighborhood of the invariant annulus A. Therefore we can expect to control a large number of their iterates as soon as a precise knowledge of the dynamics near the hyperbolic manifold is possible. To this end we derived a new version of the Sternberg conjugacy theorem, adapted to the case of noncompact normally hyperbolic manifolds in analytic systems, based on Moser’s deformation method as described in [2]. This way our system appears to be locally conjugate to the product of a neighborhood of the hyperbolic fixed point of the pendulum map with the harmonic oscillator. This brings us back to the case of compactly supported functions much as in the Gevrey category (see Lemma 2.5). Next, in order to evaluate the drift along one action axis, we introduce as in [26] a method based on the shadowing lemma of Easton, which consists in constructing small boxes localized very near the heteroclinic points and enjoying suitable intersections properties under the effect of the diffeomorphism. Here the use of that method is facilitated by the almost product structure of our system and the final situation is very similar to that of [28]. Moreover, windowing is robust enough so that we can include the remainders originating from Sternberg’s conjugacy (see Lemma 2.9), which enables us to “shadow the boxes” in the final system. Since these boxes may be chosen regularly spaced along one of the action coordinate axes, we finally easily obtain our drifting orbits and control their drifting time. In conclusion, we wish to point out that our system is also very close to an antiintegrable limit, and can be seen as an example of the methods developed by D. Treschev in [35], which could probably apply in our context to simplify the windowing control. Another remark is that more general examples could certainly be obtained using the preparation method developed in [17]. We hope to get back to that question in a subsequent paper. 1.3.2 The conjugacy theorem The conjugacy result we alluded to above deals with analytic diffeomorphisms, in a symplectic setting and along a normally hyperbolic non compact invariant submanifold. This prompted us to develop a tailor-made version of the theorem we are interested in and in so doing we were led to some observations which may be of independent interest. We hope to return to these points elsewhere, in more detail and in a more general setting. Let us be more precise. Let f0 , f1 be two symplectic diffeomorphisms of some symplectic manifold V , which preserve the submanifold M ⊂ V and are normally hyperbolic along M. All these data, namely V , M, f0 , f1 are assumed to be analytic. We wish to show that if f0 and f1 have a contact of large enough order along M, they are C ℓ
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conjugate in a neighborhood of M, for an integer ℓ ≥ 1 which we will compute. Such results originate in [32] for (germs of) diffeomorphisms of the Euclidean space near the origin, which is assumed to be a hyperbolic fixed point. In short such germs are conjugate if and only if they are formally conjugate, a phenomenon which elaborates on considerations first made by Poincar´e (see [31, 32]). For all the metamorphoses, the modern proofs are still quite close to the original one by S.Sternberg. In particular, in [10] such results are proved in a more modern and precise fashion, namely using –by now classical– fixed point theorems in Banach spaces. The symplectic setting is only briefly mentioned at the end of the book. In [13], [11, 12] (see additional references in these papers) it is shown how such conjugacy results can in principle be reduced to general invariant submanifold theorems although that reduction may not be concretely so easy or effective; it is advocated that this more abstract framework should make it possible to derive more general results. Here we will follow the strategy developed in [2], which connects this circle of problems with two classical and well-established techniques, namely the deformation method and the various theories of “normal forms,” the germs of which can be found (as usual) in Poincar´e. We refer to the clear and thorough discussion in [2] for more detail. This in particular enables one to easily incorporate the various types of geometry in the discussion; in [2] four kinds of geometry are discussed, namely general (no invariant), symplectic, volume preserving and contact diffeomorphisms (see also [3] for this last type). One also easily incorporates the continuous setting, i.e. replaces diffeomorphisms with flows. We insist that we start here from analytic data, with an invariant submanifold M which is not reduced to a point, as is the case in all papers we have mentioned so far. The output, namely the local conjugacy, is only finitely differentiable but the analyticity of the data will help to simplify the proof and it leads to interesting specific and perhaps surprising phenomena in terms of regularity properties along the invariant submanifold M (see §§4.3, 4.4). 1.3.3 Asymptotics and high dimensional diffusion To begin with let us remark that our problem is to find asymptotic estimates when the size of the perturbation tends to 0. Our construction here may be viewed as lying between the method developed in [28] and the original Arnold mechanism. Indeed, as is proved in [27], the hyperbolic annulus described aboved admits a continuous foliation by invariant circles, such that two nearby circles possess heteroclinic intersections. It is therefore possible to extract a “transition chain” from that family, and usual results ([26]) prove the existence of drifting points along such a chain. Our main difficulty was to explicitely determine the time of drift from the data, which necessitates a very precise control of all the parameters and makes it necessary to use a Sternberg type conjugacy result as explained above. The approach in [9] involves similar constructions, while the purpose is not the same: the authors produce examples of perturbations of a given completely integrable system which admit unstable orbits whose drifting time is linear with respect to the inverse of the
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perturbation. Due to the classical exponential normal form theory and stability estimates this cannot be an asymptotic mechanism. Instead, the perturbation is assumed to be not too small, typically ε ≥ exp −d, where d is the dimension of the phase space. It is then only when d → ∞ that the system can be considered as a genuine perturbation of an integrable one. While the dynamical constructions are similar, there are two main differences between these two approaches, actually aiming at different goals. The first one is that here we limit ourselves to a simple example of standard map in order to produce unstable orbits, whereas the use of Mather’s theory in [9] makes it possible to use more general examples, and as a consequence to extend the validity of the method to broader classes of unperturbed systems. The second one is that thanks to the short time needed for the orbits to drift one can first construct the perturbations in the C ∞ category and then use smoothing results in order to restore analyticity. Due to the much longer drifting times involved in the present paper (which are again unavoidable because of the stability theorems in the perturbative framework) we cannot use this more direct path, and this is precisely the reason why we had to develop the conjugacy results presented and used in this paper. We believe that a slight modification of our systems would make it possible to exhibit analytic examples of high dimensional diffusion which belong to the class constructed in [9], but we have not pursued the matter further. Finally we remark that we also could have chosen a non-convex unperturbed Hamiltonian h, of the form 2 h(r) = 12 (ℓ1 r12 + · · · + ℓn−1 rn−1 ) + ℓn rn
with (ℓ1 , · · · , ℓn ) ∈ {−1, 1}n . The necessary modifications are almost obvious, we refer to [27] for details. One should however beware of the fact that when the quadratic form is not definite (i.e. the ℓi ’s are not all equal), one can very easily construct a perturbation of h of size ε for which the action variables experience a drift with average speed ε along the isotropic planes of h (the stability theorems do not apply there). Acknowledgments: It is a pleasure to thank H.S.Dumas for a careful reading of a first version of the manuscript.
2
Speed of drift for diffeomorphisms on A2
The family (Fq )q∈N of diffeomorphisms we consider in this section was introduced in [27], to which we refer for a detailed study. When q → ∞ the maps Fq are analytic perturbations of the time-one map 1
2
2
1 2
1 2
F∗ = Φ 2 (r1 +r2 )+cos 2πθ1 = Φ 2 r1 +cos 2πθ1 × Φ 2 r2
(2.2)
so we call it an initially hyperbolic (or a priori unstable) family. Here we first briefly recall the main properties of the maps Fq , namely the existence of a normally hyperbolic
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manifold that admits an invariant foliation by invariant circles, from which one can extract a transition chain (that is a discrete subfamily of heteroclinically connected circles with minimal rotation). We then make use of that chain to construct drifting points by a windowing method due to Easton [16]. For each q large enough, we construct a countable family of small balls (B(q,k) )k∈Z (the images of the windows) located very near the heteroclinic points, such that (Fq )q (B(q,k) ) intersects B(q,k+1) in a convenient way, which will be described below. By Easton’s shadowing lemma, this yields the existence of a point ζ (q) such that the iterate (Fq )kq (ζ (q) ) belongs to B(q,k) for each integer k ∈ Z. The distance between two consecutive balls B (q,k) and B (q,k+1) is very close to 1/q, and as a consequence the number of iterates needed to make the point ζ (q) drift over an interval of length 1 is approximately q 2 ; this will enable us to estimate the time of instability as a function of the size of the perturbation in the next section.
2.1 The diffeomorphisms Fq This paragraph is devoted to a brief description of the form and geometric structure of the maps Fq . In the following we fix a positive real number σ, and we measure the C 0 -norms of our various functions over the domain Uσ (R2 ) (see the definition at the beginning of Section 1.2). The width σ will have to be chosen small enough below, in order to simplify some technical estimates. 2.1.1 We obtain the diffeomorphisms Fq by composing F∗ with the time-one map of a small Hamiltonian function. For q ≥ 1, we set 1
Fq = Φ q f
(q)
◦ F∗
(2.3)
where the function f (q) depends only on the angles θ1 and θ2 and has the product form (q) f (q) (θ1 , θ2 ) = f1 (θ1 )f2 (θ2 ), with (q) f1 (θ1 ) = (sin πθ1 )ν(q;σ) , f2 (θ2 ) = − π1 2 + sin 2π θ2 + 16 . (2.4) (q)
The exponent ν(q; σ) in the function f1 ν(q; σ) = 2
plays a crucial role in the construction. We set
h Log q 4πσ
i +1 ,
q ≥ qσ ,
(2.5)
where [x] denotes the integer part of the real number x, and where qσ is the smaller positive integer such that ν(qσ ; σ) = 2, so ν(q; σ) ≥ 2 for q ≥ qσ . Note that since ν(q; σ) is even f (q) is a well-defined 1-periodic function. (q) Remark that the function f1 has a contact of order ν(q; σ) with 0 at the point θ1 = 0, 1 (q) and that the perturbative diffeomorphism Φ q f admits the following explicit expression : 1 (q) (q) (q) Φ q f (θ1 , r1 ), (θ2 , r2 ) = θ1 , r1 − 1q f2 (θ2 )(f1 )′ (θ1 ), θ2 , r2 − 1q f1 (θ1 )f2′ (θ2 ) (2.6)
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from which one immediately deduces that the diffeomorphisms F∗ and Fq have a contact of order ν(q; σ) − 1 ≥ 1 along the submanifold of equation θ1 = 0. 2.1.2 Let us briefly depict the main invariant hyperbolic objects of the maps Fq . First consider the system F∗ , the hyperbolic properties of which come from those of the pendulum map ΦP . We denote by O = (0, 0) the hyperbolic fixed point of ΦP , and focus on the upper part of its homoclinic loop, that is the curve of equation r1 = 2 |sin πθ1 |. With a slight abuse of notation, we write W + (O, ΦP ) = W − (O, ΦP ) for that upper separatrix. For the product map F∗ , the annulus A = {O} × A is a normally hyperbolic invariant manifold, which is obviously symplectic for the canonical structure of A2 , being identified with the one-dimensional annulus A by means of the coordinates (θ2 , r2 ). In the following we are interested only in the part of its invariant manifolds corresponding to the upper separatrix of O, and we write W ± (A, F∗) = W ± (O, ΦP ) × A for these stable and unstable manifolds, which obviously coincide. On the invariant annulus A itself, with the previous identification, the restriction of the map F∗ is the integrable twist map (θ2 , r2 ) 7→ (θ2 + r2 , r2 ). The circles Cr20 = {O} × (T × {r20 }),
r20 ∈ R,
are therefore invariant and partially hyperbolic for F∗ . As above, we consider only the part of their invariant manifolds corresponding to the upper separatrix of the pendulum map, and we set W ± (Cr20 , F∗) = W ± (O, ΦP ) × (T × {r20 }). Again, they obviously coincide. As for the perturbed diffeomorphisms Fq , the contact of Fq with F∗ along {θ1 = 0} shows that the annulus A = {O} × A is still invariant and normally hyperbolic, and that the restriction of Fq to A coincides with that of F∗ . Therefore, the circles Cr20 are invariant and partially hyperbolic for Fq . The stable and unstable manifolds of A for Fq are tangent along A to those obtained for F∗ ; we denote by W ± (A, Fq ) the parts of these manifolds which are tangent to the manifolds W ± (A, F∗) defined above, and we define the invariant manifolds W ± (Cr20 , Fq ) in the same way. (q)
2.1.3 We now add some comments on the functions f1 and f2 . First note that the (q) (q) ν(q;σ) function f1 satisfies the inequality f1 (θ1 ) ≤ (πδ) for θ1 ≤ δ. We will apply this estimate in small neighborhoods of 0. To be more precise, given a ∈ ]0, 1[ and a positive integer p, one easily checks that aν(q;σ) = o (1/q p )
q→∞
when
(2.7)
provided that the width satisfies the inequality σ < σp =
|Log a| 4πp
.
(2.8)
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f1
−f2′
1
2 θ2
0 -2 0
1 2
1
θ1 (q)
Fig. 1 Graphs of the functions f1
and −f2′
It is therefore possible to think of the term aν(q;σ) as “exponentially decreasing to 0” when q → ∞, implicitly reducing the width σ as much as necessary to obtain the correct decreasing rate. This proves very useful in the following constructions. (q) (q) The other important feature of f1 is the constant value f1 ( 21 ) = 1, for all q ≥ qσ . (q) Moreover, one sees that the function f 1 converges to the constant 1 uniformly when (q) q → ∞ on intervals of the form θ1 − 21 < 1/ν(q; σ). The derivatives of f1 can also be uniformly estimated on such intervals. (q) Roughly speaking, the behaviour of the function f1 near the origin enables us to control the local invariant manifolds in the perturbed system Fq and keep them very close to those of F∗ , while the behaviour near the point θ1 = 21 is the main ingredient for creating a transverse intersection of W + (A, Fq ) and W − (A, Fq ) in the neighborhood of {θ1 = 12 }. We proved indeed in [27] that the intersection W + (A, Fq ) ∩ W − (A, Fq ) contains a two-dimensional annulus Iµ , which itself contains all the interesting homoclinic and heteroclinic objects of our system. As for the function f2 , apart from the obvious inequality |f2 | ≥ π1 (in particular f2 does not vanish on T), we will be mainly interested in the properties of the derivative f2′ , the zeroes of which correspond to homoclinic points, and of the second derivative f2′′ , which provides us with lower estimates for the splitting in the θ2 -direction. The additional property f2′ (0) = −1 allows us to produce and localize heteroclinic points in a very simple way. Finally, observe that Fq is indeed an analytic perturbation of F∗ , with the following inequality
1 (q)
f 0 ≤ kf2 kC 0 (Vσ ) √1q , (2.9) q C (Vσ )
which shows that Fq → F∗ when q → ∞ with respect to the C 0 analytic topology on Uσ (R2 ). 2.1.4 For each integer q ≥ qσ we will focus on the sequence of invariant circles (Ck/q )k∈Z . We proved in [27] that for all k ∈ Z, there exists a heteroclinic point ζ (q,k) which satisfies ζ (q,k) ∈ W − (Ck/q , Fq ) ∩ W + (C(k+1)/q , Fq ) and which is moreover located very near the point ̟ (q,k) defined by the coordinates ̟ (q,k) : (θ1 = 21 , r1 = 2, θ2 = 0, r2 = (k + 1)/q).
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r2
355
W + (Cr20 +µ , Fµ ) W + (Cr20 , Fµ) W − (Cr20 , Fµ) θ2
Fig. 2 Homoclinic and heteroclinic intersections in the annulus Iµ
To be more precise, there exists a constant d ∈ ]0, 1[ such that ζ (q,k) − ̟ (q,k) ≤ d ν(q;σ) , for q large enough and k ∈ Z. Our drifting points ζ (q) for Fq will be constructed in such a way that their orbits will pass successively extremely close to each of the heteroclinic points. The main result of this section is the following proposition. Proposition 2.1. There exists a width σ, an integer q and a constant d ∈ ]0, 1[ such that for each integer q ≥ q the diffeomorphism Fq admits a wandering point ζ (q) which satisfies
kq (q)
F (ζ ) − ̟ (q,k) ≤ d ν(q;σ) , ∀k ∈ Z. (2.10) q The remainder of this section is devoted to the proof of Proposition 2.1, which will rely on several technical lemmas.
2.2 Windows We now recall the definition and main properties of windows, following Easton [16]. Let M be a C 1 manifold of dimension d ≥ 2, and let dh , dv be two positive integers such that dh + dv = d. A (dh , dv )-window with values in M is a C 1 diffeomorphism of [−1, 1]d into M. If D is such a window, its horizontals are the partial maps D(., yv ) for yv ∈ [−1, 1]dv , and its verticals are the partial maps D(yh , .) for yh ∈ [−1, 1]dh . We denote by Ce the image of the window C. Let C and D be two (dh , dv )-windows with values in M. One says that C is aligned with D when for each yh ∈ [−1, 1]dh and yv ∈ [−1, 1]dv the vertical C(yh , .) and the horizontal D(., yv ) are transverse, and their images intersect at a unique point a = C(yh , xv ) = D(xh , yv ) which satisfies xh ∈ ] − 1, 1[dh and xv ∈ ] − 1, 1[dv . Let us examine the simple example of affine windows, which will be of interest later. Consider the two (dh , dv )-windows with values in Rd defined by C(x) = c + C x
and
D(x) = d + D x
where c, d are two points of Rd and C, D are two linear maps of Rd , that we identify with
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their matrices in the canonical basis. These admit the following block decomposition: C1 C3 D1 D3 C= D= , . C2 C4 D2 D4 Define the intermediate matrices associated with the pair (C, D) by −D1 C3 −C1 D3 M[C, D] := N[C, D] := , . −D2 C4 −C2 D4 We denote by k k∞ the product norm in Rd , and we equip the various spaces of linear maps with the induced norm. Then one easily checks that a necessary and sufficient condition for the window C to be aligned with the window D is that the matrix M is invertible, and that moreover the following inequality
χ[C, D] := Sup y∈[−1,1]d M −1 (d − c) + M −1 Ny ∞ < 1
(2.11)
is satisfied. The previous intersection points then all satisfy kak∞ ≤ χ[C, D]. We call µ(C, D) = (M[C, D])−1 and χ[C, D] the alignment parameters of the pair (C, D) of affine windows. For the sake of completeness we state the following easy lemma. Lemma 2.2. Let d1 and d2 be two integers ≥ 2 and consider the affine windows Ci (x) = ci + Ci x,
D i = d i = Di x
with values in Rdi for i ∈ {1, 2}. Assume that Ci is aligned with Di , with parameters µi and χi . Then the product window C = C1 × C2 is aligned with the product window D = D1 × D2 , with parameters µ[C, D] = Max (µ1 , µ2 )
and
χ[C, D] = Max (χ1 , χ2 ).
The following shadowing lemma was proved by Easton in [16], it will be a main ingredient for the construction of our drifting points. Lemma 2.3. Let Φ be a C 1 -diffeomorphism of a manifold M. Assume that there exists a sequence (Dk )k∈Z of (dh , dv )-windows with values in M, such that for each k ∈ Z the window Dk is aligned with the window Dk+1 . Then there exists a point z0 such that Φk (z0 ) ek of the window Dk , for each k ∈ Z. is contained in the image D Observe that if the sequence Dk satisfies the assumptions of Easton’s lemma for the diffeomorphism Φ, then it is also the case for a small enough C 1 -perturbation of Φ. One has indeed to consider only the alignment problem for any two consecutive windows of
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the sequence, which can be done using the next lemma, introduced in [26]. When C is a C 1 function from an open set V ⊂ Rd to Rd , we set
C 1 = Sup kC(x)k + Sup kDC(x)k . C (V )
x∈V
∞
x∈V
Lemma 2.4. Let C and D be two (dh , dv )-windows with values in Rd , of the following form b b C(x) = c + C x + C(x), D(x) = d + D x + D(x),
b D b two maps of class where c, d are two points of Rd , C, D two linear maps of Rd and C, C 2 from a neighborhood V of [−1, 1]d to Rd . Let C a and D a be the affine windows defined by C a (x) = c + C x and D a (x) = d + D x. Assume that the window C a is aligned with D a , with alignment parameters µ and χ,
b 1 ). Assume moreover that and set χ′ = Max ( Cb C 1 (V ) , D C (V ) 4µχ′ < 1
and
χ+
4µχ′
1 − 4µχ′
< 1.
(2.12)
Then the window C is aligned with the window D. Proof. We first prove that the verticals of C intersect the horizontals of D. Given (yh , yv ) in [−1, 1]d we denote by (xah , xav ) the unique point satisfying C a (yh , xav ) = D a (xah , yv ), and we search for solutions (xh , xv ) of the full system C(yh , xv ) = D(xh , yv ) of the form (xh , xv ) = (xah , xav ) + (zh , zv ). One easily checks that a necessary and sufficient condition for (xh , xv ) to be a solution is that z = (zh , zv ) be a solution of the equation z = F (z), where −1 b a a b F (z) = M D(xh + zh , yv ) − C(yh , xv + zv ) .
Moreover one checks that kF kC 1 (V ) ≤ 4µχ′ , so F is a contracting map by condition (2.12), and F sends the ball B ∞ (0, r) into the ball B ∞ (0, 4µχ′(1 + r)). So Banach fixed point theorem applies in the ball B ∞ (0, r) if one sets r=
4µχ′ 1 − 4µχ′
and proves the existence of a unique solution of z = F (z) in the ball of radius r. In order to ensure that the final point (xah , xav ) + (zh , zv ) belongs to the open ball ] − 1, 1[d one only needs to assume that χ + r < 1, which is exactly the second part of condition (2.12). c(x) = M[D C(x), b b As for transversality, for x ∈ [−1, 1]d consider the matrix M D D(x)]. c(x) is invertible for each solution x of the We have to check that the matrix M + M c c(x)) with previous intersection equation, which is plain since M + M(x) = M(Id + M −1 M
M −1 M c(x) ≤ 2µχ′ < 1.
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2.2.1 Proof of Proposition 2.1. It will be an easy corollary of a lemma on the existence of windows which we are now in a position to state. Lemma 2.5. There exists a width σ and an integer q such that for each q ≥ q there exists a sequence (D (q,k))k∈Z of (2, 2)-windows with values in A2 , such for each k ∈ Z the composed window C (q,k) = Fqq ◦ D (q,k) is aligned with D (q,k) . Moreover there exists d ∈]0, 1[ ˜ (q,k) is contained in the ball centered at such that for each q ≥ q and k ∈ Z the image D ν(q;σ) ̟ (q,k) of radius d . Proposition 2.1 is an immediate consequence of Lemma 2.5 and Lemma 2.3. Indeed, the former applied to the sequence (D (q,k) )k∈Z and the diffeomorphism Fqq yields the e (q,k) for each k ∈ Z, which by the last existence of a point ζ (q) satisfying Fqq (ζ (q) ) ∈ D assertion of Lemma 2.5 in turn implies that Fqq (ζ (q) ) is in the ball centered at ̟k of radius d
ν(q;σ)
.
The rest of the section is devoted to the proof of Lemma 2.5. We will first introduce a sequence (F q ) of approximations of Fq , for which one easily constructs windows and check their alignment, and we will then deduce the lemma from the closeness of F q and Fq .
2.3 The approximate maps F q (q)
To introduce the maps F q we first take advantage of the form of f1 and define a new (q) function f 1 : T → R, which satisfies (q) (q) (q) f 1 (θ1 ) = f1 (θ1 ) for θ1 − 21 ≤ 81 , f 1 (θ1 ) = 0 for |θ1 | ≤ 18 , (2.13)
and which is continued in the complement so as to be of class C ∞ on T. To fix ideas one (q) (q) can even assume that 0 ≤ f 1 ≤ f1 , although it is not necessary. We then set 1
Fq = Φqf
(q)
◦ F∗
with
f
(q)
(q)
= f 1 ⊗ f2 .
(2.14)
We will show in the rest of this section that it is possible to define domains in which the q iterate F q is explicitly determined, along with estimates of the C 1 -norm of the difference q Fqq − F q . 2.3.1 We first need to introduce suitable flow-box coordinates for the pendulum map. We write P (θ1 , r1 ) = 12 r12 + cos 2πθ1 and P = ΦP . We will work in the open domain E located above the upper separatrix defined by E = {(θ1 , r1 ) ∈ A | 0 < θ1 < 1, r1 > 2 |sin πθ1 |}. Using {θ1 = 12 } as a reference section, we define the time-energy coordinates (τ, h) of a point (θ1 , r1 ) ∈ E as Z θ1 dθ 1 2 p h(θ1 , r1 ) = 2 r1 + (cos 2πθ1 − 1), τ (θ1 , r1 ) = , 1 2(h(θ , r ) − V (θ) 1 1 2
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with V (θ) = cos 2πθ − 1. It is well-known that these coordinates are symplectic. The period of motion is given as a (decreasing) function of energy by the formula T (h) =
Z
1 2
− 12
p
dθ 2(h − V (θ))
,
(2.15)
and the range of the coordinate change is the domain E∗ = {(τ, h) ∈ R2 | h > 0, |τ | < 12 T (h)}. In the coordinates (τ, h) the flow of P is straightened out, i.e. ΦtP : (τ, h) 7→ (τ + t, h) for (τ, h) ∈ E∗ and |t| small enough. We write H = T −1 for the inverse function of T . Given an integer q ≥ 1, we define the strip S(q) = {(τ, h) ∈ E∗ | |τ | < 1, H(q + 21 ) < h < H(q − 12 )}. Assume now q ≥ 4. Then the mapping P q is well-defined in S(q) with values in E∗ , with the following explicit expression q P : (τ, h) 7→ τ + q − T (h), h . (2.16) (as usual we do no introduce a new notation for the diffeomorphisms expressed in new coordinates). 2.3.2 We can now define the domains we were looking for. For each integer q ≥ 3 we will first be interested in a neighborhood N (q) of the point a(q) with (τ, h) coordinates (0, H(q)), that is the center of the strip S(q) (in the initial coordinates, the point a(q) is the intersection of {θ1 = 21 } with the unique orbit of the pendulum which has period q and is located above the upper separatrix). We want to define the neighborhood N (q) ⊂ S(q) so as to satisfy the conditions P k (N (q) ) ⊂ {|θ1 | < 18 },
∀k ∈ {1, . . . , q − 1}.
(2.17)
Let b be the point of the upper separatrix with θ1 = 18 in the initial coordinates, therefore the h-coordinate of b is zero, and one easily checks that τ (b) < 1. Let ̺ = 1 − τ (b), and set N (q) = {(τ, h) ∈ S(q) | |τ | < ̺/4, H(q + ̺/4) < h < H(q − ̺/4)}. It is not difficult to see that N (q) satisfies our requirements (see Figure 3). Turning back to the diffeomorphism F q , we introduce the domain N (q) = N (q) × A, q in which the q th -iterate F q has the following simple expression q
1
Fq = Φqf
(q)
1
◦ F∗q = Φ q f (q)
To see this observe that the function f 1 1
diffeomorphism Φ q f
(q)
(q)
◦ F∗q .
(2.18)
vanishes on the strip {|θ1 | < 81 } and so the
reduces to the identity on that domain. Therefore the conditions
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h |θ1 | =
1 8
τ = − 12 T (h)
τ = + 12 T (h)
τ −1
1
Fig. 3 The domain E∗ and the limit curves |θ1 | = 18 .
(2.17) immediately yield the first equality, thanks to the product form of F∗ . The second (q) equality comes from the form of f 1 and the choice of the neighborhood N (q) The simple expression (2.18) now enables us to make use of Sternberg’s estimates of Section 4. This is the only (but crucial) place where we use this conjugacy result, which brings us back (up to a controlled remainder) to the approximate system F q , which is dynamically easier to handle. More precisely we use Theorem E, which yields the following lemma. Lemma 2.6. There exist q0 ∈ N and a constant δ0 ∈ ]0, 1[ such that the inequality
q
Fq − F qq 2 (q) ≤ δ0ν(q;σ) C (N ) holds for q ≥ q0 .
Proof. By Theorem E there exists an integer q0 , a constant c ∈ ]0, 1[, and for each q ≥ q0 two diffeomorphisms χq and ψq of class C 2 satisfying
χq − Id 2 (q) ≤ c ν ,
ψq − Id 2 (q) ≤ c ν , C (N ) C (N )
such that the intertwining relation
F∗ ◦ Fqq−1 ◦ ψq = χq ◦ F∗q holds true over the domain N (q) . Now over the same domain 1
Fqq = Φ q f
(q)
1
◦ (F∗ ◦ Fqq−1 ) = Φ q f
(q)
1
◦ (χq ◦ F∗q ◦ ψq−1 ) = [Φ q f
(q)
1
(q)
q
◦ χq ◦ (Φ q f )−1 ] ◦ F q ◦ ψq−1
from which one easily deduces the desired estimate, with a constant δ0 slightly larger than c, increasing q0 if necessary.
2.4 Construction of the windows and proof of Lemma 2.5 We now introduce the point ω (q,k) = a(q) × b(q,k) ∈ A2
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with b(q,k) = (0, k/q) ∈ A. We will go one step further in the simplification and replace q the approximate map F q by its first order jet at the point ω (q,k), which will allow us to easily construct the windows. q
2.4.1 The first order jet Jω1(q,k) (F q ) and affine windows. 1
(q)
Using the explicit expression (2.6) of the perturbative diffeomorphism Φ q f one immediately checks that its first order jet at the point ω (q,k) has the following product form 1
1
(q)
(q)
1
(q)
Jω1(q,k) (Φ q f ) = Ja1(q) (Φ q f2 (0)f1 ) × Jb1(q,k) (Φ q f2 ) q
Using now Equation (2.18), one sees that the first order jet of F q at ω (q,k) has in turn the product form (q) 1 q Jω1(q,k) (F q ) = Ja1(q) Φ q f2 (0)f1 ◦ P q × Jb1(q,k) Sq
where
1
1 2
Sq = Φ q f2 ◦ (Φ 2 r2 )q has the usual form of a standard map. q Taking advantage of the product form of Jω1(q,k) (F q ) we will first construct (1, 1)windows for the two factors of the annulus A, and then make use of Lemma 2.2 to get the windows we need on A2 . 1 2
1. Aligned affine windows for the standard map. Here we write S∗ = Φf2 ◦ Φ 2 r2 for the normalized standard map. The rescaled map Sq is related to the normalized one by means of the conjucacy relation Sq = σq−1 ◦ S∗ ◦ σq , where σq (θ2 , r2 ) = (θ2 , q r2 ), which is also clearly valid at the linearized level Jb1(q,k) (Sq ) = σq−1 ◦ Jb1(k) (S∗ ) ◦ σq
(2.19)
with b(k) = σq (b(q,k) ) = (0, k). We first construct a sequence of (1, 1)-windows adapted to the first order jets of S∗ . Note that S∗ (0, k) = (0, k + 1) for k ∈ Z. For each k ∈ Z we define an affine (k) (k) (1, 1)-window D2 : [−1, 1]2 → A, satisfying D2 (0, 0) = (0, k), such that the composed (k) (k+1) window (Jb1(k) (S∗ )) ◦ D2 is aligned with the window D2 for each k ∈ Z. One gets by easy computation 1 1 Db(k) S∗ = √ √ . −2π 3 1 − 2π 3 √
q
√ The matrix Db(k) S∗ (0, k) is hyperbolic, with eigenvalues λ± = (1 −π 3) ± π(3π − 2 3) (i.e. approximately λ+ ≈ −8.78 and λ− ≈ −0.11), and associated eigenvectors uh = (1, −(1 − λ− )) ≈ (1, 1.11),
uv = (1, −(1 − λ+ )) ≈ (1, 9.78).
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r2
1
0
θ2
Fig. 4 Windows for the standard map S∗
Given a positive ̺, we define for each k ∈ Z the affine window (k)
D2 (x) = b(k) + ̺ D2 x := (0, k) + ̺(x(h) uh + x(v) uv ), (k)
x = (x(h) , x(v) ) ∈ [−1, 1]2 .
(k)
We then consider the affine window C2 := Jb1(k) S∗ ◦ D2 and write (k)
C2 (x) = b(k+1) + ̺ C2 x (note that the linear part is independent of k). Thanks to the choice of the horizontal (k) (k+1) and vertical directions one immediately sees that the window C2 is aligned with D2
(k) (k+1) for each k ∈ Z. Moreover, the parameter µ2 (C2 , D2 ) = ̺−1 (M[C2 , D2 ])−1 is inde(k) (k+1) pendent of k and one checks that the parameter χ(C2 , D2 ) := χ2 is independent of k and ̺. (q) Now let q ≥ 1 be fixed. Coming back to the rescaled map Sq , we set uh = σq−1 (uh ) (h) and uv = σq−1 (uv ). We fix arbitrarily a constant d0 in the interval ]0, 1[ and for each k ∈ Z we define the window (q,k)
D2
(q)
(q,k) (x) = (0, kq ) + d0ν (x(h) uh + x(v) u(q) + d0ν (σq−1 D2 ) x, v ) = b
(2.20)
for x = (x(h) , x(v) ) ∈ [−1, 1]2 . We will prove the following lemma. (q,k)
Lemma 2.7. For each k ∈ Z the window C2 with parameters (q,k)
µ(C2
(q,k)
, D2
) ≤ µ d−ν 0 q
and
with µ = (M[C2 , D2 ])−1 and χ2 = χ(C2 , D2 ). (q,k)
Proof. For the window C2
(q,k)
= Jb1(q,k) (Sq )◦D2 (q,k)
χ(C2
(q,k)
, D2
(q,k+1)
is aligned with D2 ) = χ2
we write
(k)
C2 (x2 ) = b(q,k+1) + d0ν (σq−1 · C2 ) x so (q)
(q,k)
M2 := M[C2
(q,k)
, D2
] = d0ν M[(σq−1 · C2 ), (σq−1 · D2 )] = d0ν σq−1 · M[C2 , D2 ]
,
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which is invertible. Moreover one gets for the first parameter
(q,k) (q,k) µ2 (C2 , D2 ) = (M[C2 , D2 ])−1 d−ν 0 q
The second parameter is easily seen to be constant and equal to χ2 , which shows the alignment and concludes the proof.
2. Aligned affine windows for the perturbed pendulum. All the maps we consider here will be expressed in the (τ, h) coordinates. For q ≥ 1 we set h(q) = H(q) = T −1 (q),
Tq′ = T ′ (h(q) ).
Note that Tq′ < 0. For x = (x(h) , x(v) ) ∈ [−1, 1]2 we define a first affine window (q) D1 (x)
d0ν (q) (h) ν (q) (h) (v) := x d0 , h + (x − x ) ′ = a(q) + D1 x Tq
with (q) D1
=
d0ν
1 0 1 1 , − ′ ′ Tq Tq
(2.21)
where d0 is the constant introduced in (2.20) and ν = ν(q; σ). In the (τ, h) coordinates (q) the image of D1 is the convex hull of the four points
d0ν ,
(q)
2d0ν /T ′ (h(q) )
− h − A3 = d0ν , h(q) + 2d0ν /T ′ (h(q) ) , A1 =
,
A2 = A4 =
d0ν ,
(q)
h
,
− d0ν , h(q) .
e (q) is extremely thin and nearly “horizontal”. (see figure 5). Observe that the domain D 1 Indeed one has the well-known estimates 1 T (h) ∼0 − 2π Ln h,
h(q) = H(q) ∼∞ e−2πq ,
(2.22)
from which one easily deduces 1 2πq T ′ (h(q) ) ∼∞ − 2π e , (q)
T ′′ (h(q) ) ∼∞ 2π(T ′ (h(q) ))2 .
(2.23)
1 −2πq So the images of the horizontals of D1 are line segments with slope very close to − 2π e , ν d (q) e1 is about 0 e−2πq . and the thickness of the image D π The previous window has been chosen in order to facilitate the geometric description of the effect of the various diffeomorphisms at the linearized level. The following lemma decribes the intersection properties we need.
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(q)
(q)
Φ q f (S∗ )
h
A1
A2
A4
A3
h S∗(q)
A′4
A′1 A′2
A′3
(q)
R∗
1
(q)
(q)
(q)
(q)
Fig. 5 The domains R∗ , Φ q f (S∗ ) and S∗
Lemma 2.8. Set κ = f2 (0) = −5/2π and define the composed affine window (q)
(q)
κ
(q)
(q)
C1 (x) = a(q) + C1 x := (Ja1(q) (Φ q f1 ◦ P q )) ◦ D1 (x), (q)
(q)
(q)
Then the window C1 is aligned with D1 and the parameters µ1 (q) χ1 = χ(C1 , D1 ) satisfy the inequalities (q) µ1
≤2 κ
qd−ν 0 κ ¯ν
4q
(q)
χ1 ≤
,
(q)
κ
κ ¯ νTq′
x ∈ [−1, 1]2 .
= M([C1 , D1 ])−1 and
.
(2.24)
(q)
Proof. First notice that Ja1(q) (Φ q f1 ◦ P q ) = Ja1(q) (Φ q f1 ) ◦ (Ja1(q) (P q )) with Ja1(q) (P q )(τ, h(q) + h) = τ − Tq′ h, h(q) + h . (q)
The composed window (Ja1(q) (P q )) ◦ D1 has the explicit expression (q)
(q)
(Ja1(q) (P q )) ◦ D1 (x) = a(q) + Q1 x, with
0 (q) Q1 = d0ν
1 Tq′
1 − T1′ q
for x = (x(h) , x(v) ) ∈ [−1, 1]2 . Its image is the convex hull of the following four points (See figure 5). ′ ν (q) ν ′ ′ ν (q) A1 = d0 , h − 2d0 /Tq , A2 = d0 , h , ′ ν (q) ν ′ ′ ν (q) A3 = − d0 , h + 2d0 /Tq , A4 = − d0 , h . κ
(q)
Now let us examine the effect of the linearized perturbative map Ja1(q) (Φ q f1 ). In the initial (θ1 , r1 ) coordinates, κ (q) (q) Φ q f1 (θ1 , r1 ) = θ1 , r1 − κq (f1 )′ (θ1 ) κ
(q)
so Φ q f1 (a(q) ) = a(q) and
κ (q) 1 0 Da(q) Φ q f1 = . 2ν κπ q 1
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Let ϕ be the coordinate change ϕ(τ, h) = (θ1 , r1 ), for which an easy computation yields p (q) 2(h + 2) 0 Da(q) ϕ = −1 . p 2(h(q) + 2) 0 Therefore in the (τ, h) coordinates
1 0 κ (q) 2 (q) Da(q) Φ q f1 = 2κπ ν(h + 2) . 1 q κ
(q)
(q)
(q)
For x ∈ [−1, 1]2 , we finally get (Ja1(q) (Φ q f1 ◦ P (q) )) ◦ D1 (x) = a(q) + C1 x with (q) C1
=
d0ν
1 0 ¯ν κ ¯ν 1 , κ − ′ qTq′ q Tq
(2.25)
with κ ¯ = 2κπ 2 (h(q) + 2). We therefore obtain the intermediate matrix 1 −1 (q) (q) (q) , M1 = M[C1 , D1 ] = d0ν 1 κ ¯ ν 1 − ′ ′ Tq q Tq
from which one immediately gets the first part of first obtain 0 (q) ν N1 = d0 κ ¯ν − ′ qTq therefore
which concludes the proof.
(q)
(M1 )−1
κ ¯ν 1 − ′ −1 ν qd− Tq 0 q = , 1 κ ¯ν − ′ −1 Tq (q)
(2.24). As for the parameter χ1 , we 0 1 , Tq′
(q)
(q) 4q (q) χ1 ≤ (M1 )−1
N1 ≤ κ ¯ νTq′
q
3. Aligned affine windows for J 1 (F q ). Using Lemma 2.2, Lemma 2.8 and Lemma 2.7 we have so far proved the following result. Lemma 2.9. Denote by D (q,k) the product (2, 2)-window with values in A2 defined by (q) (h) (v) (q,k) (h) (v) (x(h) , x(v) ) 7→ D1 (x1 , x1 ), D2 (x2 , x2 )
366
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(h)
(v)
(v)
for x(h) = (x1 , x2 ) and x(v) = (x1 , x2 ). Then the window q
q
C (q,k) = Jω1(q,k) (F q ) ◦ D (q,k) = Jω1(q,k) (F q ◦ D (q,k) ) is aligned with the window D (q,k+1), with parameters −ν(q;σ)
µ(C (q,k) , D (q,k)) ≤ µ q d0
χ(C (q,k) , D (q,k)) ≤ χ2
,
(2.26)
where µ and χ2 were defined in Lemma 2.7. 2.4.2 Remainders and proof of Lemma 2.5 The next and last lemma provides us with the necessary estimates on the remainders. Lemma 2.10. For q ∈ N and k ∈ Z we set q Cb(q,k) = Fqq ◦ D (q,k) − Jω1(q,k) (F q ◦ D (q,k) ).
There exists q1 ∈ N, a constant d1 ∈ ]0, 1[ and a neighborhood V of [−1, 1]2 in R2 such that the following inequality
(q,k)
Cb
C 1 (V )
holds true for all k ∈ Z.
2ν(q;σ)
≤ C d1
Proof. It is completely elementary and relies on the mean value theorem applied to the second derivative of Cb(q,k) , for which we will obtain upper bounds using the following formula D 2 g ◦ f = (D 2 g ◦ Df ) · Df ⊗2 + (Dg ◦ Df ) · D 2 f for the composition of differentiable maps. We first write q q q Cb(q,k) = (Fqq − F q ) ◦ D (q,k) + F q ◦ D (q,k) − Jω1(q,k) (F q ◦ D (q,k) ) .
Since the window D (q,k) is linear with bounded C 1 -norm, one immediately gets the following inequality
2 q
D (F − F q ) ◦ D (q,k) 0 ≤ c1 δ 2ν (2.27) q q 0 C (V )
from Lemma 2.6, for a constant c1 > 0 large enough. (q,k) q q As for the second term C = F q ◦ D (q,k) − Jω1(q,k) (F q ◦ D (q,k) ), we now choose d0 > δ0 , q and remark that it is enough to find upper bounds for the second derivative of F q ◦ D (q,k) . q
1
(q)
We will make use of the explicit expression F q = Φ q f ◦ F∗q ◦ D (q,k) and begin with the second derivative of F∗q ◦ D (q,k) . Since both maps are direct products, their composition is a product too, and its second factor is affine, so one has to consider only the first one: dν (q,k) dν P q ◦ D1 (x) = a(q) + d0ν x(h) − T h(q) + T0′ (x(h) − x(v) ) , T0′ (x(h) − x(v) ) . q
q
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The second derivative contains only terms of the form d0ν (Tq′ )2
′′
(q)
T (h
d0ν (h) x ), Tq′
+
d0ν (Tq′ )2
T ′′ (h(q) +
d0ν (v) x ), Tq′
and usual estimates analogous to (2.23) yield 1 (Tq′ )2
′′
(q)
T (h
+
d0ν (h) x ) Tq′
∼
2π(h(q) )2 (h(q) +
d0ν (h) 2 x ) Tq′
,
so one immediately obtains the inequality D 2 F∗q ◦ D (q,k) C 0 (V ) ≤ c2 d2ν 0 for c2 large enough. Now since the perturbative diffeomorphism has bounded derivatives, one gets the final inequality
2 (q,k)
D C
0 ≤ c3 d2ν (2.28) 0 C (V ) for c3 large enough. Therefore the conclusion follows from the mean value theorem applied twice to (2.28), together with inequality (2.27), choosing d1 > d0 and q1 large enough.
Proof of Lemma 2.5. It is now an immediate consequence of Lemma 2.9, Lemma 2.4 and Lemma 2.10. One simply has to choose the width σ so as to obtain the inequality ν(q;σ) d ≤ q12 for d slightly larger than d1 , which is possible thanks to equation (2.8).
3
Proofs of Theorem A and Theorem B
This section is very similar to the corresponding one in [27]. We “add degrees of freedom” to our family (Fq ), in two different and consecutive steps. The first one is based on the coupling lemma introduced in [28], which applies to discrete systems; it makes it possible to pass from the initially hyperbolic context on A2 to the initially elliptic one on An , n ≥ 3, and to prove Theorem A. The second step is an analytic suspension to pass from discrete systems on An to continuous Hamiltonian systems on An+1 and prove Theorem B.
3.1 From initially hyperbolic to initially elliptic perturbations The base of the construction is the coupling lemma introduced in [28]. This lemma enables us to “embed” the previous family (Fq ), or more precisely a subsequence Fqj , into an initially elliptic sequence of diffeomorphisms (Ψj ) of An which converges to the elliptic 1 2 2 completely integrable diffeomorphism Φ 2 (r1 +···+rn ) when j tends to +∞. Troughout this section we split the 2n-dimensional annulus An = A2 × An−2 into two factors and adopt the following notation for the variables: x = (θ1 , θ2 , r1 , r2 ) ∈ A2 ,
b rb) = (θ3 , ..., θn , r3 , ..., rn ) ∈ An−2 . x b = (θ,
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3.1.1 The coupling lemma We refer to [28] for the proof of the following coupling lemma, which was already used in a similar context in [27]. ′
Lemma 3.1. Consider two diffeomorphisms F and G of Am and Am respectively, an ′ integer N ≥ 2, and an N-periodic point a ∈ Am for G. Let f : Am → R and g : ′ Am → R be two Hamiltonian functions which generate complete vector fields, and assume furthermore that g satisfies the following synchronization conditions: g(a) = 1;
g(Gk (a)) = 0, 1 ≤ k ≤ N − 1;
dg(Gk (a)) = 0, 0 ≤ k ≤ N.
Then, if Ψ = Φf ⊗g ◦ (F × G), the following equality ΨN (x, a) = Φf ◦ F N (x), a
(3.29)
(3.30)
holds for x ∈ Am . An immediate consequence is that the submanifold V = A × {a} is invariant under Ψ , and that, canonically identifying V with A, the restriction of ΨN to V is given by Φ = Φf ◦ F N . As a consequence, the system (V, Φ) may be seen as a subsystem of ΨN . N
3.1.2 The family Ψj We denote by (pj )j≥0 the ordered sequence of prime numbers. As in the previous section, the diffeomorphisms we now construct will be obtained by composing the time-one map of a Hamiltonian function by the time-one map of a small perturbation, but this time the Hamiltonian function is not fixed and converges to the completely integrable Hamiltonian 1 2 r . 2 To be more precise, for j ≥ n − 3 we consider the maps Ψj = ΦS
(j)
◦ ΦHj
where Hj = 21 (r12 + · · · + rn2 ) +
1 Nj2
cos 2πθ1
with
Nj = pj−(n−3) pj−(n−4) · · · pj ,
(3.31)
and where S (j) is an analytic function to be defined below, which will depend only on the angles, and the norm of which will satisfy the inequality
(j)
S 0 ≤ N12 (3.32) C (Vσ ) j
where Vσ was defined in Section 1.2, and where σ was introduced in Proposition 2.1. To obtain the function S (j) we apply the previous coupling lemma to the diffeomorphisms Fj = Φ
1 2 (r +r22 )+ 12 2 1 N j
cos 2πθ1
1
2
2
and G = Φ 2 (r3 +···+rn )
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of A2 and An−2 respectively. The role of f is played by the function 1q f (q) defined in the previous section, for a suitable q to be defined below. The characteristic period of the coupling will be the integer Nj defined in (3.31), and we choose the Nj -periodic point a(j) = (0, rb(j) ) ∈ An−2
with rb(j) = (1/pj−(n−3), · · · , 1/pj ),
for the diffeomorphism G. We then have to find an analytic function g (j) which satifies the conditions (3.29). We proceed as in [27] and introduce for p ∈ N∗ the analytic function ηp : T → R defined by P 2 ηp (θ) = 1p p−1 cos 2πℓθ , which satisfies ηp (0) = 1, ηp (k/p) = 0 for 1 ≤ k ≤ p − 1 and ℓ=0 ′ ηp (k/p) = 0 for 0 ≤ k ≤ p. We set (j)
(j)
g (j) (θ3 , · · · , θn ) = g3 (θ3 ) · · · gn(j) (θn ) with gi (θi ) = ηpj−(n−i) (θi ) for 3 ≤ i ≤ n. One easily checks that g (j) satisfies the desired conditions (3.29). Note that g (j) is an analytic function, the norm of which is easily estimated from above:
(j)
g
Finally, we set S (j) =
1 qj
f (qj ) ⊗ g (j)
C 0 (Vσ )
≤ e4πσ(n−2)pj .
with
(3.33)
qj := Nj4 [1 + e8πσ(n−2)pj ].
(3.34)
So, by equations (2.9) and (3.33), the norm of the function S (j) satisfies inequality (3.32). Now the application of the coupling lemma 3.1 immediately yields the following result. 1
Lemma 3.2. Let Φj = Φ qj
f (qj )
N
◦ Fj j . Then for (x1 , x2 ) ∈ A2 ,
N Ψj j (x1 , x2 ), a(j) = (Φj (x1 , x2 ), a(j) ).
(3.35)
N
The submanifold V (j) = A2 × {a(j) } is thus invariant under Ψj j . Moreover, if σNj θ1 , θ2 , r1 , r2 = θ1 , θ2 , Nj r1 , Nj r2 , the conjugacy relation Φj = (σNj )−1 ◦ Fqj /Nj ◦ σNj
(3.36)
holds true for all j ∈ N. −1 (qj /Nj ) Let us introduce the point u(j) = σN (ζ ), where the ζ (q) were defined in Propoj sition 2.1. Using equation (3.36) one checks that Nj (qj /Nj )2 = qj2 /Nj iterates of Φj make the r2 action of the point u(j) drift over an interval of length 1. As a consequence, if one sets z (j) = (u(j), a(j) ) ∈ An (3.37)
equation (3.35) shows that qj2 iterates of Ψj make the r2 action of z (j) drift over a length 1.
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3.1.3 Complex analytic estimatesfor Ψj and proof of Theorem A The following lemma, proved in [27], provides the necessary analytic estimates on the 1 2 distance between the perturbed diffeomorphism Ψj and the elliptic system Φ 2 r . We use the notation introduced in Section 1.2 for the complex neighborhoods. Lemma 3.3. Let ̺ = σ/6, where σ were defined in Proposition 2.1 Then there exists j0 ∈ N and c > 0 such that for j ≥ j0
c
Ψj − Φ 12 r2 0 ≤ 2. C (U̺ ) Nj
(3.38)
Proof of Theorem A. It only remains to gather together Lemma 3.2 and the definition of the drifting point z (j) , the estimate (3.32) for the norm of the function S (j) and the estimates of Lemma 3.3. The main point is to determine the relation between the pa1 2 r
2 . If j0 is large enough, rameter qj and the size of the perturbation εj = Ψj − Φ C 0 (U̺ ) 1 the Prime Number Theorem yields the inequality pj−(n−3) ≥ 2 pj for j ≥ j0 . Therefore, since εj ≤ Nc2 by Lemma 3.3, j
1
1
pj ≤ 2 Njn−2 ≤ 2 c 2(n−2)
1 εj
1 2(n−2)
.
On the other hand, by definition qj = Nj4 [1 + e−8πσ(n−2)pj ] ≤ exp κ
1 εj
1 2(n−2)
1
for κ > 16πc 2(n−2) σ(n − 2) and j ≥ j0 large enough. The proof easily follows.
3.2 Analytic suspension and proof of Theorem B We now want to pass from the discrete case to the continuous one. As in [27] we follow the approach of Kuksin and P¨oschel. Theorem ([21]). Let D be a convex bounded domain in Rn , n ≥ 1. For j ∈ N let Fj : Tn × D → An be an exact-symplectic diffeomorphism, with analytic continuation to a complex neighborhood U̺ , for some ̺ > 0 independent of j. Let h be the Hamiltonian function defined on Anc by h(r) = 12 (r12 + · · · + rn2 ) and Φh : Anc → Anc its time-one map. Let εj = Fj − Φh C 0 (U̺ ) , and assume that εj → 0 when j → ∞. Then there exists j0 ∈ N such that for all j ≥ j0 there exists a real analytic 1-periodic time dependent Hamiltonian Hj defined on Tn × D × T such that the time-one map ΦHj is well-defined on Tn × D and coincides with Fj . Moreover, there exists a constant ρ < ̺ such that each Hj , j ≥ j0 , is analytic on U ρ = Vρ (Tn+1 ) × Wρ (D) and satisfies
Hj − h 0 ≤ C εj C (U ρ )
(3.39)
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for some constant C > 0 independent of j. We can now pass to the proof of Theorem B. Let R > 0 be fixed, and let D be the ball of radius R centered at 0 in Rn . For j ≥ 0 we consider the restriction Fj = Ψj |Tn ×D , with values in Tn × Rn . As a composition of two Hamiltonian time-one maps, Fj is exact-symplectic. Since W̺ (D) ⊂ W̺ (Rn ), Lemma 3.3 shows that Fj admits an analytic continuation to Uρ , with the same estimate (3.38). So the previous suspension theorem provides us with a non-autonomous Hamiltonian function Hj : U ρ → C satisfying (3.39). To obtain an autonomous system we simply have to consider the function Hj defined on U ρ × ×C by ¯ r¯) = Hj (θ, r, θn+1 ) + rn+1 . Hj (θ, For each energy e ∈ R, the surface H−1 j (e) ∩ {θn+1 = 0} is symplectic, transverse to the flow, and admits (θ, r) as a coordinate system. In this system, the associated return map coincides with Ψj . Theorem B immediately follows.
4
Sternberg’s theorem for normally hyperbolic manifolds
In this section we develop a local conjugacy result which we then apply to our construction of drifting orbits. We found it convenient to make this section essentially self-contained, including the notation. The application to our case, which yields the crucial Lemma 2.6 above is detailed in §4.5 below.
4.1 Setup and synopsis Let f0 , f1 be two symplectic diffeomorphisms of the symplectic manifold V , which preserve the submanifold M ⊂ V and are normally hyperbolic along V . All these data, namely V , M, f0 , f1 are assumed to be analytic. We wish to show that if f0 and f1 have a contact of large enough order along M, they are C ℓ conjugate in a neighborhood of M, for an integer ℓ ≥ 1 which we will compute. We first make the setting both more precise and more restrictive. 1. The maps f0 , f1 are hyperbolic transversely to M; here we will assume a simple product structure and that their invariant manifolds have been simultaneously straightened. Namely we take V = M × E s × E u = M × E with dim(V ) = d, dim(M) = m, E s ≃ Rns , E u ≃ Rnu , m + ns + nu = d. We will not assume that M is symplectic from the start, because it turns out that the proof for a symplectic M actually uses the non symplectic case. If M is indeed symplectic, a case which is of special interest, one has nu = ns = n and the vector space E = E s ⊕ E u is endowed with the standard symplectic structure, E s and E u being Lagrangian subvector spaces. The manifold V is provided with the product symplectic structure and we regard M, identified with M × {0}, as a symplectic submanifold of V . Finally, as mentioned above, we assume that T M is trivial.
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In the applications we have in mind, M = Am ≃ T ∗ Tm is the m-dimensional infinite ring (cylinder), a symplectic manifold with trivial tangent bundle. 2. Next we assume that f0 and f1 are isotopic, that is they can be interpolated by a family fε (0 ≤ ε ≤ 1) and we refer once and for all to [2] for detail on the deformation method we will use. More precisely we assume that the family fε can be written in the form fε = ΦFε ε ◦ f0 , where Fε (0 ≤ ε ≤ 1) is a family of analytic Hamiltonians describing the deformation. In other words fε solves the evolution equation in ε: d fε = Fε ◦ fε , dε
(1)
where Fε is the vector field with Hamiltonian Fε .
3. Concerning regularity, we will work with data fε which are analytic in the space variables and we are interested in retrieving a C ℓ conjugacy g between f0 and f1 in a neighborhood of M; ℓ will depend on the data. Regularity in ε is not essential for our purpose, which is to find and study the conjugating map g. It will turn out that continuity in ε will suffice (see below for detail). 4. Let us introduce ‘coordinates’: We let xs ∈ Rns (resp. xu ∈ Rnu ) describe E s (resp. E u ) and coordinatize the points of M by means of y. Because the latter variety is not necessarily compact we will need estimates that are uniform over M, that is w.r.t. y. We write z = (y, x) = (y, xs , xu ) for a point in V . We let |xs |, |xu | and |x| denote the norm (say Euclidean norm) on E s , E u and E, with E s and E u mutually orthogonal. We will write Dy for y-derivatives, that is derivatives along M and Dx for x-derivatives, that is transverse derivatives; Ds and Du denote derivative w.r.t. xs and xu respectively. In trying to keep a manageable notation, we will (almost always implicitly) use a multiindex notation for the various tensor quantities which appear; the reader should be able to restore a fully detailed expression if need be (which will in principle not be the case). 5. The diffeomorphisms fε are always assumed to coincide on M together with their derivatives; the order of contact will in fact be assumed to be much larger. We denote by A(y) = Dfx (y, 0) the common value of this derivative in the transverse direction. We assume that W s = M × E s (resp. W u = M × E u ) is the stable (resp. unstable) invariant manifold of fε (for any ε ∈ (0, 1)). The matrix A(y) is thus block diagonal and we get a contracting endomorphism As (y) of E s , and a dilating one Au (y) on E u . More quantitatively, let Spec(As (y)) denote the spectrum of As (y) (as a finite set) and |Spec(As (y))| the list of the norms of its eigenvalues. We assume that: |Spec(As (y))| ⊂ (µs , λs ),
with 0 < µs ≤ λs < 1,
(2)
these bounds being indeed independent of y ∈ M. In the same vein, we assume that |Spec(A−1 u (y))| ⊂ (µu , λu ) with 0 < µu ≤ λu < 1.
(3)
t If M is symplectic, A(y) is a symplectic operator and A−1 u = As , so that µu = µs , λu = λs . Concerning the restriction of the system to M we only assume that the vector field is
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bounded, that is there exists ν > 0 such that for any y ∈ M: |Dy f (y, 0)| ≤ ν,
(4)
where f = fε and the restrictions of the fε coincide on M anyway. Items 1 through 5 above provide our general setting which will be refined below. We note that although the geometric setting in 1 may seem quite restrictive, one can reduce seemingly much more general situations to it, using formal constructions (see in particular [20] and [12], §3.5). We also insist that the assumptions and the conclusions will be local around M, so that one can actually work on the product M × Bρ for some ρ > 0, where Bρ denotes the ball {|x| < ρ} ⊂ E. In particular it is enough in practice to analytically straighten local invariant manifolds. Following [2] we will use the so-called deformation method in order to solve the local conjugacy problem. We refer to the latter article for a concise exposition of the method with references. Here we will confine ourselves to a bare minimum. We wish to conjugate f0 and f1 and have connected this pair by a path fε (0 ≤ ε ≤ 1). We will try to achieve more, and look for a family gε such that gε−1 ◦ fε ◦ gε = f0 for all ε ∈ (0, 1), so that g = g1 will answer the initial problem. We also require that gε be regular enough (C 1 ) in ε so that it will satisfy an evolution equation of the same form as fε , say: d gε = Gε ◦ gε , dε
(5)
with the initial condition g0 = 1 (the identity map). Finally we are also looking for a symplectic conjugacy, that is we want Gε to be a Hamiltonian vector field, with Hamiltonian Gε . In order to derive the equation for Gε , hence for Gε , one simply translates the fact that gε−1 ◦ fε ◦ gε is a constant map, namely f0 , so that the derivative of this quantity vanishes. Formal computations (see [2], §§2,3) lead to the equation satisfied by the ‘conjugating Hamiltonian’ Gε , namely: Gε − Gε ◦ fε−1 = Fε .
(E)
It is useful to write the equivalent equation obtained by composing each term with fε , which also amounts to changing fε into its inverse: Gε − Gε ◦ fε = −Fε ◦ fε .
(E ′ )
The problem of conjugating f0 to f1 has now been reduced to finding a solution of (E) or equivalently of (E ′ ), and study its regularity in the space variables. As for regularity in ε, it is enough for our purpose to be able to solve equation (5) above. By Cauchy-Lipschitz we should require that Gε be Lipschitz in z and continuous in ε, which is the time-like variable. In turn the vector field Gε is derived from the Hamiltonian Gε by taking zderivatives so that Gε and Gε have the same regularity in ε. Hence we only need Gε to be continuous in ε, something which will be obvious from the algorithms we use so that
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we will not need to return to this issue. Regularity in ε could of course be discussed if need be, much as space regularity. Note that the above discussion also applies to Fε and Fε , that is to equation (1). We now write down the formal solutions of (E) and (E ′ ) obtained by iterating these equations. We get: ∞ X Gε = Fε ◦ fε−n (F S) n=0
and G′ε = −
∞ X n=1
Fε ◦ fεn ,
(F S ′ )
which will be put to use below. This completes our description of the setting and the main characters. Let us now move to a brief exposition of the plot, which may sound a little intricate at first reading. The case we are interested in displays several specific features which we will try to accomodate as best as possible or indeed take advantage of. Namely a) the initial data, say f0 and the deformation Fε , are analytic, b) there is an invariant manifold M which not only is not reduced to a point but is also possibly not compact and c) we are working in a symplectic setting. This last feature has already been incorparted by reducing the problem to solving the ‘homological equation’ (E) (or equivalently (E ′ )). Note that these are scalar equations which are moreover linear with respect to the perturbation F . We adopt the classical overall strategy inaugurated by S.Sternberg in [32]. That is we first treat the contracting case, thus assuming nu = 0, that is V = W s = M × E s ; clearly the expanding case (V = W u ) can be treated in the same way, changing the diffeomorphisms into their inverses. Here we can take full advantage of the analyticity of the initial data. The problem is local near M and analyticity enables one to get an analytic conjugacy in a neighborhood of M which can be explicitly determined. In short analyticity in the contracting case enormously simplifies the problem and yields an effective analytic solution. However, because of b) above, that is the possible non compactness of M, we do have to add an assumption of uniform contraction along M. One then wishes to reduce the general case to the contracting one, applying the results in the contracting case for the triple (M, V, f ) to the triple (W s , V, f −1 ) in the general hyperbolic case. Here f stands for fε for some fixed ε ∈ (0, 1); it preserves W s which is an attracting invariant submanifold for f −1 . However to this end one first has to ‘prepare’ the system so that f0 and f1 acquire a contact of high order along W s (and not merely along M). This can be done by dealing with the jet of the perturbation along W s , so the traditional ‘preparation lemma’ deals with a contracting problem which is however more intricate than equation (E). But again, in the case of analytic data, it can be treated very simply and efficiently, confining oneself however to a neighborhood of M which is any case unavoidable because the end result, that is the existence of a germ of conjugacy between f0 and f1 is in essence local around M. So by using the analyticity of the data in a neighborhood of M on the stable manifold W s , we can simplify a large part of the proof
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and make it more effective, including in the presence of a nontrivial invariant manifold M. This is however not the end yet because it is (except in very special cases) not possible to prepare the system gobally along W s and indeed in a neighborhood of that invariant manifold. One in fact has to smoothly truncate using functions whith compact support, thus loosing both analyticity and uniqueness. Indeed in the contracting (not necessarily analytic) case, one gets a uniqueness result for the (germ of) conjugacy which does not hold in the general hyperbolic case, and that conjugacy is analytic if the data are analytic, which also fails in the general hyperbolic case. So after preparing the system along W s , one has to apply a result in the contracting but not analytic case. This is the reason why we also have to address the latter issue. We will give a direct treatment of that contracting but not analytic case, confining ourselves to the case of equation (E) (or (E ′ )) which is the only one we will have to use. Technically speaking we improve in that case on the results of [2] in a way which will be made precise below. We will also point out in due time a phenomemon having to do with the regularity of the solution along M which seems to have passed unnoticed as it has to do with both analyticity and the presence of a nontrivial invariant manifold.
4.2 The contracting case In this paragraph we restrict attention to W s or equivalently assume that nu = 0, V = M × E s . We consequently simplify the notation, writing x instead of xs for a point of E = E s ; similarly we write z = (y, x) ∈ V . As a rule we will in fact omit the subscript or superscript s altogether in this section. As explained above the dependency on ε will play essentially no role below, so that in order to clarify notation further we will henceforth drop the subscript ε at most places. Everything will take place at a fixed value of ε ∈ (0, 1) and as mentioned above continuity in ε, which is ultimately all we need, will be trivially satisfied. For any ρ > 0, we let Bρ ⊂ E denote the ball |x| < ρ and everything will take place inside a tubular neighborhood of M of the form Mρ = M × Bρ . More precisely we assume that there exists ρ > 0 such that fε is analytic over Mρ for all ε ∈ (0, 1) or equivalently that f0 and the deformation Hamiltonian Fε satisfy this assumption. We also assume that Fε has a zero of order at least k > 2 along M, which can be translated as Supy∈M,0<|x|<1
|x|−k |Fε (y, x)| < ∞.
(1)
This entails that f0 and f1 (or any fε , ε ∈ (0, 1)) have a contact of order at least k − 1 > 1 along M and can be expressed in the language of weighted conical norms used in [10] and other papers. This condition already contains a uniformity assumption along M, but we will actually need somewhat more, namely (3) below. Next we assume of course that the diffeomorphisms are uniformly contracting along M. Because k > 2, A(y)(= As (y)) = Dx fε (y, 0), that is the derivative of fε along E = E s ,
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does not depend on ε and it is assumed to satisfy (2) of §1, that is: |A(y)v| ≤ λ|v|
for
y ∈ M, v ∈ E,
(2)
and some λ = λs < 1. Analyticity supplemented by (1) and (2) would suffice in the case of a compact invariant manifold M. Here we will need to reinforce these assumptions in order to make them uniform in a tubular neighborhood of M. We assume first that there is a constant Kρ > 0 such that for z ∈ Mρ one has: |F (z)| ≤ Kρ |x|k .
(3)
Next let us write f (z) = (fy (z), fx (z)) ∈ V = M × E. We assume that for z ∈ Mρ one has: |Dx fx (z)| ≤ λρ < 1 (4) for some λρ (λ ≤ λρ < 1). This should hold of course for all fε , ε ∈ (0, 1). Under the above assumptions we get the following: Proposition 4.1 (Analytic contracting case). Assume that f0 and the deformation Fε are defined and analytic (w.r.t z) over Mρ = M × Bρ for some ρ > 0; assume moreover that they satisfy (3) and (4) above on that domain. Then there is a unique germ of continuous conjugacy g between f0 and f1 which is the identity on M. It is actually defined and analytic over Mρ and it has a contact of order k with the identity along M. Se we not only get existence and uniqueness but we also have an explicit domain over which the conjugacy is defined. Using continuation to the complex domain and the Cauchy formula in a standard way, one can then estimate derivatives. We note again that if M is compact, (1) and (2) imply that (3) and (4) hold true for some ρ > 0. The statement above is also purely local, and the existence of the data is actually required only on Mρ . Proving the above statement is equivalent to showing that (E) and (E ′ ) have a unique solution which vanishes on M, that it is actually analytic in Mρ and that it vanishes on M at order k. We will do just that presently and this is the way in which we will cast the analogous statements in the sequel. The proof in the present analytic contracting case is quite straightforward. First note that (4) implies that for any positive integer n: |fxn (z)| ≤ λnρ |x|,
(5)
where we write fxn = (f n )x for simplicity. By iterating (E ′ ) to order N, we find that any solution G satisfies: N X G(z) = − F ◦ f n (z) + G(f N +1 (z)). (6) n=1
If we require G to vanish on M, we see using (3) and (5) that the last term goes to 0 as N increases to infinity and G thus has to coincide with the formal solution (F S ′ ). The
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convergence of that series, which we now simply call G, is also obvious since by (3) and (5) the general term is dominated by Kρ (λnρ |x|)k . We find that there is indeed a unique solution G of (E) vanishing on M, that it is analytic on Mρ because the convergence is uniform on that domain. We moreover get the estimate: X |G(z)| ≤ Kρ (λnρ |x|)k ≤ Cρ |x|k , (7) n≥1
with Cρ = λkρ (1 − λkρ )−1 Kρ . This confirms that G vanishes on M at order k. The ‘preparation lemma’ leading to the general hyperbolic case will require solving an equation a little more complicated than (E). Let Q = Q(z) be a square matrix depending on z ∈ Mρ . We are now interested in solving: G(z) − Q(z)G ◦ f (z) = F (z).
(8)
Here G is now a vector function (of the same size as Q) and we have denoted the vector perturbation simply by F (compare (E ′ ) in §1). It turns out that in the analytic category it is essentially as easy to study (8) as (E) or (E ′ ). We have: Proposition 4.2 (Analytic contracting case with a cocycle). Assume that f , F and Q are defined and analytic over Mρ , that f and F satisfy (3) and (4) and that the norm of Q = Q(z) is bounded by µρ ≥ 0 on that domain. Then if λkρ µρ < 1 there is a unique solution of (8) which vanishes on M. It is analytic on Mρ and vanishes on M at order k. The proof is essentially the same as above. The candidate formal solution vanishing on M reads: X m=n−1 Y G(z) = Q(f m (z)) F (f n (z)). (9) n≥0
m=0
The general term is now dominated on Mρ by Kρ µnρ (λnρ |x|)k and one concludes as in the proof of Proposition 4.1. Once again the assumptions and the conclusion are local around M. If one assumes that as ρ decreases to 0, λρ tends to λ = λs and µρ tends to a value µ = µ0 , one finds that provided λk µ < 1, one gets the conclusion on some neighborhood of M. On the other hand, as soon as λρ < 1, the conclusion will hold true on Mρ for k large enough, that is in the original problem if one requires a contact of high enough order between the original diffeomorphisms. We now turn to the smooth setting and will treat only the case of equation (E). We use a direct method which enables us to get quite precise results but would be substantially more difficult to apply in the case of equation (8). More abstract approaches, using classical fixed point results for contracting maps are naturally less sensitive but also less precise. We refer to [2] for a sketch of the proof in a similar setting (see Lemma 5.5
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there; we note that we will implement below the Remark at the end of that statement). So from now on we consider data which are defined and of class C r (k < r ≤ ∞) on a tubular neighborhood of M, of the form Mρ = M × Bρ for some ρ. Everything is still local around M and although it is useful to keep r as a free parameter, the reader may think primarily of the smooth case r = ∞. We always keep the same letter ρ in the various assumptions but needless to say, it only implies the existence of a value ρ > 0 such that the assumption at hand is satisfied. Until now we did not have to mention the nature of the diffeomorphisms when restricted to the invariant manifold M. This is quite remarkable in fact, because that spectrum (Lyapunov exponents) will indeed play an important role in the sequel. This is the phenomenon we alluded to at the end of §1. In the analytic case one gets an analytic solution in all variables; but in the smooth case, transverse regularity and regularity along M will appear to be quite different questions and we will discover the anisotropic character of the problem. Recall the pieces of notation Dx and Dy , as well as condition (4) in §1, stating that the Lyapunov exponents of the restricted diffeomorphisms are bounded by ν on M (for all ε ∈ (0, 1)). Let us reinforce it in the usual way, assuming that in fact, possibly at the expense of shrinking Mρ , one has: |Dy f (z)| ≤ νρ ,
(10)
for z ∈ Mρ and some νρ ≥ ν ≥ 1. The last inequality is by convention; we may increase ν and replace it by max(1, ν), which we do for convenience. We will start with a sample statement which will subsequently be generalized. We include it for illustrative purposes as it displays the seeds of the main phenomena. By (1) we have that Dxj F = 0 for j < k; we can take the derivative in y, permute the derivatives and find that Dxj (Dy F ) = 0 for j < k. Note that here we are using of course the standard multiindex notation and j < k actually means that the length of j is strictly smaller than the integer k. We hope that this simplified notation, which we also apply to tensor quantities will help clarify the text without causing misinterpretations. Let us stick for the time being to the case j = 1. We now assume the analog of (3) at order 1, namely (1) that there exists a constant Kρ such that for z ∈ Mρ one has: |F (z)| ≤ Kρ(1) |x|k ,
|Dx F (z)| ≤ Kρ(1) |x|k−1,
|Dy F (z)| ≤ Kρ(1) |x|k .
(11)
It is as usual understood that this holds for all Fε ; moreover this assumption will automatically be fulfilled if M is compact. Note the fact that the x derivative vanishes on M at order k − 1, whereas the y derivatives vanishes at order k. Under these assumptions we have the following: Proposition 4.3. Assume that f and F are defined over Mρ = M × Bρ for some ρ > 0 and that they are of class C r (w.r.t. z) on that domain, for some integer r with 0 < k ≤ r ≤ ∞. Assume that the data satisfy (4), (10) and (11) on Mρ . Then there is a unique continuous solution G of (E) vanishing on M; it can be extended to a function on Mρ which is transversely (i.e. with respect to x) of class C 1 . It is of class C 1 along M (i.e.
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with respect to y) provided the inequality λkρ νρ < 1 obtains (in which case G is of class C 1 on Mρ ) . The proof begins as that of Proposition 4.1. Following the latter proof quite literally, it ensures in our case the uniqueness of the solution G and that it exists and is continuous over Mρ . It remains to investigate its regularity. To start with, let us differentiate the P defining formula (cf. (F S ′) in §1): G = − n≥1 F ◦ f n . We get: DG(z) = −
X
DF (f n (z))Df (f n−1 (z)) . . . Df (f (z))Df (z),
(12)
n≥1
where D stands for either Dx or Dy . We immediately encounter an important difference between x and y differentiation, that is between transverse and longitudinal regularity. For D = Dx , the n-th term of the sum in (12) can be estimated on Mρ in the operator norm using, (5) and (11): |Dx F (f n (z))Dx f (f n−1(z)) . . . Dx f (f (z))Dx f (z)| ≤ Kρ(1) (λnρ |x|)k−1λnρ . This in turn yields: |Dx G(z)| ≤
X n≥1
k−1 Kρ(1) |x|k−1λkn , ρ = c|x|
(13)
(14)
where the constant c is easily computable (we will use the letter c for the ‘generic’ constants, possibly making comments on their nature). This shows that under the above assumptions G is C 1 w.r.t. x with an explicit estimate on Mρ ; moreover Dx G vanishes at order k − 1 on M. Anticipating a little, it will turn out that in fact there is no loss in x regularity with respect to the data. But regularity along M, that is w.r.t. y, rests on a different mechanism. In the n-th term of (12) with D = Dy , we get a product of n terms of the form Dy f (f j (z)) which is asymptotically governed by the Lyapunov exponents of the restriction of f to M, namely: |Dy F (f n (z))Dy f (f n−1 (z)) . . . Dy f (f (z))Dy f (z)| ≤ Kρ(1) (λnρ |x|)k νρn , and from there: |Dy G(z)| ≤
X n≥1
Kρ(1) |x|k (λkρ νρ )n = c|x|k ,
(15)
(16)
with again an easily computable constant c. This finishes the proof of the Proposition and shows that Dx G (resp. Dy G) vanishes at order k − 1 (resp. k) on M. Note that of course this vanishing property does not presuppose the existence of higher derivatives.
We have just seen the first manifestation of the fact that when working along an invariant submanifold rather than in the neighborhood of a point, and whatever the characteristics of the induced flow, they can be compensated for by requiring a higher order contact of the initial diffeomorphisms f0 , f1 , that is by increasing k. The instructions
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for use of such statements of course depend on which parameters are considered free. One can for instance look at what happens on M and its normal bundle, that is determine ν and λ, pick k such that λk ν < 1 and find ρ small enough so that one still has λkρ νρ < 1, assuming of course that νρ and λρ are continuous functions of ρ (νρ ≥ ν ≥ 1, λ ≤ λρ < 1). Again if M happens to be compact, global uniformity along M prevails and assumptions over a tubular neighborhood follow automatically, e.g. (11) is a consequence of (1). We now explore higher regularity. Existence and uniqueness of a solution G of (E) and (E ′ ) vanishing on M are proved as above, and in fact as in Proposition 4.1, together with the fact that it coincides with the formal series (F S ′) of §1, and we will use direct differentiation of that expression. We start from the obvious: X |D ℓ G| ≤ |D ℓ (F ◦ f n )|. (17) n≥1
Here ℓ is any multiindex and for the time being we do not distinguish between the transverse (Dx ) and longitudinal (Dy ) factors. We will prove as usual the existence of D ℓ G by showing that the series converges locally uniformly and indeed uniformly over Mρ under certain assumptions. We will in fact also obtain fairly explicit estimates for that quantity. Let us first recall (see e.g. [10], Appendice 1) the formula for the successive derivative of a composition of maps, namely: D ℓ (F ◦ h) =
ℓ X X q=1
m
σm (D q F ◦ h)D m1 h . . . Dmq h.
(18)
This is the so-called ‘Faa-di Bruno formula’, whose proof is formal and which is valid for any composition of maps (here denoted F and h) of Banach spaces. The second sum runs over the set of indices m = (m1 , . . . , mq ) such that mj ≥ 1 for any j (1 ≤ j ≤ q) and m1 + . . . mq = ℓ. The σ’s are integers which can be defined recursively (see [10] or [3]). We will only retain the obvious fact that they can be bounded with a bound which depends on ℓ only and so can the number of terms in the sum (18). Our task now consists in estimating (17) using (18) with h = f n ; we are however in an anisotropic setting, so that we need to distinguish between the x and y derivatives and study a kind of weighted form of (18). Very roughly speaking, any factor Dx f contributes a converging factor λ, whereas Dy f contributes a possibly diverging factor ν ≥ 1. Let ℓ ≤ k; we wish to study the existence and continuity of D ℓ G, to which end it is enough ′ ′′ to investigate derivatives of the form Dxℓ Dyℓ G, with ℓ′ + ℓ′′ = ℓ. All multiindices will now be split according to their x and y content, using primes for the first and double primes for the second set. ′ For q ≤ k, we have from (1) that |x|q −k D q F is bounded as |x| goes to 0. Here again only the number q ′ of x derivatives comes in: Taking y derivatives does not let the order of contact decrease. So for given ℓ ≤ k it is sensible to require the higher order analog of (11) over Mρ ; it is once again a consequence of (1) for compact M. So we assume the (ℓ) existence of a constant Kρ such that for z ∈ Mρ and q ≤ ℓ one has the following bound: ′
|D q F (z)| ≤ Kρ(ℓ) |x|k−q .
(19)
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We also have to assume that the successive derivatives of f are bounded over Mρ , i.e. (ℓ) there exists a constant Dρ such that for z ∈ Mρ and q ≤ ℓ: |D q f | ≤ Dρ(ℓ) .
(20)
Under these assumptions, our goal is now to prove the following: Proposition 4.4 (Smooth contracting case). Assume that f and F are defined over Mρ = M × Bρ for some ρ > 0 and that they are of class C r (w.r.t. z) on that domain for some integer r with 0 < k ≤ r ≤ ∞. Assume that the data satisfy (4), (10) and (19) on Mρ where ℓ ≤ k and the inequality λkρ νρℓ < 1 is satisfied. Then: i) If the derivatives of f are bounded on Mρ to order ℓ, that is if (20) holds, there is a unique solution G of (E) vanishing on M and it can be extended to a function of class C ℓ on Mρ ; ii) If the derivatives of f of order ≤ r are bounded on Mρ , the function G is transversely (i.e. with respect to x) of class C r on Mρ . If r = ∞ no uniformity with respect to the length of the multiindex is required in the last boundedness assumption on the derivatives of f . So if M is compact this assumption is automatically fulfilled (as well as (20) a fortiori). As usual one can then replace (4), (10) and (19) by the corresponding infinitesimal assumptions: (4) is implied by (2), (10) follows from (4) in §1 and (19) is a consequence of (1). We need only prove the regularity assertions. We will show i) and ii) at one go but one could give a simpler direct proof of ii). It may be useful to briefly explain why. The point is that when taking transverse derivatives there are two sources of convergence, one being the contact along M (order of vanishing of F ), and the other being contraction. This is enough to ensure that the solution G is transversely as smooth as the data, provided the necessary derivatives are bounded, as recorded in ii). By contrast, as already illustrated in Proposition 4.3, in the case of i) a high order of contact has to compensate for the possible divergence originating from the possibly large Lyapunov exponents (ν) of the flow on M. First putting (19) and (5) together, we find that: ′
|D q F (f n (z))| ≤ Kρ(ℓ) (λnρ |x|)k−q ,
(21)
still for q ≤ ℓ and z ∈ Mρ . Looking back at (18) with h = f n , we see that the first term (k−q ′ )n in each factor provides a converging factor λρ . It remains to investigate the other m n factors, of the form D f for m ≤ ℓ. We may and do assume that m = m′ + m′′ and that ′ ′′ in fact D m = Dxm Dym . The necessary technical but elementary properties are contained in the following ′
′′
Proposition 4.5. For m ≥ 1 and n ≥ 1, D m f n = Dxm Dym f n satisfies the following properties: i) It is a sum of at most (m − 1)!nm−1 terms;
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ii) Each term is a product of at most mn factors of the form D i f ◦ f j , with 1 ≤ i ≤ m, 0 ≤ j ≤ n − 1; iii) In each term there enter at least n − m factors with i = 1 and at most m with i > 1; iv) If m′ ≥ 1 and m′′ ≥ 1, there enter in each term at least n − m and at most m′ n (resp. m′′ n) factors of the form Dx f ◦ f j (resp. Dy f ◦ f j ). The first three items are isotropic and constitute Lemma 5.4 of [3]. The first two are proved by a straightforward induction, using the product and chain rules in order to bound respectively the number of terms and the number of factors in each term (we corrected a typo in [3]: The number of terms is indeed bounded by (m − 1)!nm−1 , not just m!nm ; in particular it is 1 for m = 1. This does not play any role in the sequel). The third assertion is proved by inspection and we prove iv) much in the same way. The statement is symmetric in x and y, and because it is formal we may swap the x and y ′ ′′ derivatives in the proof. In other words it is enough to prove it for Dxm Dym f n (in this order), and the y derivatives. By ii), there enter at least n − m′′ terms of the form Dy f ◦ f j in the expression of each ′′ term of Dym f n , and at most m′′ n terms that are not of this form, that is involve higher ′ derivatives. One then applies the operator Dxm to this expression. The number of terms of the form Dy f ◦ f j cannot increase, and in fact by the product rule, an application of Dx to any factor lets the number of such terms decrease by at most 1. So the number of terms Dy f ◦ f j in the end result is at least n − m′ − m′′ = n − m per factor and is also at most m′′ n. This finishes the proof of iv) and thus of the lemma. Returning to the proof of i) in Proposition 4.4 we first note that by iii) of the lemma, the number of terms in each factor involving higher derivatives is at most m. Now for q > 1 we know nothing about D q f and can only use the a priori estimate (20). We did not try to distinguish there between x and y derivatives as it does not seem useful since we have in general no information in either direction. We can now estimate D m f n over Mρ by: ′′ |D m f n | ≤ cℓ nm−1 (Dρ(ℓ) )m λn−m νρnm , (22) ρ an estimate which is valid for any m ≥ 1 (including if m′′ = 0). Here we used of course Lemma 5 and have absorbed the factor (m − 1)! appearing in i) of that lemma in the combinatorial constant cℓ ; the term nm−1 actually plays no role either, being polynomial (ℓ) in n. The crux of the matter is that Dρ is raised to the power m ≤ ℓ (independently of n) and that the divergence originating from ν has been controlled in an essentially optimal way. We note that we have de facto implemented the Remark following Lemma 5.5 in [2], leading to the perhaps optimal exponents appearing in the inequality connecting k (the order of contact) and ℓ (the regularity of the conjugacy), which here takes the form of the condition λkρ νρℓ < 1 occuring in the statement of the proposition (for comparison k here should be shifted to k + 1 in the notation of [2]). There remains to return to (17) and (18) and collect estimates. Let us compute the powers of λρ and νρ appearing in the estimate for each term of (18), with h = f n . By
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n(k−q ′ )
(19) the first factor yields a power λρ , which originally comes from the high order contact of f0 and f1 along M and the contracting character of the maps: This is in fact our only source of convergence here. Then from the product of terms of the form D mi f n P in (18) one gets a factor λaρ νρb , where using (22) we can take: a = i (n−mi ) = qn−ℓ and P b = i nm′′i = nm′′ . Since q ≥ q ′ and using some obvious inequalities we can combine the above into: k ℓ n |D ℓ (F ◦ f n )(z)| ≤ cℓ Kρ(ℓ) |x|k−ℓ (Dρ(ℓ) )ℓ nℓ λ−ℓ ρ (λρ νρ ) , ′
′′
(23)
which is again valid for z ∈ Mρ . In order to evaluate D ℓ G (or of course any D q G with q ≤ ℓ) and show the local uniform convergence of its formal expression, it simply remains to sum over n ≥ 1. By (23) one gets a geometrically convergent series provided ′′ λkρ νρℓ < 1 and one finds that convergence is determined by a factor involving ℓ′′ ≤ ℓ, that is the number of longitudinal derivatives, which proves both i) and ii) (in the latter case ℓ′′ = 0). It actually yields somewhat more: In particular the derivative D q G (q ≤ ℓ) actually vanishes to order k − q ′ on M, where q ′ is the number of transverse derivatives. It is also plain from the above that one could devise variants of Proposition 4.4 mixing assertions i) and ii) but we will not go into that. This completes our study of the contracting case, in particular of the conjugacy problem on W = W s . Obviously the case of W u is dealt with by changing f into its inverse, and this will be put to use in order to treat the general hyperbolic case in section 4 below.
4.3 The preparation lemma A key remark due to S.Sternberg in his original paper ([32]) is that the general hyperbolic case can be reduced to the contracting case. In order to achieve this, one has to replace M by W = W s , viewing the latter as a repulsive invariant submanifold. That is one would like to apply the results of section 2 to the triple (W, V, fε−1 ) instead of (M, V, fε ) (clearly the roles of W s and W u could be switched all along, replacing as usual f by f −1 ). Note that if M is symplectic W is not (being then actually Lagrangian) which is one of the reasons not to confine oneself to a symplectic M in the contracting case. Now in order to apply the results of the contracting case, one needs the fε to have a contact of high order along W , not just along M. In order to achieve this one ‘prepares’ the original system, that is in our case, performs a preliminary conjugating transform which will ensure a contact of high order of the transformed diffeomorphisms along W . This in turn cannot in general be done globally around W in the analytic setting. But a key point in our case is that we can first take advantage of the fact that the data are analytic and only then perform a cutoff which destroys analyticity. Let us turn to more specific matters; we will be a little less detailed than in section 2 and will leave some routine operations or translations to the good will of the reader. We insist however that no new technical estimates are needed here.
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So starting from the assumptions of §1 that the fε have a contact of high order (= k−1) along M, that their stable manifolds coincide and have moreover been straightened (at least in a neighborhood of M), we should ‘prepare’ the system further in order to ensure a contact of high enough order along W . The results of the last section applied to W = W s provide a contact of order 0. This means that if one considers G = Gε as in section 2 and the associated vector field Gε , then solves equation (5) in §1 and conjugates fε by the solution gε on W s , one gets a family which is constant over M × Bρs = Wρ ⊂ W , that is over a neighborhood of M in W ; here Bρs denotes the ball {|xs | < ρ}. We now have to examine the behaviour of the jets transverse to W and the contracting case can be seen as the 0-th order of that procedure. We write as in §1 z ∈ V with z = (y, xs , xu ) and introduce ws = (y, xs ), parametrizing the points of the stable manifold W s , so that a point of V can also appear as z = (ws , xu ). Everything here is again local around M and in fact takes place on an infinitesimal neighborhood of Wρ where ρ > 0 will be made precise later and is related to the assumptions made in §2. Recall that the ultimate goal, to be achieved in the next section, is to solve (E) in a neighborhood of M. Here we will solve it in an infinitesimal neighborhood of Wρ , actually only to a finite order. Let G be a putative solution and expand it formally around W , writing: X G(ws , xu ) = Gi (ws )xiu . (1) i≥0
We adopt as usual a simplified system of notation; for instance i is a multiindex, we make no notational distinction between multiindices and their lengths, we could write i x⊗i u instead of xu etc. As for the diffeomorphisms fε we leave out the index ε as usual (and ditto for F , G etc.) and write f = (f s , f u ) with f s ∈ W s = W and f u ∈ E u . We expand these components around W as: X s,u f s,u (ws , xu ) = fi (ws )xiu . (2) i≥0
We know that f0u = 0, that is f u (ws , 0) = 0 for any ws ∈ Wρ simply because W is invariant under f . We write f1u (ws ) = Du f u (ws , 0) = Au (ws ) where Du denotes of course the derivatives w.r.t. xu . This notation extends the one in §1 (cf. item 5 there) which was introduced for z ∈ M, that is xs = 0 (ws = (y, 0)). We write out equation (E ′ ), expanding both sides around W (xu = 0). So we need to expand the composition G ◦ f (as well as F ◦ f ), which is no more and no less than the Taylor expansion of a composition of maps. One can of course spell out explicit expressions but we will not actually make use of them. We simply write: X X G ◦ f (z) = Gi (f s (ws , xu ))(f u (ws , xu ))i = Hi (ws )xi . (3) i
i≥0
Let us say a word again about notation which may become a little misleading. What we actually want to do is simply solve (E ′ ) recursively, order by order. To that end we regroup the terms of the same order, corresponding to multiindices of the same length.
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From now on Gi (resp. Hi ) with integer i will accordingly denote the operators which correspond to the terms of order i, that is to the multilinear map Dui G (resp. Dui (G ◦ f )) (this same shift of notation occurs implicitly in [2], Lemma 5.4). The first two terms read: H0 = G0 ◦ f0s , H1 = G1 ◦ f0s · f1u + ((Dw G0 ) ◦ f0s ) · f1s , (4) where the argument is ws and if a quantity φ is defined near W we write φ(ws ) for the restriction φ(ws , 0). In particular, in the formula above f1u = Au . We expand the perturbation F ◦ f in a similar way and for integer i ≥ 0 we denote by E i (ws ) the i-th order term of the expansion. A little contemplation yields the following two pieces of information about the Hi and Ei , which are defined in a neighborhood Wρ of M inside W: i) Hi can be decomposed as Hi = Gi ◦ f0s · Aiu + Hi′ , where Hi′ involves only the Gj for j < i; ii) Ei is analytic and vanishes on M at order k − i. We can now rewrite (E ′ ) along W as an infinite system: Gi (ws ) − Gi ◦ f0s (ws ) · Aiu = −Ei − Hi′
(5)
which we wish to solve to a finite order ℓ ≤ k, to be determined below. The important point is that we are now in a position to apply Proposition 4.2 in §1 recursively, with Q = Aiu . We apologize at this point that the matrix Q in that proposition stands to the left of the unknown whereas here it stands to the right. It made the writing easier in both cases but it should be plain that this does not actually alter the statement or proof. Here we fully benefit from the analyticity of the data, which enables us to give a much shorter proof and get a much more effective statement than in the smooth case. The latter does not seem to have been treated in the presence of a nontrivial invariant manifold. Again we encounter the fact that the dynamics on M will not play any role in our present analytic setting as it certainly would in the smooth case, via its Lyapunov exponents. In other words no quantity of the type ν or νρ occurs in this section, as it did in Propositions 3,4 and will again in the next section. Let us make sure that the assumptions in Proposition 4.2 can be met and that it applies recursively to yield the Gi ’s for 0 ≤ i ≤ ℓ. Looking back at the statement we find that we now have f = f0s describing the dynamics on the stable manifold near M. We write Ds f0s (ws ) = Ds f s (ws , 0) = As (ws ) (with Ds the derivative w.r.t. xs ) thus again extending the notation As (y) of §1 to a neighborhood of M inside W s (As (y) = As (y, 0)). By assumptions (2) and (3) of §1, As (y, xs ) = Ds f s (y, xs , 0) and Au (y, xs ) = Du f u (y, xs , 0) are bounded on M, that is for xs = 0, by λs < 1 and µ−1 u ≥ 1 respectively. We can now strengthen these assumptions as in the statement of Proposition 4.2, assuming the existence of ρ > 0 such that: |As (y, xs )| ≤ λs,ρ < 1,
|Au (y, xs )| ≤ µ−1 u,ρ ,
(6)
−1 for ws = (y, xs ) ∈ Wρs , that is simply for |xs | < ρ. Here of course µ−1 u,ρ ≥ µu ≥ 1 and λs ≤ λs,ρ < 1 and there always exists such a ρ > 0 if M is compact. We remark that
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these assumptions can be expressed either in terms of spectral radiuses or of norms of matrices because even in the not necessarily semisimple case one can adapt the norm on the space so that the norm of the relevant matrix is arbitrarily close to its spectral radius. Finally we can leave the assumptions on the perturbation in Proposition 4.2 as is, that is we assume that there exists Kρ > 0 such that on Wρ : |F (ws )| ≤ Kρ |xs |k ,
(7)
with F (ws ) = F (ws , 0). We may now state the following: Proposition 4.6 (Analytic preparation lemma). Assume that f and F are defined and analytic over Wρ = M × Bρs (Bρs = {|xs | < ρ} ⊂ E s ), that they satisfy (6) and (7) −ℓ and that λk−ℓ s,ρ µu,ρ < 1 for an integer ℓ ≥ 0. Then the homological equation (E) can be solved uniquely for the jet of order ℓ of G on Wρ . The solution is analytic on Wρ and vanishes on M at order k. In other words there exists a unique G(ℓ) (z) which is analytic in ws ∈ Wρ and polynomial of degree ℓ in xu such that: Duj G(ℓ) − G(ℓ) ◦ f + F ◦ f = 0 for 0 ≤ j ≤ ℓ. (8) P In shorthand one can write G(ℓ) (z) = ℓi=0 Gi (ws )xiu where i denotes here again a multiindex and the Gi ’s we have worked and will be working with regroup these for a given length i of the multiindex. The proof consists indeed in an iterative application of Proposition 4.2 where at the i-th step we apply it with k − i instead of k, f0s (the restriction of f to W s ) G = Gi , Q = Aiu and the right-hand side Di = −Ei − Hi′ which depends on F and the Gj ’s for j < i. Step 0 is just Proposition 4.2 or actually Proposition 4.1 and yields the initial term G(0) = G0 (ws ). The assumptions are readily seen to carry over by induction, namely Gi and Ei are analytic over Wρ , they vanish at order k − i on M and Ei satisfies the analog −i of (7). Finally the inequality λk−i s,ρ µu,ρ < 1 holds true since i ≤ ℓ. One uses essentially the fact that Ei is analytic on Wρ and vanishes on M at order k − i and that Proposition 4.2 provides a solution which is also analytic over Wρ and vanishes on M at order k (which is replaced by k − i at step i). The fact that Hi also vanishes at order k − i on M is then purely formal. −ℓ If M is compact the existence of ρ > 0 such that the inequality λk−ℓ s,ρ µu,ρ < 1 is satisfied −ℓ results from the corresponding condition on M itself, that is λk−ℓ s µu < 1. If moreover M is symplectic, one has λs = λu = λ, µs = µu = µ (cf. §1) and this can be rewritten as λk−ℓ < µℓ . Technical Note: We corrected above what seems to be an overly optimistic assertion in [2], Lemma 5.4. One finds there a short sketch of proof in the smooth case with an invariant manifold reduced to a point. However the contact with M at step i is taken to be k and not k − i (see also [13], Lemma 4.1 on that point). This results in an overestimate of ℓ,
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−ℓ replacing k − ℓ by just k in the defining inequality: λk−ℓ s,ρ µu,ρ < 1. We also point out that this factor k − ℓ is of a quite different nature from the one appearing in the statement of Lemma 5.5, still in [2]. The latter factor can indeed be improved to k as suggested by the remark there, and this is precisely what we did in Proposition 4.4 above. Although these observations may look quite technical they actually reflect rather simple geometric phenomena.
4.4 The general analytic normally hyperbolic case We now explain how to use the above ‘preparation lemma’ in order to bring back the general hyperbolic case to the contracting case. This is where we will lose both analyticity and uniqueness by using arbitrary smooth cut-off functions. This cannot be avoided: In the hyperbolic but non contracting case there is in general no (germ of) analytic conjugacy for analytic data. So there is no solution to the conjugacy problem in the analytic category. By contrast one can find solutions which are differentiable to a high order (as we will proceed to demonstrate presently) but they are far from being unique: Germs of hyperbolic analytic diffeomorphisms usually have huge centralizers (cf. [10]). Note finally that linearization results requiring diophantine arithmetic conditions in the style of the celebrated Siegel theorem and its variants are not really relevant in our setting, if only because of the presence of an invariant manifold. In this section we will again favor readability in the sense that we will prove a fairly simple istotropic statement without insisting on explicit estimates. The reader who follows the by now simple proof will immediately perceive that we show actually slightly more than what is stated and that most steps can be made fairly explicit. In this field statements cannot usually be applied as stated and the potential user may find it easier to modify the statement rather than the proof. The general setting is as in Section 4.1. We start from a family (fε ) (0 ≤ ε ≤ 1) of analytic diffeomorphisms which are obtained from f0 by a deformation using the Hamiltonian Fε . The fε have a contact of large order (= k − 1) along M; equivalently Fε vanishes on M at order k. We are looking for a germ of conjugacy gε between f0 and fε ; gε will be defined and differentiable to some order m in a tubular neighborhood of M. Equivalently we are looking for a solution Gε of the homological equation (E) of class C m in a neighborhood of M. The diffeomorphism g1 , obtained by integrating (5) of Section 4.1 gives the answer to the original conjugacy problem between f0 and f1 . From now on we drop again the index ε for the most part; as usual everything will be continuous (and in fact much more) in ε which suffices for our needs. Note also that ε varies over the closed unit interval so that all estimates are de facto uniform in ε. We take up the notation of the last section. For ρ > 0 we write Bρs = {|xs | < ρ} ⊂ E s , Bρu = {|xu | < ρ} ⊂ E u . For x = (xs , xu ) ∈ E we use the norm |x| = max(|xs |, |xu |) for convenience. In particular Bρ = {|x| < ρ} = Bρs × Bρu ⊂ E. We denote by Wρs = M × Bρs ⊂ W s the local stable manifold and we now write Mρ = M × Bρ ⊂ V for a tubular neighborhood of M in V . The data and the conclusions are all local near M, that
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is over Mρ for some ρ > 0. We will not try to really keep track of an explicit value but the reader can check that a (ridiculously small) explicit value could be extracted from the procedure described below. We write z = (y, xs , xu ) = (ws , xu ) and decompose f as f (z) = (f i , f s , f u ) ∈ M × E s × E u . We remark –better late than never– that we do not notationally distinguish strongly stable from stable manifolds (and ditto for unstable) but this should not cause confusion. Here f i described the M-component, that is the motion along the invariant manifold (f i (y, 0, 0) = f (y) ∈ M is the induced dynamics). For z ∈ Mρ we define As,u (z) = Ds,u f s,u (z), extending to a neighborhood of M in V the notation of the last section where attention was confined to Wρs . On M (z = (y, 0, 0)) this again extends the notation of §1. We reintroduce the quantities λs,u and µs,u of §1 and will need to control the spectrum of f −1 on M (which is independent of ε). So we introduce ν as in (4) of §1: |Dy f (y)| < ν and also define ν¯ such that |Dy f −1 (y)| < ν¯. Equivalently: ν¯−1 < |Dy f (y)| < ν for y ∈ M. As usual if M is compact these quantities are known to exist and if not we assume that they do. As usual again we actually assume more, that is that these quantities λs,u , µs,u and ν, ν¯ can be continued to a tubular neighborhood Mρ of M into quantities which we denote as before with an index ρ (λs,ρ etc.) which bound the spectra of As,u and Dy f i respectively over Mρ . Again if M is compact this is not an assumption but just a matter of notation. Let us now gather together our assumptions on the family fε . We assume that there exists ρ > 0 such that: i) All fε and Fε are analytic over Mρ and indeed can be continued into the complex domain to a strip of constant width in all variables; ii) The quantities λs,ρ, etc. bounding the derivatives of fε exist over Mρ with λs , ρ < 1 and λu,ρ < 1; iii) Fε satisfies the inequality: |Fε (z)| ≤ Kρ |x|k for z ∈ Mρ and a constant Kρ > 0.
We remark that all these assumptions reduce to the existence and analyticity of fε and Fε in a neighborhood of M if the latter is compact. Because of i) Cauchy formula estimates ensure that the higher derivatives of the fε and of the perturbation Fε satisfy the assumptions of Proposition 4.4, that is (19) and (20) of §3. We now state a rough version of our final result:
Theorem C. Assume that fε and Fε satisfy i), ii) and iii) above and that k > 0 is large enough. Then the homological equation (E) has a solution G of class C m in a tubular neighborhood Mr = M × Br of the invariant manifold M for some r > 0. One can take m = [ck] ([x] is the integral part of x) for some constant c > 0; G vanishes at order m on M. As mentioned above we actually show more than what is stated above and will gather part of that information after the proof. Returning to the original conjugacy problem, the above ensures the existence of a C m conjugacy between f0 and f1 in a tubular neighborhood of M (of constant width), whose jet of order m − 1 coincides with that of the
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identity along M. As explained at the end of §4.1, because we cannot control what happens globally along the invariant manifolds, we now have to modify the original data using a smooth truncation, thereby giving up analyticity. We first modify the family fε so as to make it constant (w.r.t. ε) and linear hyperbolic at infinity. Namely denote by A the dynamics on the tangent bundle of M, that is: A(z) = (f (y), As (y), Au(y)) where of course f (y) = f (y, 0, 0) describes the induced dynamics. The diffeomorphism A is analytic and independent of ε as we assume k > 2. Let χ : R+ → R+ denote a truncating function: χ ∈ C ∞ is monotone decreasing, χ(r) = 1 for r ≤ 1 and χ(r) = 0 for r ≥ 2 (say). The precise shape of χ is immaterial for our purpose. For R > 0, χR will denote the rescaled function: χR (r) = χ(r/R). We now replace fε with f˜ε which interpolates between fε and A. Specifically and dropping the subscript ε, we define f˜ = (f˜i , f˜s , f˜u ) by: f˜i (y, x) = f i (y, χR (|x|)x);
f˜s,u (z)) = χR (|x|)f s,u(z) + (1 − χR (|x|))As,u (y),
so that f˜ε coincides with fε on MR and with A outside of M2R . We pick R < ρ/2 where ρ is the value occuring in assumptions i), ii), iii) above. This implies that for the new quantities f˜ and F˜ assumptions ii) and iii) still obtain and that indeed the higher derivatives of f˜ and F˜ still satisfy estimates of type (19) and (20) in §3, simply because of their definition and the smoothness of χ. We drop the tildes from now on but remember the crucial fact that the new data coincide with the old ones over MR , so in particular are analytic there. It should perhaps be noticed at this point that in a practical case one can start from a local situation in some Mρ and the above can be used in order to extend the situation to the whole of M × E. We now apply Proposition 4.6 on WRs . We note that the values of λs,ρ and µs,ρ occuring there refer to Wρs but for simplicity we can a fortiori use the values that occur in assumption ii) above (using also that R < ρ). This is one of the several places where our present assumptions could be weakened if necessary. So we solve (E) for the jet along WRs of order ℓ satisfying: −ℓ λk−ℓ (1) s,ρ µu,ρ < 1, the values of the constants being as in assumption ii) above. We then extend that solution G(ℓ) to the whole of W s = M × E s using the functional equation it satisfies, namely (E) (More precisely we are actually extending the Gi (ws ), i = 0, . . . , ℓ; cf. (1) in §3). Recall that G(ℓ) is actually the unique solution vanishing on M; it is also analytic on WRs . Finally all fε coincide with A outside of M2R , so that Fε is in fact constant outside that ball and the extension there is completely explicit (this will not be needed in the sequel). Let us now replace G(ℓ) with G1 (z) = χR (|xu |)G(ℓ) , that is localize around W s (this explicit truncation is not really necessary and is more a matter of psychological comfort). The function G1 solves near W s an equation of type (E) with a right-hand side F1 such that the order ℓ jets of F and F1 coincide on W s . Consequently the function G−G1 in the strip W s ×BRu solves an equation with right-hand side F −F1 which vanishes at order ℓ on W s . One can actually say more: Because of the analyticity of the original perturbation
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and the way it was truncated we are now in a position to apply Proposition 4.4. We should replace there f with f −1 (V.Arnold once remarked that the ‘stable manifold’ derives its name from the fact that it is unstable), M with W s , k with ℓ. We also set r = ∞ and the derivatives of f are bounded to all orders (in other words the assumption of the second statement are satisfied). As for the constants λρ and νρ occuring in the statement of Proposition 4.4, they should now be interpreted as follows. First λρ is to be replaced simply by λu,ρ < 1 as in assumption ii) of the Theorem. This can be seen easily, recalling that we patched the original f with its linear part A along M, which has unstable exponents (µu , λu ) in the notation of (3), §1. Second, we should replace νρ with −1 µ−1 ¯ρ ). This number indeed controls the expansion for f −1 inside the strip W,s = max(µs,ρ , ν W s × BRu . Note that this is the first and last time here where we need to worry about the dynamics on M because we used the analytic preparation lemma, in which it does not appear (nor does it appear in Propositions 1 and 2). The integer m in the statement should thus be small enough so as to satisfy the inequality: λℓu,ρ µ−m W,ρ < 1.
(2)
Under this condition we can apply Proposition 4.4 and find a solution G as in the statement of the theorem. Let us finish with some elucidations and complements, first repeating one last time that if M is compact the only assumption becomes the existence and analyticity of the data, say f0 and the deformation Hamiltonian Fε in the vicinity of M. Given k, which is defined by the fact that the diffeomorphisms fε have a contact of order k − 1 along M, we get a C m conjugacy with m controlled by (1) and (2) (concerning the exponent k − ℓ in (1), see the technical note at the end of §3). If M is symplectic λu = λs = λ, µu = µs = µ ≤ λ < 1, ν¯ = ν ≥ 1. So the best possible value of m, possibly at the expense of a very small ρ is given by the inequalities: λk−ℓ < µℓ , λℓ < µm W . If moreover the dynamics on M is elliptic, that is if ν = ν¯ = 1, one has µW = µ and one gets the inequalities λk−ℓ < µℓ , λℓ < µm (recall that µ ≤ λ < 1). Concerning the order of contact along M of the conjugating diffeomorphism with the identity, or what amounts to the same the order of vanishing of the solution G of the homological equation on M, one can say more. It is indeed to be expected that G vanishes at order k but we only proved that it is C m , vanishing on M at order m < k. However we note that we first applied the analytic preparation lemma which yields a solution G(ℓ) which is analytic near M and remains so after truncation. Moreover it does vanish on M along W s at order k, according to Proposition 4.6. We then applied Proposition 4.4, which leaves untouched the restriction of G to W s . According to the second part of Proposition 4.4, which we may apply with r = ∞ as noted above, the solution G is C ∞ with respect to xu and does vanish to order k on M in that direction. In other words we actually showed that the higher order derivatives Dsq G and Duq G exist for all q and do vanish for q ≤ k. We did not however study the existence and vanishing of the mixed Ds /Du transverse derivatives along M of orders between m and k.
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4.5 Application to the case at hand We will now check that the above result does apply to the system which is the subject matter of the present paper. Indeed, after some notational translation Theorem D below will be a direct application of Theorem C above. So we are again interested in the family Fq of symplectic diffeomorphisms of A2 introduced in §2. We write F∗ for the unperturbed diffeomorphism (that is when q = ∞), again as in that paragraph. Explicitly we have: 1
2
2
1 2
1 2
F∗ = Φ 2 (r1 +r2 )+cos 2πθ1 = Φ 2 r1 +cos 2πθ1 × Φ 2 r2
The diffeomorphism Fq is the perturbation of F∗ explicitly given by (2.3). Implicit is the choice of a width σ as in (2.8) which will play no role whatsoever in what follows. Finally recall that Fq is √1q -close to F∗ for large q (see (2.9)). As for the conjugacy problem, we proceed as follows: Fixing q large enough, we regard Fq as given by the (ε) time-one map of the autonomous (i.e. ε-independent) Hamiltonian ε (q) f . In other words we deform along a straight line in the space of Hamiltonian functions. q It is then easily checked that the analog of Theorem C above is valid uniformly for q large (q) enough. In fact, in view of the expression of f (q) and especially f1 (f2 is independent of q; cf. (2.4)) we find that (writing z for θ1 ) the q-dependence enters through the sequence of functions (z 7→ z ν ) on a small disc near the origin (see (2.5); ν = ν(q; σ)). It is as well-behaved as can be; in particular for any given N it converges to 0 in the C N -topology as q increases to infinity. The reader can then easily check that all the constructions of the previous sections are uniform with respect to this added parameter. The annulus A which is invariant and normally hyperbolic for Fq will play the role of M. Recall that A = {O} × A ≃ A where O denotes the hyperbolic fixed point of the pendulum P , with coordinates θ1 = r1 = 0. Let Bρ denote, for ρ > 0, the neighborhood of O in A defined by: Bρ = {(θ1 , r1 ) ∈ A, |θ1 | + |r1 | < ρ}. We let Aρ = Bρ × A denote the corresponding tubular neighborhood of A in A2 . Our first and main goal is to prove the following local conjugacy statement: Theorem D (Local conjugacy around the invariant annulus). For q0 large enough, there exist ρ, ρ′ , with 0 < ρ′ < ρ such that for q ≥ q0 , Fq (A2ρ′ ) ⊂ Aρ and the following holds: There exists a C k diffeomorphism φq (k ≥ 1) defined on Aρ with Aρ′ ⊂ φq (A2ρ′ ) ⊂ Aρ and such that on Aρ′ : φq ◦ F ∗ = F q ◦ φq . Moreover there is a constant a (0 < a < 1) such that: ν(q) ||φ±1 , q − Id|| ≤ a
for the C k norm on Aρ′ . Here and below we abbreviate ν(q; σ) (cf. (2.5)) to ν(q). The above estimates are uniform in q in the sense that the constants ρ, ρ′ and a are independent of q ≥ q0 . The
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regularity index k can in fact be taken to be large as q goes to infinity and indeed one can ensure k = k(q) = [cν(q)] (of order Log q) for some constant c > 0. This fact, that the local conjugacy actually gets smoother as q approaches infinity will not be needed in the sequel. Let us now see how Theorem C implies Theorem D. The annulus A is symplectic and the map induced by Fq is elliptic, actually an integrable twist so that in the notation of the previous sections we have ν = ν¯ = 1. We also have λu = λs = λ, µu = µs = µ and there is only one transverse exponent, so that λ = µ. Finally this number is nothing but the stable exponent of the time-one map of the pendulum at the point O, namely λ = e−2π , the exact value being however irrelevant for our present purpose. Because A is not compact we will have to check some uniformity property w.r.t. to the variable r2 which comes readily from the fact that the perturbation depends on the angles only (see below). Before we do that however we go on with a few simple reductions. The main parameters in the statement are ρ and a, which are independent of q. Once we have found ρ, ρ′ is determined by the condition that Fq (Aρ′ ) ⊂ Aρ . Roughly speaking we should take 2ρ′ ∼ λρ (λ = e−2π as above), so that for q large enough we may actually set ρ′ = λρ/4. We also note that, if ||φq − Id|| ≤ aν(q) the same estimate holds for φ−1 q , perhaps at the expense of increasing a slightly. Below we will not deal in detail with domain and inversion problems. They involve as usual the effective application of the classical implicit function theorem and in our context this has been detailed in [27] (see especially §5.2 there). We first need to dispose of a preliminary step, namely the straightening of the stable and unstable manifolds. We are working locally around A and uniformly in q for q large enough on a domain Aρ . We wish to symplectically conjugate Fq to a diffeomorphism ± F q near A so that the local manifolds Wloc (A, F q ) of F q coincide with the planes of our coordinate system, thus providing the product structure which is required in the setting of §1 above. In order to achieve this, the main ingredient is the result of [27] (§5) which ± asserts that the local manifolds Wloc (A, Fq ) can be represented as graphs of analytic ± functions vq on suitable domains, after having performed a linear transformation on the first factor in order to let the axes coincide with the eigendirections at the hyperbolic point O. We refer to [27] for details and the proof. Given the existence of this analytic graph parametrization, one constructs a local conjugacy hq between Fq and F q , as explained in [23], §1.9. Moreover, Proposition 4.5.1 of [27] provides an estimate of the form: ||vq± − v0 || ≤ cν(q) ; here c < 1 is a constant and the norm is the sup-norm over a complex domain which contains a thickening of Aρ for some ρ > 0 (v0 = v0+ = v0− ). As a result we get the ν(q) estimate: ||h± after slightly increasing c if necessary. q − h0 || < c We wish to apply Theorem C to the family F q ; this will provide us with a local −1 conjugacy φ¯q between F q and F ∗ : F q = φ¯q ◦ F ∗ ◦ φ¯−1 q . Given that Fq = hq ◦ F q ◦ hq , we will get φq as: φq = hq ◦ φ¯q ◦ h−1 0 . In view of the exponential estimate of the difference ± hq − h0 recalled above, this shows that in order to secure the estimate in the statement of Theorem D, it is enough to get one of the same form for φ¯q .
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We are now reduced to applying Theorem C to F q for fixed large enough q, considering that it is obtained from F ∗ by a straight line deformation governed by the Hamiltonian 1 ∗ h (f (q) ). Looking back at conditions i), ii) and iii) before the statement of Theorem C, q q we find that i) is clearly fullfilled. Turning to ii), we have seen above that on A the only parameters are ν = ν¯ = 1 and λ = µ < 1. In order to extend this to Aρ , for ρ small enough, into estimates of the needed kind, we must check that the deformation is uniform w.r.t. the variable r2 , that is in the noncompact direction along the invariant manifold A. But this is clear, because firstly the initial deformation f (q) is independent of r2 and secondly the straightening diffeomorphism hq is uniformly close to h0 , the latter being in turn independent of r2 because F∗ has a simple product structure. Finally iii) holds with k tending to infinity as q goes to infinity, with a value of ρ which is uniform in q for q large enough. This yields Theorem D and slightly more; namely first as mentioned above the regularity index k can be made to tend to infinity as q approaches infinity and second if one considers the constant a as a function of the radius ρ, one may in fact pick a = a(ρ) = ρc for ρ small enough and a (possibly very small) constant c > 0. Having obtained a local conjugacy in a tubular neighborhood of the annulus A, we now would like to extend it along the invariant manifolds of A, much as in [27], §5.2. We use the pieces of notation Bρ and Aρ as above and we choose ρ and ρ′ as given by Theorem D. Let Q be the midpoint of the separatix of the pendulum, with coordinates Q = (r1 , θ1 ) = (2, 1/2). We let Qσ denote the ball with center Q and radius σ (i.e. the tubular neighborhood of Q with thickness σ) in the pendulum plane (or annulus) (θ1 , r1 ). Let A˜ denote the annulus Q × A where A denotes as usual the (θ2 , r2 ) annulus. Finally let A˜σ = Qσ × A denote the tubular neighborhood of A˜ with thickness σ. If ΦP is as usual the time-one map associated with the flow of the pendulum, we let m be a positive integer such that (ΦP )m (Q) ∈ Bρ′ , that is the point Q enters Bρ′ after m iterations (or less; we do not require that m be minimal). By symmetry the same property holds if we iterate backward, that is we also get that (ΦP )−m (Q) ∈ Bρ′ . Although the numbers ρ (and ρ′ ) as well as m will be kept fixed in what follows we note that for small 1 ρ the integer m is on the order of 2π ln(1/ρ). We now remark that for fixed ρ and m we can choose σ small enough so that for q large enough Fq±m (A˜σ ) ⊂ Aρ′ . Indeed for q = 0 this comes simply from the continuity of F∗m with respect to the initial conditions and we can extend this to large enough q’s because the difference Fq − F∗ vanishes as q tends to infinity, being actually of order √1q . Again σ will be held fixed in what follows (one may think of σ as being on the order of λm ρ). Let n ≥ 2m be a positive integer, to be thought of as ‘large’, actually much larger than m. For a point ̟ ∈ A2 , let us write the following formal equality, of which we will subsequently make sense for ̟ in a certain domain: m n −1 Fqn (̟) = Fqm ◦ Fqn−2m ◦ Fqm (̟) = Fqm ◦ φq ◦ F∗n−2m ◦ φ−1 q ◦ Fq (̟) = ϕq ◦ F∗ ◦ ψq (̟), (1)
with ϕq = Fqm ◦ φq ◦ F∗−m and ψq = Fq−m ◦ φq ◦ F∗m ; here φq denotes of course the local conjugacy whose existence is asserted by Theorem D. The maps ϕq and ψq are welldefined C k diffeomorphisms on A˜σ , with k as in Theorem D. They are also √1q -close to
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the identity map, so that we can find σ ′ , with 0 < σ ′ < σ such that A˜σ′ ⊂ ϕq (A˜2σ′ ) ⊂ Aσ and ditto for ψq (see again §5.2 of [27] for quantitative estimates in a similar setting). The following statement is now within easy reach: Proposition 4.7. Let m, ρ, ρ′ , σ and σ ′ be as above (0 < σ ′ < σ < ρ′ < ρ). For q large enough (i.e. q ≥ q0 ) there exist two C k diffeomorphisms ϕq and ψq (with k as in Theorem D) which are defined on A˜σ and are √1q -close to the identity map as q tends to infinity, such that for any integer n ≥ 2m and any point ̟ ∈ A˜σ′ ∩ F −n (A˜σ′ ) the ∗
following intertwining relation holds:
Fqn ◦ ψq (̟) = ϕq ◦ F∗n (̟). Indeed assume that ̟ ∈ A˜σ′ ∩ F∗−n (A˜σ′ ), i.e. it is a point whose first projection is very close to Q and whose unperturbed orbit returns there at time n. Equation (1) now makes good sense for ̟. The main point is that if ̟m = F∗m (̟), we find that ̟m ∈ Aρ and F∗n−2m (̟m ) ∈ Aρ′ , the latter property coming from the fact that F∗n−m (̟) = F∗n−2m (̟m ) ∈ F∗−m (A˜σ′ ) ⊂ Aρ′ . We also note that since m is kept fixed Fqm (̟) and F∗m (̟) are √1q -close and we may pick q large enough so that F∗n−2m ◦ Fqm (̟) ∈ Aρ . Proposition 4.7 yieds a nice intertwining relation but the intertwining maps are a priori only √1q -close to the identity map. Our final result, which was applied in §2, provides a slighly less natural intertwining relation, but ensures that the intertwing maps are very close to identity. As will be apparent from the proof it is an immediate consequence of Proposition 4.7 supplemented by an easy observation. We first state: Theorem E. Let m, ρ, ρ′ , σ and σ ′ be as above (0 < σ ′ < σ < ρ′ < ρ) and set χq = F∗ ◦ Fqm−1 ◦ φq ◦ F∗−m ,
ψq = Fq−m ◦ φq ◦ F∗m .
For q large enough (i.e. q ≥ q0 ) χq and ψq are two C k diffeomorphisms (with k as in Theorem D) which are defined on A˜σ and are cν -close to the identity map as q tends to infinity, for some constant c (0 < c < 1). Moreover for any integer n ≥ 2m and any point ̟ ∈ A˜σ′ ∩ F∗−n (A˜σ′ ) the following intertwining relation holds: F∗ ◦ Fqn−1 ◦ ψq (̟) = χq ◦ F∗n (̟). In order to prove this statement, let us analyze the maps ϕq and ψq a little more closely, first recalling their respective definitions, namely ϕq = Fqm ◦ φq ◦ F∗−m and ψq = Fq−m ◦ φq ◦ F∗m . By Theorem D we have that φq is actually cν -close to identity, where here and below we use the letter c to denote a generic constant satisfying 0 < c < 1 and ν = ν(q). Moreover the difference Fq − F∗ is of order √1q on the whole of A2 , but it is in fact of order cν outside of A˜σ for any fixed σ > 0.
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Upon examining the definition of ϕq and ψq we find that the only place where the perturbed map Fq is applied in a region where the perturbation is significant occurs in the very last factor of ϕq (reading of course from right to left). So we modify this by simply replacing Fq with F∗ in that factor. This leads to the definition of χq and finishes the proof because now in the definition of χq and ψq the factors Fq are all applied outside of A˜σ , which yields an ‘exponential’ estimate as in the statement. As a final remark, we emphasize that the threshold of validity q0 in Theorem E (as well as in Proposition 4.7) is independent of n, an important uniformity feature which is actually used in our application of the result.
References [1] V.I. Arnold: “Instability of dynamical systems with several degrees of freedom”, Dokl. Akad. Nauk SSSR, Vol. 156, (1964), pp. 9–12; Soviet Math. Dokl., Vol. 5, (1964), pp. 581–585. [2] A. Banyaga, R. de la Llave and C.E. Wayne: “Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem”, J. Geom. Anal., Vol. 6, (1996), pp. 613–649. [3] A. Banyaga, R. de la Llave and C.E. Wayne: “Cohomology equations and commutators of germs of contact diffeomorphisms”, Trans. AMS, Vol. 312, (1989), pp. 755–778. [4] G. Benettin and G. Gallavotti: “Stability of motions near resonances in quasiintegrable Hamiltonian systems”, J. Phys. Stat., Vol. 44, (1986), Vol. 293–338. [5] G. Benettin, L. Galgani and A. Giorgilli: “A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems”, Celestial Mech., Vol. 37, (1985), pp. 1–25. [6] U. Bessi: “An approach to Arnold’s diffusion through the calculus of variations”, Nonlinear Anal. TMA, Vol. 26, (1996), pp. 1115–1135. [7] U. Bessi: “Arnold’s example with three rotators”, Nonlinearity, Vol. 10, (1997), pp. 763–781. [8] J. Bourgain: “Diffusion for Hamiltonian perturbations of integrable systems in high dimensions”, (2003), preprint. [9] J. Bourgain and V. Kaloshin: “Diffusion for Hamiltonian perturbations of integrable systems in high dimensions”, (2004), preprint. [10] M. Chaperon: “G´eom´etrie diff´erentielle et singularit´es de syst`emes dynamiques”, Ast´erisque, Vol. 138-139, (1986). [11] M. Chaperon: “Invariant manifolds revisited”, Tr. mat. inst. Steklova, Vol. 236, (2002), Differ. Uravn. i Din. Sist., pp. 428–446. [12] M. Chaperon: “Stable manifolds and the Perron-Irwin method”, Erg. Th. and Dyn. Syst., Vol. 24, (2004), pp. 1359–1394. [13] M. Chaperon and F. Coudray: “Invariant manifolds, conjugacies and blow-up”, Erg. Th. and Dyn. Syst., Vol. 17, (1997), pp. 783–791.
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[14] B.V. Chirikov: “A universal instability of many-dimensional oscillator systems”, Phys. Reports, Vol. 52, (1979), pp. 263–379. ´ Norm. [15] R. Douady: “Stabilit´e ou instabilit´e des points fixes elliptiques”, Ann. Sc. Ec. Sup., Vol. 21, (1988), pp. 1–46. [16] R. Easton and R. McGehee: “Homoclinic phenomena for orbits doubly asymptotic to an invariant three-sphere”, Ind. Univ. Math. Journ., Vol. 28(2), (1979). [17] E. Fontich and P. Mart´ in: “Arnold diffusion in perturbations of analytic integrable Hamiltonian systems”, Discrete and Continuous Dyn. Syst., Vol. 7(1), (2001), pp. 61–84. [18] G. Gallavotti: “Quasi-integrable mechanical systems”, In: K. Osterwalder and R. Stora (Eds.): Ph´enom`enes critiques, syst`emes al´eatoires, th´eories de jauge, part II (Les Houches 1984), North-Holland, Amsterdam New York, 1986, pp. 539–624. [19] M. Herman: Notes de travail, December, 1999, manuscript. [20] M.W. Hirsch, C.C. Pugh and M. Shub: Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer Verlag, 1977. [21] S. Kuksin and J. P¨oschel: “Nekhoroshev estimates for quasi-convex Hamiltonian systems”, Math. Z., Vol. 213, (1993), pp. 187–216. [22] P. Lochak: “Canonical perturbation theory via simultaneous approximation”, Usp. Mat. Nauk., Vol. 47, (1992), pp. 59–140; Russian Math. Surveys, Vol. 47, (1992), pp. 57–133. [23] P. Lochak, J.-P. Marco and D. Sauzin: “On the splitting of invariant manifolds in multi-dimensional near-integrable Hamiltonian systems”, Memoirs of the Amer. Math. Soc., Vol. 163, (2003). [24] P. Lochak and A.I. Neishtadt: “Estimates in the theorem of N. N. Nekhorocheff for systems with a quasi-convex Hamiltonian”, Chaos, Vol. 2, (1992), pp. 495–499. [25] P. Lochak, A.I. Neishtadt and L. Niederman: “Stability of nearly integrable convex Hamiltonian systems over exponentially long times”, In: Proc. 1991 Euler Institute Conf. on Dynamical Systems, Birkha¨ user, Boston, 1993. [26] J.-P. Marco: “Transition le long des chaines de tores invariants pour les syst`emes hamiltoniens analytiques”, Ann. Inst. H. Poincar´e, Vol. 64(2), (1996), pp. 205–252. [27] J.-P. Marco: “Uniform lower bounds of the splitting for analytic symplectic systems”, Ann. Inst. Fourier, submitted to. [28] J.-P. Marco and D. Sauzin: “Stability and instability for Gevrey quasi-convex nearintegrable Hamiltonian systems”, Publ. Math. I.H.E.S., Vol. 96, (2003), pp. 77. [29] N.N. Nekhoroshev: “An exponential estimate of the time of stability of nearly integrable Hamiltonian systems”, Usp. Mat. Nauk., Vol. 32, (1977), pp. 5–66; Russian Math. Surveys, Vol. 32, (1977), pp. 1–65. [30] J. P¨oschel: “Nekhoroshev estimates for quasi-convex Hamiltonian systems”, Math. Z., Vol. 213, (1993), pp. 187–216. [31] S. Sternberg: “Local contractions and a theorem of Poincar´e”, Amer. J. Math., Vol. 79, (1957), pp. 809–824. [32] S. Sternberg: “On the structure of local homeomorphisms II”, Amer. J. Math., Vol. 80, (1958), pp. 623–631.
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[33] S. Sternberg: “On the structure of local homeomorphisms III”, Amer. J. Math. , Vol. 81, (1959), pp. 578–604. [34] S. Sternberg: “Infinite Lie groups and the formal aspects of dynamical systems”, J. Math. Mech., Vol. 10, (1961), pp. 451–474. [35] D.V. Treshchev: “Evolution of slow variables in near-integrable Hamiltonian systems”, Progress in nonlinear science, Vol. 1, (2001), pp. 166–169.
CEJM 3(3) 2005 398–403
On regular polynomial endomorphisms of C2 without bounded critical orbits Malgorzata Stawiska∗ Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607, USA
Received 9 November 2004; accepted 31 March 2005 Abstract: We study conditions involving the critical set of a regular polynomial endomorphism f : C2 7→ C2 under which all complete external rays from infinity for f have well defined endpoints. c Central European Science Journals. All rights reserved.
Keywords: Regular polynomial maps, hyperbolic invariant sets, external rays, landing MSC (2000): 32H50, 37F15, 34M45
Among polynomial maps on Ck , k ≥ 2, regular polynomial endomorphisms are those whose dynamical behavior most resembles that of polynomials in one complex variable. Specifically, a regular polynomial endomorphism f : Ck 7→ Ck is a mapping f = (f1 , ..., fk ), such that f1 , ..., fk are polynomial maps of degree d ≥ 2 and −1 −1 fˆ1 (0) ∩ ... ∩ fˆk (0) = {0} (i.e., fˆj have no common components) , where fˆj is the homogeneous part of degree d of fj , j = 1, ..., k. Such a map extends holomorphically to Pk . With f we can associate the following objects: 1 log+ kf n (z)k, z ∈ Ck , n→∞ dn 1 c T = dd G, 2π J = supp(T ∧ ... ∧ T )(k times ),
G(z) = lim
K = {z ∈ Ck : {z, f (z), f 2 (z), ...}is bounded}, Recall that G is a continuous plurisubharmonic function in Ck satisfying the equation G(f (z)) = d · G(z) for all z ∈ Ck , T is a positive closed (1, 1)-current in Ck extending ∗
E-mail:
[email protected]
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trivially to Pk , µ := T ∧ ... ∧ T (k times) is an f -invariant Borel finite measure and K is a compact set in Ck with J ⊂ ∂K. (cf. [1], [2], [6], [9], [10], [12], [20]) Let C denote the critical set of f , i.e., C = {z ∈ Ck : det(Df (z)) = 0}. Critical sets play an important role in the dynamics of non-invertible maps. For example, it is well known that if f : C 7→ C is a polynomial such that K ∩C = ∅, then f is uniformly expanding on K (see e.g. [16], theorem 19.1; a recent paper [11] gives an algorithmic construction of an expanding metrics). It turns out that this result can be extended to regular polynomial endomorphisms of Ck ([8], [19]). Recall first the following notion, coming from [17]:
Definition 1. Let K be a compact subset of a smooth Riemannian manifold M such that f (K) = K for a C 1+α -map f : M 7→ M. K is a mixing repeller for f if it satisfies the following three conditions: (i) there exists an open neighborhood V of K (called a basin) such that K = {x ∈ V : f n (x) ∈ V for all n ≥ 0}; (ii) there exists c > 0 and λ > 1 such that kDfxn vk ≥ cλn kvk for all x ∈ K, v ∈ Tx M and n ≥ 1 (with respect to a Riemannian metric on M), i.e., f is uniformly expanding on K; (iii) for any open set U intersecting K there is a natural number N > 0 such that K ⊂ f N (U). Theorem 2. Let f : Ck 7→ Ck be a regular polynomial endomorphism, C its critical set and K its set of bounded forward orbits. If C ∩ K = ∅, then K is a mixing repeller for f . Proof. (cf. [19], theorem 4.0.1): (i)If K is as above, then K = f −1 (K), so any bounded open neighborhood V of K can be taken as a basin. (ii) We will work with a special kind of basin. Note that by the definition of the function G, K = G−1 (0). By the continuity of G, {{G < ε} : ε > 0} is a neighborhood base for K. Let ǫ > 0. Each component of the open set V = {G < ε} is a bounded domain, hence it admits a hyperbolic (e.g., Kobayashi) metric κε (see [13] for background on complex hyperbolic geometry). We will show that f strictly expands this metric. Note that by the maximum principle and functional equation for the Green function G we have ∂{G ≤ ε} = ∂{G < ε} = {G = ε} and f −1 ({G = ε}) = {G = ε/d} = ∂{G ≤ ε/d}.
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Therefore, {G < ε/d} lies strictly inside {G < ε}. Since C ∩ K = ∅, we can choose ε > 0 so that f : {G < ε/d} 7→ {G < ε} is a covering map (recall that f is proper, being a regular polynomial endomorphism). Hence κε (f (z), Df (z)(v)) = κε/d (z, v) for z ∈ {G < ε}. Since {G < ε/d} lies within a positive Euclidean distance from the boundary of {G < ε}, there is a constant λ > 1 such that κε/d (z, v) ≥ λκε (z, v). To prove (iii), recall that by [3] repelling periodic points of f are dense in J (which is here equal to K; see [8] for the proof), so U can be taken to satisfy U ⊂ f (U). Take a R smooth test function 0 ≤ ϕ ≤ 1, which is positive on U, 0 outside U and has ϕdµ > 0. P Note that for all n ≥ 0, 0 ≤ An ϕ ≤ 1, where (Aϕ)(x) = d1k f (y)=x ϕ(y). By a theorem of Ueda [20], {An ϕ} has a subsequence that converges uniformly on compact subsets of some basin V (which can be chosen so that C ∩ V = ∅) to a continuous function h. S S∞ n n Observe that h(z) = 0 for z ∈ / ∞ f (U). Suppose there is a z ∈ J \ 0 n=0 n=0 f (U). Then W = {z : h(z) < 1/2} is an open neighborhood of z0 . Furthermore, Z Z Z 1 n∗ n n (χW ◦ f )ϕ dµ = (χW ◦ f )ϕ d nk f µ = χW An ϕ dµ d Since the measure µ is f -mixing (see [6]), the integrals in the above equation converge R to µ(W ) ϕdµ as n → ∞. On the other hand, by Lebesgue dominated convergence, the R limit over some subsequence of n’s is equal to W hdµ, which does not exceed (1/2)µ(W ). Hence µ(W ) = 0. But W is a neighborhood of z0 ∈ supp µ, which gives a contradiction. From now on we consider only k = 2. The holomorphic extension of f to P2 acts as a rational map on the complex line Π at infinity. Let JΠ denote the Julia set of the restricted map f |Π . We have the following: Proposition 3. If f : P2 7→ P2 is a holomorphic map arising from a regular polynomial endomorphism of C2 such that K ∩ C = ∅ and f |Π is uniformly expanding on JΠ , then f satisfies Axiom A. Proof. Let Ωf be the nonwandering set for f . Note that by condition (ii) of theorem 2, K is the intersection of Ωf with the affine space C2 . Under the assumptions of the present proposition, K equals J, hence Ωf = S0 ∪JΠ ∪J, where S0 is the set of attracting periodic points of f |Π in Π (see [16], problem 19-a, for the characterization of the nonwandering set for a rational map on P1 ). This describes the decomposition of Ωf into subsets with unstable index 0, 1 and 2 respectively, so f is hyperbolic on Ωf . Periodic points of f are dense in Ωf by [3]. For a regular polynomial endomorphism f of C2 Bedford and Jonsson introduced the set of external rays E and the endpoint map e : E 7→ ∂K, in analogy to one-dimensional theory developed in [5](cf. [1], theorem 7.3 and the preceding discussion). They also studied conditions under which e is a continuous surjection.(This property is useful in
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investigating the topology of J.) They proved that e maps E H¨older continuously onto J if f is Axiom A, f −1 (S2 ) = S2 (where S2 is the subset of Ωf with unstable index 2) and W s (JΠ ) ∩ C = ∅, where W s (JΠ ) = {z ∈ P2 : dist(f n (z), JΠ ) → 0 as n → ∞}. Below we prove an analogous result for the maps satisfying the assumptions of Proposition 3. We do this in two steps. First we show that the maps satisfying the assumptions of Proposition 3 do not satisfy W s (JΠ ) ∩ C = ∅ (unlike the maps studied in [1]). Then we show that all complete external rays for such maps have uniquely defined endpoints.
Proposition 4. Let f be as in Proposition 3. Then W s (JΠ ) ∩ C 6= ∅. Proof. If f |Π is uniformly expanding on JΠ , then by [1], corollary 5.2, W s (JΠ ) = supp T ∩ P2 \ K. The closure C of C in P2 is an algebraic set (cf. [15], theorem VII.6.3), so the set reg C of nonsingular points of C supports a positive closed (1, 1)-current (the current of integration). By [7], theorem 4.4, supp T ∩ C 6= ∅. If K ∩ C = ∅, then C ⊂ P2 \ K, so W s (JΠ ) ∩ C = supp T ∩ C 6= ∅. It remains to show this intersection is not a subset of Π. But if f |Π is expanding on JΠ , then W s (JΠ ) ∩ Π = JΠ . On the other hand, C ∩ Π consists of critical points of f |Π , which cannot belong to JΠ if f |Π is expanding there. Hence W s (JΠ ) ∩ C 6= ∅. Before formulating the next theorem, let us briefly recall the construction of external rays for a regular polynomial endomorphism f . A detailed exposition can be found in [1], an overview in [2]. (For the theory and applications in dimension one, see [16] and the references given there.) One starts with constructing Riemann surfaces Wa , a ∈ JΠ , by pasting together connected components of successive preimages by f of local stable manifolds of points in JΠ . Corollary 6.8 in [1] states that each Wa is a simply connected Riemann surface and the topology of Wa as a manifold coincides with its topology as a subspace of P2 \ K. The restriction of G to Wa is a harmonic function with a logarithmic pole at a. One defines external rays for f in Wa as gradient lines of G | Wa (i.e., trajectories of the vector field grad G | Wa ). The set E is the union of all external rays in all Wa , a ∈ JΠ . The endpoint map e is defined as follows: for γ ∈ E and r > 0, er (γ) := γ ∩ {G = r} and e(γ) = limr→0+ er (γ). For each a, the set of the external rays in Wa for which er cannot be defined is finite. Moreover, by theorem 7.4 in [1], if f is expanding on JΠ , the set of points contained in incomplete gradient lines is closed and nowhere dense in W s (JΠ ). Let us consider the set E ′ of external rays γ such that er (γ) exists for all r > 0. Then the following holds:
Theorem 5. If f is as in Proposition 3 and γ ∈ E ′, then e(γ) exists. Proof. Recall the following result due to Ueda [20] (reformulated in the present version
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in [18], Remarques 3.2.3(ii)): let f be a holomorphic map of degree d ≥ 2 on CPk and let a belong to a hyperbolic set for f . Let W s (a) = {z : dist (f n (z), f n (a)) → 0 as n → ∞}. ˜ t) = ˜ |π−1 (W s (a)) is pluriharmonic, where G(z, ˜ t) = limn→∞ d−n log |f˜n (z, t)|, f(z, Then G d d (t f (z/t), t ) and π(z, t) = [z : t]. In our setting, a ∈ JΠ , which by assumption is hyperbolic (of stable index 1), so s W (a) \ {a} = W s (a) ∩ C2 , and W s loc (a) is a 1-dimensional complex manifold. Also, ˜ 1) = G(z), z ∈ C2 . G(z, ˜ being harmonic, is a real analytic function on Wa ⊂ W s (a). W s (a), a ∈ Note that G, loc JΠ are plaques of a Riemann surface lamination with a metric g introducing its complex structure (see [4] for background on laminations) By a result of [14], there is a neighborhood of a in W s (a) with respect to the metric g (which is equivalent to the Fubini- Study ˜ has finite metric on CP2 near JΠ ), in which each trajectory of the gradient field of G length. In fact, the lengths of external rays for f will be uniformly bounded in a neighborhood U of JΠ , since the local stable manifolds depend continuously on a and JΠ is a compact set. Extending the complete external rays we see that there is a uniform upper bound on Euclidean lenghts of pieces of all rays in each set {ρ′ ≤ G ≤ ρ}, 0 < ρ′ < ρ: Observe that a piece of an external ray not contained in U is a smooth image of a compact interval in R; the Fubini- Study lengths of external rays above the level {G = ρ′ } are uniformly bounded, and away from Π the Fubini- Study metric is equivalent to the Euclidean one. We need to choose suitable ρ, ρ′ > 0, since the bound on lenghts depends on these numbers. Take ρ > 0 so that in the neighborhood V = {G < ρ} of K we have kDf (z)(v)k ≥ λkvk for all z ∈ V, v ∈ Tz C2 (we can adjust the expanding metric from Theorem 2 so that c = 1; this metric is equivalent to the Euclidean one in a neighborhood of K). Let ρ′ = ρ/d and let L be the upper bound on lengths of all pieces of external rays in {ρ′ ≤ G ≤ ρ}. The lenghts of f - preimages of these rays are uniformly bounded above by L/λ. Since G(f (z)) = d · G(z), we can represent each complete external ray in V as a union of preimages of other rays by successive iterates of f . Summing the lenghts of these preimages gives a uniform upper bound L/(λ − 1) on the length of each piece of an external ray in V . Hence each complete external ray has exactly one limit point on K. (The part of the argument involving infinite summation of geometrically decreasing lengths of rays was used in [5] to prove landing of external rays for a polynomial in C that is subhyperbolic on K.)
Acknowledgment I thank Dmitry Novikov for helpful discussions on gradient vector fields.
References [1] E. Bedford and M. Jonsson: “Dynamics of regular polynomial endomorphisms of Ck ”, Amer. J. Math., Vol. 122, (2000), pp. 153–212. [2] E. Bedford and M. Jonsson: “Potential theory in complex dynamics: regular
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polynomial mappings of Ck ”, In: Complex analysis and geometry (Paris 1997), Progr. Math., Vol. 188, Birkh¨auser, Basel, 2000, pp. 203–211. [3] J.-Y. Briend and J. Duval: “Exposants de Liapounoff et distribution des points p´eriodiques d’un endomorphisme de CPk ”, Acta Math., Vol. 182(2), (1999), pp. 143– 157. ´ Norm. Sup., [4] A. Candel: “Uniformization of surface laminations”, Ann. Scient. Ec. Seria 4, Vol. 26, (1993), pp. 489–516. ´ [5] A. Douady and J.H. Hubbard: “Etude dynamique des polynomes complexes I”, Publ. Math. Orsay, (1984). [6] J.E. Fornaess and N. Sibony: “Complex dynamics in higher dimension. Notes partially written by Estela A. Gavosto”, In: NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 439, Complex Potential Theory (Montreal, PQ, 1993), Kluwer Acad. Publ., Dordrecht, 1994, pp. 131–186. [7] J.E. Fornaess and N. Sibony: “Oka’s inequality for currents and applications”, Math. Ann., Vol. 301, (1998), pp. 339–419. [8] J.E. Fornaess and N. Sibony: “Dynamics of P2 (Examples)”, In: Laminations and foliations in dynamics, geometry and topology, Contemporary Mathematics 269, AMS, 2001, pp. 47–87. [9] S. Heinemann: “Julia sets for endomorphisms of Cn ”, Ergod. Th. & Dynam. Sys., Vol. 16, (1996), pp. 1275– 1295. [10] J.H. Hubbard and P. Papadopol: “Superattractive fixed points in Cn ”, Indiana Univ. Math. J., Vol. 43(1), (1994), pp. 321–365. [11] S.L. Hruska: “Constructing an expanding metrics for the dynamical systems in one complex variable”, Nonlinearity, Vol. 18(1), (2005), pp. 81–100. [12] M. Klimek: “Metrics associated with extremal plurisubharmonic functions”, Proc. Amer. Math. Soc., Vol. 123(9), (1995), pp. 2763–2770. [13] S. Lang: Introduction to complex hyperbolic spaces, Springer- Verlag, New York, 1987. [14] S. Lojasiewicz: “Sur les trajectoires du gradient d’une fonction analytique”, Univ. Stud. Bologna, (1983), pp. 115–117. [15] S. Lojasiewicz: Introduction to complex analytic geometry, Translated from the Polish by Maciej Klimek, Birkh¨auser Verlag, Basel, 1991. [16] J. Milnor: Dynamics in one complex variable. Introductory lectures, Friedr. Vieweg & Sohn, Braunschweig, 1999. [17] D. Ruelle: “Repellers for real analytic maps”, Ergod. Th. & Dynam. Sys., Vol. 2, (1982), pp. 99–107. [18] N. Sibony: “Dynamique des applications rationnelles de Pk ”, In: Dynamique et g´eom´etrie complexes, Panoramas et Syntheses, Vol. 8, SMF, 1999, pp. 97–185. [19] M. Stawiska: Repellers for regular polynomial endomorphisms of Ck , Thesis (PhD), Northwestern University, 2001. [20] T. Ueda: “Critical orbits of holomorphic maps on projective spaces”, J. Geom. Anal., Vol. 8(2), (1998), pp. 319–334.
CEJM 3(3) 2005 404–411
Rank 4 vector bundles on the quintic threefold Carlo Madonna∗ Dipartimento di Matematica, Universit`a degli Studi di Roma ”La Sapienza”, P.le A.Moro 1, 00185 Roma, Italia
Received 3 February 2005; accepted 7 April 2005 Abstract: By the results of the author and Chiantini in [3], on a general quintic threefold X ⊂ P4 the minimum integer p for which there exists a positive dimensional family of irreducible rank p vector bundles on X without intermediate cohomology is at least three. In this paper we show that p ≤ 4, by constructing series of positive dimensional families of rank 4 vector bundles on X without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class Ext1 (E, F ), for a suitable choice of the rank 2 ACM bundles E and F on X. The existence of such bundles of rank p = 3 remains under question. c Central European Science Journals. All rights reserved.
Keywords: ACM bundles, quintic threefold MSC (2000): 14J60
1
Introduction
Let X ⊂ P4 be a smooth quintic hypersurface and let E be a rank 2 vector bundle without intermediate cohomology, i.e. such that hi (X, E(n)) = 0
(1)
for all n ∈ Z and i = 1, 2. In [6] we found all the possible Chern classes of an indecomposable rank 2 vector bundle satisfying condition (1). Moreover in [3] we showed, when X is general, if such bundles exist then they are all infinitesimally rigid, i.e. Ext1 (E, E) = 0. On the other hand, the existence of infinitely many isomorphism classes of irreducible vector bundles without intermediate cohomology on any smooth hypersurface Xr of degree ∗
E-mail:
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r ≥ 3 in P4 was shown in [2]. One may check that in the case of a general hypersurface the rank of these bundles is 23 . Hence we introduced in [3] the number BGS(Xr ) defined as the minimum positive integer p for which there exists a positive dimensional family of irreducible rank p vector bundles without intermediate cohomology on Xr . Then combining the above quoted results we get, on a general quintic X, that 3 ≤ BGS(X) ≤ 8.
(2)
In this paper we show the following: Theorem 1.1. If X is general then BGS(X) ≤ 4. We should then answer the following: Question 1.2. Let X be a general quintic hypersurface in P4 . Could it be that BGS(X) = 3? To show our main result we give examples of rank 4 vector bundles without intermediate cohomology, which are not infinitesimally rigid. The examples are constructed by means of extension classes 0 → E1 → E → E2 → 0
(3)
i.e. elements in Ext1 (E2 , E1 ), where E1 and E2 are rank 2 bundles on X. When the bundles E1 and E2 are not split then E has not trivial summand. Moreover for a suitable choice of bundles E1 and E2 , there exists a non trivial extension class such that the rank 4 bundle E which corresponds to this class does not split as a direct sum of two rank 2 bundles, because of Chern classes. Of course if E1 and E2 have no intermediate cohomology then neither does E. We then conclude with direct calculations in order to determine the appropriate choice of bundles E1 and E2 .
2
Generalities
We shall work over the complex numbers C and we denote by X ⊂ P4 a smooth hypersurface of degree 5 in P4 . Since P ic(X) ∼ = Z[H] is generated by the class of a hyperplane section, given the vector bundle E we identify c1 (E) with the integer number c1 which corresponds to c1 (E) under the above isomorphism. We identify c2 with deg c2 (E) = c2 (E) · H. If E is a rank k vector bundle on X we denote by E(n) = E ⊗ OX (n). Definition 2.1. A rank k vector bundle E is called arithmetically Cohen-Macaulay (ACM for short) if E has no intermediate cohomology, i.e. hi (E(n)) = 0
(4)
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for all i = 1, 2, and n ∈ Z. Theorem 1.1 will follow by: Proposition 2.2. Let X be a smooth quintic hypersurface in P4 . Then, there exist indecomposable rank 2 vector bundles E1 and E2 on X without intermediate cohomology such that there exists an open subset of a positive dimensional projective space parameterizing extension classes Ext1 (E2 , E1 ) which correspond to infinitely many isomorphism classes of irreducible rank 4 vector bundles E on X without intermediate cohomology. A proof of the previous proposition will be given in the next section. We will frequently use the following version of the Riemann-Roch theorem for vector bundles: Theorem 2.3. If E is a rank 2 vector bundle on a smooth hypersurface X ⊂ P4 of degree 5 with Chern classes ci (E) = ci ∈ Z for i = 1, 2, then 5 1 25 (5) χ(E) = c31 − c1 c2 + c1 6 2 6
3
The examples
In this section we will give a proof of Proposition 2.2 which is a direct consequence of Proposition 3.1 and Theorem 3.4 below. As in [3] given a rank 2 vector bundles E we introduce the non-negative integer b(E) = max{n | h0 (E(−n)) 6= 0}.
(6)
We say that the vector bundle E is normalized if b(E) = 0. Notice that changing E by E(−b) we may always assume that E is normalized. The rank two bundle E is semistable if 2b − c1 ≤ 0. If 2b − c1 < 0 then E is stable. All the possible Chern classes of rank 2 irreducible ACM bundles are listed in the following proposition (also see [6] and [3]): Proposition 3.1. Let E be a normalized and irreducible rank 2 ACM bundle on a smooth quintic X. Then (c1 , c2 ) ∈ A ∪ B where
A = {(−2, 1), (−1, 2), (0, 3), (0, 4), (0, 5), (1, 4), (1, 6), (1, 8), (4, 30)}
and B = {(2, α), (3, 20)} with α = 11, 12, 13, 14. When X is general, all of the cases in A arise on X and moreover for all the pairs (c1 , c2 ) ∈ A ∪ B the corresponding rank 2 ACM bundles are infinitesimally rigid i.e. Ext1 (E, E) = 0. Below we shall construct examples of rank 4 bundles G as extensions of type 0 → F (m) → G → E → 0,
(7)
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where m ≤ 0, and F and E are indecomposable and normalized rank 2 ACM bundles on X with Chern classes as in Proposition 3.1. Such nontrivial extensions G will exist whenever the extension space Ext1 (E, F (m)) has positive dimension, i.e. h1 (F (m) ⊗ E ∨ ) > 0. By the long exact sequence of cohomology of (7), any such extension G has vanishing intermediate cohomology since F and E are ACM. Lemma 3.2. Let E and F be two normalized and indecomposable rank 2 ACM bundles on the smooth quintic X, and suppose that h0 (F ∨ (c1 (E)−m)) = 0 (hence c1 (E)−c1 (F )−m < 0 since F is normalized). Then for any zero-locus C ⊂ X of a global section of E h0 (IC (c1 (E)) ⊗ F ∨ (−m)) = 0. Moreover, if h0 (F ∨(−m)) = 0 (hence −m − c1 (F ) < 0 since F is normalized) then h3 (F (m) ⊗ E ∨ ) = h0 (E ⊗ F ∨ (−m)) = 0. Proof. Tensoring by F ∨ (−m + c1 (E)) the ideal sheaf sequence of C ⊂ X: 0 → IC (c1 (E)) ⊗ F ∨ (−m) → F ∨ (c1 (E) − m) → OC (c1 (E)) ⊗ F ∨ (−m) → 0 we get h0 (IC (c1 (E)) ⊗ F ∨ (−m)) ≤ h0 (F ∨ (c1 (E) − m)) = 0. The rank 2 bundle E fits in the exact sequence 0 → OX → E → IC (c1 (E)) → 0; (8) and after tensoring (8) by F ∨ (−m) we get 0 → F ∨ (−m) → E ⊗ F ∨ (−m) → IC (c1 (E)) ⊗ F ∨ (−m) → 0. Therefore, since h0 (F ∨(−m)) = h0 (IC (c1 (E)) ⊗ F ∨ (−m)) = 0 then h0 (E ⊗ F ∨ (−m)) = 0, and by duality h3 (F (m) ⊗ E ∨ ) = 0.
Remark 3.3. Let E and F be as in (7), and suppose that χ(F (m) ⊗ E ∨ ) < 0. Then by the above lemma, the space of extensions (7) will be non-empty since h1 (F (m) ⊗ E ∨ ) = h0 (F (m) ⊗ E ∨ ) + h2 (F (m) ⊗ E ∨ ) − χ(F (m) ⊗ E ∨ ) > 0. More generally the argument used here works whenever h3 (F (m) ⊗ E ∨ ) < −χ(F (m) ⊗ E ∨ ). In the following table we summarize the cases pertaining to our interest depending on the Chern classes of the bundles E and F . To get the value of χ(F (m) ⊗ E ∨ ) we used the Schubert package (see [5]), and then by the Lemma we derived the lower bound for d.
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(c1 (E), c2 (E)) χ(F (m) ⊗ E ∨ )
Case
(c1 (F ), c2(F ))
m
d
(1)
(4,30)
(1,8)
−14
0
> 14
(2)
(4,30)
(0,3)
−6
−1
>6
(3)
(4,30)
(0,4)
−8
−1
>8
(4)
(4,30)
(0,5)
−10
−1
> 10
(5)
(1,8)
(0,3)
−1
0
>1
(6)
(1,8)
(0,4)
−2
0
>2
(7)
(1,8)
(0,5)
−3
0
>3
We are now ready to show the following: Theorem 3.4. Let X be a smooth quintic in P4 , and let E, F, m, d be as in the above table. Then in each of the cases (1) − (7) there exists a d–dimensional parameter space of extensions (7), with a general element G an indecomposable rank 4 vector bundle on X without intermediate cohomology. Proof. For F, E as in the above table, the dimension d = dim Ext1 (E, F (m)) is always d > 1. Therefore for such F, E there exist nontrivial extensions given by (7), and let G be one of them. Since E and F are ACM then by the cohomology sequence of (7) G is without intermediate cohomology, and by Remark 3.3 we need only to show that G is indecomposable. Suppose the contrary, i.e. that G splits. Then either or
(i) G = OX (a)⊕G1 for a ∈ Z and G1 a rank 3 bundle without intermediate cohomology, (ii) G = G1 ⊕ G2 for two rank 2 ACM bundles G1 and G2 .
We show that under the conditions of the theorem both cases (i) and (ii) are impossible. Let us start with case (i). In this case the exact sequence (7) reads as f
g
0 −−−→ F (m) −−−→ OX (a) ⊕ G1 −−−→ E −−−→ 0.
(9)
We use the following (see below for a proof).
Lemma. Under the above conditions either h0 E(−a) = 0 or h0 F (−m − c1 (F ) + a) = 0. Suppose h0 (E(−a)) = 0. Then by the exact sequence (9) tensored by OX (−a) we have h0 (F (m − a)) > 0. Let s be a non trivial global section of F (m − a), then we have a map s : OX (a) → F (m).
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Let j : OX (a) ⊕ G1 → OX (a) be the projection. Then we have the composition map ϕ := j ◦ f ◦ s : OX (a) → F (m) → OX (a). Then ϕ ∈ H 0 OX ∼ = C and hence it is either the identity map or the zero map. If this map is the identity then j ◦ f is surjective and hence ker(j ◦ f ) ∼ = OX (b) for some b ∈ Z. Then we have the exact sequence 0 → OX (b) → F (m) → OX (a) → 0 and F (m), and hence also F , splits since dim Ext1 (OX (a), OX (b)) = 0, which is absurd. Now suppose ϕ is zero. Then j ◦ f is zero. Thus the image of F (m) in the exact sequence (9) is contained in G1 . Then the kernel of g is contained in G1 , being equal to the image of f . Let i : OX (a) → OX (a) ⊕ G1 be the inclusion. By the assumption the map g ◦ i : OX (a) → E is the zero map, which means that ker g is not contained in G1 , which is absurd. Suppose now that h0 F (−m − c1 (F ) + a) = 0 and consider the dual exact sequence of exact sequence (9) 0 → E ∨ → OX (−a) ⊕ G1∨ → F ∨ (−m) → 0.
(10)
Set c = c1 (F ) and c′ = c1 (E). Since E ∨ ∼ = F (−c − m) the above = E(−c′ ) and F ∨ (−m) ∼ exact sequence reads as 0 → E(−c′ ) → OX (−a) ⊕ G1∨ → F (−c − m) → 0. This exact sequence tensored by OX (a) reads as 0 → E(−c′ + a) → OX ⊕ G1∨ (a) → F (−c − m + a) → 0. Then h0 E(−c′ + a) > 0 and a non-trivial global section s of E(−c′ + a) gives a non zero map s : OX (−a) → E(−c′ ). Arguing as above this implies that E splits, which is absurd. Then to finish the proof that case (i) cannot arise, we have to prove the lemma. Proof of the Lemma. If h0 E(−a) > 0, since E is normalized then −a ≥ 0 i.e. a ≤ 0. Suppose that h0 F (−m − c + a) > 0. Since F is normalized then −m − c + a ≥ 0. Then from conditions −m − c + a ≥ 0 and −a ≥ 0 we derive condition c + m ≤ 0 which is absurd since by hypotheses we have condition c + m > 0 (see the table). To show the theorem it remains now to consider the case (ii) i.e. when G has an indecomposable summand which is ACM of rank equal to 2, i.e. when G = G1 ⊕ G2
(11)
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with both Gi ACM of rank equal to 2. Of course, we may assume that Gi are both indecomposable otherwise we reduce to the case (i) above. Then we have non trivial extension class f
g
0 −−−→ F (m) −−−→ G2 ⊕ G1 −−−→ E −−−→ 0.
(12)
The extension class (12) is non trivial by assumption. Moreover one has c1 (Gi ) ∈ / {c1 (F (m), c1 (E)}
(13)
for i = 1, 2, by the corollary to Lemma 1.2.8 in [7]. Indeed, suppose that c1 (Gi ) ∈ {c1 (F (m)), c1 (E)} for at least on i = 1, 2. Here we note that at least one of the bundles F (m) and E is stable. Hence Gi ’s are semistable and one of these is always stable. Then from the above exact sequence we have a map between semistable bundles of the same rank with the same first Chern class where at least one is stable. Therefore, this map is an isomorphism and hence the extension class is trivial, which is absurd. Then to show that the splitting of (11) cannot arise, we will use Proposition 3.1 and a direct computation on the Chern classes. It will show that the only possibility is that the extension class (7) is trivial, which is absurd, since by assumption, G is represented by a non-trivial class in Ext1 (E, F (m)). To begin, we notice that the bundle G of (9) is normalized because F and E are, and m ≤ 0. In particular G1 and G2 are also normalized. Then we consider all the possible splitting types of G under condition (13) in all the cases (1)-(7) of the table. Some of these decompositions are easy to show to be impossible, so we give here only the cases, which require some more computations. Case (1). In this case (see the table) (c1 (G), c2 (G)) = (5, 58). By Proposition 3.1 and by condition (13) if G ∼ = G1 ⊕ G2 splits then c2 (G1 ) = 20 and c2 (G2 ) = α, with α = 11, 12, 13, 14, are the only possible cases. A direct calculation on the Chern classes shows in these cases (c1 (G), c2 (G)) 6= (5, 58). Case (2). In this case (c1 (G), c2 (G)) = (2, 18). By Proposition 3.1 and by condition (13) if G ∼ = G1 ⊕G2 then we have only two possibilities: either (c1 (G1 ), c2 (G1 )) = (2, 14) and (c1 (G2 ), c2 (G2 )) = (0, 4) or (c1 (G1 ), c2 (G1 )) = (2, 13) and (c1 (G2 ), c2 (G2 )) = (0, 5). The first case is impossible since by the Riemann-Roch theorem we have h0 F (−1) + h0 (E) = 1 < h0 (G1 ) + h0 (G2 ) = 2. The second case is also impossible since one computes h0 (G1 ) + h0 (G2 ) = 3. Case (3). In this case we have (c1 (G), c2 (G)) = (2, 19). If G ∼ = G1 ⊕ G2 by Proposition 3.1 and by condition (13) we could have possible cases (c1 (G1 ), c2 (G1 )) = (1, 6) and (c1 (G2 ), c2 (G2 )) = (1, 8) or (c1 (G1 ), c2 (G1 )) = (2, 14) and
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(c1 (G2 ), c2 (G2 )) = (0, 5). In the first case by the Riemann-Roch we compute 0 = h0 F (−1)+ h0 E < h0 G1 + h0 G2 = 3. In the second case one concludes in a similar way, since h0 G1 + h0 G2 = 1, and similarly as well for the other cases. Case (4). In this case we have (c1 (G1 ), c2 (G2 )) = (2, 20). If G ∼ = G1 ⊕ G2 by Proposition 3.1 and by condition (13) we could have possible cases (c1 (G1 ), c2 (G1 )) = (1, α) and (c1 (G2 ), c2 (G2 )) = (1, α′) with α, α′ = 4, 6, 8 and α + α′ = 15 which is impossible. Cases (5)–(7). In this case we have (c1 (G1 ), c2 (G2 )) = (1, α + 8) where α = 3, 4, 5. If G ∼ = G1 ⊕ G2 by Proposition 3.1 and by condition (13), the conclusion follows.
References [1] E. Arrondo and L.Costa: “Vector bundles on Fano 3-folds without intermediate cohomology”, Comm. Algebra, Vol. 28, (2000), pp. 3899–3911. [2] R.O. Buchweitz, G.M. Greuel and F.O. Schreyer: “Cohen-Macaulay modules on hypersurface singularities II”, Invent. Math., Vol. 88, (1987), pp. 165–182. [3] L. Chiantini and C. Madonna: “ACM bundles on a general quintic threefold”, Matematiche (Catania), Vol. 55, (2000), pp. 239–258. [4] R. Hartshorne: “Stable vector bundles of rank 2 on P3 ”, Math. Ann., Vol. 238, (1978), pp. 229–280. [5] S. Katz and S. Stromme: Schubert, a Maple package for intersection theory and enumerative geometry, from website http://www.mi.uib.no/schubert/ [6] C.G. Madonna: “ACM bundles on prime Fano threefolds and complete intersection Calabi Yau threefolds”, Rev. Roumaine Math. Pures Appl., Vol. 47, (2002), pp. 211– 222. [7] C. Okonek, M. Schneider and H. Spindler: “Vector bundles on complex projective spaces”, Progress in Mathematics, Vol. 3, (1980), pp. 389.
CEJM 3(3) 2005 412–429
Generalized interval exchanges and the 2 − 3 conjecture Shmuel Friedland1∗ , Benjamin Weiss2† 1
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA 2 Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
Received 19 February 2005; accepted 18 May 2005 Abstract: We introduce the notion of a generalized interval exchange φA induced by a measurable k-partition A = {A1 , ..., Ak } of [0, 1). φA can be viewed as the corresponding restriction of a nondecreasing function fA on R with fA (0) = 0, fA (k) = 1. A is called λ-dense if λ(Ai ∩ (a, b)) > 0 for each i and any 0 ≤ a < b ≤ 1. We show that the 2 − 3 Furstenberg conjecture is invalid if and only if there are 2 and 3 λ-dense partitions A and B of [0, 1), such that fA ◦ fB = fB ◦ fA . We give necessary and sufficient conditions for this equality to hold. We show that for each integer m ≥ 2, such that 3 ∤ 2m + 1, there exist 2 and 3 non λ-dense partitions A and B of [0, 1), corresponding to the interval exchanges on 2m intervals, for which fA and fB commute. c Central European Science Journals. All rights reserved.
Keywords: Generalized interval exchange, entropy, 2-3 conjecture MSC (2000): 37A05, 37A35
1
Introduction
Let Σ be the σ-algebra of measurable sets in R with respect to the Lebesgue measure λ. Let k ∈ N and J ∈ Σ. A := {A1 , ..., Ak } is called a partition (or k-partition) of J if A1 , ..., Ak are pairwise disjoint measurable sets whose union is J. Let I = [0, 1). Then a ∗ †
E-mail:
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k-partition A of I induces the following partition {I1 , ..., Ik } of I to k intervals: Ij = [βj−1 , βj ),
j = 1, ..., k,
β0 = 0, βj =
j X
λ(Aj ), j = 1, ..., k.
(1)
i=1
A is called regular if λ(Aj ) > 0 for j = 1, ..., k. For A ⊂ R let χA (x) be the characteristic function of A. Then the partition A induces the following generalized k-interval exchange φA : I → I: Z x φA : Aj → I j , φA (x) = βj−1 + χAj dλ, x ∈ Aj , j = 1, ..., k. (2) 0
φA : I → I is a measure preserving transformation of (I, Σ(I), λ). If each Aj is a finite union of intervals then φA is an orientation preserving interval exchange. See [1] for other generalizations of interval exchange maps. Let A ⊂ R be the following measurable set induced by A: A ∩ [m − 1, m) = Ai + m − 1 Define fA (x) :=
Z
for m ∈ Z with m ≡ i mod k.
(3)
x
χA dλ,
0
x ∈ R.
(4)
Clearly fA is a continuous nondecreasing function on R with the properties fA (0) = 0,
fA (x + k) = fA (x) + 1, x ∈ R.
(5)
A measurable set T ⊂ [s, t] is called λ-dense if λ(T ∩ (a, b)) > 0 for all s ≤ a < b ≤ t. A is called λ-dense if each Aj is λ-dense in I. Then fA is increasing on R if and only if A is λ-dense. Assume that fA is increasing on R. Let FA be the inverse function of fA . Then FA (0) = 0 and FA (1) = k. Furthermore FA = F is expansive: y − x < F (y) − F (x),
for all x < y.
(6)
Let S 1 = R/Z. Then FA induces an expansive orientation preserving k-covering map F˜A : S 1 → S 1 , which fixes 0 and preserves λ. Furthermore F˜A is λ-invertible. The λ-inverse of FA is φA . Hence the entropy hλ (φA ) is 0 if A is λ-dense. (We prove that hλ (φA ) = 0 for any partition A of I.) ˜ k , where Gk (x) = We show that F˜A is conjugate to the standard k-covering map G kx, x ∈ R. λ is conjugate to a nonatomic probability measure ω on I whose support is S 1 . ˜ k preserves ω and G ˜ k is ω invertible. Vice versa, a nonatomic G ˜ k -invariant probability G measure, ω whose support is S 1 and which is invertible with respect to ω, is conjugate to F˜A for some λ-dense k-partition A. Recall the 2 − 3 conjecture of Furstenberg [2]: Let ω be a nonatomic probability ˜2, G ˜ 3 . Then ω = λ. measure on S 1 which is invariant for G
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Furstenberg’s conjecture is one of the long standing open problems in dynamics, and its proof or a counterexample would be very interesting. Moreover, its generalization by Margulis [5] have strong connections to other areas of mathematics. The decision for the circle would shed some light on that conjecture too. Furstenberg showed that the support of ω is S 1 . Rudolph [7] proved the 2−3 conjecture ˜ 2 ) or hω (G ˜ 3 ) are positive. Thus it is left to consider the 2 − 3 conjecture if either hω (G ˜ 2 ) = hω (G ˜ 3 ) = 0. This is equivalent to the ω invertibility of G ˜ 2 and G ˜ 3. in the case hω (G We show Theorem 1.1. The 2 − 3 conjecture is false if and only if there exist 2 and 3 λ-dense partitions A and B of I respectively such that FA ◦ FB = FB ◦ FA .
(7)
Clearly the condition (7) yields the condition fA ◦ fB = fB ◦ fA ,
(8)
φA ◦ φB = φB ◦ φA .
(9)
which in turn implies We give necessary and sufficient conditions for the equality (8) for any 2 and 3-partitions A and B respectively. A k-partition C is called a k-n-partition if it is induced by the partition of I to n equal length intervals. (C is not λ-dense.) Assume that A and B are 2-n and 3-n-partitions of I respectively. Then φA , φB induce permutation σ, η respectively on the set < n >:= {1, ..., n}. Assume that (8) holds. Then σ and η are two commuting permutations. The equality (8) gives the precise structure of σ and η. We show that for n ≤ 3 there are no regular 2-n and 3-n-partitions for which (8) holds. For n = 4 there are unique regular 2-4 and 3-4-partitions which satisfy (8) 1 1 3 1 1 3 1 3 1 3 1 1 A = {[ , ) ∪ [ , 1), [0, ) ∪ [ , )}, B = {[ , ), [0, ) ∪ [ , 1), [ , )}. 4 2 4 4 2 4 2 4 4 4 4 2
(10)
It is possible to extend this example in a trivial way to any n ≥ 5, by letting σ and η to fix a few first and last integers in the interval [1, n]. For each integer m ≥ 2, where 3 ∤ 2m+ 1, the maps G2 , G3 induce regular 2 −2m and 3 −2m partitions which satisfy (8). It seems that the non-validity of the 2 − 3 conjecture is closely related to the existence of other type 2-n and 3-n-partitions which satisfy (8). We now summarize briefly the contents of the paper. Section 2 is devoted to the discussion of the connection between k-λ dense partitions and a nonatomic invariant ˜ k whose support is S 1 . In Section 3 we discuss the map φA for any kmeasure of G partition of I. In particular we show that the λ entropy of φA is zero. In Section 4 we discuss the conditions on 2 and 3 partitions A and B of I which satisfy the condition (8). In the last section we discuss the combinatorial conditions on 2-n and 3-n-partitions of I which satisfy (8). In particular we show that the example (10) is the first nontrivial
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example of 2-4 and 3-4-partitions of I satisfying (8). This example is a particular case of the examples of 2 − 2m and 3 − 2m partitions (3 ∤ 2m + 1) satisfying (8), induced by the maps G2 , G3 . We remark that the Furstenberg 2-3 conjecture generalizes to the p − q case, where p, q ∈ N\{1} are not integer powers of a fixed r ∈ N. See for example [4] and [3]. It is straightforward to generalize the results of this paper to this case too. We decided to restrict ourselves to the 2-3 case to make our exposition simpler.
2
Covering maps of S 1
Let F : I → R be a continuous function such that F (0) = 0, F (1) = k for some 1 ≤ k ∈ Z. We then extend F to R F (0) = 0,
F (x + 1) = F (x) + k for all x ∈ R.
(11)
Then F induces the map F˜ : S 1 → S 1 where the degree of F˜ is k. F˜ is a k-covering map if and only if F is increasing on R. We call F expansive if (6) holds. Theorem 2.1. Let F : R → R be a continuous increasing function on R satisfying (11) for an integer k ≥ 2. Assume that F is expansive. Then there exist a unique continuous increasing function H : R → R satisfying (11) with k = 1 such that F ◦ H = H ◦ Gk ,
(12)
˜ k on S 1 . where Gk (x) = kx. In particular F˜ is conjugate to G Proof. Observe that (11) implies that F (j) = jk for j ∈ Z. Let 1 ≤ m ∈ Z and define F ◦m = F · · ◦ F}. Then F ◦m (1) = k m . Observe that F ◦m is also expansive. For | ◦ ·{z m
i ∈ [0, k m ] ∩ Z let x(i, m) ∈ [0, 1] be the unique solution of F ◦m (x(i, m)) = i. Clearly, if i = i′ k then x(i′ , m − 1) = x(i, m). Moreover 0 = x(0, m) < x(1, m) < ... < x(k m , m) = 1. m
k We claim that the set T := ∪∞ m=1 ∪i=0 {x(i, m)} is dense in I. This is equivalent to the statement that for any 0 ≤ x < y ≤ 1 there exists x(i, m) such that x < x(i, m) < y. Assume to the contrary that there exist 0 ≤ x < y ≤ 1 such that for any m ≥ 1 and i ∈ [0, k m ]∩Z the condition x(i, m) 6∈ (x, y) holds. Hence 0 < F ◦m (y) −F ◦m (x) < 1, m = 1, ... Choose x′ , y ′ such that x < x′ < y ′ < y. As F ◦m is expansive
F ◦m (y ′) − F ◦m (x′ ) =
F ◦m (y) − F ◦m (x) − (F ◦m (y) − F ◦m (y ′)) − (F ◦m (x′ ) − F ◦m (x)) <
1 − ǫ,
ǫ = (y − y ′ + x′ − x) > 0.
Since F is expansive it follows that 0 < F ◦m (y ′) − F ◦m (x′ ) < F ◦(m+1) (y ′) − F ◦(m+1) (x′ ) < 1 − ǫ,
m = 0, 1, ...
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Hence lim F ◦m (y ′) − F ◦m (x′ ) = a,
m→∞
0 < a ≤ 1 − ǫ.
Let pm := ⌊F ◦m (x′ )⌋, um := F ◦m (x′ ) − pm ∈ [0, 1), vm := F ◦m (y ′ ) − pm ,
m = 0, 1, ...
Choose a subsequence umj , j = 1, ... which converges to u ∈ I. Then vmj , j = 1, ... converges to u + a. Observe that F (v) − F (u) =
lim F (vmj ) − F (umj ) = lim F (F ◦mj (y ′ ) − pmj ) − F (F ◦mj (x′ ) − pmj ) =
j→∞
j→∞
lim F (F
j→∞
◦mj
′
(y )) − pmj k − (F (F ◦mj (x′ )) − pmj k) =
lim F ◦(mj +1) (y ′) − F ◦(mj +1) (x′ ) = a = v − u.
j→∞
This contradicts the expansiveness of F . Define H on the following dense countable set i km S := ∪∞ m=1 ∪i=0 { k m }: H(
i ) = x(i, m), km
i = 0, ..., k m , m = 1, ...
(13)
′
i Note that if i = i′ k then H( kim ) = H( km−1 ) = x(i′ , m − 1) = x(i, m). So H is well defined on S. Furthermore H is an increasing function on S. As S and T are dense in I H has a unique continuous extension to I. Clearly the function H is increasing on I with H(0) = 0, H(1) = 1. Extend H to R by (11). For i ∈ [0, k m ] ∩ Z such that i = j + ij k m−1 with j ∈ [0, mk−1] ∩ Z, ij ∈ [0, k] ∩ Z we have
H(Gk (
i i j j )) = H( m−1 ) = H( m−1 + ij ) = H( m−1 ) + ij = x(j, m − 1) + ij . m k k k k
Observe next that F (H( kim )) = F (x(i, m)). We claim that F (x(i, m)) = x(j, m − 1) + ij . Indeed F ◦(m−1) (x(j, m − 1) + ij ) = F ◦(m−1) (x(j, m − 1)) + ij k m−1 =
j + ij k m−1 = i = F ◦(m−1) (F (x(i, m)).
Hence (12) holds on S. Since S is dense in I (12) holds on I. Use the ”periodic” properties of F, Gk , H to deduce (12) on R. It is left to show that H is unique. Recall that H is the identity map on Z. Assume ◦m that (12) holds. Then H ◦ G◦m ◦ H. Clearly k = F F ◦m (H(
i i ◦m )) = H(G◦m (x(i, m)), k ( m )) = H(i) = i = F m k k
Hence H( kim ) = x(i, m).
i ∈ [0, k m ] ∩ Z.
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Theorem 2.2. Let F be a continuous increasing function on R satisfying (11) for an integer k ≥ 2. Let f be the inverse function of F . Then the orientation preserving kcovering map F˜ : S 1 → S 1 preserves the Lebesgue measure λ if and only if there exists k nonnegative measurable functions p1 , ..., pk such that Z b 0< pi dλ for all 0 ≤ a < b ≤ 1, i = 1, ..., k, a
k X
pi (x) = 1,
i=1
f (x + i − 1) =
a.e. in I = [0, 1), Z
0
x
pi dλ +
i−1 Z X j=0
0
(14)
1
pj dλ,
p0 (x) = 0, x ∈ I, i = 1, ..., k.
In particular, F˜ is λ-preserving and is invertible with respect to λ if and only if there exists a k-λ-dense partition A = {A1 , ..., Ak } of I such that pi = χAi a.e. for i = 1, ..., k. In this case φA is the λ inverse of F˜ . Proof. Since F is a continuous increasing function on R satisfying (11), it follows that F induces the orientation preserving k-covering map F˜ : S 1 → S 1 . Let 0 ≤ x < y ≤ 1. Clearly F˜ −1 (x, y) = ∪ki=1 (f (x + i − 1), f (y + i − 1)). (15) Then F˜ is λ-preserving if and only if λ(F˜ −1 (x, y)) = y − x. Assume first that F˜ is λ-preserving. That is k X i=1
f (y + i − 1) − f (x + i − 1)) = y − x for any 0 ≤ x < y ≤ 1.
(16)
Hence for each i ∈< k > one has 0 < f (y + i − 1) − f (x + i − 1) < y − x. Thus, f (x) is an increasing Lipschitz function on R. Therefore the derivative p(x) := f ′ (x) exists (λ-)a.e., and p(x) ∈ [0, 1] almost everywhere. Let pi (x) := p(x + i − 1) = f ′ (x + i − 1) for i ∈< k >. Then the last equality of (14) follows immediately. Since f (x) is increasing in the interval [0, k] we deduce the first equality of (14). Divide (16) by y − x and let y → x to deduce the second equality of (14). Assume now that F˜ preserves λ and F˜ has a λ inverse ψ. As F˜ −1 (x) = ∪ki=1 f (x + i − 1), the existence of ψ implies the partition of I to k measurable pairwise distinct sets A1 , ..., Ak , such that for ψ(x) = f (x + i − 1) x ∈ Ai . Let B be a measurable subset of Ai . Since F˜ preserves λ the first equality of (17) implies Z Z −1 ˜ λ(B) = λ(F (B)) = λ(ψ(B)) = λ(f (B + i − 1)) = pi dλ ≤ dλ = λ(B). B
B
Hence pi |B = 1 a.e.. The second condition of (14) yields pi = χAi a.e. for i = 1, ..., k. The first condition of (14) implies that A = {A1 , ..., Ak } is a k-λ-dense partition of I. Vice versa, assume that we are given k nonnegative measurable functions p1 , ..., pk which satisfy the first two conditions of (14). Define f : [0, k] → R by the last condition
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of (14). Then f is an increasing function which maps [0, k] on I. Let F : I → [0, k] be the inverse of f . Then F˜ is an orientation preserving k-covering of S 1 which preserves λ. Note that for any set B ⊂ I, which is a finite union of intervals, the last equality of (14) and (15) yield λ(f (B + i − 1)) =
Z
pi dλ, i = 1, ..., k,
λ(B) = λ(F˜ −1 (B)) =
B
k X i=1
λ(f (B + i − 1)). (17)
Hence the above equalities hold for any measurable set B ⊂ I, i.e. F˜ preserves λ. Suppose furthermore that pi (x) = χAi a.e. for some measurable set Ai ⊂ I for i = 1, ..., k. The first two conditions of (14) are equivalent to the assumption that A = {A1 , ..., Ak } can be chosen to be a k-λ-dense partition. (17) yields Z −1 λ(F˜ (B)) = χAi dλ, for any measurable set B ⊂ Ai , i ∈< k > . (18) B
Hence F˜ has the λ inverse φA given by φA (x) = f (x + i − 1) for x ∈ Ai ,
i ∈< k > .
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Theorem 2.2 was inspired by Parry’s paper [6]. Theorem 2.3. Let A = {A1 , ..., Ak } be a k-λ-dense partition with k ≥ 2. Let fA be given by (4) and FA be the inverse of fA . Then FA is expansive, F˜A is an orientation preserving k covering of S 1 which preserves λ. The generalized interval exchange φA given by (2) is the λ inverse of F˜A . Furthermore hλ (F˜A ) = hλ (φA ) = 0.
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Proof. Assume that x, y ∈ [j − 1, j], j ∈ Z and x < y. Let j ≡ i mod k for some i ∈< k >. Since A is λ-dense Z y Z y k Z y X y−x= dλ = χp dλ > χi dλ = f (y) − f (x). x
p=1
x
x
Hence F (v) − F (u) > v − u for any v > u. The proof of Theorem 2.2 and the definitions of fA and φA yield that F˜A is λ preserving and φA is the λ inverse of F˜A . As FA is ˜k. λ expansive by Theorem 2.1 FA is conjugate to Gk . In particular F˜A is conjugate to G ˜kis conjugate to a nonatomic probability measure ω, whose support is I and which is G ˜ k has the standard Markov partition Mi = [ i−1 , i ), i = 1, ..., k, we deduce invariant. As G k k that F˜A is equivalent to complete Z+ shift on k symbols. Let M = {H(M1 ), ..., H(Mk )}) ˜ −i the Markov partition for F˜A . Then F = ∨∞ i=0 F M is the σ-subalgebra generated by the cylinders, which is equivalent to the Borel algebra for any nonatomic probability measure ν. Since F˜ is λ invertible it follows that hλ (F˜ ) = 0 (cf.[9, Cor. 4.18.1]).
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In the next section we show that for any k-partition A of I hλ (φA ) = 0. Problem 2.4. Let A = {A1 , ..., Ak } be a k-partition of I. When is φA ergodic? Corollary 2.5. Let A = {A1 , ..., Ap }, B = {B1 , ..., Bq } be two p, q-λ-dense partitions of I with p, q ≥ 2. Then hλ (φA ◦ φB ) = hλ (F˜B ◦ F˜A ) = 0. (21) Proof. F := FB ◦ FA is a continuous increasing expansive function on R satisfying (11) for k = pq. Furthermore F˜ preserves λ. Theorem 2.2 implies that F = FC for some k-λ-dense partition of I. Hence (21) holds. Problem 2.6. Let A = {A1 , ..., Ap }, B = {B1 , ..., Bq } be two p and q-λ dense partitions of I with p, q ≥ 2. Estimate from above −1 hλ (φ−1 A ◦ φB ) = hλ (φB ◦ φA ).
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Theorem 2.7. Let F : R → R be a measurable function satisfying (11) a.e. for some k ∈ Z. Assume that m is a fixed integer satisfying |m| ≥ 2. Suppose that F ◦ Gm = Gm ◦ F.
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Then F = Gk = kx a.e.. Proof. Let E(x) = F (x) − kx. Then E(x + 1) = E(x) a.e. in R. Let j be a positive integer. Since F and Gk commute with Gm it follows that E ◦ Gmj = Gmj ◦ E. Hence mj E(x) = E(mj x) = E(mj x + 1) = E(mj (x + E(x +
1 ) = E(x). mj
1 1 )) = mj E(x + j ) ⇒ j m m
Since j is an arbitrary positive integer it follows that E is constant a.e.. The condition E(mx) = mE(x) yields that E = 0 a.e.. The above theorem is related to a theorem (unpublished) of Jean-Paul Thouvenot: Theorem 2.8. Let p, q ∈ Z\{−1, 0, 1} and assume that p and q are multiplicatively independent, i.e. p and q are not integer powers of some integer r. Let T : S 1 → S 1 be ˜ p and G ˜ q . Then T = G ˜ k for measurable λ-preserving. Assume that T commutes with G ∗ some k ∈ Z := Z\{0}. Proof of Theorem 1.1. Suppose first that there exist 2 and 3-λ-dense partitions A and B of I such that (7) holds. Theorem 2.3 yields that FA is expansive. Theorem 2.1 yields that H −1 ◦ FA ◦ H = G2 . Let F := H −1 ◦ FB ◦ H. Then F is a continuous function on R satisfying (11) with k = 3 which commutes with G2 . Theorem 2.7 yields that
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˜ 2, G ˜ 3 preserve the F = G3 . As F˜A , F˜B preserve the Lebesgue measure λ it follows that G −1 ∗ probability measure ω = (H ) λ, which is nonatomic and whose support is I. As F˜A , F˜B ˜ 2, G ˜ 3 are ω-invertible. Hence ω 6= λ, which contradicts are λ-invertible (Theorem 2.3), G the 2 − 3 conjecture. Assume now that 2 − 3 conjecture is false. Then there exists a nonatomic probability ˜ 2, G ˜ 3 invariant. According to [2] the support of ω is I. Rudolph’s measure ω which is G ˜ 2 ) = hω (G ˜ 3 ) = 0. Hence G ˜2, G ˜ 3 are ω-invertible (cf.[9, Cor. theorem [7] claims that hω (G 4.14.3]). Let Z x
H(x) =
dω,
0
x ∈ I.
Then H(x) is strictly increasing function on I with H(0) = 0, H(1) = 1. Extend H to R using (11) with k = 1. Let Fk = H ◦ Gk ◦ H −1 , k = 2, 3. Then F2 ◦ F3 = F3 ◦ F2 . Furthermore F˜2 , F˜3 preserve λ and are λ invertible. Theorem 2.2 implies that F2 = FA and F3 = FB for some 2 and 3-λ-dense partitions A and B of I.
3
hλ (φA ) = 0
Let F : R → R is be a nondecreasing function, which may be discontinuous. Then F has a countable number of points of discontinuity. We will assume the normalization that F is right continuous. Assume now that F is an increasing function on R which is not bounded from below and above. Then there exists a unique continuous nondecreasing function f : R → R, which is unbounded from below and above, such that f ◦ F = Id. We call f the inverse of F . Vice versa, if f : R → R is a continuous nondecreasing function, which is not bounded from below and above, then there exists a unique increasing function F : R → R such that f ◦ F = Id. We call F the inverse of f . Let k ∈ N and assume that F is an increasing function on R which is continuous at the integer points Z and satisfies (11). Then we can define a measurable map F˜ : S 1 → S 1 . We call F˜ an almost k-covering map. Theorem 3.1. Let F be an increasing function on R continuous on Z and satisfying (11) for an integer k ≥ 2. Let f be the inverse function of F . Then the almost k-covering map F˜ : S 1 → S 1 preserves the Lebesgue measure λ if and only if there exists k nonnegative measurable functions p1 , ..., pk such that k X
pi (x) = 1,
i=1
f (x + i − 1) =
a.e. in I, Z
0
x
pi dλ +
i−1 Z X j=0
0
1
pj dλ,
p0 (x) = 0, x ∈ I, i = 1, ..., k.
In particular, F˜ is λ-preserving and is invertible with respect to λ if and only if there exists a k-partition A = {A1 , ..., Ak } of I such that pi = χAi a.e. for i = 1, ..., k. In this case φA is the λ inverse of F˜ .
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The proof of this theorem follows from simple modifications of the proof of Theorem 2.2 and is left to the reader. Let U, V ∈ Σ. In what follows we use the notation: U ∼ V ⇐⇒ λ(U∆V ) = 0,
U 6∼ V ⇐⇒ λ(U∆V ) > 0.
Let J ⊂ R be an interval of positive Lebesgue measure (open, closed or half open). Let A = {A1 , ..., Ak } and B = {B1 , ..., Bm } be two partitions of J. Recall that A and B are equivalent if there exist permutations µ :< k >→< k >, ν :< m >→< m > and a positive integer p such that Aµ(i) ∼ Bν(i) , i = 1, ..., p, Aµ(i) ∼ Bν(j) ∼ ∅ for i > p and j > p. Theorem 3.2. Let k ≥ 1 and assume that A = {A1 , ..., Ak } is a partition of I = [0, 1). Let fA be the continuous nondecreasing function defined by (3-4). Let FA : R → R be the increasing function which is the inverse of fA . Let F˜A be an almost k-covering of S 1 preserving λ and whose λ inverse is φA . Let 0 = β0 ≤ β1 ≤ ... ≤ βk = 1 be defined in (1). Let B = {[β0 , β1 ), [β1 , β2 ), ..., [βk−1 , βk )} be a partition of S 1 to k intervals. Then the partition Bn := B ∨ φA˜B ∨ ... ∨ φnA˜B is equivalent to a partition of [0, 1) to intervals Cn := {Jn,1, ..., Jn,ℓ(n) } with the following properties: (a) ℓ(0) = k, J0,j = [βj−1 , βj ), j = 1, ..., k. (b) Cn is obtained from Cn−1 by subdividing each interval Jn−1,j to a finite number of subintervals for each n ∈ N. Moreover one of the following conditions holds: (c) The partitions Cn , n = 0, 1, ..., separate points on S 1 . (d) The partitions Cn , n = 0, 1, ..., do not separate points on S 1 . Then there exists a nonempty countable J ⊂ N with the following properties. For each j ∈ J there exist mj ∈ N pairwise disjoint open intervals Ij,1, ..., Ij,mj ⊂ S 1 of equal lengths such that φA acts on {Ij,1, ..., Ij,mj } as an orientation preserving cyclic interval exchange up to a set of zero measure: φA (Ij,p) ⊂ I j,p+1,
Ij,p+1 ∼ φA (Ij,p ), p = 1, ..., mj , (Ij,mj +1 = Ij,1), for any j ∈ J , ′
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′
Ij,p ∩ Ij ′,p′ = ∅ for any j 6= j and p ∈< mj >, p ∈< mj ′ > . m
j Let X = ∪j∈J ∪p=1 I j,q . Then the restriction of the partitions Cn , n = 0, 1, ... to S\X separate the points in S\X. Hence in both of the cases the measure entropy hλ (φA ) equals to zero.
Proof. For k = 1 F˜A = Id and the theorem is trivial. Without a loss of generality we may assume that k ≥ 2 and λ(Ai ) > 0 for i = 1, ..., k. Let J ⊂ R be an interval. From the definition of fA it follows that fA (J) is an interval. Let J ⊂ [0, 1). Define Ii = fA (J + i − 1) ∩ [βi−1 , βi ) for i = 1, ..., k. Then I1 , ..., Ik are pairwise distinct intervals, which may be empty or consisting of one point. From the
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definition of φA it follows that φA (J) ∼ ∪ki=1 Ii . Hence Bn is equivalent to a partition Cn of [0, 1) to disjoint intervals. Furthermore Cn is the refinement of Cn−1 . Hence (a) and (b) hold. Assume first that the partitions Cn , n = 0, 1, ..., separate points. Hence ∨∞ n=0 Cn is 1 n equivalent to the Borel σ-algebra on S up to sets of zero measure. Therefore ∨∞ n=0 φA B is equivalent to the Borel σ-algebra on S 1 up to sets of zero measure. As F˜A−1 = φA we deduce that hλ (F˜A ) = 0, e.g. [9, Cor.4.18.1], which implies that hλ (φA ) = 0. Assume now that Cn , n = 0, 1, ..., do not separate points. Then there is at least one nested set of intervals J1,q1 ⊃ J2,q2 ⊃ ... such that ∩∞ i=1 J i,qi = K = K o , Ko = (a, b), 0 ≤ 1 1 a < b ≤ 1. Note that for each i ≥ 2 there exists Ji−1,q1 such that Ji,qi \φA (Ji−1,q ) ∼ ∅. 1 i−1
Then
1 J1,q 1 1
⊃
1 J2,q 1 2
⊃ ... is nested set of intervals such that
1 ∩∞ i=1 J i,qi1
i−1
1
= K is a closed 1
interval in S. Clearly λ(K\φA (K 1 )) = 0. Hence λ(K 1 ) ≥ λ(K), i.e. K 1 = K o , Ko1 = (a1 , b1 ), 0 ≤ a1 < b1 ≤ 1, b1 − a1 ≥ b − a. Since K and K 1 are intersections of nested sequences of the intervals in the partitions Cn , n = 1, ..., it follows that either Ko = Ko1 or Ko ∩ Ko1 = ∅. Repeating this argument we obtain for each integer p ≥ 2 a sequence p p p p of nested intervals J1,q ⊃ ... such that ∩∞ p ⊃ J i=1 J i,qip = K is a closed interval in S. 2,q 2 1
2
p
Furthermore λ(K p−1\φA (K p )) = 0. Hence K p = K o , Kop = (ap , bp ), 0 ≤ ap < bp ≤ 1, bp − ap ≥ bp−1 − ap−1 for p = 2, 3, ...,. Let K 0 = K. Then for any 0 ≤ r < p either Kor = K0p or Kor ∩Kop = ∅. Consider the sequence of open intervals Ko0 , Ko1 , Ko2 , ... in (0, 1), whose lengths form a nondecreasing sequence. Then it is impossible that all these open intervals are pairwise disjoint. So assume that Kor ∩ Kop 6= ∅ for some 0 ≤ r < p. Hence Kor = Kop . If Kor+1 = Kor we choose p = r + 1. Otherwise we can assume without loss of generality that Koj 6= Kor for j = r + 1, ..., p − 1. Clearly λ(K j ) = λ(K r ). Therefore up a zero measure φA acts the orientation preserving interval exchange Kor = Kop → Kop−1 → ... → Kor of p − r distinct open intervals in (0, 1). Obviously Ko0 appears among this p − r intervals. Clearly all maximal open intervals Ko whose points are not separated by Cn , n = 0, 1, ..., form a countable set of pairwise disjoint intervals of (0, 1). If we group each Ko with the other p − r − 1 intervals as above, we obtain a countable set J of such groups mj I j,q . Then φA (X) = X (up to zero as described in the theorem. Let X = ∪j∈J ∪p=1 1 measure sets). Clearly hλ (φA |X ) = 0. Then Y = S \X is a φA invariant set (up to a set of zero measure). Cn ∩ Y, n = 0, 1, ..., separates the points on Y . The arguments in the beginning of the proof of the theorem yield that hλ (φA |Y ) = 0. Hence hλ (φA ) = 0.
4
The condition fA ◦ fB = fA ◦ fB
Lemma 4.1. Let A, B be 2 and 3 partitions of I = [0, 1) respectively. Then fA ◦ fB = fC ,
fB ◦ fA = fD
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for some 6-partitions C, D of I. Suppose furthermore that A and B are λ-dense partitions. Then C and D are λ-dense partitions.
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Proof. Recall that A = {A1 , A2 } and B = {B1 , B2 , B3 }. Clearly fA′ = χA ,
fB′ = χB ,
(fA ◦ fB )′ = χf −1 (A) χB , B
(fB ◦ fA )′ = χf −1 (B) χA , A
fA ◦ fB (x + 6) = fA ◦ fB (x) + 1,
fB ◦ fA (x + 6) = fB ◦ fA (x) + 1.
Let Bi,j := {x ∈ Bi :
Aj,i := {x ∈ Aj :
fB (i − 1 + x) ∈ Aj }, for i = 1, 2, 3, j = 1, 2,
fA (j − 1 + x) ∈ Bi }, for i = 1, 2, 3, j = 1, 2,
(26)
We claim that C := {B1,1 , B2,1 , B3,1 , B1,2 , B2,2 , B3,2 },
D := {A1,1 , A2,1 , A1,2 , A2,2 , A1,3 , A2,3 }
(27)
are 6-partitions of I and (25) holds. Since B is a partition of I Bi,j ∩ Bp,q = ∅ for i 6= p. As A is a partition of I Bi,j ∩ Bi,p = ∅ for j 6= p. As fB ([0, 3]) = [0, 1] and fB (B ∩ [0, 3]) has measure 1 it follows that C is a 6-partition of I. Similar arguments show that D is a 6 partition of I. Let C, D ⊂ R be the induced sets by C, D respectively. The definition of C and a straightforward calculation shows that (fA ◦ fB )′ = χC . As fA ◦ fB (0) = 0 we deduce the first equality of (25). The second equality of (25) follows similarly. Suppose now that A and B are λ-dense partitions. Then fA and fB are increasing. Hence fA ◦ fB and fB ◦ fA are also increasing. The equalities (25) yield that C and D are λ-dense partitions. For a set A ⊂ R we denote A(s, t) := A ∩ [s, t], A(t) := A ∩ [0, t],
s ≤ t,
0 ≤ t.
Let A = {A1 , ..., Ak } and A′ = {A′1 , ..., A′k } be two k-partitions of [0, 1). We say that A and A′ are strongly equivalent, and denote it by A ∼ A′ if Ai ∼ A′i for i = 1, ..., k. Lemma 4.2. Let A and B be 2 and 3 partitions of [0, 1] respectively. Let A, B ∈ Σ be defined by A, B using (3) respectively. Then the following are equivalent (a) fA ◦ fB = fB ◦ fA . (b) The partitions C and D given in (27) are both strongly equivalent to the partition A · B := {A1 ∩ B1 , A2 ∩ B2 , A1 ∩ B3 , A2 ∩ B1 , A1 ∩ B2 , A2 ∩ B3 }.
(28)
(c) A(fB (s), fB (t)) ∼ fB (A(s, t) ∩ B(s, t)), B(fA (s), fA (t)) ∼ fA (A(s, t) ∩ B(s, t)),
for all s ≤ t, for all s ≤ t.
(29)
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Proof. Assume (a). Then (25) implies that C ∼ D. Furthermore C ∼ D ⊂ A ∩ B. A straightforward argument yields that A ∩ B is induced by a partition A · B. As 1 = λ(C(6)) = λ(A ∩ B ∩ [0, 6]) we deduce that C ∼ A ∩ B and C ∼ D ∼ A · B. Assume (b). Then (25) implies (a). Assume (a) and (b). Use the definition of C and the condition C ∼ A ∩ B to deduce the first condition in (29) with s = 0 and t ≥ 0. Hence the first condition of (29) holds for any 0 ≤ s ≤ t. Use the the condition (5) for fB with k = 3 to deduce the condition of (29) for any s ≤ t. The second condition in (29) is derived similarly. Assume (c). Recall that fB maps any measurable set E ⊂ B to a set E ′ of the same measure. Furthermore the complement of B (B c ) is mapped to a set of zero measure. Hence fB (B(t)) ∼ fB ([0, t]) = [0, fB (t)] ⇒ λ(A(t) ∩ B(t)) = λ(fB (A(t) ∩ B(t)). Similar conditions hold for fA ([0, t]). Assume first that (29) holds for s = 0 and any t ≥ 0. Then fA (fB (t)) = λ(A(fB (t))) = λ(fB (A(t) ∩ B(t))) = λ(A(t) ∩ B(t)) =
λ(fA (A(t) ∩ B(t))) = λ(B(fA (t))) = fB (fA (t)).
Hence (8) holds for any t ≥ 0. Since the two functions appearing in (8) satisfy (5) we deduce (8) for all t ∈ R. It is straightforward to show that the condition (8) yields the condition (9). In the next section we show that the condition (9) is sometimes weaker than (8).
5
Interval exchanges
In this section we consider only partitions of the interval I = [0, 1) induced by the partition of I to n intervals of equal length n1 . Let J := {J1 , ..., Jn } be a partition of I to n ≥ 2 half closed intervals of length n1 arranged in an increasing order. Let 2 ≤ k ≤ n and let Ω1 , ..., Ωk be a partition of < n > to k disjoint (possibly empty) sets. Set Aj = ∪l∈Ωj Jl ,
j = 1, ..., k.
Then A = {A1 , ..., Ak } is called a k-n-partition of I. A is a regular k-n-partition of I if and only if each Ωj is a nonempty set. Then φA is an interval exchange. φA induces the following permutation σ :< n >→< n >: φA (Ji ) = Jσ(i) ,
i = 1, ..., n.
σ maps the nonempty set Ωj to the set [γj−1 + 1, γj−1 + |Ωj |] ∩ Z monotonically for j = 1, ..., k. Here j X γ0 = 0, γj = |Ωl |, j = 1, ..., k. l=1
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Any k-n-interval partition A induces a unique regular m-n-interval partition A′ with 1 ≤ m ≤ n, by discarding the empty sets. Clearly, φA = φA′ , that is A and A′ induce the same interval exchange on I. Equivalently, A and A′ induce the same permutation σ :< n >→< n >. Any permutation σ on < n > we identify with the ordered set of the elements of < n >: {i1 , i2 , ..., in } = {σ −1 (1), σ −1(2), ..., σ −1 (n)}.
(30)
It is easy to show that σ given in the above form is induced by a unique minimal regular m-n-interval partition, where m is exactly the number of j ≤ n − 1 for which ij > ij+1 . Lemma 5.1. Let A and B be 2-n-interval and 3-n-interval regular partitions of I respectively. Assume that the condition (8) holds. Suppose furthermore that the induced permutations σ, η either both fix 1 or both fix n. Then there exist 2-(n − 1)-interval and 3-(n − 1)-interval partitions A′ and B′ satisfying the condition (8). Proof. Since B is a regular 3-n partition of I we obtain that n ≥ 3. Let Γ1 := {1 ≤ i1 < i2 < ... < ip },
Γ2 = {1 ≤ ip+1 < ip+2 < ... < in },
1 ≤ p < n, Γ1 ∪ Γ2 =< n >,
∆1 := {1 ≤ j1 < j2 < ... < jq },
∆2 = {1 ≤ jq+1 < jq+2 < ... < jq′ },
∆3 = {1 ≤ jq′ +1 < jq′ +2 < ... < jn },
1 ≤ q < q ′ < n, ∆1 ∪ ∆2 ∪ ∆3 =< n >, m−1 m m−1 m , ), i = 1, 2, Bj = ∪m∈∆j [ , ), j = 1, 2, 3. Ai = ∪m∈Γi [ n n n n
(31)
Assume first that σ, η both fix 1. Then i1 = j1 = 1. Let Γ′1 = {i2 − 1, ..., ip − 1},
Γ′2 = {ip+1 − 1, ..., in − 1},
∆′1 = {j2 − 1, ..., jq − 1},
∆′2 = {jq+1 − 1, jq+2 − 1, ..., jq′ − 1},
∆′3 = {jq′ +1 − 1, jq′ +2 − 1, ..., jn − 1}. Let A′ , B′ be induced by {Γ′1 , Γ′2 }, {∆′1 , ∆′2 , ∆′3 } respectively. A straightforward argument using Lemma 4.2 shows that fA ◦ fB = fB ◦ fA ⇒ fA′ ◦ fB′ = fB′ ◦ fA′ .
(32)
(Another way to deduce the above implication is to collapse each interval [m, m + n1 ) ⊂ R, m ∈ Z to a point to obtain R. Then (8) holds also on R, which is equivalent to fA′ ◦ fB′ = fB′ ◦ fA′ .)
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Assume now that σ, η both fix n. Then ip′ = jq′′ = n. Let Γ′2 = Γ2 \{n}, ∆′3 = ∆3 \{n}. Let A′, B′ be induced by {Γ1 , Γ′2 }, {∆1 , ∆2 , ∆′3 } respectively. Then (32) holds. Lemma 5.2. Let A and B be regular 2-n and 3-n-partitions induced by the regular 2-n and 3-n-partitions of < n > given in (31). Let the partition C = A · B, given by (28), be induced by Ω1 = Γ1 ∩ ∆1 = {k1 , ...kr11 },
r11 ≥ 0,
Ω2 = Γ2 ∩ ∆2 = {kr11 +1 , ..., kr22 },
r22 ≥ r11 ,
Ω4 = Γ2 ∩ ∆1 = {kr13 +1 , ..., kr21 },
r21 ≥ r13 ,
Ω3 = Γ1 ∩ ∆3 = {kr22 +1 , ..., kr13 },
Ω5 = Γ1 ∩ ∆2 = {kr21 +1 , ..., kr12 }, Ω6 = Γ2 ∩ ∆3 = {kr12 +1 , ..., kr23 },
r13 ≥ r22 , r12 ≥ r21 ,
n = r23 ≥ r12 .
(33)
Assume that (8) holds. Then q = r22 ≤ p = r13 ≤ q ′ = r21 .
(34)
ku = iju = jiu ,
(35)
u = 1, ..., n.
jr11 ≤ p < jr11 +1 ≤ jq ,
jq+1 ≤ jp ≤ p < jp+1 ≤ jq′ , jq′ +1 ≤ jr12 ≤ p < jr12 +1 ,
(36)
ir11 ≤ q < ir11 +1 ≤ iq ≤ q ′ < iq+1 ≤ ip ,
ip+1 ≤ iq′ ≤ q < iq′ +1 ≤ ir12 ≤ q ′ < ir12 +1 .
(37)
If one the below equalities hold 0 = r11 , r11 = q, q = p, p = q ′ , q ′ = r12 , r12 = n, then the above corresponding inequalities do not apply. Proof. Lemma 4.2 yields q r22 = λ(fA (A(2) ∩ B(2))) = λ(B1 ) = , n n fA (A(4) ∩ B(4)) = B(2) = B1 ∪ (1 + B2 ) ⇒ r21 q′ = λ(fA (A(2) ∩ B(2))) = λ(B1 ) + λ(B2 ) = , n n r13 p fB (A(3) ∩ B(3)) = A(1) = A1 ⇒ = λ(fB (A(3) ∩ B(3))) = λ(A1 ) = . n n
fA (A(2) ∩ B(2)) = B(1) = B1 ⇒
(38)
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Hence (34) holds. Let σ, η be the permutations of < n > induced by {Γ1 , Γ2 }, {∆1 , ∆2 , ∆3 } respectively. Consider ku , ∈ Γi ∩ ∆j for some i ∈< 2 >, j ∈< 3 >. Then ku = il for l ∈< p > if i = 1 and l > p if i = 2. ku corresponds to the interval [tu − n1 , tu ] ∈ A(2mij ) ∩ B(2mij ) for the smallest integer mij ∈< 3 >. Then fA (A(tu ) ∩ B(tu )) = B(fA (tu )) is of total length u . So the interval [tu − n1 , tu ) is mapped on the interval [mij − 1 + jun−1 , mij − 1 + jnu ) ∈ n mij − 1 + Bmij . Hence σ(il ) = l = ju . This proves the first equality in (35). Observe next that ku = jv . Use the identity fB (A(tu ) ∩ B(tu )) = A(fB (tu )) to deduce the the equality v = iu . If r11 > 0 then kr11 ∈ Ω1 ⊂ Γ1 . As kr11 = ijr11 it follows that jr11 ≤ p. If q = r22 > r11 then kr11 +1 ∈ Ω2 ⊂ Γ2 . As kr11 +1 = ijr11 +1 it follows that jr11 +1 > p. If q = r22 < r13 = p then Ω3 6= ∅. Then kq+1 , kp ∈ Γ1 . As kq+1 ) = iq+1 , kp = ijp it follows that jq+1 ≤ jp ≤ p. If p = r13 < r21 = q ′ then Ω4 6= ∅. Then kp+1 ∈ Γ2 . As kp+1 = ijp+1 it follows that jp+1 > p. If q ′ = r21 < r12 then Ω5 6= ∅. Then kq′ +1 , kr12 ∈ Γ1 . As kq′ +1 = iq′ +1 , kr12 = ijr12 it follows that jq′ +1 ≤ jr12 ≤ p. If r12 < r23 then Ω6 6= ∅. Then kr12 +1 ∈ Γ2 . As kr12 +1 = ijr12 +1 it follows that jr12 +1 > p. These arguments prove (36). Recalling that Ωi is also a subset of the corresponding ∆j and combining the above arguments with the equality ku = jiu we deduce (37). Corollary 5.3. Let the assumptions of Lemma 5.2 hold. Then q + q ′ = 2p, r11 = q − q ′ + p, r12 = q ′ − q + p,
1 ≤ q < q ′ < n, q ≤ p ≤ q ′ , 2q ≥ p, 3p − 2q ≤ n.
(39)
Corollary 5.4. Let A and B be 2-n and 3-n-partitions which are not of the form A = {[0, t), [t, 1)}, B = {[0, t), ∅, [t, 1)} for t ∈ [0, 1]. Assume that n ≤ 3. Then (8) does not hold. Let n = 3 and assume that σ is the cyclic permutation on < 3 >. Let η = σ 2 . A straightforward calculation shows that for A = {A1 , A2 } and B = {B1 , B2 , B3 }: A1 = {J2 , J3 }, A2 = {J1 },
B1 = J3 , B2 = J1 , B3 = J2 ,
φA and φB are inducing the permutations σ and η of < 3 > respectively. Hence (9) holds. In view of Corollary 5.4 (8) does not hold. Lemma 5.5. The following regular 2-4 and 3-4-interval partitions A = {{J2 , J4 }, {J1 , J3 }},
B = {{J3 }, {J1 , J4 }, {J2 }}
(40)
are the unique regular 2-4 and 3-4-interval partitions for which (8) holds. The induced permutations σ, η are cyclic permutation with η = σ −1 . The proof of the lemma is left to the reader. Combine Lemma 5.1 with Lemma 5.5 to obtain:
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Corollary 5.6. Let p, n be nonnegative integers such that 0 ≤ p ≤ n − 4. Then the following regular 2-n and 3-n-partitions satisfy (8): p p+1 p+2 p+3 p+4 p p+1 p+2 p+3 p+4 A := {{[0, ), [ , ), [ , )}, {[ , ), [ , ), [ , 1)}}, n n n n n n n n n n p p+2 p+3 p p+1 p+3 p+4 p+1 p+2 p+4 B := {{[0, ), [ , )}, {[ , )}, {[ , )}, {[ , ), [ , 1)}}. n n n n n n n n n n The corresponding permutations σ, η satisfy η = σ −1 . For n = 2m with m ≥ 2 and 3 ∤ 2m + 1, there exist regular 2 − n and 3 − n partitions of I, induced by the commuting maps G2 , G3 , for which (8) holds. Lemma 5.7. Let m ≥ 2 be an integer and assume that 2m + 1 is not divisible by 3. Let σ1 , η1 :< 2m >→< 2m > are given by the maps x → 2x, x → 3x modulo 2m + 1 restricted to < 2m >. Then σ1 and η1 commute. Let A2m , B2m be the regular 2−2m, 3−2m partitions induced by Γ1 := {σ1 (1), σ1 (2), . . . , σ1 (m)}, Γ2 := {σ1 (m + 1), σ1 (m + 2), . . . , σ(2m)}, 2m + 1 4m + 2 2m + 1 ∆1 = {η1 (1), . . . , η1 (⌊ ⌋)}, ∆2 = {η1 (⌊ ⌋ + 1), . . . , }, η1 (⌊ ⌋)}, 3 3 3 4m + 2 ⌋ + 1), . . . , η1 (2m)}. ∆3 = {η1 (⌊ 3 Then φA2m ◦ φB2m = φB2m ◦ φA2m . The proof is left to the reader. Note that lim φA2m (x) =
m→∞
x = G−1 2 (x), 2
lim φB2m (x) =
m→∞
x = G−1 3 (x). 3
Thus Lemma 5.7 does not give in the limit a contradiction to the 2-3 conjecture. We do not know for which m ≥ 3 the converse to Lemma 5.7 holds. That is, assume that m ≥ 3, 3 ∤ 2m + 1 and A = {A1 , A2 }, B = {B1 , B2 , B3 } are regular 2 − 2m, 3 − 2m partitions. Suppose furthermore that J2m ∈ A1 , J1 ∈ A2 and (8) holds. Are A, B equal to A2m , B2m respectively? Another way to find a counterexample to the 2-3 conjecture is to study the ergodic ˜ 2, G ˜ 3 , which are supported on a finite number of points. It measures invariant under G is straightforward to show that a such measure is equi-distributed on an orbit of the action of the permutations σ1 , η1 given in Lemma 5.7. It seems that this approach is not straightforward related to the problem of the converse to Lemma 5.7 we discussed above.
Acknowledgment We thank the two referees for their remarks.
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References [1] P. Arnoux, D.S. Ornstein and B. Weiss: “Cutting and stacking, interval exchanges and geometric models”, Israel J. Math., Vol. 50, (1985), pp. 160–168. [2] H. Furstenberg: “Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation”, Math. Sys. Theory, Vol. 1, (1967), pp. 1–49. [3] B. Host: “Nombres normaux, entropie, translations”, Israel J. Math., Vol. 91, (1995), pp. 419–428. [4] A. Johnson and D.J. Rudolph: “Convergence under ×q of ×p invariant measures on the circle”, Adv. Math., Vol. 115, (1995), pp. 117–140. [5] G. Margulis: “Problems and conjectures in rigidity theory”, In: Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 161–174. [6] W. Parry: “In general a degree two map is an automorphism”, Contemporary Math., Vol. 135, (1992), pp. 219–224. [7] D. Rudolph: “×2 and ×3 invariant measures and entropy”, Ergodic Theory & Dynamical Systems, Vol. 10, (1990), pp. 395–406. [8] Jean-Paul Thouvenot: private communication. [9] P. Walters: An Introduction to Ergodic Theory, Springer-Verlag, 1982.
CEJM 3(3) 2005 430–474
Dual Pairs and Kostant–Sekiguchi Correspondence. II. Classification of nilpotent elements Andrzej Daszkiewicz1∗ , Witold Kra´skiewicz1† , Tomasz Przebinda2‡§ 1
Faculty of Mathematics, Nicholas Copernicus University, Chopina 12, 87-100 Toru´ n, Poland 2 Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA
Received 11 January 2005; accepted 17 May 2005 Abstract: We classify the homogeneous nilpotent orbits in certain Lie color algebras and specialize the results to the setting of a real reductive dual pair. For any member of a dual pair, we prove the bijectivity of the two Kostant–Sekiguchi maps by straightforward argument. For a dual pair we determine the correspondence of the real orbits, the correspondence of the complex orbits and explain how these two relations behave under the Kostant–Sekiguchi maps. In particular we prove that for a dual pair in the stable range there is a Kostant–Sekiguchi map such that the conjecture formulated in [6] holds. We also show that the conjecture cannot be true in general. c Central European Science Journals. All rights reserved.
Keywords: Dual pairs, nilpotent orbits, Lie color algebras, Kostant-Sekiguchi correspondence MSC (2000): 20G05, 17B75, 22E45
1
Introduction
Let G be a real reductive group with a maximal compact subgroup K and corresponding Cartan decomposition g = k ⊕ p of the Lie algebra g of G. As is well known, see [21], understanding the structure of the nilpotent G-orbits in g is essential in the search for a complete description of the unitary dual of G. Due to a theorem of Kostant and ∗ † ‡ §
E-mail:
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[email protected] Third author supported in part by NSF grant DMS 0200724.
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Sekiguchi, [19], these (real) orbits are in one to one correspondence with the (complex) nilpotent KC -orbits in pC , which are amenable to the tools of the complex algebraic geometry. Furthermore, by the works of Barbasch-Vogan, Howe and Rossmann, Schmid and Vilonen, culminating in [20], the wave front of an irreducible admissible representation of G coincides with the associate variety of the corresponding Harish-Chandra module, by the Kostant-Sekiguchi correspondence of the orbits. Let G0 and G1 be an irreducible real reductive dual pair acting on a symplectic vector space W (see [10, 11]). For readers convenience we recall that these are pairs (G0 , G1 ) of type I (Sp2n (R), O(p, q)), (O(p, q), Sp2n(R)), (Sp2n (C), O(p)), (O(p), Sp2n (C)), (Up,q , Ur,s ), ∗ ∗ (Spp,q , O2n ), (O2n , Spp,q ),
and pairs of type II (GLm (D), GLn (D)), (D = R, C, or H). Let K0 ⊆ G0 , K1 ⊆ G1 be maximal compact subgroups with the complexifications K0,C , K1,C . The groups K0 , K1 centralize a positive definite, compatible complex structure J on W . Let WC+ denote an i-eigenspace of J in WC , the complexification of W . Let g0 , g1 denote the Lie algebras of G0 and G1 , with the Cartan decompositions g0 = k0 ⊕ p0 , g1 = k1 ⊕ p1 . Let ν0 : W → g0 , ν1 : W → g1 be the moment maps, as defined in [11] (see also [6] and [7]), and let µ0 : WC → g0,C , µ1 : WC → g1,C be the analogous moment maps of the complexifications. Then µ0 (WC+ ) ⊆ p0,C , µ1 (WC+ ) ⊆ p1,C , and we have the following pair of diagrams ν
ν
µ0
µ1
1 g0 ←−0−− W −−− → g1
p0,C ←−−− WC+ −−−→ p1,C
(1)
We call an element w ∈ W nilpotent if ν0 (w) is nilpotent in g0 (equivalently if ν1 (w) is nilpotent in g1 ). Similarly we define nilpotent elements in WC as elements mapped by either of µ0 , µ1 onto nilpotent elements of the complexified Lie algebras. Howe’s correspondence provides a convenient tool for a construction of irreducible representations of classical groups with small wave front set. In fact the wave front set of a representation of the larger group may be estimated, and in some cases computed, in terms of the wave front set of the corresponding representation of the smaller one via the moment maps (1), (see [17]). Also, in some cases the associate varieties of the corresponding Harish-Chandra modules have been computed in [13]. Furthermore, as explained in [17], a pair of representations in Howe’s correspondence, is determined by a temperate distribution, which lives on W . This distribution has its own wave front set, a union of nilpotent orbits, under the action of both members of the dual pair. We attempt
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to look for an analog of the Kostant-Sekiguchi Theorem for the (real) orbits in W and the (complex) orbits in WC+ . As the reader will see below, there is no direct analog, but there is some hope of finding a geometric link of the Kostant-Sekiguchi correspondences for the two groups, through W and WC+ . In this paper we provide a description of the set of the nilpotent G0 × G1 -orbits in W and the set of the nilpotent K0,C × K1,C -orbits in WC+ . Both classification problems, as well as similar problems such as the problem of classifying nilpotent orbits in g0 ,g1 and in p0,C ,p1,C arise as special instances of a more general problem of classifying homogeneous nilpotent orbits in certain Lie color algebras. We solve this problem in Section 3 using the methods of Burgoyne and Cushman by checking that the ideas of [2] carry over to the situation we consider. In the context of Lie algebras of reductive groups the standard approach to the classification of nilpotent orbits uses the Jacobson-Morozov theorem and the representation theory of sl2 . We are not aware, however, of any analogue of the Jacobson-Morozov theorem for the Lie algebras studied in this paper. A reader interested in the history of the problem of classification of nilpotent orbits in classical Lie algebras may consult the introduction to [2] and the book [4]. In particular in [4], one can find a traditional, diagrammatic presentation of classification results, used also by Ohta in [15], [16] in his study of nilpotent orbits for classical symmetric pairs. We use a different approach, which seems more natural for our purposes. In Section 4 we show how one can apply the general results of Section 3 to the classification problems described above in the case of a dual pair of type II, and in Section 5 we do the same for pairs of type I. ¯ see Proposition 6.6), referred to There are two bijections, S and S¯ (sometimes S = 6 S, as Kostant-Sekiguchi correspondences, from the set of nilpotent G0 -orbits in g0 onto the set of nilpotent K0,C -orbits in p0,C , and similarly for G1 ([19]). In Section 6 we compute ¯ in terms of our parametrization of orbits in g and in pC for dual both maps, S and S, pairs of type I. As a main tool we use the description of the Cayley transform of a Cayley triple in g by the conjugation by a special element of the complex group. In [6], the following conjecture was stated. Conjecture 1.1. Let O ⊆ g1 be a nilpotent orbit. Then S(ν0 ν1−1 (O)) = µ0 µ−1 1 (S(O)).
(2)
We provide a counterexample in Section 7 which proves that this conjecture is false, as stated. In some cases however, for instance for pairs in the stable range (see Section 8) the conjecture holds for an appropriate choice of the Kostant-Sekiguchi map. The determination of all orbits O for which the equality (2) holds might be of interest, but is beyond the scope of this paper. The last author wants to express his gratitude to the Nicolas Copernicus University for the hospitality and support during his frequent visits to Torun, during the last six years, where most of the results of this paper were developed.
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Also, we would like to thank the referee for a quick review and for several corrections.
2
Sesqui-linear Forms
Let D = R, C or H (the quaternions), and let ι be a possibly trivial anti-involution on D. (Notice that for a commutative field, an anti-involution is the same as an involution.) Let V be a left vector space over D. A sesqui-linear form on V is a map τ : V × V → D such that for all u, v, u′, v ′ ∈ V and for all a ∈ D, τ (au, v) = aτ (u, v), τ (u, av) = τ (u, v)ι(a) τ (u + u′, v) = τ (u, v) + τ (u′ , v), τ (u, v + v ′ ) = τ (u, v) + τ (u, v ′ ). For σ = ±1, we will say that the form τ is σ-hermitian if τ (u, v) = σι(τ (v, u)) for all u, v ∈ V . We shall say that two subspaces V ′ , V ′′ ⊆ V are orthogonal (with respect to τ ) if τ (v ′ , v ′′ ) = 0 for all v ′ ∈ V ′ and all v ′′ ∈ V ′′ . In this case we shall write V ′ ⊥ V ′′ . We shall say that the space V , or more precisely, the pair (V, τ ) is decomposable if there are two non-zero subspaces V ′ , V ′′ ⊆ V such that V = V ′ ⊕ V ′′ and V ′ ⊥ V ′′ . Otherwise the space V , or the pair (V, τ ) is called indecomposable. The form τ is called non-degenerate if the following two implications hold: if τ (u, v) = 0 for all v ∈ V, then u = 0, and if τ (u, v) = 0 for all u ∈ V, then v = 0.
Every σ-hermitian non-degenerate formed space (V, τ ) is a direct sum of mutually orthogonal indecomposable spaces. Classification of indecomposable σ-hermitian forms is well known. We collect them in Table 1. If D = R or ι is nontrivial we define the signature of a form as follows. If the form τ is degenerate, then by the signature of τ , sgn(τ ), we mean the signature of the quotient form of τ on V /Rad(τ ). Suppose τ is nondegenerate. The signature of τ , sgn(τ ), is a pair of two nonnegative integers sgn(τ ) = (n+ , n− ) defined as follows. If the form is hermitian then n+ (resp. n− ) is the dimension of any maximal subspace of V on which the restriction of τ is positive (resp. negative) definite. If D = C and τ is skew-hermitian, then the form −iτ is hermitian and by definition sgn(τ ) = sgn(−iτ ). If D = R or H and τ is skew-hermitian, then n+ = n− = 12 dimD (V ). Fix a positive integer n and let V = V0 ⊕ V1 ⊕ .. ⊕ Vn−1
(3)
be a direct sum of left D-vector spaces V0 , V1 , ..., Vn−1 . We view V as a Z/nZ-graded vector space, via the above decomposition, where Z/nZ is realized as the set {0, 1, 2, ..., n−
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(D, ι)
σ = −1
σ=1
(R, id)
(R, +) (x, y) 7→ xy (R, −) (x, y) 7→ −xy
(C, id) (C, ¯·) (H, ¯·)
(C, sym) (x, y) 7→ xy
(R2 , sk) (x, y) 7→ x
h
0 1 −1 0
i
yt
(C2 , sk) (x, y) 7→ x
h
0 1 −1 0
i
yt
(C, +) (x, y) 7→ x¯ y
(C, +i) (x, y) 7→ xi¯ y
(C, −) (x, y) 7→ −x¯ y
(C, −i) (x, y) 7→ −xi¯ y
(H, +) (x, y) 7→ x¯ y (H, −) (x, y) 7→ −x¯ y
(H, sk) (x, y) 7→ xj y¯
Table 1 The list of indecomposable σ-hermitian forms.
1} with addition modulo n. The subspace Va ⊆ V is called the subspace of degree a (= 0, 1, 2, ..., n − 1). The dimension vector of V is the sequence dim(V ) = (dimD (V0 ), dimD (V1 ), . . . , dimD (Vn−1 )). We shall say that the space V , or more precisely, the pair (V, τ ) is decomposable if there are two non-zero graded subspaces V ′ , V ′′ ⊆ V such that V = V ′ ⊕ V ′′ and V ′ ⊥ V ′′ . Otherwise the space V , or the pair (V, τ ) is called indecomposable. In this paper, we will consider sesquilinear forms τ on V for which there exists an involution ∗ of Z/nZ and a function Z/nZ ∋ b 7→ σb ∈ {±1} such that σb = σb∗ for every b ∈ Z/nZ and the following conditions are satisfied. (1) Vb ⊥ Vc unless c = b∗ . (2) For every b = 0, 1, 2, ..., n − 1, the form τ provides a non-degenerate pairing between Vb and Vb∗ , (3) The form τ restricted to Vb if b = b∗ , or to Vb ⊕ Vb∗ if b 6= b∗ , is σb -hermitian. Two graded formed spaces (V, τ ) and (V ′ , τ ′ ) are called isometric if there is a D-linear bijection g : V → V ′ such that τ (Va ) ⊆ Va′
(a = 0, 1, 2, ..., n − 1),
τ (u, v) = τ ′ (gu, gv)
(u, v ∈ V ).
In that case we shall write (V, τ ) ≈ (V ′ , τ ′ ) and say that the map g is a graded isometry. Lemma 2.1. Every graded formed space (V, τ ) satisfying (1)-(3) above is isometric to an orthogonal direct sum of indecomposable graded formed spaces. Every indecomposable graded formed space either is concentrated in one degree b with b = b∗ and coincides with one of the spaces listed in Table 1, or is equal to the sum Vb ⊕ Vb∗ with b 6= b∗ hwhere i dimD (Vb ) = dimD (Vb∗ ) = 1 and in a suitable basis the form τ is given by the matrix σ0b 10 . The following lemma is an easy consequence of the classification of (non-graded) σhermitian forms.
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Lemma 2.2. Let ∗ be an involution of Z/nZ and let (V, τ ) and (V ′ , τ ′ ) be two Z/nZgraded formed spaces satisfying conditions (1)-(3) above. Then (V, τ ) ≈ (V ′ , τ ′ ) if and only if the following two conditions hold. (1) dim(V ) = dim(V ′ ). (2) sgn(τ |Vb ) = sgn(τ ′ |Vb′ ) for every b such that b = b∗ .
3
Classification of homogeneous nilpotent elements in certain Lie color algebras
Let n be an even positive integer and let V = V0 ⊕ V1 ⊕ · · · ⊕ Vn−1 be a Z/nZ-graded left vector space over D. For a ∈ Z/nZ let End(V )a = {X ∈ End(V ); X(Vb ) ⊆ Va+b for b ∈ Z/nZ }. Define a bilinear bracket [−, −] : End(V ) × End(V ) → End(V ) by the formula [X, Y ] = XY − (−1)ab Y X, for X ∈ End(V )a , Y ∈ End(V )b .
(4)
Then End(V ) becomes a Lie color algebra with respect to the symmetric bicharacter β(a, b) = (−1)ab on the group Z/nZ (see [1]). Fix a ∈ Z/nZ and let N ∈ End(V )a be nilpotent. Recall that the height of N, or the height of (N, V ), is the integer m ≥ 0 such that N m 6= 0 and N m+1 = 0. We shall write m = ht(N) = ht(N, V ). The pair (N, V ) of height m is called uniform if Ker(N m ) = NV . Lemma 3.1. Suppose the pair (N, V ) is uniform of height m. Then for any graded subspace E ⊆ V complementary to Ker(N m ), V = E ⊕ NE ⊕ N 2 E ⊕ ... ⊕ N m E. Moreover, dim(E) = dim(NE) = ... = dim(N m E). Proof. By the choice of E, we have V = E ⊕ NV . Hence, NV ⊆ NE + N 2 V . Thus V ⊆ E + NE + N 2 V . But N 2 V ⊆ N 2 E + N 3 V . Hence, inductively, V ⊆ E + NE + N 2 E + ... + N m E. If for some i0 < i1 < . . . < ik the intersection N i0 E ∩ (N i1 E + . . . + N ik E) were nonzero, then N i0 e0 = N i1 e1 + . . . + N ik ek for some ej ∈ E, with N i0 e0 6= 0, but then
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N m e0 = N m (e0 − N i1 −i0 e1 − . . . − N ik −i1 ek ) = 0, hence e0 = 0 by the injectivity of N m on E. Thus V = E ⊕ NE ⊕ N 2 E ⊕ ... ⊕ N m E. It remains to show the equality of dimensions. Suppose v ∈ E, 0 ≤ i ≤ m − 1 and NN i v = 0. Then N m v = 0. Hence v = 0, and therefore N i v = 0. Thus the linear map N i E ∋ u 7→ Nu ∈ N i+1 E is injective. Since this map is obviously surjective, we are done.
It is clear that if E is a graded subspace satisfying the conditions of Lemma 3.1 then we can reconstruct the dimension vector of V from the dimension vector of E, since dim(NE) is obtained from dim(E) by a shift in grading by a. More precisely, we can identify dimension vectors with functions on Z/nZ and let η be the left regular representation of Z/nZ i.e. (ηb f )(c) = f (c − b) (b, c ∈ Z/nZ).
(5)
dim(V ) = (1 + ηa + ηa2 + . . . + ηam )dim(E).
(6)
Then
3.1 A general linear Lie color algebra The group GL(V )0 = GL(V ) ∩ End(V )0 acts by conjugation on nilpotent elements in End(V )a for every a ∈ Z/nZ. In this subsection we will classify the nilpotent orbits of this action. The result can also be found in [12]. We include the proof for completeness and to introduce some notation used in the following sections. Lemma 3.2. Let N, N ′ ∈ End(V )a be nilpotent. Assume that the pairs (N, V ), (N ′ , V ) are uniform of height m. Then the elements N, N ′ are in the same GL(V )0 -orbit if and only if dim(V /Ker(N m )) = dim(V /Ker(N ′m )). Proof. There is only one non-trivial implication which requires proof. Suppose dim(V /Ker(N m ))a = dim(V /Ker(N ′m ))a for all a = 0, 1, 2, ..., n − 1. According to Lemma 3.1 we have decompositions V = E ⊕ NE ⊕ N 2 E ⊕ ... ⊕ N m E,
V = E ′ ⊕ N ′ E ′ ⊕ N ′2 E ′ ⊕ ... ⊕ N ′m E ′ . Since by our assumption dim(Ea ) = dim(Ea′ ) for all a, there is a graded linear isomorphism g : E → E ′ . We extend g to a graded linear isomorphism g : V → V by gN i v = N ′i gv Clearly gNg −1 = N ′ .
(v ∈ E; i = 0, 1, 2..., m).
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Lemma 3.3. If the pair (N, V ) is uniform of height m, then there exist graded Ninvariant subspaces V j ⊆ V such that V = V 1 ⊕ V 2 ⊕ ..., where each pair (N, V j ) is uniform, and dim(V j /Ker(N|V j )m ) = 1 for each j. Proof. This is clear from Lemma 3.1 via a decomposition of E into one dimensional subspaces. Lemma 3.4. Let (N, V ) be a pair of height m. Let U ⊆ V be a graded subspace such that (a) U is N-invariant, (b) (N, U) is uniform of height m. Then there is a graded N-invariant subspace U ′ ⊆ V such that (c) V = U ⊕ U ′ . Proof. We proceed by induction on m. If m = 0 then N = 0 and (c) is obvious. Suppose that m ≥ 1. We know from Lemma 3.1 that there is a graded subspace E such that U = E ⊕ NE ⊕ N 2 E ⊕ ... ⊕ N m E. Hence, U ∩ Ker(N m ) = NE ⊕ N 2 E ⊕ ... ⊕ N m E = NU. This space is N-invariant. The pair (N, U ∩ Ker(N m )) is uniform of height m − 1. Moreover, U ∩ Ker(N m ) ∩ Ker(N|Ker(N m ) )m−1 = U ∩ Ker(N m ) ∩ Ker(N m−1 ) = U ∩ Ker(N m−1 ) = N 2 E ⊕ N 3 E ⊕ ... ⊕ N m E 6= U ∩ Ker(N m ).
Hence, U ∩ Ker(N m ) * Ker(N|Ker(N m ) )m−1 . Thus the pair of spaces U ∩Ker(N m ) ⊆ Ker(N m ) satisfy conditions (a) and (b), but the height of (N, Ker(N m )) is m − 1. Therefore, by induction, there exists a graded N-invariant subspace U ′ ⊆ Ker(N m ) such that Ker(N m ) = U ∩ Ker(N m ) ⊕ U ′ . Hence, U + Ker(N m ) = U ⊕ U ′ .
(7)
If V = U + Ker(N m ), we are done. Otherwise, choose a graded subspace F ⊆ V such that V = F ⊕ (U + Ker(N m )). Let W = F + NF + N 2 F + ... + N m F. Then W is N-invariant and ht(N, W ) = m. We claim that U ∩ W = 0.
(8)
Indeed, if this is not the case then there exist e ∈ E \ {0} and u ∈ N i+1 U for some i, 0 ≤ i ≤ m, such that N i e + u ∈ W . Hence N m e ∈ W . Since NN m e = 0, there exists f ∈ F
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such that N m e = N m f . Hence, f ∈ U + Ker(N m ) and therefore f ∈ F ∩ (U + Ker(N m )). But this last space is zero. Thus 0 = f = N m f = N m e = e, a contradiction. Therefore (8) holds. If V = U ⊕ W , we are done. Otherwise, notice that V = (U ⊕ W ) + Ker(N m ). Moreover, the pair of spaces U ⊕ W ⊂ V satisfy conditions (a) and (b). Hence, by (7), V = U ⊕ (W ⊕ U ′ ). Lemma 3.5. If the pair (N, V ) is indecomposable, then it is uniform. Proof. In order to avoid trivialities we assume N 6= 0 and V 6= 0. Let E be a graded subspace of V such that V = E ⊕ Ker(N m ), where m = ht(N, V ). Let U = E ⊕ NE ⊕ N 2 E ⊕ ... ⊕ N m E. It is easy to see that the subspace U ⊆ V satisfies conditions (a) and (b) of Lemma 3.4. Hence, there is an N-invariant graded subspace U ′ ⊆ V such that V = U ⊕ U ′ . Since (N, V ) is indecomposable, U ′ = 0. Thus (N, V ) = (N, U) is uniform. Corollary 3.6. If V 6= 0 and if the pair (N, V ) is indecomposable, then it is uniform and dim(V /Ker(N m )) = 1, where m = ht(N, V ). Theorem 3.7. Let N ∈ End(V )a be nilpotent. Then there exist graded N-invariant subspaces V j ⊆ V such that (a) V = V 1 ⊕ V 2 ⊕ · · · ⊕ V s , (b) each (N, V j ) is indecomposable, (c) ht(N, V 1 ) ≥ ht(N, V 2 ) ≥ .... The decomposition (a) having the properties (b) and (c) is unique up to the action of GL(V )N 0 , the centralizer of N in GL(V )0 . Thus the above decomposition determines the GL(V )0 -orbit of N in End(V )a . Proof. Let E ⊆ V be a graded subspace complementary to the kernel of N m , where m is the height of (N, V ). Set U = E⊕NE⊕N 2 E⊕...⊕N m E. Then the pair (N, U) is uniform. If V = U, the theorem follows from Lemmas 3.2 and 3.3. Otherwise, notice that the spaces U ⊆ V satisfy the conditions (a), (b) of Lemma 3.4. Moreover, V = U ⊕ Ker(N m ). Hence, as in (7), there exists a graded N-invariant subspace U ′ ⊆ Ker(N m ) such that V = U ⊕ U ′ . In particular ht(N, U ′ ) < ht(N, V ). So we may proceed inductively. For b ∈ Z/nZ, let D[b] denote Z/nZ-graded space which is concentrated in homogeneous degree b and isomorphic to D as a non-graded space. If (N, V ) is indecomposable with N of degree a and height m then V = D[b] ⊕ ND[b] ⊕ · · · ⊕ N m D[b]
= D[b] ⊕ D[b + a] ⊕ · · · ⊕ D[b + ma]
for some b ∈ Z/nZ. Corollary 3.8. Nilpotent orbits of the group GL(V )0 in End(V )a are parametrized by
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sequences of pairs (b1 , m1 ), . . . , (bs , ms ) such that (1) m1 ≥ m2 ≥ · · · ≥ ms ≥ 0, mj ∈ N, bj ∈ {0, 1, . . . , n − 1}; P m (2) dim(V ) = sj=1 (1 + ηa + ηa2 + · · · + ηa j )dim(D[bj ]); (3) if mj = mj+1 then bj ≤ bj+1 .
3.2 The Lie color algebra of a formed space Here we consider hermitian analogues of the orthosymplectic Lie color algebras of [1]. For the remainder of this section fix σ = ±1. Define a map S ∈ End(V )0 by S(v) = (−1)a v
(v ∈ Va ; a ∈ Z/nZ),
(9)
and let τ be a non-degenerate sesqui-linear form on V such that (u, v ∈ V ).
τ (u, v) = σι(τ (v, Su))
(10)
We assume that the form τ provides a non-degenerate pairing between Vb and V−b for each b ∈ Z/nZ, and that Vb ⊥ Vc if b + c 6= 0, so that the involution ∗ of Section 2 is given by b∗ = −b. Let g(V, τ )a = {X ∈ End(V )a ; τ (Xu, v) + τ (S a u, Xv) = 0, u, v ∈ V } and let g(V, τ ) =
M
g(V, τ )a .
a∈Z/nZ
Then g(V, τ ) is closed under the bracket defined in (4) and it is a Lie color subalgebra of End(V ). Let G(V, τ )0 denote the isometry group G(V, τ )0 = {g ∈ End(V )0 ; τ (gu, gv) = τ (u, v), u, v ∈ V }.
(11)
The goal of this subsection is to classify the orbits of the group G(V, τ )0 in the set of nilpotent elements in each homogeneous component g(V, τ )a of the algebra g(V, τ ). The following lemma states several easy to check properties of homogeneous elements of g(V, τ ). Lemma 3.9. Let X ∈ g(V, τ )a . Then SX = (−1)a XS, τ (u, Sv) = τ (Su, v), a
τ (Xu, v) = −τ (u, S Xv),
τ (u, Xv) = −τ (XS a u, v).
(12)
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Define δ(k) = (−1)k(k−1)/2 . Then for k, l ≥ 0 (SX)k = δ(k)a S k X k , S l X k = (−1)akl X k S l , k
k
a
(13) ak
k
τ (X u, v) = (−1) δ(k) τ (u, S X v), τ (u, X k v) = (−1)k δ(k + 1)a τ (S ak X k u, v). Suppose henceforth that N ∈ g(V, τ )a be nilpotent. Lemma 3.10. Let m = ht(N, V ). Then the formula τ˜(˜ u, v˜) = τ (u, N m v) (˜ u = u + Ker(N m ), v˜ = v + Ker(N m ); u, v ∈ V ) defines a non-degenerate sesqui-linear form on the Z/nZ-graded space V˜ = V /Ker(N m ) and τ˜(˜ u, v˜) = (−1)m δ(m + 1)a σι˜ τ (S ma+1 v˜, u ˜), ma
τ˜(S u˜, v˜) = (−1)
τ˜(˜ u, S˜ v),
V˜b and V˜c are τ˜-orthogonal, unless b + c + ma = 0. Proof. The fact that the form is well defined and the statements (14), (15) and follow easily from (10) and (13). We will show that the form τ˜ is non-degenerate. to (15), it is enough to show that for every non-zero v˜ ∈ V˜ there exists u˜ ∈ V˜ such τ˜(˜ u, v˜) 6= 0. Let v˜ = v + Ker(N m ). Then N m v 6= 0 and there exists u ∈ V such τ (u, N m v) 6= 0 and for u˜ = u + Ker(N m ) we have τ˜(˜ u, v˜) 6= 0.
(14) (15) (16) (16) Due that that
Theorem 3.11. Let (N, V ) be uniform of height m. Then there is a graded subspace F ⊆ V , complementary to Ker(N m ), such that V = F ⊕ NF ⊕ N 2 F ⊕ ... ⊕ N m F
(a)
N k F ⊥ N l F for k + l 6= m.
(b)
and
Remark 3.12. It is clear from (13) that (b) is equivalent to F ⊥ N k F for 0 < k ≤ m − 1. Proof of Theorem 3.11. We will define inductively a sequence F (0) , F (1) , . . . , F (m−1)
(b′ )
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of subspaces of V such that for all k = 0, 1, . . . , m − 1 the following conditions hold: (i) F (k) is graded and V = F (k) ⊕ NF (k) ⊕ N 2 F (k) ⊕ ... ⊕ N m F (k) , (ii) F (k) ⊥ N m−k F (k) + N m−k+1 F (k) + . . . + N m−1 F (k) . Then F = F (m−1) satisfies the conditions of the theorem. It follows from Lemma 3.1 that for F (0) we can take any graded subspace complementary to Ker(N m ). Assume that k > 0 and that the space F (k−1) has already been constructed. Set E = F (k−1) and let E ∗ = HomD (E, D). Define two maps τˆ0 , τˆ : E → E ∗ ,
τˆ0 (v)(u) = τ (u, N m v),
τˆ(v)(u) = τ (u, N m−k v),
We know from Lemma 3.10 that τˆ0 is a bijection. Notice that τ (u, N m τˆ0−1 τˆ(v)) = τ (u, N m−k v)
(u, v ∈ E).
(u, v ∈ E).
(17)
Indeed, the left hand side of (17) is equal to τ (u, N m τ0−1 τˆ(v)) = τˆ0 τˆ0−1 τˆ(v)(u) = τˆ(v)(u) = τ (u, N m−k v). Let
1 (18) ρ = ρk = τˆ0−1 τˆ 2 and define the space F (k) as F (k) = (1 − N k ρ)F (k−1) . First we will show that the space F (k) is graded. We see from (17) that for any v ∈ E, N k τˆ0−1 τˆ(v) coincides with the unique element x ∈ N k E such that for all u ∈ E, τ (u, N m−k (x − v)) = 0. Hence, if v ∈ Eb := E ∩ Vb , then τ (u, N m−k x) = 0 for all P u ∈ c6=−c0 Ec , where c0 = (m − k)a + b. Thus, N m−k x ∈ (N m E) ∩ =
\
c6=−c0
m
(N E) ∩
Ec⊥
X
Ec
c6=−c0
=
\
c6=−c0
!⊥
= (N m E) ∩
m
(N E) ∩
Vc⊥
\
Ec⊥
c6=−c0
= (N m E) ∩ V(m−k)a+b .
P Let us write x = xb + x′ , where xb ∈ Vb ∩ N k E and x′ ∈ c6=b Vc ∩ N k E. Since N m−k x ∈ V(m−k)a+b , we have N m−k x = N m−k xb . Therefore, by the uniqueness of x, x′ = 0. Hence, x ∈ Vb ∩ N k E. Thus N k ρ(Eb ) ⊆ Vb ∩ N k E for all b ∈ Z/nZ and the space F (k) = (1 − N k ρ)E is graded. It follows from the construction that dim(F (k) ) ≤ dim(E) and E ⊂ F (k) +N k E. Since N m+1 F (k) = 0, we have V = E ⊕ NE ⊕ N 2 E ⊕ ... ⊕ N m E ⊂ F (k) + NF (k) + N 2 F (k) + · · · + N m F (k) , so the sum on the right hand side is also a direct sum. It remains to prove that F (k) satisfies the orthogonality property (ii). For u, v ∈ E τ ((1 − N k ρ)u, N m−k (1 − N k ρ)v) = τ (u, N m−k v) − τ (u, N m ρv) − τ (N k ρu, N m−k v), (19)
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because N k N m = 0. Furthermore, by (17), 1 τ (u, N m ρv) = τ (u, N m−k v), 2 and by (10), (13) and (17)
τ (N k ρu, N m−k v) = σι τ (N m−k v, SN k ρu) = (−1)m−k δ(m − k)a σι τ (v, S (m−k)a N m−k SN k ρu)
= (−1)(m−k)(a+1) δ(m − k)a σιτ (S (m−k)a+1 v, N m ρu)
= 12 (−1)(m−k)(a+1) δ(m − k)a σιτ (S (m−k)a+1 v, N m−k u) = 21 (−1)(m−k)(a+1) δ(m − k)a τ (N m−k u, S (m−k)a v)
= 12 (−1)(m−k)a τ (u, S (m−k)a N m−k S (m−k)a v) = 12 τ (u, N m−k v).
Hence (19) is equal to zero and F (k) is orthogonal to N m−k F (k) . Notice also that for i > 0 τ ((1 − N k ρ)u, N m−k+i (1 − N k ρ)v)
= τ (u, N m−k+i v) − τ (u, N m+i ρv) − τ (N k ρu, N m−k+i v) = τ (u, N m−k+i v).
Hence F (k) is also orthogonal to N l F (k) , (l = m − k + 1, . . . , m − 1), due to condition (ii) for F (k−1) . Corollary 3.13. Let the pairs (N, V ), (N ′ , V ) be uniform of height m with N, N ′ homogeneous of the same degree. Then the elements N, N ′ are in the same G(V, τ )0 orbit if and only if the spaces (V /Ker(N m ), τ˜), (V /Ker(N ′m ), τ˜) are isometric. Proof. It is easy to check that if N, N ′ are conjugate by an element of G(V, τ )0 then the above two graded spaces are isometric. Conversely, suppose these two spaces are isometric. Let N, N ′ ∈ g(V, τ )a and let F, F ′ ⊆ V be as in Theorem 3.11 for N, N ′ respectively. By assumption, we have a graded bijection g : F → F ′ such that τ (gu, N ′m gv) = τ (u, N m v)
(u, v ∈ F ).
Set g(N k v) = N ′k g(v)
(v ∈ F ; k = 0, 1, 2, ..., m − 1).
Then g ∈ End(V ) is bijective and intertwines N and N ′ . Furthermore, for u, v ∈ F and
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for k = 0, 1, 2, ..., m − 1 we have τ (gN k u, gN m−k v) = τ (N ′k gu, N ′m−k gv) = (−1)k δ(k)a τ (gu, S ak N ′m gv) = (−1)k+akm δ(k)a τ (gu, N ′m gS ak v) = (−1)k+akm δ(k)a τ (u, N m S ak v) = (−1)k+ak δ(k)a τ (u, N k S ak N m−k v) = (−1)ak δ(k)a δ(k + 1)a τ (S ak N k u, S ak N m−k v) = τ (N k u, N m−k v). Hence, g ∈ G(V, τ )0 .
Definition 3.14. The pair (N, V ) is called indecomposable if V does not have any nontrivial orthogonal N-invariant direct sum decomposition into graded subspaces. Otherwise the pair (N, V ) is called decomposable. Proposition 3.15. If the pair (N, V ) is indecomposable then it is uniform. Proof. Let m = ht(N, V ) and let E ⊆ V be a graded subspace complementary to Ker(N m ). Set U = E + NE + N 2 E + ... + N m E. Then U is a graded subspace of V preserved by N and it is easy to see that U is uniform. We will show that the restriction of the form τ to U is non-degenerate. Since, U ⊥ is N-invariant (by (12)) this will complete the proof. Let 0 ≤ k ≤ i ≤ m and let ui ∈ E. Suppose N k uk + N k+1 uk+1 + ... + N m um ⊥ U. Then N k uk ⊥ N m−k E. Hence, by (13), uk ⊥ N m E. But N m E = N m V . Thus uk ⊥ N m V . Therefore N m uk ⊥ V . Hence, uk ∈ Ker(N m )∩E = {0}. Similarly, uk+1 = uk+2 = ... = um = 0. Proposition 3.16. Let the pair (N, V ) be uniform of height m. Then (N, V ) is indecomposable if and only if the formed space (V /Ker(N m ), τ˜) is indecomposable. Proof. Clearly if (N, V ) is decomposable then so is (V /Ker(N m ), τ˜). Conversely, suppose (V /Ker(N m ), τ˜) is decomposable. Choose a subspace F ⊆ V as in Theorem 3.11 and let τm (u, v) = τ (u, N m v) (u, v ∈ F ). Then (F, τm ) is isometric to (V /Ker(N m ), τ˜), and hence is decomposable. Thus there exist two non-zero graded τm -orthogonal subspaces F ′ , F ′′ ⊆ F such that F = F ′ ⊕ F ′′ . Let V ′ = F ′ + NF ′ + N 2 F ′ + ... + N m F ′ and let V ′′ = F ′′ + NF ′′ + N 2 F ′′ + ... + N m F ′′ . Since the spaces F ′ , F ′′ are τm -orthogonal, we have F ′ ⊥ N m F ′ and F ′′ ⊥ N m F ′′ , with respect to τ . Hence it is easy to see that V = V ′ ⊕ V ′′ and V ′ ⊥ V ′′ .
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It follows from Corollary 3.13 and Proposition 3.16 that in order to classify indecomposable nilpotent elements of height m in g(V, τ )a up to the action of G(V, τ )0 , we have to classify (up to grading-preserving isometry) indecomposable Z/nZ-graded formed spaces (V˜ , τ˜). Proposition 3.17. Let (V˜ , τ˜) be an indecomposable Z/nZ-graded formed space. Let a ∈ Z/nZ and let m ∈ N. If (V˜ , τ˜) satisfies (14), (15) and (16) then dim(V˜ ) = 1 or 2. Moreover one of the following two conditions holds. (1) V˜ = V˜b for some b ∈ Z/nZ with 2b + ma = 0 (which implies that ma is even) and (V˜b , τ˜) is nondegenerate indecomposable as a (nongraded) formed space. (2) V˜ = V˜b ⊕ V˜−b−ma for some b ∈ Z/nZ with 2b + ma 6= 0. In this case both summands are τ˜-isotropic of dimension one and τ˜ provides a pairing between them. In both cases the form τ˜ is σ ˜ -hermitian, where σ ˜=σ ˜ (a, b, m) = (−1)m (−1)(ma+1)b δ(m + 1)a σ.
(20)
Any formed space as above will be called (a, m)-admissible. Proof. Let V˜even = V˜0 ⊕ V˜2 ⊕ . . . ⊕ V˜n−2 , V˜odd = V˜1 ⊕ V˜3 ⊕ . . . ⊕ V˜n−1 . Assume first that 2|ma. Then (15) is equivalent to the condition that V˜even and V˜odd are τ˜-orthogonal. Indecomposability of (V˜ , τ˜) forces V˜even = 0 or V˜odd = 0. Then condition (14) says that τ˜ is σ ˜ -hermitian and once again from the indecomposability we see that we are in the case (1) or (2) of the proposition. If 2 ∤ ma then (15) is equivalent to the condition that V˜even and V˜odd are τ˜-isotropic. Indecomposability and conditions (14) and (16) guarantee that we are in case (2). Theorem 3.18. Let N ∈ g(V, τ )a be a nilpotent. Then there exist a sequence (F (1) , F (2) , . . . , F (s) ) of graded subspaces of V and a sequence m1 ≥ m2 ≥ . . . ≥ ms ≥ 0 of nonnegative integers such that (1) for every i = 1, 2, . . . , s, the space F (i) with the form τ˜(i) given by the formula τ˜(i) (u, v) = τ (u, N mi v),
(u, v ∈ F (i) )
is an (a, mi )-admissible space (as defined in Proposition 3.17); L (2) V = si=1 F (i) ⊕ NF (i) ⊕ · · · ⊕ N mi F (i) .
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Let N ′ ∈ g(V, τ )a be another nilpotent element and let (F ′(1) , F ′(2) , . . . , F ′(s ) ) and m′1 ≥ m′2 ≥ . . . ≥ m′s′ be the sequences corresponding to N ′ . Then N and N ′ are G(V, τ )0 conjugate if and only if s = s′ , mi = m′i for every i = 1, 2, . . . , s and, up to a permutation of indices i preserving the sequence (mi )si=1 , the graded formed spaces (F (i) , τ˜(i) ) and ′ (F ′(i) , τ˜(i) ) are isometric. ′
Proof. Let E ⊆ V be a graded subspace complementary to Ker(N m ), where m = ht(N, V ). Set U = E + NE + N 2 E + ... + N m E. As in the proof of Proposition 3.15 we verify that the restriction of τ to U is non-degenerate. Notice that U ⊥ ⊆ (N m E)⊥ = (N m V )⊥ = Ker(N m ). Hence, ht(N, U ⊥ ) < ht(N, V ). After a finite number of steps we obtain V = U (1) ⊕ U (2) ⊕ ... ⊕ U (r) , where the spaces U (j) are graded, N-invariant, mutually orthogonal, each pair (N, U (j) ) is uniform and ht(N, U (1) ) > ht(N, U (2) ) > ... > ht(N, U (r) ). Now we split each (N, U (j) ) into indecomposables and obtain V = V (1) ⊕ V (2) ⊕ ... ⊕ V (s) . Let mi = ht(N, V (i) ) and let F (i) be a graded subspace of V (i) satisfying conditions of Theorem 3.11 for the restriction of N to V (i) . Then the sequences (F (1) , F (2) , . . . , F (s) ) and (m1 , m2 , . . . , ms ) satisfy conditions (1) and (2) of the theorem. Consider a nilpotent element N ′ in the G(V, τ )0 -orbit of N and assume that the ′ sequences (F ′ (1) , F ′(2) , . . . , F ′ (s ) ) and m′1 ≥ m′2 ≥ · · · ≥ m′s′ ≥ 0 satisfy conditions (1) and (2). Then m1 = ht(V, N) = ht(V, N ′ ) = m′1 . Denote this number by m and let ℓ = max{i : mi = m}, ℓ′ = max{i : m′i = m}. Then the spaces V /N m V ≈ F (1) ⊕ F (2) ⊕ · · · ⊕ F (ℓ) , m
V /N ′ V ≈ F ′
(1)
⊕ F′
(2)
⊕ ··· ⊕ F′
(ℓ′ )
are isomorphic and the isomorphism becomes an isometry when we equip the spaces with ′ ′ ⊕· · ·⊕ τ˜(ℓ′ ′ ) , respectively. Hence ℓ = ℓ′ and, up ⊕ τ˜(2) the forms τ˜(1) ⊕ τ˜(2) ⊕· · ·⊕ τ˜(ℓ) and τ˜(1)
′ ) are isometric for to permutation of indices i, the formed spaces (F (i) , τ˜(i) ) and (F ′ (i) , τ˜(i) i = 1, 2, . . . , ℓ. Now, the necessity of the condition follows by induction on the dimension of V . It is clear from Corollary 3.13 that the above argument may be reversed. Hence the proof is complete.
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Corollary 3.19. Nilpotent orbits of the group G(V, τ )0 in g(V, τ )a are parameterized by sequences (F (i) , τ˜(i) , mi ),
i = 1, . . . , s,
such that (1) the space (F (i) , τ˜(i) ) is (a, mi )-admissible for every i = 1, 2, . . . , s; P (2) dim(V ) = si=1 (1 + ηa + ηa2 + · · · + ηami )dim(F (i) ); (3) for b = 0 and for b = n/2, m m s i i X X X (i) sgn(τ |Vb ) = dim Fb−ka · (1, 1) + sgn((−1)k+ka(b+1) τ˜(i) |F (i) ) . i=1
k=0 (mi −2k)a6=0
b−ka
k=0 (mi −2k)a=0
′ Two such sequences (F (i) , τ˜(i) , mi ), i = 1, . . . , s, and (F ′(i) , τ˜(i) , m′i ), i = 1, . . . , s′ , determine the same orbit if and only if s = s′ and there exists a permutation π of the set {1, 2, . . . , s} such that for i = 1, 2, . . . , s the graded spaces (F (i) , τ˜(i) ) and (F ′(πi) , τ˜(πi) ) are isometric and mi = m′πi . Wherever convenient, we shall identify the orbit with the sequence (or with the formal direct sum) of triples (F (i) , τ˜(i) , mi ).
Proof. By Theorem 3.18, it remains to show that for every pair of sequences (F (i) , τ˜(i) , mi ), i = 1, . . . , s, satisfying conditions 1.–3. of the corollary, there exists a corresponding nilpotent element N ∈ g(V, τ )a . For an a ∈ Z/nZ and a Z/nZ-graded vector space W , let ηa W be a copy of W shifted in grading by a. For i = 1, 2, . . . , s, let V′
(i)
= F (i) ⊕ ηa F (i) ⊕ ηa2 F (i) ⊕ · · · ⊕ ηamj F (i)
and V′ = V′
(1)
⊕V′
(2)
(s)
⊕···⊕V′ .
We equip V ′ with a sesquilinear form τ ′ such that the spaces V ′ (i) are mutually orthogonal (i) and for u ∈ F (i) , v ∈ Fb τ ′ (ηak u, ηal v) =
( (−1)k (−1)ak(mi +b) δ(k)a τ˜(i) (u, v), if k + l = mi , 0,
otherwise.
(21)
Then it follows from (13) that the formed spaces (V, τ ) and (V ′ , τ ′ ) are isometric. Defining a nilpotent endomorphism N ′ of V ′ by ( ηak+1 (u), for k < mi , N ′ (ηak u) = (u ∈ F (i) ), (22) 0, for k = mi , we obtain the orbit corresponding to the sequence (F (i) , τ˜(i) , mi ), i = 1, . . . , s.
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4
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Dual pairs of type II
Let V = V0 ⊕ V1 be a Z/2Z-graded vector space over D. The group GL(V )0 = GL(V ) ∩ End(V )0 of degree zero linear automorphisms of V is isomorphic to the direct product GL(V0 ) × GL(V1 ). The results of section 3.1 give the classification of nilpotent orbits of GL(V )0 in End(V )0 = End(V0 ) × End(V1 ), which reduces to the well known classification of nilpotent orbits of GL(Vj ) in End(Vj ) via the Jordan normal form, and the classification of nilpotent orbits of GL(V )0 in End(V )1 = Hom(V0 , V1 ) ⊕ Hom(V1 , V0 ), which is also well known (see [6], Section 2). The formula h(A, B), (A′ , B ′ )i = trD/R (AB ′ ) − trD/R (BA′ ) ′
(23)
′
(A, A ∈ Hom(V0 , V1 ), B, B ∈ Hom(V1 , V0 )) defines a non-degenerate symplectic form on the real vector space W = Hom(V0 , V1 ) ⊕ Hom(V1 , V0 ). The action of the group GL(V0 ) × GL(V1 ) on W preserves this form, hence the groups GL(V0 ), GL(V1 ) form a dual pair of type II in the symplectic group Sp(W ). The maps End(V )1 ∋ N → N 2 |V0 ∈ End(V0 ), End(V )1 ∋ N → N 2 |V1 ∈ End(V1 )
(24) (25)
coincide with the moment maps νk : W −→ End(Vk ),
(26)
Hom(V0 , V1 ) ⊕ Hom(V1 , V0 ) ∋ (A, B) → AB ∈ End(V1 ).
(28)
Hom(V0 , V1 ) ⊕ Hom(V1 , V0 ) ∋ (A, B) → BA ∈ End(V0 ),
4.1
(27)
Nilpotent orbits in gln (D)
Theorem 3.7 and Corollary 3.8 give the following classification of nilpotent orbits of GL(V )0 in End(V )0 = End(V0 ) × End(V1 ) in terms of the sizes of the blocks of the Jordan normal form of the restriction of a nilpotent endomorphism of V to Vk : Corollary 4.1. The nilpotent orbits of the group GL(V )0 in End(V )0 = End(V0 ) × End(V1 ) are parametrized by sequences of pairs (b1 , m1 ), . . . , (bs , ms ) such that (1) m1 ≥ m2 ≥ · · · ≥ ms ≥ 0, mj ∈ N, bj = 0, 1; P (2) dim(Vk ) = bj =k (mj + 1); (3) if mj = mj+1 then bj ≤ bj+1 ; where for each j = 1, . . . , s, mj + 1 is the size of the appropriate block in Vaj . In particular, the nilpotent orbits of GL(Vk ) in End(Vk ) are parametrized by sequences P m1 ≥ m2 ≥ · · · ≥ ms > 0 satisfying j (mj + 1) = dim(Vk ).
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Nilpotent orbits in W
Theorem 3.7 and Corollary 3.8 give the classification of nilpotent orbits of the group GL(V )0 = GL(V0 ) × GL(V1 ) in W = End(V )1 in terms of the parameters of the Jordan normal form. For an alternate description of the parametrization of orbits the reader may consult [6], Section 2. Let m + 1 + δk,b dk (b, m) = , k = 0, 1, (29) 2 where δk,b is the Kronecker delta and [x] is the largest integer less than or equal to x. Corollary 4.2. The nilpotent orbits of the group GL(V )0 in W = End(V )1 are parametrized by sequences of pairs (b1 , m1 ), . . . , (bs , ms ) such that (1) m1 ≥ m2 ≥ · · · ≥ ms ≥ 0, mj ∈ N, bj = 0, 1; P (2) dim(Vk ) = j dk (bj , mj ); (3) if mj = mj+1 then bj ≤ bj+1 . Now we describe the action of the moment maps (26) on nilpotent orbits in terms of the parameters introduced above. Corollary 4.3. Let O ⊆ W be the nilpotent orbit which corresponds to the sequence (b1 , m1 ), . . . , (bs , ms ), then the image νk (O) is equal to the nilpotent orbit in End(Vk ) corresponding to the sequence (dk (b1 , m1 ), . . . , dk (bs , ms )). We end this section by giving an explicit description of all non-zero nilpotent indecomposable elements (N, V ), N ∈ End(V )1 . Proposition 4.4. The following is a complete list of all non-zero nilpotent indecomposable elements (N, V ), N ∈ End(V )1 . V =
m M k=0 k
Dvk , veven ∈ V0 , vodd ∈ V1 ,
(a)
vk = N v0 6= 0, 0 ≤ k ≤ m, Nvm = 0.
V =
m+1 M k=1
Dvk , veven ∈ V0 , vodd ∈ V1 ,
(b)
vk+1 = N k v1 6= 0, 0 ≤ k ≤ m, Nvm+1 = 0.
4.3
Nilpotent orbits in pC
Let gln (D) = k ⊕ p be a Cartan decomposition of the general Lie algebra. The complexification KC of the maximal compact subgroup K ⊆ GLn (D) acts on the complexification
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pC of p. For D = C the action of KC on pC may be identified with the conjugation action of GLn (C) on gln (C), so in this case the description of nilpotent orbits is as in Section 4.1. For D = R, H the classification of the nilpotent orbits is well known ([18]). Both in the case D = R and in the case D = H a nilpotent KC -orbit in pC is uniquely determined by the Jordan canonical form of any of its element, hence by the corresponding partition of n if D = R and of 2n if D = H. In the first case all partitions arise, in the second case a partition arises if and only if each of its parts occurs with an even multiplicity.
4.4
Nilpotent orbits in WC+
The complete classification of the nilpotent orbits in WC+ was given in [6] (in fact, as noted in [6], in the context of symmetric spaces it was described earlier by Ohta). It turns out that it can also be obtained as a special case of the classification of Section 3. Consider first the case D = R. Let U = U0 ⊕ U1 ⊕ U2 ⊕ U3 be the Z/4Z-graded complex vector space defined by U0 = V0 ⊗ C, U2 = V1 ⊗ C, U1 = U3 = 0, endowed with a nondegenerate symmetric form ϕ with U0 orthogonal to U2 . Then the group G(U, ϕ)0 of homogeneous isometries of U, equal to O(U0 ) × O(U2 ), is isomorphic to KC × KC′ , and its conjugation action on g(U, ϕ)2 can be identified with the action of KC × KC′ on WC+ (see Section 3 of [6]). It is easy to see that Theorem 3.18 and Corollary 3.19 describe a classification of nilpotent orbits in WC+ equivalent to Theorem 3.6 in [6]. The case D = H is similar. Let U = U0 ⊕ U1 ⊕ U2 ⊕ U3 be the Z/4Z-graded complex vector space, defined by U0 = U2 = 0, U1 = V0 |C , U3 = V1 |C , with the complex structures being the restriction of the structures of vector spaces over H. Let ϕ be a nondegenerate skew-symmetric form on U with U1 orthogonal to U3 . Then g(U, ϕ)2 can be identified with WC+ . The group G(U, ϕ)0 of homogeneous isometries of U is equal to Sp(U1 ) × Sp(U3 ), and it is isomorphic to KC × KC′ . Its conjugation action on g(U, ϕ)2 can be identified with the action of KC × KC′ on WC+ (see Section 4 of [6]). Theorem 3.18 and Corollary 3.19 give a classification of nilpotent orbits in WC+ equivalent to Theorem 4.5 in [6].
5
Dual Pairs of type I
Now we consider the case of the Lie color algebra of a Z/2Z-graded formed space V = V0 ⊕ V1 . Let τ be the form considered in (10) with σ = 1. Then τ = τ0 ⊕ τ1 , where τ0 is a non-degenerate hermitian form on V0 and τ1 is a non-degenerate skew-hermitian form on V1 . The group G(V, τ )0 , defined in (11), is isomorphic to the direct product G(V0 , τ0 ) × G(V1 , τ1 ), by restriction. Similarly, the component of degree 0 of g(V, τ ) is a Lie algebra isomorphic to the direct sum of the corresponding Lie algebras g(V, τ )0 = g(V0 , τ0 ) ⊕ g(V1 , τ1 ). Hence the classification of G(V, τ )0 orbits in g(V, τ )0 is equivalent to the classification of G(V0 , τ0 ) orbits in g(V0 , τ0 ) and G(V1 , τ1 ) orbits in g(V1 , τ1 ). We consider this case in
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Section 5.1. in Section 5.2 we explain how the action of G(V, τ )0 on g(V, τ )1 may be viewed in terms of dual pairs of type I.
5.1 Nilpotent orbits in the Lie algebra of an isometry group In order to describe G(V, τ )0 orbits in g(V, τ )0 we begin with a classification of (0, m)admissible spaces. It follows from Proposition 3.17 that if the space (V˜ , τ˜) is (0, m)admissible, then V˜ = V˜b for some b = 0, 1 and τ˜ is σ ′ -hermitian nondegenerate indecomposable form with σ ′ = (−1)b+m . Such forms are well known and are tabulated in Table 1. A complete classification of the orbits is given by Theorem 3.18 (with a = 0). The closure relations between nilpotent orbits in the Lie algebras of isometry groups of formed spaces have been described by Djokovic. If X, Y ∈ g(Vb , τb ), b = 0, 1, are nilpotent, then by Theorem 6 in [8], OY ⊂ OX if and only if sgn τb ( , Y k ) ≤ sgn τb ( , X k ),
k = 1, 2, . . . ,
(30)
where “(m, n) ≤ (m′ , n′ )” means “m ≤ m′ and n ≤ n′ ”. If a nilpotent orbit O ∈ N Og(Vb , τb ) is identified with the sequence (F (i) , τ˜(i) , mi ), i = 1, 2, . . . , s, as above, then for X ∈ O and k = 0, 1, . . . sgn τb (−, X k −) = +
X 1 mi −k (mi − k)(fi , fi ) + sgn(−1) 2 τ˜(i) 2 k≤m
i 2|mi −k
X 1 (mi − k + 1)(fi , fi ) 2 k≤m
(31)
i 2∤mi −k
where fi = dimD F (i) .
5.2
Nilpotent orbits in W
Let W be the real vector space W = HomD (V0 , V1 ). The groups G(V0 , τ0 ), G(V1 , τ1 ) act on the space W by the formula: g0 (w) = wg0−1, g1 (w) = g1 w
(32)
(g0 ∈ G(V0 , τ0 ), g1 ∈ G(V1 , τ1 ), w ∈ W ). Define a map Hom(V0 , V1 ) ∋ w → w ∗ ∈ Hom(V1 , V0 ) by τ1 (wv0 , v1 ) = τ0 (v0 , w ∗v1 )
(v0 ∈ V0 , v1 ∈ V1 ).
(33)
(w, w ′ ∈ Hom(V0 , V1 ))
(34)
Then the formula hw, w ′i = − trD/R (w ′∗ w)
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defines a non-degenerate symplectic form on the real vector space Hom(V0 , V1 ). It is easy to see that the action (32) preserves the form (34). Hence the groups G(V0 , τ0 ), G(V1 , τ1 ) form a dual pair of type I in the symplectic group defined by the form (34). Furthermore, the maps ν0 : g(V, τ )1 ∋ N 7→ N 2 |V0 ∈ g(V0 , τ0 ),
(35)
2
ν1 : g(V, τ )1 ∋ N 7→ N |V1 ∈ g(V1 , τ1 ) coincide with the moment maps Hom(V0 , V1 ) ∋ w 7→ −w ∗ w ∈ g(V0 , τ0 ),
(36)
∗
Hom(V0 , V1 ) ∋ w 7→ −ww ∈ g(V1 , τ1 ). Lemma 5.1. The map g(V, τ )1 ∋ N 7→ N|V0 ∈ Hom(V0 , V1 ) is an R-linear bijection which intertwines the adjoint action of G(V, τ )0 on g(V, τ )1 with the action of G(V0 , τ0 ) × G(V1 , τ1 ) on Hom(V0 , V1 ) given by (32). Proof. Let N ∈ g(V, τ )1 . Then for v0 ∈ V0 and v1 ∈ V1 τ1 (Nv0 , v1 ) = τ (Nv0 , v1 ) = τ (v0 , SNv1 ) = −τ0 (v0 , Nv1 ). Hence, N|V1 = −(N|V0 )∗ . Thus the R-linear map N → N|V0 is bijective. For g ∈ G(V, τ )0 , let g0 = g|V0 and let g1 = g|V1 . Then (gNg −1 )|V0 = g1 (N|V0 )g0−1. Notice that in terms of Lemma 5.1, the symplectic form (34) coincides with the graded trace 1 hN, N ′ i = trD/R ([SN, N ′ ]), (N, N ′ ∈ g(V, τ )1 ), (37) 4 where [−, −] is the bracket defined in (4) i.e. [SN, N ′ ] = SNN ′ + N ′ SN ∈ g(V, τ )0 .
(38)
We will say that an element w ∈ Hom(V0 , V1 ) is nilpotent if the element w + w ∗ is a nilpotent endomorphism of V0 ⊕V1 . Observe that w ∈ Hom(V0 , V1 ) is nilpotent if and only if its image by the moment map ν0 (equivalently ν1 ) is nilpotent. We see from Lemma 5.1 that the problem of classifying the nilpotent orbits in the symplectic space Hom(V0 , V1 ) under the action of the dual pair G(V0 , τ0 ), G(V1 , τ1 ) is equivalent to the problem of classifying the nilpotent G(V, τ )0 -orbits in g(V, τ )1 , considered in section 2.2. In order to obtain a classification of the orbits we need the list of all (1, m)-admissible spaces. This list is provided in Table 2. The following proposition gives an explicit description of indecomposable graded nilpotent morphisms. The proposition follows directly from Theorem 3.18 and Table 2.
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Case 1. m even (D, ι)
(R, id)
(C, id)
(C,¯· )
(H,¯· )
m≡0 (mod 4)
m≡2 (mod 4)
(R[0], +)
(R2 [0], sk)
(R[0], −)
(R[1], +)
(R2 [1], sk)
(R[1], −)
(C[0], sym) (C2 [0], sk) (C2 [1], sk) (C[1], sym) (C[0], +)
(C[0], i)
(C[0], −)
(C[0], −i)
(C[1], i)
(C[1], +)
(C[1], −i)
(C[1], −)
(H[0], +)
(H[0], sk)
(H[0], −)
(H[1], +)
(H[1], sk)
(H[1], −)
Case 2. m odd m≡1
m≡3
(mod 4)
V˜ = D[0] ⊕ D[1]
(mod 4)
V˜ = D[0] ⊕ D[1]
τ˜(x, y) = x · J2′ · ι(y)t τ˜(x, y) = x · J2 · ι(y)t h i 0 1 where J2 = −1 where J2′ = 01 10 0 Table 2 The list of Z/2Z-graded (1, m)-admissible spaces.
Proposition 5.2. For a 1-hermitian form α with sgn α = (n+ , n− ) let sign(α) = n+ −n− . The following is a complete list of all non-zero nilpotent indecomposable elements (N, V ), N ∈ g(V, τ )1 . m ∈ 4Z; P V = m k=0 Dvk , veven ∈ V0 , vodd ∈ V1 ; (a)
vk = N k v0 6= 0, 0 ≤ k ≤ m, Nvm = 0; τ (vk , vl ) = 0
if l 6= m − k, τ (vk , vm−k ) = (−1)k δ(k)δ( m2 )sgn(τ0 ), where s = 1 if D = C and ι = 1, and s = sign(τ0 ) otherwise;
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m ∈ 4Z, D 6= R, ι 6= 1; P V = m+1 k=1 Dvk , veven ∈ V0 , vodd ∈ V1 ;
vk+1 = N k v1 6= 0, 0 ≤ k ≤ m, Nvm+1 = 0;
(b) τ (vk , vl ) = 0 if l 6= m + 2 − k, τ (vk , vm+2−k ) = δ(k − 1)τ (v1 , vm+1 ), τ (v1 , vm+1 ) = i sign(−iτ1 )δ(1 +
m ) 2
if D = C;
τ (v1 , vm+1 ) = j if D = H;
m ∈ 4Z, D 6= H, ι = 1; P ′ ′ ′ V = m+1 k=1 (Dvk ⊕ Dvk ), veven , veven ∈ V0 , vodd , vodd ∈ V1 ;
′ vk+1 = N k v1 6= 0, vk+1 = N k v1′ 6= 0, 0 ≤ k ≤ m, Nvm+1 = 0, ′ (c) Nvm+1 = 0;
τ (vk , vl ) = τ (vk′ , vl′ ) = 0, 1 ≤ k, l ≤ m + 1, τ (vk , vl′) = τ (vk′ , vl ) = 0, l 6= m + 2 − k, ′ τ (vk , vm+2−k ) = −τ (vk′ , vm+2−k ) = δ(k − 1), 1 ≤ k ≤ m + 1;
m ∈ 2Z \ 4Z, D 6= R, ι 6= 1; P V = m k=0 Dvk , veven ∈ V0 , vodd ∈ V1 ;
(d) vk = N k v0 6= 0, 0 ≤ k ≤ m, Nvm = 0;
τ (vk , vl ) = 0 if l 6= m − k, τ (vk , vm−k ) = δ(k − 1)i sign(−iτ1 ), (here − iτ1 is hermitian);
m ∈ 2Z \ 4Z; P V = m+1 k=1 Dvk , veven ∈ V0 , vodd ∈ V1 ; (e)
vk+1 = N k v1 6= 0, 0 ≤ k ≤ m, Nvm+1 = 0;
τ (vk , vl ) = 0 if l 6= m + 2 − k, τ (vk , vm+2−k ) = δ(k)τ (v1 , vm+1 ), τ (v1 , vm+1 ) = −δ(1 +
m )sign(τ0 ); 2
453
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m ∈ 2Z \ 4Z, D 6= H, ι = 1; P ′ ′ ′ V = m k=0 (Dvk ⊕ Dvk ), veven , veven ∈ V0 , vodd , vodd ∈ V1 ;
vk = N k v0 6= 0, vk′ = N k v0′ 6= 0, 0 ≤ k ≤ m, Nvm = 0, ′ (f) Nvm = 0;
τ (vk , vl ) = τ (vk′ , vl′ ) = 0, 0 ≤ k, l ≤ m, τ (vk , vl′ ) = τ (vk′ , vl ) = 0, l 6= m − k, ′ τ (vk , vm−k ) = −τ (vk′ , vm−k ) = δ(k − 1), 0 ≤ k ≤ m;
m ∈ 2Z + 1; P ′ ′ ′ V = m k=0 (Dvk ⊕ Dvk+1 ), veven , veven ∈ V0 , vodd , vodd ∈ V1 ;
′ vk = N k v0 6= 0, vk+1 = N k v1′ 6= 0, 0 ≤ k ≤ m, Nvm = 0,
(g)
′ Nvm+1 = 0; ′ ′ τ (vk , vl ) = τ (vk+1 , vl+1 ) = 0, 0 ≤ k, l ≤ m, ′ ′ τ (vk , vl+1 ) = τ (vk+1 , vl ) = 0, l 6= m − k, ′ ′ τ (vk , vm+1−k ) = δ(k)δ(m), τ (vk+1 , vm−k ) = δ(k − 1),
0 ≤ k ≤ m. Now we will describe the action of moment maps ν0 and ν1 , defined by (35), on the nilpotent orbits in W . Since moment maps respect decomposition and group action, it is enough to treat the case of indecomposable elements in W . Let (V, τ ) be a Z2 -graded formed space and let N ∈ g(V, τ )1 be indecomposable of height m. Then the orbit of N is uniquely determined by the isometry class of the space (V /NV, τ˜) defined in Lemma 3.10. Recall (Proposition 3.17) that τ˜ is σ-hermitian for some σ = ±1. For k = 0, 1, the image νk (N) = N 2 |Vk of N under the moment map νk is a nilpotent uniform element of g(Vk , τk ). Let mk be its height. In order to identify the orbit of νk (N), we have to consider a formed space (Vk /N 2 Vk , τˆk ), where τˆk (ˆ u, vˆ) = τ (u, N 2mk v), (u, v ∈ Vk )
(39)
and, for u ∈ V , uˆ denotes the coset of u in V /N 2 V . Lemma 5.3. With the above notation, the form τˆk is σk -hermitian for a suitable σk = ±1. Specifically, there are three possibilities. (1) 2mk = m. Then Vk /N 2 Vk = V /NV , τˆk (ˆ u, vˆ) = τ˜(˜ u, v˜) and σk = σ. 2 (2) 2mk = m − 1. Then Vk /N Vk = Vk /(NV ∩ Vk ) ⊕ NVk+1 /(N 2 V ∩ NVk+1 ) and if u ∈ Vb , c = τ˜(e c uˆ) = (−1)b τ˜(e v ∈ Vk+1 then τˆk (ˆ u, Nv) u, e v ), τˆk (Nv, v, u e). In particular, σk = (−1)k σ.
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(3) 2mk = m−2. Then Vk /N 2 Vk = NV /N 2 V = NVk+1 /N 2 Vk+1 and if u = Nu′ , v = Nv ′ , where u′, v ′ ∈ Vk+1 , then τˆk (ˆ u, vˆ) = (−1)m (−1)k τ˜(ue′ , ve′). In particular, σk = σ.
Using Lemma 5.3, one can easily describe how the moment maps act on indecomposable orbits in W . The detailed description is given in Table 3. The first column contains the parameter (V /NV, τ˜) describing an orbit in W and its height m, the second column contains the parameters of the image of that orbit under the projection ν0 and the third, the image under the projection ν1 . For the details on the last column see (74).
5.3
Nilpotent orbits in pC
We assume that if D = C then the involution ι is nontrivial. Lemma 5.4. Up to conjugation by G(V, τ )0 , there is a unique element T ∈ G(V, τ )0 such that T 2 = S and such that the form τ (T u, v) (u, v ∈ V ), is hermitian and positive definite. In particular θ = Ad(T ) is a Cartan involution on g(V, τ )0 . Furthermore, the following diagram g(V, τ )1 ∋ N Ty
−−−→
N|V0 ∈ Hom(V0 , V1 ) Jy
g(V, τ )1 ∋ T NT −1 −−−→ (T NT −1 )|V0 ∈ Hom(V0 , V1 ) defines a positive compatible complex structure J on the symplectic space Hom(V0 , V1 ) (i.e. J preserves the symplectic form h− ,− i defined in (34), J 2 = −1 and the form hJ− ,− i is symmetric and positive definite). If we define a map g(V, τ )1 ∋ N → N † ∈ g(V, τ )1 by τ (T Nu, v) = τ (T u, N † v), then J(N) = SN † . Proof. We shall give an explicit construction of T . For integers p ≥ 0, q ≥ 0 (p + q > 0) and for n ≥ 1, let 0 In Ip 0 Ip,q = . , J2n = −In 0 0 −Iq There are three cases to consider. D = R, V0 = Rp+q , V1 = R2n , (a)
τ0 (u, v) = ut Ip,q v
(u, v ∈ V0 ),
τ1 (u, v) = ut J2n v
(u, v ∈ V1 ),
T |V0 := Ip,q , T |V1 := J2n ;
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(V /NV, τ˜)
m
(V0 /N 2 V0 , τˆ0 )
m0
(V1 /N 2 V1 , τˆ1 )
m1
(R, −) (R, +) (R2 , sk) (R, +) (R, −) (R2 , sk) (R2 , sk) (R, +) ⊕ (R, −)
2s − 1 2s − 1 2s 2s + 1 2s + 1 2s 2s 2s + 1
S S S, S¯ S¯ S¯ S, S¯
(C, sym) (C2 , sk) (C2 , sk) (C, sym) (C2 , sk) (C, sym)⊕2
2s − 1 2s 2s 2s + 1 2s 2s + 1
S, S¯ S, S¯ S, S¯ S, S¯ S, S¯ S, S¯
(D, ι) = (R, id)
(R[0], +) (R[0], −) (R2 [1], sk) (R[1], +) (R[1], −) (R2 [0], sk) (R[0] ⊕ R[1], J2′ ) (R[0] ⊕ R[1], J2 )
4s 4s 4s 4s + 2 4s + 2 4s + 2 4s + 1 4s + 3
(R, +) 2s (R, −) 2s 2 (R , sk) 2s − 1 (R, +) 2s (R, −) 2s (R2 , sk) 2s + 1 (R, +) ⊕ (R, −) 2s 2 (R , sk) 2s + 1 (D, ι) = (C, id)
(C[0], sym) (C2 [1], sk) (C2 [0], sk) (C[1], sym) (C[0] ⊕ C[1], J2′ ) (C[0] ⊕ C[1], J2 )
4s 4s 4s + 2 4s + 2 4s + 1 4s + 3
(C, sym) (C2 , sk) (C2 , sk) (C, sym) (C, sym)⊕2 (C2 , sk)
2s 2s − 1 2s + 1 2s 2s 2s + 1
(D, ι) = (C, ·)
(C[0], +) (C[0], −) (C[1], +i) (C[1], −i) (C[0], +i) (C[0], −i) (C[1], +) (C[1], −) (C[0] ⊕ C[1], J2′ ) (C[0] ⊕ C[1], J2 )
4s 4s 4s 4s 4s + 2 4s + 2 4s + 2 4s + 2 4s + 1 4s + 3
(C, +) 2s (C, −) 2s (C, +i) 2s − 1 (C, −i) 2s − 1 (C, +i) 2s + 1 (C, −i) 2s + 1 (C, +) 2s (C, −) 2s (C, +) ⊕ (C, −) 2s (C, +i) ⊕ (C, −i) 2s + 1
(C, −) 2s − 1 (C, +) 2s − 1 (C, +i) 2s (C, −i) 2s (C, −i) 2s (C, +i) 2s (C, +) 2s + 1 (C, −) 2s + 1 (C, +i) ⊕ (C, −i) 2s (C, +) ⊕ (C, −) 2s + 1
S S S¯ S¯ S S S¯ S¯ S, S¯ S, S¯
(D, ι) = (H, ·)
(H[0], +) (H[0], −) (H[1], sk) (H[0], sk) (H[1], +) (H[1], −) (H[0] ⊕ H[1], J2′ ) (H[0] ⊕ H[1], J2 )
4s 4s 4s 4s + 2 4s + 2 4s + 2 4s + 1 4s + 3
(H, +) 2s (H, −) 2s (H, sk) 2s − 1 (H, sk) 2s + 1 (H, +) 2s (H, −) 2s (H, +) ⊕ (H, −) 2s ⊕2 (H, sk) 2s + 1
(H, −) (H, +) (H, sk) (H, sk) (H, +) (H, −) (H, sk)⊕2 (H, +) ⊕ (H, −)
Table 3 Moment maps on indecomposable orbits in W .
2s − 1 2s − 1 2s 2s 2s + 1 2s + 1 2s 2s + 1
S S S, S¯ S, S¯ S¯ S¯ S, S¯ S, S¯
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D = C, V0 = Cp+q , V1 = Cr+s , (b)
τ0 (u, v) = ut Ip,q ι(v)
(u, v ∈ V0 ),
τ1 (u, v) = ut iIr,s ι(v)
(u, v ∈ V1 ),
T |V0 := Ip,q , T |V1 := −iIr,s ; D = H, V0 = Hp+q , V1 = Hn ,
(c)
τ0 (u, v) = ut Ip,q ι(v)
(u, v ∈ V0 ),
τ1 (u, v) = ut jIn ι(v)
(u, v ∈ V1 ),
T |V0 := Ip,q , T |V1 := right multiplication by j −1 , where j ∈ H is such that H = C ⊕ jC, ι(j) = −j = j −1 and jzj −1 = ι(z) for z ∈ C. For the last statement we notice that for u, v ∈ V and for N ∈ g(V, τ )1 we have τ (T Nu, v) = τ (T NT −1 T u, v) = τ (T u, ST NT −1 v). Hence, N † = ST NT −1 , and therefore J(N) = T NT −1 = SN † , as claimed. Via a case by case analysis we see that the condition T 2 = S determines T up to a sign, and this sign is determined by the positivity of the form τ (T u, v). Thus there is a one to one correspondence between the elements T and the maximal compact subgroups. Hence T is unique up to conjugation. Let K = G(V, τ )T0 and let k = g(V, τ )T0 . Since the form τ (T u, v) (u, v ∈ V ) is hermitian and positive definite, K is a maximal compact subgroup of G(V, τ )0 corresponding to the Cartan involution θ and k is the Lie algebra of K. Let p be the (-1)-eigenspace of θ on g(V, τ )0 so that g(V, τ )0 = k ⊕ p
(40)
is the corresponding Cartan decomposition. Set p0 = p ∩ g(V0 , τ0 ), k0 = k ∩ g(V0 , τ0 ), and similarly for p1 , k1 . Then we have the Cartan decompositions g(V0 , τ0 ) = k0 ⊕ p0 , g(V1 , τ1 ) = k1 ⊕ p1 ,
(41) (42)
with maximal compact subgroups K0 , K1 . In order to describe the complexifications pC and KC of p and K let us define U = V ⊗ R C if D = R, and U = V |C if D = C or H.
(43)
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Then U is a vector space over C and the element T constructed in Lemma 5.4 acts on U. Since T 4 = S 2 = 1, T has at most four eigenvalues: 1, i, −1, −i. Let Uk = {u ∈ U; T u = ik u}
(k = 0, 1, 2, 3).
(44)
Then U = U0 ⊕ U1 ⊕ U2 ⊕ U3
(45)
and U is a Z/4Z-graded vector space over C. Let us first analyze the case of D = C. In this case pC = End(U)2 and KC = GL(U0 ) × GL(U1 ) × GL(U2 ) × GL(U3 ). Since End(U)2 = Hom(U0 , U2 ) ⊕ Hom(U2 , U0 ) ⊕ Hom(U1 , U3 ) ⊕ Hom(U3 , U1 ), the problem of the classification of nilpotent KC -orbits in pC is equivalent to the analogous problem for the nilpotent orbits of K0,C = GL(U0 ) × GL(U2 ) in p0,C = Hom(U0 , U2 ) ⊕ Hom(U2 , U0 ) and the nilpotent orbits of K1,C = GL(U1 )×GL(U3 ) in p1,C = Hom(U1 , U3 )⊕ Hom(U3 , U1 ). This is exactly the problem studied in Section 4.2. As an immediate application of Corollary 4.2 we get the description of the indecomposable nilpotents in pC in this case. Proposition 5.5. If D = C and the anti-involution ι is nontrivial, then there are two orbits of indecomposable nilpotent elements (N, U0 ⊕ U2 ) in p0,C , determined by the ˜ = C[b], b = 0, 2, and there are two orbits of indecomposable nilpotent condition U ˜ = C[b], b = 1, 3. elements (N, U1 ⊕ U3 ) in p1,C , determined by the condition U Until stated otherwise, suppose that D = R or H. We shall define a sesqui-linear form φ on the space U, as follows. If D = R we let φ be the unique complex linear extension of the form τ . Then φ|U0+U2 is symmetric, and φ|U0 , φ|U2 are non-degenerate. Furthermore, φ|U1 +U3 is skew-symmetric, and φ|U1 = 0, φ|U3 = 0. If D = H we define 1 φ(u, v) = (iτ (u, v) + τ (u, v)i) − τ (u, v) j (u, v ∈ U). (46) 2i Lemma 5.6. The form φ defined in (46) is a C-valued, C-bilinear non-degenerate form on U. The restricted form φ|U0 +U2 is skew-symmetric, and φ|U0 , φ|U2 are non-degenerate. Similarly, φ|U1 +U3 is symmetric, and φ|U1 = 0, φ|U3 = 0. Moreover, for x ∈ EndH (V )0 and for u, v ∈ V , we have τ (xu, v) = −τ (u, xv) if and only if φ(xu, v) = −φ(u, xv). Proof. Notice that for any quaternion q = a + bj, where a, b ∈ C, we have 1 (iq + qi) − q j = b. 2i
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Hence φ takes values in C. Since zj = jz, for z ∈ C, it is easy to check that φ is bilinear over C. Furthermore, ι(q) = a − bj, so 1 (iι(q) + ι(q)i) − ι(q) j = −b. 2i Thus, if a restriction of τ to a subspace U ′ ⊂ U is hermitian then φ|U ′ is skew-symmetric, and if τ |U ′ is skew-hermitian then φ|U ′ is symmetric. Suppose φ(u, v) = 0 for all v ∈ W . Then τ (u, v) ∈ C for all v ∈ W . But then τ (u, jv) = −τ (u, v)j ∈ Cj. Hence τ (u, v) = 0 for all v ∈ U. Thus u = 0, so φ is non-degenerate. The last claim follows from equation (46) defining φ in terms of τ and from the following, easy to check, equation expressing τ in terms of φ: τ (u, v) = −φ(u, jv) + φ(u, v)j
(u, v ∈ U).
In both cases the form φ is non-degenerate and bilinear over C. A straightforward argument shows that g(U, φ)0 coincides with the complexification kC of k = g(V, τ )T0 (the centralizer of T in g(V, τ )0 ), and that K = G(V, τ )T0 is a maximal compact subgroup of G(V, τ )0 . Thus KC = G(U, φ)T0 = {g ∈ G(U, φ); gT = T g}
= {g ∈ G(U, φ); g(Uk ) = Uk for all k},
(47)
pC = g(U, φ)2 = {x ∈ g(U, φ); x(Uk ) ⊆ Uk+2 for all k}. As explained in Theorem 3.18, the orbits of KC in pC are determined by (2, m)-admissible ˜ According to Proposition 3.17, they are as follows. ˜ φ). spaces (U, Proposition 5.7. If D = R then φ satisfies condition (10) with σ = 1 and (2, m)˜ are of the form: ˜ φ) admissible spaces (U, (1) U˜ = C[b], where b + m is even and φ˜ is symmetric; (2) U˜ = C[b] ⊕ C[b + 2], where b + m is odd and φ˜ is skew-symmetric. ˜ ˜ φ) If D = H then φ satisfies condition (10) with σ = −1 and (2, m)-admissible spaces (U, are of the form: (1) U˜ = C2 [b], where b + m is even and φ˜ is skew-symmetric; (2) U˜ = C[b] ⊕ C[b + 2], where b + m is odd and φ˜ is symmetric with maximal isotropic subspaces C[b] and C[b + 2].
5.4
Nilpotent orbits in WC+
By definition, the space WC+ is equal to the i-eigenspace of J acting on WC , the complexification of W . In the case of D = C the space WC can be identified with the the direct sum Hom(V0 , V1 ) ⊕ Hom(V1 , V0 ) with the complex structure iX = X ∗ and we have
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WC+ = End(U)1 . The problem of classification of KC -orbits in WC+ is equivalent to the problem of classification of GL(U)0 -orbits in End(U)1 . As in Section 4.2 the classification follows from Corollary 3.8. We leave the details to the reader. In the case of D = R or H we have WC+ = g(U, φ)1 and KC -orbits in WC+ are G(U, φ)0orbits in g(U, φ)1 . In order to understand them in terms of Theorem 3.18, we need to list (1, m)-admissible spaces. ˜ are Proposition 5.8. With σ as in Proposition 5.7, all (1, m)-admissible spaces (U˜ , φ) as follows: (1) if 4 | 2b + m then U˜ = U˜b and φ˜ is σ-symmetric; ˜b ⊕ U ˜−b−m , where 2b + m 6= 0, and φ˜ is σ ′ -symmetric with (2) if 4 ∤ 2b + m then U˜ = U σ ′ = (−1)(m+1)b δ(m)σ. Table 4 lists all the indecomposable orbits in WC+ and the corresponding orbits obtained via the moment maps µ0 and µ1 , in terms of the notation explained above.
6
The Cayley transform and the Kostant-Sekiguchi correspondence
We begin by recalling Kostant-Sekiguchi bijections S and S¯ [19]. Let G be a real reductive Lie group with the Lie algebra g and let gC be the complexification of g. Recall that a standard triple (e, f, h) in g is a triple of elements e, f, h ∈ g satisfying conditions [e, f ] = h, [h, e] = 2e, [h, f ] = −2f. Fix a Cartan involution θ of g, with Cartan decompositions g = k ⊕ p, gC = kC ⊕ pC . A standard triple (e, f, h) in g is a Cayley triple, if f = −θ(e), h = −θ(h). (To the best of our knowledge, such triples appeared for the first time in published literature in [9, Section 14].) By definition, the Cayley transform of a Cayley triple (e, f, h) is the standard triple ′ (e , f ′ , h′ ) in gC defined by 1 (e + f − ih), 2 1 f ′ = (e + f + ih), 2 h′ = −i(e − f ). e′ =
(48)
The Kostant-Sekiguchi map S sends the G-orbit of e in g into the KC -orbit of e′ in pC , ¯ equal to the where KC is the complexification of K. The Kostant-Sekiguchi map S, composition of S with the complex conjugation of gC over the real form g ⊆ gC , maps
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O
µ0 (O)
461
µ1 (O)
D=R (C[0], sym, 4k) (C[2], sym, 4k) (C[1], sym, 4k+2) (C[3], sym, 4k+2) (C[1] ⊕ C[3], sk, 4k) (C[0] ⊕ C[3], sym, 4k+1) (C[1] ⊕ C[2], sym, 4k+1) (C[0] ⊕ C[2], sk, 4k+2) (C[0] ⊕ C[1], sk, 4k+3) (C[2] ⊕ C[3], sk, 4k+3) (C[0], (C[1], (C[2], (C[3],
(C[0], sym, 2k) (C[2], sym, 2k) (C[2], sym, 2k) (C[0], sym, 2k) (C[0] ⊕ C[2], sk, 2k−1) (C[0], sym, 2k)⊕2 (C[2], sym, 2k)⊕2 (C[0] ⊕ C[2], sk, 2k+1) (C[0] ⊕ C[2], sk, 2k+1) (C[0] ⊕ C[2], sk, 2k+1)
(C[1], sym, 2k−1) (C[3], sym, 2k−1) (C[1], sym, 2k+1) (C[3], sym, 2k+1) (C[1] ⊕ C[3], sk, 2k) (C[1] ⊕ C[3], sk, 2k) (C[1] ⊕ C[3], sk, 2k) (C[1] ⊕ C[3], sk, 2k) (C[1], sym, 2k+1)⊕2 (C[3], sym, 2k+1)⊕2
D = C. (C[0], m 2 ) m−1 (C[2], 2 ) m (C[2], 2 ) (C[0], m−1 ) 2
m) m) m) m)
(C[1], m−1 ) 2 m (C[1], 2 ) ) (C[3], m−1 2 m (C[3], 2 )
D = H.
(C2 [0], sk, 4k) (C2 [2], sk, 4k) (C2 [1], sk, 4k+2) (C2 [3], sk, 4k+2) (C[1] ⊕ C[3], sym, 4k) (C[0] ⊕ C[3], sk, 4k+1) (C[1] ⊕ C[2], sk, 4k+1) (C[0] ⊕ C[2], sym, 4k+2) (C[0] ⊕ C[1], sym, 4k+3) (C[2] ⊕ C[3], sym, 4k+3)
(C2 [0], sk, 2k) (C2 [2], sk, 2k) (C2 [2], sk, 2k) (C2 [0], sk, 2k) (C[0] ⊕ C[2], sym, 2k−1) (C2 [0], sk, 2k) (C2 [2], sk, 2k) (C[0] ⊕ C[2], sym, 2k+1) (C[0] ⊕ C[2], sym, 2k+1) (C[0] ⊕ C[2], sym, 2k+1)
(C2 [1], sk, 2k−1) (C2 [3], sk, 2k−1) (C2 [1], sk, 2k+1) (C2 [3], sk, 2k+1) (C[1] ⊕ C[3], sym, 2k) (C[1] ⊕ C[3], sym, 2k) (C[1] ⊕ C[3], sym, 2k) (C[1] ⊕ C[3], sym, 2k) (C2 [1], sk, 2k+1) (C2 [3], sk, 2k+1)
Table 4 Moment maps on indecomposable orbits in WC+ .
the G-orbit of e into the KC -orbit of f ′ . We shall focus on the map S. The computations for S¯ are parallel, with i replaced by −i. Proposition 6.1. Let (e, f, h) be a standard triple in gC , and let
π C = exp(i ad(e + f )) ∈ Aut(gC ), 4 π ¯ C = exp(−i ad(e + f )) ∈ Aut(gC ). 4
(49) (50)
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Then ¯ ) = 1 (e + f − ih), C(e) = C(f 2 1 ¯ = (e + f + ih), C(f ) = C(e) 2 ¯ C(h) = −C(h) = −i(e − f ).
(51)
In particular, if (e, f, h) is a Cayley triple in g, then the triple (C(e), C(f ), C(h)) is equal to the Cayley transform of (e, f, h). Proof. The following formulas are easy to check: ad(e + f )2k e = 22k−1 (e − f ) ad(e + f )
2k+1 2k
2k
e = −2 h
ad(e + f ) f = −2 ad(e + f )
2k+1 2k
2k−1 2k
f =2 h 2k
ad(e + f ) h = 2 h ad(e + f )
2k+1
h = −2
(k ≥ 1),
(k ≥ 0),
(e − f )
(k ≥ 0),
(k ≥ 0),
2k+1
(k ≥ 1),
(e − f )
(k ≥ 0).
Hence, for any z ∈ C, exp(z ad(e + f ))e = cosh2 (z)e − sinh2 (z)f − cosh(z) sinh(z)h,
exp(z ad(e + f ))f = − sinh2 (z)e + cosh2 (z)f + cosh(z) sinh(z)h, exp(z ad(e + f ))h = cosh(2z)h − sinh(2z)(e − f ).
Substitution of z = ±i π4 into this formulas completes the proof.
If G is a complex group then there are identifications pC = g and KC = G such that the KC orbits in pC become the G orbits in g and S = S¯ is the identity. Let V and τ be as in Section 5.1, with additional assumption that either V = V0 or V = V1 . Let T be as in Section 5.3, let θ be the Cartan involution on g(V, τ ) equal to the conjugation by T and assume that e, f, h ∈ g(V, τ ) form a Cayley triple. Then i(e + f ) is in the complexification of g(V, τ ) which coincides with g(U, φ) as in section 5.4. Set π c = exp(i (e + f )) ∈ G(U, φ), 4 π c¯ = exp(−i (e + f )) ∈ G(U, φ). 4
(52) (53)
Then a standard Lie theory argument shows that the automorphism C ∈ Aut(g(V, τ )C ) = Aut(g(U, φ)) defined by (49) is equal to conjugation by c. In particular Proposition 6.1 implies that the Kostant-Sekiguchi map S sends the orbit of e to the orbit of cec−1 . Similarly, the Kostant-Sekiguchi map S¯ sends the orbit of e to the orbit of c¯e¯ c−1 .
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6.1 The Kostant-Sekiguchi correspondence for indecomposable nilpotents ¯ = c¯e¯ In this section we will compute the maps e 7−→ C(e) = cec−1 , e 7−→ C(e) c−1 for indecomposable nilpotents e ∈ g(V, τ ). We will do this by a case-by-case analysis, according to the classification of nilpotent orbits in g(V, τ ) described in Section 5.1. In each case we will • • • • •
define the space V , describe the action of e on an explicit basis of V , define the form τ in this basis, define the map T : V −→ V in this basis, compute the elements c, c¯ ∈ G(U, φ) in the basis of U induced by the chosen basis of V , ˜ corresponding to the nilpotent elements (cec−1 , U), • compute the formed spaces (U˜ , φ) (¯ ce¯ c−1 , U) as in section 5.3. We begin with a general construction, which will be used in all cases. Let ξ be a non-degenerate symplectic form on R2 , and let ǫ1 , ǫ2 ∈ R2 be a basis such that ξ(ǫ1 , ǫ1 ) = ξ(ǫ2 , ǫ2 ) = 0, ξ(ǫ1 , ǫ2 ) = 1. (54) We extend the form ξ to the tensor algebra of R2 , and restrict to the subspace S m R2 of symmetric homogeneous tensors of degree m = 0, 1, 2, .... Then a straightforward calculation shows that (−1)m−k k!(m − k)! , if l = m − k, k m−k l m−l ξ(ǫ1 ǫ2 , ǫ1 ǫ2 ) = (55) m! 0, otherwise.
Let V = S m R2 . We set
τ (u, v) = (−1)m ξ(u, v)
(u, v ∈ V).
Let vk = ǫk1 ǫm−k , 0 ≤ k ≤ m. Then (55) may be rewritten as 2 (−1)k k!(m − k)! , if l = m − k, τ (vk , vl ) = m! 0, otherwise.
(56)
(57)
Let T : V → V be a linear map defined by
Tvk = (−1)m−k vm−k
(0 ≤ k ≤ m).
(58)
Then (57) implies that T ∈ G(V, τ ) and that the form τ (T− ,− ) is symmetric and positive definite. Let E, F, H ∈ g(R2 , ξ) (a Lie algebra isomorphic to sl2 (R)) be defined by E(ǫ1 ) = 0, F (ǫ1 ) = ǫ2 , H(ǫ1 ) = ǫ1 , E(ǫ2 ) = ǫ1 , F (ǫ2 ) = 0, H(ǫ2 ) = −ǫ2 .
(59)
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These elements act on V and it is easy to check that E(vk ) = (m − k)vk+1 , F (vk ) = kvk−1 , H(vk ) = (−m + 2k)vk .
(60)
The formulas (58) and (60) imply that the elements E, F and H form a Cayley triple in g(V, τ ) with respect to the Cartan involution θ = Ad(T). The point of this construction is to avoid direct calculation of the exponential in (52). Instead we proceed as follows. Let U = C ⊗ V = S m C2 be the complexification of V. The complexification of the group G(R2 , ξ) (isomorphic to SL(2, C)) acts on the space U. Consider E, F as elements of C2 . It is easy to check that for z ∈ C, exp(z(E + F ))(ǫ1 ) = cosh(z)ǫ1 + sinh(z)ǫ2 exp(z(E + F ))(ǫ2 ) = sinh(z)ǫ1 + cosh(z)ǫ2 , and hence, as an endomorphism of U, k m−k X X k m − k m−k+l−j sinh(z)k−l+j vl+j . exp(z(E + F ))(vk ) = cosh(z) l j l=0 j=0 Therefore, by taking z = i π4 , c(vk ) = 2−m/2
k m−k X X l=0 j=0
k m − k k−l+j vl+j . i l j
(61)
The formulas (58) and (61) imply that T(c(vk )) = im−2k c(vk )
(0 ≤ k ≤ m).
(62)
Similarly, T(¯ c(vk )) = (−i)m−2k c¯(vk )
(0 ≤ k ≤ m).
(63)
We are now ready for the computation of the Kostant-Sekiguchi maps for all indecomposable orbits. Consider first the case D = R. According to our analysis in Section 5.1 there are six possibilities for the formed space (V˜ , τ˜) corresponding to an indecomposable nilpotent element e ∈ g(V, τ ) of height m, namely (R[b], +), (R[b], −) with b + m even, (R2 [b], sk) with b + m odd, b = 0, 1, where as usual Rk [b] denotes the graded vector space concentrated in degree b. For an explicit realization of the first case, let (V, τ ) = (V[b], τ ), T = T and e = E. The space V has a decomposition V = Rv0 ⊕ Rv1 ⊕ ... ⊕ Rvm ,
(64)
τ (v0 , em (v0 )) = m!,
(65)
with e(Rvk ) = Rvk+1 and
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which implies that the space (V˜ , τ˜) is of the form (R[b], +). The complexification U of V (equal to U in this case) has a decomposition U = Cc(v0 ) ⊕ Cc(v1 ) ⊕ ... ⊕ Cc(vm ),
(66)
with cec−1 (Cc(vk )) = Cc(vk+1 ). Moreover, by (62), T (c(v0 )) = im c(v0 ). ˜ is of the form (C[m], φ) ˜ with φ˜ symmet˜ , φ) This implies that the formed space (U ric, m = m mod 4, so this is the formed space corresponding to the nilpotent element (cec−1 , U) ∈ g(U, φ)2 . Similarly, by (63) T (¯ c(v0 )) = i−m c(v0 ), ˜ is now of the form (C[−m], φ) ˜ with φ˜ symmetric, ˜ φ) This implies that the formed space (U, so this is the formed space corresponding to the nilpotent element (¯ ce¯ c−1 , U) ∈ g(U, φ)2 . In the same way, if we consider the space (V, τ ′ ) = (V[b], −τ ) with T = −T, the indecomposable formed space corresponding to (e, V ) will be of type (R[b], −), the inde˜ with composable formed space corresponding to (cec−1 , U) will be of type (C[m + 2], φ) φ˜ symmetric (the eigenvalue of T on c(v0 ) is equal to −im = im+2 in this case), and the ˜ indecomposable formed space corresponding to (¯ ce¯ c−1 , U) will be of type (C[−m + 2], φ) with φ˜ symmetric. Finally we consider the space V = V ⊗ R2 with the form τ ⊗ ξ, where ξ is the skewsymmetric form on R2 defined in (54). Let R be the endomorphism of R2 defined by R(ǫ1 ) = ǫ2 , R(ǫ2 ) = −ǫ1 . Then the form ξ(R , ) is symmetric and positive definite. Let T = T ⊗ R : V −→ V . The form (τ ⊗ ξ)(T− ,− ) is symmetric and positive definite. Moreover, T ∈ G(V ⊗ R2 , τ ⊗ ξ), and it is clear that the elements e = E ⊗ 1, f = F ⊗ 1, h = H ⊗ 1 form a Cayley triple in g(V ⊗ R2 , τ ⊗ ξ) with respect to the Cartan involution Ad(T ). Let U = V ⊗ C. We have (τ ⊗ ξ)(v0 ⊗ ǫµ , em (v0 ⊗ ǫν )) = τ (v0 , E m (v0 ))ξ(ǫµ , ǫν ) = m!ξ(ǫµ , ǫν ), c(vk ⊗ ǫν ) = 2−m/2
k m−k X X l=0 r=0
k m − k k−l+r vl+r ⊗ ǫν , i l r
T (c(vk ⊗ ǫν )) = im−2k c(vk ⊗ R(ǫν )). Hence, in particular, T (c(v0 ⊗ ǫ1 ) + ic(v0 ⊗ ǫ2 )) = −im+1 (c(v0 ⊗ ǫ1 ) + ic(v0 ⊗ ǫ2 )), T (c(v0 ⊗ ǫ1 ) − ic(v0 ⊗ ǫ2 )) =
im+1 (c(v0 ⊗ ǫ1 ) − ic(v0 ⊗ ǫ2 )).
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It follows that T has two eigenvalues im+1 , im+3 on the space U˜ , so the formed space ˜ is of type (C[m + 1] ⊕ C[m + 3], φ). ˜ Similarly, we have (U˜ , φ) T (¯ c(v0 ⊗ ǫ1 ) + i¯ c(v0 ⊗ ǫ2 )) = −i−m+1 (¯ c(v0 ⊗ ǫ1 ) + i¯ c(v0 ⊗ ǫ2 )), T (¯ c(v0 ⊗ ǫ1 ) − i¯ c(v0 ⊗ ǫ2 )) =
−m+1
i
(68)
(¯ c(v0 ⊗ ǫ1 ) − i¯ c(v0 ⊗ ǫ2 )),
˜ is of type (C[−m + 1] ⊕ C[−m + 3], φ) ˜ in this case. hence the formed space (U˜ , φ) We summarize our computations in the following proposition. Proposition 6.2. The Kostant-Sekiguchi correspondences for the orbits of indecomposable nilpotent elements in Lie algebras of isometries of formed spaces over R map the orbit O ⊆ g(V, τ ) of a nilpotent element of height m corresponding to a formed space ˜ as ¯ ˜ φ) (V˜ , τ˜) into the orbits S(O), S(O) ⊆ g(U, φ) corresponding to the formed spaces (U, in the following table. m
O
even (R[0], +)
S(O)
¯ S(O)
(C[m], sym)
(C[m], sym)
even (R[0], −) (C[m + 2], sym) (C[m + 2], sym) odd (R2 [0], sk) (C[0] ⊕ C[2], sk) (C[0] ⊕ C[2], sk) odd (R[1], +)
(C[m], sym)
odd (R[1], −) (C[m + 2], sym)
(C[m + 2], sym) (C[m], sym)
even (R2 [1], sk) (C[1] ⊕ C[3], sk) (C[1] ⊕ C[3], sk) Now assume that D = C or H. Let τD be a hermitian form on D such that τD (1, 1) = 1. Consider the left vector space V = D ⊗ V over D, with the form τD ⊗ τ . Then the map T = 1 ⊗ T belongs to G(V, τD ⊗ τ ) and the form (τD ⊗ τ )(T− ,− ) is hermitian and positive definite. Furthermore, it is clear that the elements e = 1 ⊗ E, f = 1 ⊗ F , h = 1 ⊗ H form a Cayley triple in g(V, τD ⊗ τ ) with respect to Ad(T ). The equality (65) implies that the indecomposable formed space (V˜ , τ^ D ⊗ τ ) is of the form (D[b], +). Moreover, the map c defined in (52), can be written as k m−k X X k m − k k−l+r ⊗ vl+r , c(1 ⊗ vk ) = 2−m/2 i r=0 l r l=0 and therefore T (c(1 ⊗ vk )) = im−2k c(1 ⊗ vk )
(0 ≤ k ≤ m).
(69)
T (¯ c(1 ⊗ vk )) = i−m−2k c¯(1 ⊗ vk )
(0 ≤ k ≤ m).
(70)
Similarly,
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Assume that D = C with ι the complex conjugation. By the results of Section 5.1 there are two possibilities for the formed space corresponding to a indecomposable nilpotent element of height m, namely (C, +) and (C, −) for even m and (C, +i) and (C, −i) for odd m (we assume that C is in degree zero in all cases). Similarly, by Proposition 5.5 the formed space U˜ corresponding to an orbit of an indecomposable nilpotent element in pC is of the form C[k], k = 0, 2. In the above construction the space (V˜ , τ^ C ⊗ τ ) is of the ˜ form (C, +), and the space U is equal to C[m]. When we replace the form τC by ik τC and the map T by i−k T , we will get the remaining cases of the following proposition. Proposition 6.3. The Kostant-Sekiguchi correspondences for the orbits of indecomposable nilpotent elements in Lie algebras of isometries of formed spaces over C map the orbit O ⊆ g(V, τ ) of a nilpotent element of height m corresponding to a formed space (V˜ , τ˜) ¯ into the orbits S(O), S(O) ⊆ g(U, φ) corresponding to the space U˜ as in the following table. ¯ O S(O) S(O) (C[0], +, 2s)
(C[2s], 2s)
(C[2s], 2s)
(C[0], −, 2s)
(C[2s + 2], 2s)
(C[2s + 2], 2s)
(C[0], +i, 2s + 1)
(C[2s], 2s + 1)
(C[2s + 2], 2s + 1)
(C[0], −i, 2s + 1) (C[2s + 2], 2s + 1)
(C[2s], 2s + 1)
(C[1], +, 2s + 1) (C[2s + 1], 2s + 1) (C[2s + 3], 2s + 1) (C[1], −, 2s + 1) (C[2s + 3], 2s + 1) (C[2s + 1], 2s + 1) (C[1], +i, 2s)
(C[2s + 3], 2s)
(C[2s + 3], 2s)
(C[1], −i, 2s)
(C[2s + 1], 2s)
(C[2s + 1], 2s)
Now assume that D = H, so our formed space is equal to (V, τH ⊗ τ ). It follows that T (c(1 ⊗ v0 )) = im c(1 ⊗ v0 ), and similarly T (c(j ⊗ v0 )) = im c(j ⊗ v0 ), where j is the element of the standard basis {1, i, j, k} of H, hence the formed space ˜ is of the form (C2 [m], φ) ˜ with φ˜ skew-symmetric. (U˜ , φ) If we replace the form τ by −τ and the map T by −T , we will obtain the nilpotent element (e, V ) whose corresponding indecomposable formed space (V˜ , τ^ H ⊗ τ ) is of the −1 form (D[b], −), and the nilpotent element (cec , U) whose corresponding indecomposable ˜ with φ˜ skew-symmetric. ˜ is of the form (C2 [m + 2], φ) ˜ φ) formed space (U, The last case to consider is the case (V˜ , τ˜) of type (H[b], sk) with b + m odd. We proceed as follows.
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For x, y ∈ H set τH′ (x, y) = xjι(y) ∈ H. Then τH′ is a skew-hermitian form on the left vector space H. Consider the space V = H ⊗ V with the form τH′ ⊗ τ . Let e = 1 ⊗ E, as before. Since (τH′ ⊗ τ )(1 ⊗ v0 , em (1 ⊗ v0 )) = τH′ (1, 1)τ (v0 , E m (v0 )) = m! j, ′ the formed space (V˜ , τ^ H ⊗ τ ) is of the form (H[b], sk). Let Rj −1 denote right multiplication by j −1 on H. Then the map T = Rj −1 ⊗ T belongs to G(H ⊗ V, τH′ ⊗ τ ) and the form (τH′ ⊗ τ )(T− ,− ) is hermitian and positive definite. Furthermore the elements e = 1 ⊗ E, f = 1 ⊗ F , h = 1 ⊗ H form a Cayley triple in g(H ⊗ V, τH′ ⊗ τ ) with respect to the Cartan involution Ad(T ). A straightforward calculation using (58) and (61) shows that
T (c(1 ⊗ v0 )) = −im c(j ⊗ v0 ),
(71)
im c(1 ⊗ v0 ),
T (c(j ⊗ v0 )) = hence
T (c(1 ⊗ v0 ) + ic(j ⊗ v0 )) = im+1 (c(1 ⊗ v0 ) + ic(j ⊗ v0 )), m+3
T (c(1 ⊗ v0 ) − ic(j ⊗ v0 )) = i
(72)
(c(1 ⊗ v0 ) − ic(j ⊗ v0 )),
so the restriction of T to the space c(H ⊗ v0 ) has two eigenvalues im+1 and im+3 . The eigenspaces are isotropic with respect to the form φ˜ if φ is the form defined in (46) with ˜ ˜ φ) τ in (46) replaced by τH′ ⊗ τ . It follows that the indecomposable formed space (U, ˜ with φ˜ symmetric. corresponding to cec−1 is of the form (C[m + 1] ⊕ C[m + 3], φ) We summarize our computations in the following proposition. Proposition 6.4. The Kostant-Sekiguchi correspondences for the orbits of indecomposable nilpotent elements in Lie algebras of isometries of formed spaces over H map the orbit O ⊆ g(V, τ ) of a nilpotent element of height m corresponding to a formed space ˜ as ¯ ˜ φ) (V˜ , τ˜) into the orbits S(O), S(O) ⊆ g(U, φ) corresponding to the formed spaces (U, in the following table. S(O)
¯ S(O)
even (H[0], +)
(C2 [m], sk)
(C2 [m], sk)
even (H[0], −)
(C2 [m + 2], sk)
(C2 [m + 2], sk)
m
O
odd (H[0], sk) (C[0] ⊕ C[2], sym) (C[0] ⊕ C[2], sym) odd (H[1], +)
(C2 [m], sk)
(C2 [m + 2], sk)
odd (H[1], −)
(C2 [m + 2], sk)
(C2 [m], sk)
even (H[1], sk) (C[1] ⊕ C[3], sym) (C[1] ⊕ C[3], sym)
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We will say that an orbit O ⊆ g(V, τ ) is indecomposable if for every (or equivalently some) element N ∈ O the element (N, V ) is indecomposable. Similarly we define indecomposable orbits in g(U, φ). Corollary 6.5. The image of an indecomposable nilpotent orbit in g(V, τ ) under a Kostant-Sekiguchi correspondence is also indecomposable.
6.2 The Kostant-Sekiguchi correspondences for general nilpotent elements Let N ∈ g(V, τ ) be an arbitrary nilpotent element, and let (V, τ ) = (V (1) , τ (1) ) ⊕ . . . ⊕ (V (s) , τ (s) )
(73)
be an orthogonal decomposition such that each of the spaces V (k) is N-invariant and each of the restrictions N (k) = N|V (k) is an idecomposable nilpotent element in g(V (k) , τ (k) ). Let U (k) = V (k) ⊗C if D = R, and U (k) = V (k) |C if D = C, H. By the results of the previous subsection for each k there exists T (k) ∈ G(V (k) , τ (k) ) such that the form τ (k) (T (k) − ,− ) is hermitian and positive definite, and that e(k) = N (k) , f (k) = −T (k) e(k) (T (k) )−1 , h(k) = [e(k) , f (k) ] is a Cayley triple in g(V (k) , τ (k) ) with respect to the Cartan involution θ(k) = Ad(T (k) ). Moreover, if c(k) denotes the element c(k) = exp(i π4 (e(k) + f (k) )) ∈ G(U (k) , φ(k) ) (with φ(k) defined by τ (k) as in (46)), then the G(U (k) , φ(k) )-orbit through c(k) e(k) (c(k) )−1 is the image of the G(V (k) , τ (k) )-orbit through e(k) by the Kostant-Sekiguchi correspondence S. Define T ∈ G(V, τ ) as T = T (1) ⊕ . . . ⊕ T (s) , then the form τ (T− ,− ) is hermitian and positive definite, and the triple e = N, f = −T eT −1 , h = [e, f ] is a Cayley triple in g(V, τ ) with respect to the Cartan involution θ = Ad(T ) = θ(1) ⊕ . . . ⊕ θ(s) . The element c ∈ G(U, φ) defined in (52) is equal to the product of commuting elements c = c(1) ·. . .·c(s) and the nilpotent element (cec−1 , U) ∈ g(U, φ)2 , whose orbit in g(U, φ)2 is equal to the image of the G(V, τ )-orbit of e under the Kostant-Sekiguchi correspondence S, is equal to the orthogonal sum (cec−1 , U) = (c(1) e(1) (c(1) )−1 , U (1) ) ⊕ . . . ⊕ (c(s) e(s) (c(s) )−1 , U (s) ). We can conclude that the Kostant-Sekiguchi correspondence S is compatible with orthogonal decompositions of nilpotent elements in g(V, τ ). The same statement holds for ¯ S. A simple analysis of Propositions 6.2, 6.3 and 6.4 verifies the following statement. Proposition 6.6. The maps S,S¯ are not equal if the corresponding group G(Vb , τb ) is ∗ isomorphic to Sp2n (R), O2n , or Up,q (with p, q > 0). For the remaining isometry groups ¯ and for the general linear groups S = S.
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A counterexample
For two nilpotent orbits O0 ⊂ g0 and O1 ⊂ g1 we shall write O0 ∼W O1 if and only if there is an orbit O ⊂ W such that O0 = ν0 (O) and O1 = ν1 (O).
Similarly, O0 ∼W + O1 if and only if there is an orbit O ⊂ WC+ such that C
O0 = µ0 (O) and O1 = µ1 (O). The relation ∼W is determined by Table 3 and the relation ∼W + is determined by Table C 4. For an indecomposable orbit O ⊂ W , the last column of the corresponding row of Table 3 contains S if and only if Sµ0 (O) ∼W + Sµ1 (O) C
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¯ In particular we see that (except for complex groups) there is no choice and similarly for S. of the Kostant–Sekiguchi map S or S¯ for which the two relations would be compatible. Below we provide a simple example (beyond the stable range) where the conjecture 1.1 is not true no matter which of the two possible Kostant-Sekiguchi maps is chosen. We consider the dual pair (O3,1 (R), Sp4 (R)). Let O be the nilpotent orbit in o3,1 ¯ is the union of two orbits: O and corresponding to (R, +, 2) ⊕ (R, +, 0). The closure O (R, +, 0)⊕3 ⊕ (R, −, 0) (the zero orbit). In these terms ¯ = {(R, +, 1) ⊕ (R, −, 1), (R, +, 3), (R, +, 1)⊕2 , ν1 ν0−1 (O) (R, +, 1) ⊕ (R2 , sk, 0), (R2 , sk, 0)⊕2} .
Then by Proposition 6.2 we have ¯ = {(C[3], sym, 1) ⊕ (C[1], sym, 1), (C[3], sym, 3), S(ν1 ν0−1 (O)) (C[1], sym, 1)⊕2 , (C[1] ⊕ C[3], sk, 0)⊕2 , (C[1], sym, 1) ⊕ (C[1] ⊕ C[3], sk, 0)}.
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On the other hand ¯ = {(C[2], sym, 2) ⊕ (C[0], sym, 0), (C[0], sym, 0)} S(O) and, by Table 4, ⊕2 ¯ µ1 µ−1 0 (S(O)) = {(C[3], sym, 1) , (C[1], sym, 3),
(C[1], sym, 1) ⊕ (C[3], sym, 1),
(C[1] ⊕ C[3], sk, 0) ⊕ (C[3], sym, 1),
(C[1] ⊕ C[3], sk, 0) ⊕ (C[1], sym, 1), (C[1] ⊕ C[3], sk, 0)⊕2 }.
(76)
In particular ¯ 6= µ1 µ−1 (S(O)) ¯ = µ1 µ−1 (SO), S(ν1 ν0−1 (O)) 0 0
the equality holds because S commutes with orbit closures ([3], [16, p. 177]). Replacing the Kostant-Sekiguchi map S by S¯ results in exchanging degrees 1 and 3 in (75) and in (76) (see Proposition 6.2), and these sets remain different.
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471
Dual pairs of type I in the stable range
Recall that a dual pair G(V0 , τ0 ), G(V1 , τ1 ) is in the stable range with G(V1 , τ1 ) - the smaller member if the dimension of V1 is less than or equal to the Witt index of the form τ0 . The goal of this section is to verify the following special case of Conjecture 1.1. Theorem 8.1. Let G(V0 , τ0 ), G(V1 , τ1 ) be a dual pair in the stable range with G(V1 , τ1 ) - the smaller member. Then for every orbit O ∈ N Og(V1 , τ1 ) there is a unique orbit Omax ∈ N Og(V0 , τ0 ) such that ν0 ν1−1 (O) = Omax .
(77)
µ0 µ−1 1 (S(O)) = S(Omax ).
(78)
S(ν0 ν1−1 (O)) = µ0 µ−1 1 (S(O)).
(79)
Moreover, Furthermore ¯ The same holds for τ0 skew-hermitian, τ1 hermitian and S replaced by S. Notice that, since Sekiguchi correspondence commutes with orbit closures, (77) and (78) imply (79): −1 S(ν0 ν1−1 (O)) = S(Omax ) = S(Omax ) = µ0 µ−1 1 (S(O)) = ν0 ν1 (SO)).
For a nilpotent O ∈ N Og(V1 , τ1 ) we define Omax ∈ N Og(V0 , τ0 ) as follows. Let (F , τ˜i , mi ), i = 1, 2, . . . s, be the sequence corresponding to O. Consider the sequence (F (i) , −˜ τi , mi + 1), i = 1, 2, . . . s. This sequence defines an orbit O′ ∈ N Og(V0′ , τ0′ ), where (i)
X 1 mi −1 (mi + 1)(fi , fi ) + sgn(−1) 2 τ˜(i) 2 i:2|mi +1 X 1 + (mi + 2)(fi , fi) 2 i:2|mi X X1 mi −1 (mi + 1)(fi , fi ) + sgn(−1) 2 τ˜(i) = 2 i i:2|mi +1 X 1 + (fi , fi ). 2
sgn(τ0′ ) =
i:2|mi
Notice that Moreover
P
i (mi
+ 1)fi = dimV1 and, for mi + 1 even, sgn(τi ) equals (1, 0) or (0, 1). 2#{i : 2|mi + 1} +
X
i:2|mi
fi ≤ dim(V1 ).
Hence sgn τ0′ ≤ (dim(V1 ), dim(V1 )). Therefore there exists a formed space (V0′′ , τ0′′ ) such that (V0′ , τ0′ ) ⊕ (V0′′ , τ0′′ ) is isometric to (V0 , τ0 ). Define Omax ∈ N Og(V0 , τ0 ) as the orbit through the extension by zero on V0′′ of any X ′ ∈ O′ .
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Let X ∈ O and Xmax ∈ Omax . Then k sgn τ0 ( , Xmax ) = sgn(−τ1 )( , X k−1 ),
k = 1, 2, . . . .
(80)
The following lemma is a simple consequence of (80). ′ Lemma 8.2. Suppose O, O′ ∈ N Og(V1 , τ1 ) and O′ ⊂ O. Then Omax ⊂ Omax .
The next lemma (which holds even withe stable range assumption) may be verified by a computation based on Lemma 3.9. Lemma 8.3. Let O ∈ N Og(Vb , τb ) and O′ ∈ N Og(Vb+1 , τb+1 ) with O ∼W O′ . If X ∈ O′ and Y ∈ O then sgn τb ( , Y k ) ≤ sgn(−1)b τb+1 ( , X k−1 ),
k = 1, 2, . . . .
(81)
Lemma 8.4. Suppose O ∈ N Og(V1 , τ1 ) and O′ ∼W O. Then O′ ⊂ Omax . Proof. Let X ∈ O, Xmax ∈ Omax and X ′ ∈ O′ . By (80) and Lemma 8.3 k sgn τ1 ( , Xmax ) = sgn(−τ0 )( , X k−1 ) ≥ sgn τ1 ( , X ′k )
for k = 1, 2, . . . .
The following lemma verifies (77). Lemma 8.5. For any O ∈ N Og(V1 τ1 ) ν0 ν1−1 (O) = Omax . ′ and, by Proof. Suppose that O′ ⊂ O and O′′ ∼W O′ . Then by Lemma 8.4, O′′ ⊂ Omax ′ ′′ Lemma 8.2, Omax ⊂ Omax . Hence O ⊂ Omax . Thus
ν0 ν1−1 (O) ⊂ Omax . For the reverse inclusion, we proceed as follows. Consider a nilpotent orbit O′′ ⊂ Omax . Let (F (i) , τ˜i , mi ), i = 1, 2, . . . s, be the sequence corresponding to O′′ . Suppose mi > 0 for 1 ≤ i ≤ t and mi = 0 for i > t. The sequence (F (i) , −˜ τi , mi − 1), i = ′ ′ ′ 1, 2, . . . t, defines a nilpotent orbit O ∈ N Og(V1 , τ1 ) for the apropriate formed space (V1′ , τ1′ ) which is skew-hermitian. Therefore there is a formed space (V1′′ , τ1′′ ) such that ′′ the space (V1′ ⊕ V1′′ , τ1′ ⊕ τ1′′ ) is isometric to (V1 , τ1 ). Let Omin ∈ N Og(V1 , τ1 ) be the orbit ′′ ′ through the extension by zero on V1 of any element of O . ′′ As in (80), for Y ∈ O′′ and Ymin ∈ Omin , we have k k+1 sgn τ1 (− , Ymin − ) = sgn(−τ0 )(− , Y − ),
k = 1, 2, . . . .
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Let X ∈ O and Xmax ∈ Omax . Since O′′ ⊂ Omax . we have k+1 ). sgn(−τ0 )( , Y k+1 ) ≤ sgn(−τ1 )( , Xmax
By (80), k+1 sgn(−τ1 )( , Xmax ) = sgn τ0 ( , X k ).
We obtain O′′ ⊂ ν0 ν1−1 (O) which completes the proof.
Now we consider KC -orbits in pC . In terms of (47) let p0,C = g(U0 ⊕ U2 , φ)2 and p1,C = g(U1 ⊕ U3 , φ)2 .
Let O ⊂ p0,C be a nilpotent orbit corresponding to a sequence (F (i) , φ˜i , mi ), i = 1, 2, . . . , s, as in Corollary 3.19. For a graded space F let η−1 F be the same space with the grading shifted by −1. The sequence (η−1 F (i) , φ˜i , mi + 1), = 1, 2, . . . , s, defines a nilpotent orbit O′ ⊂ g(U1′ ⊕ U3′ , φ′)2 , where (U1′ ⊕ U3′ , φ′ ) is of the same type as (U1 ⊕ U3 , φ) and there exists (U1′′ ⊕ U3′′ , φ′′ ) such that (U1′ ⊕ U3′ , φ′ ) ⊕ (U1′′ ⊕ U3′′ , φ′) is isometric to (U1 ⊕ U3 , φ). Let Omax ⊂ p1,C be the orbit through the extension by zero on U1′′ ⊕ U3′′ of any element of O′ . By Lemmas 10 and 14 in [16], , (see also [14]), Omax = µ0 µ−1 1 (O).
(82)
Our description of the Kostant-Sekiguchi correspondences in Section 6 and definitions of Omax in both cases imply S(Omax ) = (SO)max . (83) The last two equations prove (78). The proof of the theorem for τ0 skew-hermitian and τ1 hermitian is completely analogous.
References [1] G. Benkart, Ch.L. Shader and A. Ram: “Tensor product representations for orthosymplectic Lie superalgebras”, J. Pure Appl. Algebra, Vol. 130(1), (1998), pp. 1–48. [2] N. Burgoyne and R. Cushman: “Conjugacy Classes in Linear Groups”, Journal of Algebra, Vol. 44, (1975), pp. 339–362. [3] D. Barbasch and M. Sepanski: “Closure ordering and the Kostant-Sekiguchi correspondence”, Proc. Amer. Math. Soc., Vol. 126(1), (1998), pp. 311–317. [4] D. Collingwood and W. McGovern: Nilpotent orbits in complex semisimple Lie algebras, Reinhold, Van Nostrand, New York, 1993.
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[5] A. Daszkiewicz, W. Kra´skiewicz and T. Przebinda: “Nilpotent Orbits and Complex Dual Pairs”, J. Algebra, Vol. 190, (1997), pp. 518–539. [6] A. Daszkiewicz, W. Kra´skiewicz and T. Przebinda: “Dual Pairs and Kostant– Sekiguchi Correspondence. I.”, J. Algebra, Vol. 250, (2002), pp. 408–426. [7] A. Daszkiewicz and T. Przebinda: “The Oscillator Character Formula, for isometry groups of split forms in deep stable range”, Invent. Math., Vol. 123, (1996), pp. 349– 376. ˇ Djokoviˇc: Closures of Conjugacy Classes in Classical Real Linear Lie Groups, [8] D. Z. Lecture Notes in Mathematics, Vol. 848, Springer Verlag, 1980, pp. 63–83. [9] Harish-Chandra: “Invariant distributions on Lie algebras”, Amer. J. Math., Vol. 86, (1964), pp. 271–309. [10] R. Howe: “θ-series and invariant theory”, Proc. Symp. Pure. Math., Vol. 33, (1979), pp. 275–285. [11] R. Howe: A manuscript on dual pairs, preprint. [12] G. Kempken: Eine Darstellung des K¨ ochers A˜k . Dissertation, Rheinische FriedrichWilhelms-Universit¨at Bonner Mathematische Schriften, Vol. 137, Bonn, 1981. [13] Nishiyama, Kyo, Zhu and Chen-Bo: “Theta lifting of unitary lowest weight modules and their associated cycles”, Duke Math. Jour., Vol. 125(3), (2004), pp. 415–465. [14] Nishiyama, Kyo, Zhu and Chen-Bo: “Theta lifting of nilpotent orbits for symmetric pairs”, Trans. Amer. Math. Soc. to appear. [15] T. Ohta: “The singularities of the closures of nilpotent orbits in certain symmetric pairs”, Tˆohoku Math. J., Vol. 38, (1986), pp. 441–468. [16] T. Ohta: “The closures of nilpotent orbits in the classical symmetric pairs and their singularities”, Tˆohoku Math. J., Vol. 43, (1991), pp. 161–211. [17] T. Przebinda: “Characters, dual pairs, and unitary representations”, Duke Math. J., Vol. 69(3), (1993), pp. 547–592. [18] J. Sekiguchi: “The nilpotent subvariety of the vector space associated to a symmetric pair”, Publ. RIMS, Vol. 20, (1984), pp. 155–212. [19] J. Sekiguchi: “Remarks on real nilpotent orbits of a symmetric pair”, J. Math. Soc. Japan, Vol. 39, (1987), pp. 127–138. [20] W. Schmid and K. Vilonen: “Characteristic cycles and wave front cycles of representations of reductive Lie groups”, Ann. of Math. 2, Vol. 151(3), (2000), pp. 1071–1118. [21] D. Vogan: Representations of reductive Lie groups. A plenary address presented at the International Congress of Mathematicians held in Berkeley, California, August 1986. Introduced by Wilfried Schmid., ICM Series, American Mathematical Society, Providence, RI, 1988.
CEJM 3(3) 2005 475–495
Analytic Feynman integrals of transforms of variation of cylinder type functions over Wiener paths in abstract Wiener space Myung Jae Kim∗ Department of Mathematics, Kyonggi University, Suwon 443-760, Korea
Received 23 March 2005; accepted 15 May 2005 Abstract: In this paper, we evaluate various analytic Feynman integrals of first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transform of cylinder type functions defined over Wiener paths in abstract Wiener space. We also derive the analytic Feynman integral of the conditional Fourier-Feynman transform for the product of the cylinder type functions which define the functions in a Banach algebra introduced by Yoo, with n linear factors. c Central European Science Journals. All rights reserved.
Keywords: Conditional first variation, conditional Fourier-Feynman transform, conditional Wiener integral, Feynman integral, first variation, Fourier-Feynman transform, Wiener integral, Wiener paths in abstract Wiener space MSC (2000): 28C20
1
Introduction and preliminaries
Let C0 [0, T ] be the space of all continuous paths x on [0, T ] with x(0) = 0 which is known as the classical Wiener space. The concept of an L1 analytic Fourier-Feynman transform for the functions on this space was introduced by Brue in [1]. In [3], Cameron and Storvick introduced an L2 analytic Fourier-Feynman transform, and in [13] Johnson and Skoug developed an Lp analytic Fourier-Feynman transform theories for 1 ≤ p ≤ 2 that extended the results in [1, 3] and gave various relationships between the L1 and L2 theories. ∗
E-mail:
[email protected]
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On the other hand, in [2], Cameron obtained the Wiener integral of first variation of a function F in terms of the Wiener integral of the product of F with a linear factor. In [16], Park, Skoug and Storvick found the Fourier-Feynman transform of the product of a function with n linear factors from the Banach algebra S which was introduced by Cameron and Storvick in [4]. In [8], Chang, Song and Yoo expressed the analytic Feynman integral of the first variation of a function F in terms of analytic Feynman integrals of the product of F with a linear factor on abstract Wiener space. And then, using the recurrence relation, they derived the Fourier-Feynman transform for the product of a function in the Fresnel class with n linear factors. In [18], over Wiener paths in abstract Wiener space, Yoo introduced a Banach algebra ′′ SB which corresponds to the Banach algebra S ′′ of Cameron and Storvick in [4]. In [6], Chang, Cho and Yoo introduced the concept of conditional analytic Feynman integral on the space and in [7], they evaluated the integrals for the functions in the Banach algebra SB′′ . In [5], Chang, Cho, Kim, Song and Yoo introduced a concept of conditional FourierFeynman transform and in [9], Cho evaluated conditional Fourier-Feynman transforms of functions in SB′′ . And also, in [10], he introduced a concept of conditional first variation over Wiener paths in abstract Wiener space and in [11], found relationships between first variation and Fourier-Feynman transform of various functions on the space. In [12], he investigated several properties of first variations, conditional first variations and conditional Fourier-Feynman transforms and, using these properties, found relationships among first variation, conditional first variation, Fourier-Feynman transform and conditional FourierFeynman transform of cylinder type functions on the space. And then, he derived the conditional Fourier-Feynman transform for the product Gn of the cylinder type functions which define the functions in SB′′ , with n linear factors. In this paper, using the results of [12], we evaluate various analytic Feynman integrals of first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transform of the cylinder type functions over Wiener paths in abstract Wiener space. We then derive the analytic Feynman integrals of the Fourier-Feynman transform and conditional Fourier-Feynman transform of Gn . Let (Ω, A, P ) be a probability space, let B be a real normed linear space and let B(B) be the Borel σ-field on B. Let X : (Ω, A, P ) → (B, B(B)) be a random variable and let F : Ω → C be an integrable function. Let PX be the probability distribution of X on (B, B(B)) and let D be the σ-field {X −1 (A) : A ∈ B(B)}. Let PD be the probability measure induced by P , that is, PD (E) = P (E) for E ∈ D. By the definition of conditional expectation, there exists a D-measurable function E[F |X] (the conditional expectation of F given X) defined on Ω such that the relation Z Z E[F |X](ω) dPD (ω) = F (ω) dP (ω) E
E
holds for every E ∈ D. But there exists a PX -integrable function ψ defined on B which is unique up to PX -a.e. such that E[F |X](ω) = (ψ ◦ X)(ω) for PD -a.e. ω in Ω. ψ is also called the conditional expectation of F given X and without loss of generality, it is denoted by E[F |X](ξ) for ξ ∈ B. Throughout this paper, we will consider the function
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ψ as the conditional expectation of F given X. Let (H, B, m) be an abstract Wiener space([15]). Let {ej : j ≥ 1} be a complete orthonormal set in the real separable Hilbert space H such that ej ’s are in B∗ , the dual space of real separable Banach space B. For each h ∈ H and y ∈ B, define the stochastic inner product (h, y)∼ of h and y by limn→∞ Pn hh, ej i(y, ej ), if the limit exists; j=1 ∼ (h, y) = 0, otherwise,
where (·, ·) denotes the dual pairing between B and B∗ ([14]). Note that for each h(6= 0) in H, (h, ·)∼ is a Gaussian random variable on B with mean zero and variance |h|2 ; also (h, y)∼ is essentially independent of choice of the complete orthonormal set used in its definition and further, (h, λy)∼ = (λh, y)∼ = λ(h, y)∼ for all λ ∈ R. It is well-known that if {h1 , h2 , · · · , hn } is an orthogonal set in H, then the random variables (hj , ·)∼ ’s are independent. Moreover, if both h and y are in H, then (h, y)∼ = hh, yi where h·, ·i denotes the inner product on H. Now, we introduce a useful integral formula which appears in the proofs of several results. The proof easily follows from the fact that (h, ·)∼ is normally distributed with mean 0 and variance |h|2 if h 6= 0. Lemma 1.1. Let (H, B, m) be an abstract Wiener space and let h ∈ H. Suppose that α is pure imaginary. Then we have 2 2 Z α |h| ∼ exp{α(h, x1 ) }dm(x1 ) = exp . 2 B
2
Wiener paths in abstract Wiener space
Let C0 (B) denote the space of all continuous functions on [0, T ] into B which vanish at 0. Then C0 (B) is a real separable Banach space with the norm kxkC0 (B) ≡ sup0≤t≤T kx(t)kB . The minimal σ-field making the mapping x → x(t) measurable is B(C0 (B)), the Borel σfield on C0 (B). Further, the Brownian motion in B induces a probability measure mB on (C0 (B), B(C0 (B))) which is mean-zero Gaussian([17]). Now we begin with introducing a concrete form of mB . Let ~t = (t1 , t2 , · · · , tk ) be given with 0 = t0 < t1 < t2 < · · · < tk ≤ T . Let T~t : Bk → Bk be given by ! k X p √ √ √ T~t(x1 , x2 , · · · , xk ) = t1 − t0 x1 , t1 − t0 x1 + t2 − t1 x2 , · · · , tj − tj−1 xj . j=1
We define a set function ν~t on B(Bk ) by ν~t(B) =
k Y 1
!
m
T~t−1 (B)
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for B ∈ B(Bk ). Then ν~t is a Borel measure. Let f~t : C0 (B) → Bk be the function defined by f~t(x) = (x(t1 ), x(t2 ), · · · , x(tk )). Q For Borel subsets B1 , B2 , · · · , Bk of B, f~t−1 ( kj=1 Bj ) is called the I-set with respect to B1 , B2 , · · · , Bk . Then the collection I of all I-sets is a semi-algebra. We define a set function mB on I by !! ! k k Y Y −1 mB f~t Bj = ν~t Bj . j=1
j=1
Then mB is well-defined and countably additive on I. Using Carath´eodory extension process, we have a Borel measure mB on C0 (B). A function defined on C0 (B) is said to be Wiener measurable if it is measurable and the function is said to be Wiener integrable if it is integrable. Definition 2.1. Let F : C0 (B) → C be Wiener integrable and let X : (C0 (B), B(C0 (B)), mB ) → (B, B(B)) be a random variable, where B is a real normed linear space with the Borel σ-field B(B). The conditional expectation E[F |X] of F given X defined on B is called the conditional Wiener integral of F given X. Now, we introduce Wiener integration theorem without proof. For the proof, see [17]. Theorem 2.2 (Wiener integration theorem). Let ~t = (t1 , t2 , · · · , tk ) be given with 0 = t0 ≤ t1 ≤ t2 ≤ · · · ≤ tk ≤ T and let f : Bk → C be a Wiener measurable function. Then Z f (x(t1 ), x(t2 ), · · · , x(tk )) dmB (x) C0 (B) ! Z k Y ∗ = (f ◦ T~t)(x1 , x2 , · · · , xk ) d m (x1 , x2 , · · · , xk ), Bk
j=1
∗
where by = we mean that if either side exists, then both sides exist and they are equal. A subset E of C0 (B) is called a scale-invariant null set if mB (λE) = 0 for any λ > 0. A property is said to hold scale-invariant almost everywhere (in abbreviation, s-a.e.) if it holds except for a scale-invariant null set. Let F be defined on C0 (B) and let 1 F λ (x) = F (λ− 2 x) for λ > 0. Suppose that E[F λ ] exists for any λ > 0 and it has the analytic extension Jλ∗ (F ) on C+ ≡ {λ ∈ C : Re λ > 0}. Then we call Jλ∗ (F ) the analytic Wiener integral of F over C0 (B) with parameter λ and it is denoted by E anwλ [F ] = Jλ∗ (F ). Moreover, if for a non-zero real q, E anwλ [F ] has a limit as λ approaches to −iq through C+ , then it is called the analytic Feynman integral of F over C0 (B) with parameter q and
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denoted by E anfq [F ] = lim E anwλ [F ]. λ→−iq
Let τ : 0 = t0 < t1 < · · · < tk = T be a partition of [0, T ] and let x be in C0 (B). Define the polygonal function [x] of x on [0, T ] by
k X
t − tj−1 [x](t) = χ(tj−1 ,tj ] (t) x(tj−1 ) + (x(tj ) − x(tj−1 )) t − t j j−1 j=1
(1)
~ be the polygonal function of ξ~ on [0, T ] for t ∈ [0, T ]. For ξ~ = (ξ1 , · · · , ξk ) ∈ Bk , let [ξ] given by (1) with replacing x(tj ) by ξj for j = 0, 1, · · · , k (ξ0 = 0). The following lemmas are useful to define and evaluate the conditional analytic Wiener and Feynman integrals over C0 (B). For detailed proofs, see [6]. Lemma 2.3. If {x(t) : 0 ≤ t ≤ T } is the Wiener process on C0 (B) × [0, T ], then {x(t) − [x](t) : tj−1 ≤ t ≤ tj }, for j = 1, · · · , k, are stochastically independent. Lemma 2.4. Let F be defined and integrable on C0 (B). Let Xτ : C0 (B) → Bk be a random variable given by Xτ (x) = (x(t1 ), · · · , x(tk )). Then for every Borel measurable subset B of Bk , we have Z Z ~ dPXτ (ξ) ~ F (x) dmB (x) = E[F (x − [x] + [ξ])] Xτ−1 (B)
B
where PXτ is the probability distribution of Xτ on (Bk , B(Bk )). By the definition of conditional Wiener integral (Definition 2.1) and Lemma 2.4, we have ~ = E[F (x − [x] + [ξ])] ~ ~ E[F |Xτ ](ξ) for PXτ - a.e. ξ.
(2)
1 ~ exists. From For λ > 0 let Xτλ (x) = Xτ (λ− 2 x) and for ξ~ ∈ Bk suppose that E[F λ |Xτλ ](ξ) (2) we have 1
~ = E[F (λ− 2 (x − [x]) + [ξ])] ~ E[F λ |Xτλ ](ξ) for PXτλ -a.e. ξ~ ∈ Bk where PXτλ is the probability distribution of Xτλ on (Bk , B(Bk )). If 1 ~ has the analytic extension J ∗ (F )(ξ) ~ on C+ , then we call J ∗ (F )(ξ) ~ E[F (λ− 2 (x−[x]) + [ξ])] λ λ the conditional analytic Wiener integral of F given Xτ over C0 (B) with parameter λ and it is denoted by ~ = J ∗ (F )(ξ). ~ E anwλ [F |Xτ ](ξ) λ For a non-zero real q, if the limit ~ lim E anwλ [F |Xτ ](ξ)
λ→−iq
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exists, where λ approaches to −iq through C+ , then it is called the conditional analytic Feynman integral of F given Xτ over C0 (B) with parameter q and denoted by ~ = lim E anwλ [F |Xτ ](ξ). ~ E anfq [F |Xτ ](ξ) λ→−iq
3
Analytic Feynman integrals of transforms of variations
For a given extended real number p with 1 < p ≤ ∞, suppose that p and p′ are related by p1 + p1′ = 1(possibly p′ = 1 if p = ∞). Let Kn and K be measurable functions such that, for each γ > 0, Z ′ lim |Kn (γx) − K(γx)|p dmB (x) = 0. n→∞
C0 (B)
Then we write ′
l.i.m.(wsp )(Kn ) ≈ K n→∞
and call K the scale-invariant limit in the mean of order p′ . A similar definition is understood when n is replaced by a continuously varying parameter. Now, we define the Fourier-Feynman transform of functions over Wiener paths in abstract Wiener space. Definition 3.1. Let F be defined on C0 (B) and for λ ∈ C+ let Tλ (F )(y) = E anwλ [F (· + y)] for s-a.e. y ∈ C0 (B) if it exists. For a non-zero real q, we define the L1 analytic Fourier(1) Feynman transform Tq (F ) of F by the formula Tq(1) (F )(y) = E anfq [F (· + y)] if it exists for s-a.e. y ∈ C0 (B) and for 1 < p ≤ ∞ we define the Lp analytic Fourier(p) Feynman transform Tq (F ) of F by the formula ′
Tq(p) (F ) ≈ l.i.m. (wsp )(Tλ (F )) λ→−iq
where λ approaches to −iq through C+ . Definition 3.2. Let F be a Wiener measurable function defined on C0 (B) and let w ∈ C0 (B). The derivative ∂ F (x + tw)|t=0 ∂t for x ∈ C0 (B) if it exists, is called the first variation of F at x in the direction of w and denoted by δw F (x) =
∂ F (x + tw)|t=0 . ∂t
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Let H be an infinite dimensional separable real Hilbert space and let ∆n = {(s1 , · · · , sn ) : 0 < s1 < · · · < sn ≤ T } for n ∈ N. For ~s = (s1 , · · · , sn ) ∈ ∆n and ~h = (h1 , · · · , hn ) ∈ Hn ,
(3)
X n ∼ ~ Fn (x, ~s, h) = exp i (hj , x(sj ))
(4)
let
j=1
for x ∈ C0 (B). Further, for w ∈ C0 (B) with w(sj ) ∈ H (j = 1, 2, · · · , n), let Gn (x, w, ~s, ~h) = Fn (x, ~s, ~h)
n X
(w(sj ), x(sj ))∼
(5)
j=1
for x ∈ C0 (B) where Fn is given by (4). Let M′′n = M′′n (∆n × Hn ) be the class of all complex Borel measures on ∆n × Hn and ′′ ′′ let kµn k = varµn , the total variation of µn in M′′n . Let Sn,B = Sn,B (∆n × Hn ) be the space of functions of the form Z Hn (x) = Fn (x, ~s, ~h)dµn (~s, ~h) (6) ∆n ×Hn
for s-a.e. x ∈ C0 (B), where µn is in the class M′′n and Fn is given by (4). Here we take kHn k′′n = inf{kµn k}, where the infimum is taken over all µn ’s so that Hn and µn are P related by (6). Let M′′ = M′′ ( ∆n × Hn ) be the class of all sequences {µn } of measures P P ′′ ′′ such that each µn is in M′′n with ∞ ∆n × Hn ) be the n=1 kµn k < ∞. Let SB = SB ( space of functions on C0 (B) of the form H(x) =
∞ X
Hn (x)
(7)
n=1
P ′′ ′′ ′′ where each Hn is in Sn,B with ∞ n=1 kHn kn < ∞. The norm of H is defined by kHk = P∞ inf{ n=1 kHn k′′n }, where the infimum is taken over all representations of H given by (7). Note that SB′′ is a Banach algebra([18]). In convenience, for j1 + j2 + · · · + jk = n and for α = 1, · · · , k, let ~sα = (sα,1 , · · · , sα,jα ),
~hα = (hα,1 , · · · , hα,j ) α
and ~sn = (~s1 , · · · , ~sk ) ∈ ∆n ,
(8)
~hn = (~h1 , · · · , ~hk ) ∈ Hn .
(9)
The following lemma is useful, but we can easily prove it from the definition of first variation.
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Lemma 3.3. Let Fn , ~sn , ~hn be given by (4), (8), (9), respectively, and let w ∈ C0 (B). Then, for x ∈ C0 (B), δw Fn (x, ~sn , ~hn ) exists and is given by δw Fn (x, ~sn , ~hn ) = iFn (x, ~sn , ~hn )
jα k X X (hα,β , w(sα,β ))∼ .
(10)
α=1 β=1
Lemma 3.4. Let Fn , ~sn , ~hn be given by (4), (8), (9), respectively. Let q1 be a non-zero real number. Then we have E anfq1 [Fn (·, ~sn , ~hn )] = Qn (−iq1 , ~sn , ~hn )
(11)
where, for λ ∈ C∼ + ≡ {λ ∈ C : Re λ ≥ 0} − {0}, Qn is given by ju k 1 XX 2 ~ ~ Qn (λ, ~sn , hn ) = exp − γu,v |Bu,v (hn )| 2λ u=1 v=1 with for u = 1, · · · , k − 1, s1,0 = 0, su+1,0 = su,ju , and, for u = 1, · · · , k; v = 1, · · · , ju , γu,v = su,v − su,v−1 , Bu,v (~hn ) =
ju X
hu,β +
β=v
(12)
jα k X X
hα,β .
(13)
α=u+1 β=1
Proof. For λ > 0, we have 1 E[Fn (λ− 2 ·, ~sn , ~hn )] X Z jα k X − 21 ∼ = exp i (hα,β , λ x(sα,β )) dmB (x)
C0 (B)
=
Z
α=1 β=1
exp iλ
Bn
− 12
jα k X X
hα,β ,
α=1 β=1
ju α−1 X X √
γu,v xu,v +
u=1 v=1
β X √
γα,v xα,v
v=1
∼
dmn (~xn )
by Theorem 2.2, where ~xn = (x1,1 , · · · , x1,j1 , x2,1 , · · · , x2,j2 , · · · , xk,1, · · · , xk,jk ) and γu,v is given by (12). Then we have 1 E[Fn (λ− 2 ·, ~sn , ~hn )] X ∼ Z ju ju jα k X k X X X √ − 12 exp iλ γu,v hu,β + hα,β , xu,v dmn (~xn ) =
Bn
=
Z
Bn
u=1 v=1
exp iλ
− 12
= Qn (λ, ~sn , ~hn )
ju k X X u=1 v=1
β=v
√
α=u+1 β=1
γu,v (Bu,v (~hn ), xu,v )
∼
dmn (~xn )
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by Lemma 1.1. By Morera’s theorem, we have the result for λ ∈ C+ . Letting λ → −iq1 through C+ , we have the lemma. Theorem 3.5. Let Fn , ~sn , ~hn be given by (4), (8), (9), respectively. Let q1 , q2 ∈ R − {0} and let 1 ≤ p ≤ ∞. Then we have E anfq2 [Tq(p) (Fn (·, ~sn , ~hn ))] = E anfq1 [Tq(p) (Fn (·, ~sn , ~hn ))] 1 2 = E anfq1 [Fn (·, ~sn , ~hn )]E anfq2 [Fn (·, ~sn , ~hn )] k ju i 1 1 XX 2 ~ = exp − + γu,v |Bu,v (hn )| 2 q1 q2 u=1 v=1 where γu,v and Bu,v are given by (12) and (13), respectively. Proof. For λ > 0, we have 1 1 Tλ (Fn (·, ~sn , ~hn ))(y) = E[Fn (λ− 2 · +y, ~sn , ~hn )] = Fn (y, ~sn , ~hn )E[Fn (λ− 2 ·, ~sn , ~hn )]
for s-a.e. y ∈ C0 (B) and hence, for p = 1, we have Tq(p) (Fn (·, ~sn , ~hn ))(y) = Fn (y, ~sn, ~hn )E anfq1 [Fn (·, ~sn , ~hn )] 1
(14)
by Lemma 3.4. For 1 < p ≤ ∞, we can prove (14), easily. Hence we have E anfq2 [Tq(p) (Fn (·, ~sn , ~hn ))] = E anfq1 [Fn (·, ~sn , ~hn )]E anfq2 [Fn (·, ~sn , ~hn )] 1 and the other result follows from Lemma 3.4. Corollary 3.6. Let the assumptions and notations be given as in Theorem 3.5. Moreover, let w be in C0 (B). Then we have E anfq2 [δw Tq(p) (Fn (·, ~sn , ~hn ))] = E anfq2 [Tq(p) (δw Fn (·, ~sn , ~hn ))] 1 1 = E anfq1 [Fn (·, ~sn , ~hn )]E anfq2 [δw Fn (·, ~sn , ~hn )] = E anfq1 [δw Fn (·, ~sn , ~hn )]E anfq2 [Fn (·, ~sn , ~hn )] jα k X X anfq2 (p) ~ = iE [Tq1 (Fn (·, ~sn , hn ))] (hα,β , w(sα,β ))∼ α=1 β=1
(p) where E anfq2 [Tq1 (Fn (·, ~sn , ~hn ))] is given as in Theorem 3.5.
Proof. By (14) and Lemma 3.3, we have δw Tq(p) (Fn (·, ~sn , ~hn ))(y) = δw Fn (y, ~sn , ~hn )E anfq1 [Fn (·, ~sn , ~hn )] 1 = iFn (y, ~sn , ~hn )E anfq1 [Fn (·, ~sn , ~hn )]
jα k X X α=1 β=1
= Tq(p) (δw Fn (·, ~sn , ~hn ))(y) 1
(hα,β , w(sα,β ))∼
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for s-a.e. y ∈ C0 (B). On the other hand, for λ > 0, we have 1
1
E[δw Fn (λ− 2 ·, ~sn , ~hn )] = iE[Fn (λ− 2 ·, ~sn , ~hn )]
jα k X X
(hα,β , w(sα,β ))∼
α=1 β=1
and hence, by Lemma 3.4, we also have E
anfq2
[δw Fn (·, ~sn , ~hn )] = iE anfq2 [Fn (·, ~sn , ~hn )]
jα k X X
(hα,β , w(sα,β ))∼ .
α=1 β=1
Now, the results follow from Theorem 3.5. Theorem 3.7. Let Hn , H be given by (6), (7), respectively, let q1 , q2 ∈ R − {0} and let 1 ≤ p ≤ ∞. Moreover, for a partition 0 < t1 < · · · < tk = T of [0, T ] and for j1 + · · · + jk = n, let ∆′n;j1 ,··· ,jk = {~sn ∈ ∆n : 0 < s1,1 < · · · < s1,j1 ≤ t1 < s2,1 < · · · < s2,j2 ≤ t2 < · · · ≤ tk−1 < sk,1 < · · · < sk,jk ≤ tk = T }
where ~sn is given by (8). Then we have X Z anfq2 (p) E [Tq1 (Hn )] = ∆′n;j
j1 +···+jk =n
1 ,··· ,jk
×Hn
E anfq2 [Tq(p) (Fn (·, ~sn , ~hn ))]dµn (~sn , ~hn ) 1
and E
anfq2
[Tq(p) (H)] 1
=
∞ X
(Hn )] E anfq2 [Tq(p) 1
n=1
(p)
where ~hn is given by (9) and E anfq2 [Tq1 (Fn (·, ~sn , ~hn ))] is given as in Theorem 3.5. Proof. By [11, Theorem 3.4], it is not difficult to show that X Z (p) Tq1 (Hn )(y) = Tq(p) (Fn (·, ~sn , ~hn ))(y)dµn (~sn , ~hn ) 1 j1 +···+jk =n
∆′n;j
1 ,··· ,jk
×Hn
and Tq(p) (H)(y) 1
=
∞ X
Tq(p) (Hn )(y) 1
n=1
for s-a.e. y ∈ C0 (B). Now, for λ > 0, we have 1
=
E[Tq(p) (Hn )(λ− 2 ·)] 1 Z X Z
j1 +···+jk =n
=
X
j1 +···+jk =n
Z
C0 (B)
∆′n;j
∆′n;j
1 ,··· ,jk
1 ,··· ,jk
×Hn
×Hn
1 Tq(p) (Fn (·, ~sn , ~hn ))(λ− 2 x)dµn (~sn , ~hn )dmB (x) 1
1 E[Tq(p) (Fn (·, ~sn , ~hn ))(λ− 2 ·)]dµn (~sn , ~hn ) 1
(15)
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by Fubini’s theorem and 1 E[Tq(p) (H)(λ− 2 ·)] 1
=
∞ X
1
E[Tq(p) (Hn )(λ− 2 ·)] 1
n=1
by the dominated convergence theorem. By Morera’s theorem, we have the results for λ ∈ C+ . By Theorem 3.5 and the dominated convergence theorem, letting λ → −iq2 through C+ , we have the theorem. Corollary 3.8. Let the assumptions and notations be given as in Theorem 3.7. Moreover, for s-a.e. w ∈ C0 (B), suppose that X ∞ Z X n ∼ (hj , w(sj )) dkµn k(~s, ~h) < ∞ (16) n=1
∆n ×Hn j=1
where both ~s and ~h are given by (3). Then, for s-a.e. w ∈ C0 (B), we have X Z anfq2 (p) E E anfq2 [δw Tq(p) [δw Tq1 (Hn )] = (Fn (·, ~sn , ~hn ))]dµn (~sn , ~hn ) 1 n
j1 +···+jk =n ∆n;j1 ,··· ,jk ×H E anfq2 [Tq(p) (δw Hn )] 1 ′
=
and E anfq2 [δw Tq(p) (H)] = 1
∞ X
E anfq2 [δw Tq(p) (Hn )] = E anfq2 [Tq(p) (δw H)] 1 1
n=1
(p) where E anfq2 [δw Tq1 (Fn (·, ~sn , ~hn ))] is given as in Corollary 3.6.
Proof. From Definition 3.2, it is not difficult to show that X Z δw Hn (x) = δw Fn (x, ~sn , ~hn )dµn (~sn , ~hn )
(17)
×Hn 1 ,··· ,jk
∆′n;j
j1 +···+jk =n
and δw H(x) =
∞ X
δw Hn (x)
(18)
n=1
for s-a.e. x ∈ C0 (B). Now, by Corollary 3.6 and an application of Theorem 3.7 with (17), we have X Z anfq2 (p) E [Tq1 (δw Hn )] = E anfq2 [Tq(p) (δw Fn (·, ~sn , ~hn ))]dµn (~sn , ~hn ) 1 j1 +···+jk =n
= =
X
∆′n;j
Z
1 ,··· ,jk
×Hn
∆′n;j ,··· ,j ×Hn 1 k
j1 +···+jk =n E anfq2 [δw Tq(p) (Hn )] 1
E anfq2 [δw Tq(p) (Fn (·, ~sn , ~hn ))]dµn (~sn , ~hn ) 1
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and hence E
anfq2
[δw Tq(p) (H)] 1
=
∞ X
E anfq2 [δw Tq(p) (Hn )] 1
=
∞ X
E anfq2 [Tq(p) (δw Hn )] = E anfq2 [Tq(p) (δw H)] 1 1
n=1
n=1
by (16) and (18). Now, the proof is completed. Remark 3.9. For the first result in Corollary 3.8, the condition (16) is surplus. In this case, it can be replaced by n Z X (hj , w(sj ))∼ dkµn k(~s, ~h) < ∞. (19) n ∆n ×H
j=1
Theorem 3.10. Let Gn , ~sn , ~hn be given by (5), (8), (9), respectively. Let q1 , q2 ∈ R−{0} and let 1 ≤ p ≤ ∞. Then, for s-a.e. w ∈ C0 (B), we have E anfq2 [Tq(p) (Gn (·, w, ~sn, ~hn ))] 1 ju k X X 1 1 anfq2 (p) ~ =− + E [Tq1 (Fn (·, ~sn , hn ))] γu,v hBu,v (~hn ), Bu,v (w ~ n )i q1 q2 u=1 v=1
with the existence of each integral, where γu,v , Bu,v are given by (12), (13), respectively, (p) E anfq2 [Tq1 (Fn (·, ~sn , ~hn ))] is given as in Theorem 3.5, and w ~ n = (w(s1,1), · · · , w(s1,j1 ), w(s2,1 ), · · · , w(s2,j2 ), · · · , w(sk,1), · · · , w(sk,jk )). Proof. It is not difficult to show E anfq2 [Gn (·, w, ~sn , ~hn )] = −
ju k X X 1 anfq2 E [Fn (·, ~sn , ~hn )] γu,v hBu,v (~hn ), Bu,v (w ~ n )i (20) q2 u=1 v=1
by [11, Theorem 2.11] and for s-a.e. y ∈ C0 (B) Tq(p) (Gn (·, w, ~sn , ~hn ))(y) 1
=E
anfq1
~ [Fn (·, ~sn , hn )] Gn (y, w, ~sn, ~hn )
ju k X X 1 ~ ~ γu,v hBu,v (hn ), Bu,v (w ~ n )i − Fn (y, ~sn , hn ) q1 u=1 v=1 by [11, Theorem 3.8]. Then, by (20) and Theorem 3.5, we have E anfq2 [Tq(p) (Gn (·, w, ~sn, ~hn ))] 1 anfq1 ~ =E [Fn (·, ~sn , hn )] E anfq2 [Gn (·, w, ~sn, ~hn )] ju k X X 1 anfq2 − E [Fn (·, ~sn , ~hn )] γu,v hBu,v (~hn ), Bu,v (w ~ n )i q1 u=1 v=1 ju k X X 1 1 anfq2 (p) ~ + E [Tq1 (Fn (·, ~sn , hn ))] γu,v hBu,v (~hn ), Bu,v (w ~ n )i =− q1 q2 u=1 v=1
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which is the desired result.
4
Analytic Feynman integrals of conditional transforms of variations
In this section, we derive the analytic Feynman integrals of conditional transforms of variations. For this purpose, we define the conditional first variation and conditional Fourier-Feynman transform over Wiener paths in abstract Wiener space. Definition 4.1. Let F be a function defined on C0 (B) and let F (· + x) be integrable for x ∈ C0 (B). Let w ∈ C0 (B), let B be a real linear normed space and let X : C0 (B) → B be a random variable. Let PX be the probability distribution of X on (B, B(B)). For x ∈ C0 (B), if the derivative ∂ E[F (· + x + tw)|X](ξ)|t=0 ∂t exists for PX -a.e. ξ ∈ B, then it is called the conditional first variation of F given X at x in the direction of w and denoted by δw E[F |X](x, ξ) =
∂ E[F (· + x + tw)|X](ξ)|t=0. ∂t
Definition 4.2. Let F be defined on C0 (B) and let Xτ be given as in Lemma 2.4. For λ ∈ C+ and for s-a.e. y ∈ C0 (B) let ~ = E anwλ [F (· + y)|Xτ ](ξ) ~ Tλ [F |Xτ ](y, ξ) for a.e. ξ~ ∈ Bk if it exists. For a non-zero real q and for s-a.e. y ∈ C0 (B), we define (1) the L1 analytic conditional Fourier-Feynman transform Tq [F |Xτ ] of F given Xτ by the formula ~ = lim Tλ [F |Xτ ](y, ξ) ~ Tq(1) [F |Xτ ](y, ξ) λ→−iq
for a.e. ξ~ ∈ Bk if it exists and for 1 < p ≤ ∞ we define the Lp analytic conditional (p) Fourier-Feynman transform Tq [F |Xτ ] of F given Xτ by the formula ~ ≈ l.i.m. (w p′ )(Tλ [F |Xτ ](·, ξ)) ~ Tq(p) [F |Xτ ](·, ξ) s λ→−iq
for a.e. ξ~ ∈ Bk , where λ approaches to −iq through C+ and
1 p
+
1 p′
= 1.
Lemma 4.3. Let Fn , ~sn , ~hn be given by (4), (8), (9), respectively, and let Xτ be given as in Lemma 2.4. Let q1 be a non-zero real number and suppose that 0 < s1,1 < · · · < s1,j1 ≤ t1 < s2,1 < · · · < s2,j2 ≤ t2 < · · · ≤ tk−1 < sk,1 < · · · < sk,jk ≤ T.
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Then, we have ~ = Fn ([ξ], ~ ~sn , ~hn )Γn (−iq1 , ~sn , ~hn ) E anfq1 [Fn (·, ~sn , ~hn )|Xτ ](ξ)
(21)
for a.e. ξ~ ∈ Bk , where for λ ∈ C∼ + , Γn is given by k jX α +1 X 1 2 Γn (λ, ~sn , ~hn ) = exp − lα,v |Av (τ, ~sα , ~hα )| 2λ α=1 v=1
(22)
with t0 = s1,0 = 0, sk,jk +1 = tk = T, tα = sα+1,0 = sα,jα+1 for α = 1, · · · , k − 1 and lα,v = sα,v − sα,v−1 ,
Av (τ, ~sα , ~hα ) =
v−1 X tα−1 − sα,β β=1
tα − tα−1
hα,β
jα X tα − sα,β + hα,β tα − tα−1 β=v
(23)
(24)
for α = 1, · · · , k; v = 1, · · · , jα + 1. Proof. For λ > 0 and for a.e. ξ~ ∈ Bk , by Lemmas 2.3 and 2.4, we have 1 ~ ~sn , ~hn )] E[Fn (λ− 2 (x − [x]) + [ξ], Z jα k X X 1 ∼ − ~ ~sn , ~hn ) = Fn ([ξ], exp iλ 2 (hα,β , x(sα,β ) − [x](sα,β )) dmB (x)
C0 (B)
~ ~sn , ~hn ) = Fn ([ξ],
α=1 β=1
k Z Y
α=1
C0 (B)
exp iλ
− 12
jα X hα,β , x(sα,β ) − x(tα−1 ) β=1
∼ sα,β − tα−1 − (x(tα ) − x(tα−1 )) dmB (x) . tα − tα−1
Let lα,v be given by (23) and ~xα = (x1 , · · · , xjα +1 ) for α = 1, · · · , k. Then, by Lemma 1.1
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and Wiener integration theorem(Theorem 2.2), we have 1 ~ ~sn , ~hn )] E[Fn (λ− 2 (x − [x]) + [ξ], jα β k Z Y X X p sα,β − tα−1 − 21 ~ ~ = Fn ([ξ], ~sn , hn ) exp iλ hα,β , lα,v xv − tα − tα−1 Bjα +1 α=1 v=1 β=1 ∼ jα +1 Xp jα +1 × lα,v xv dm (~xα )
v=1
k Z Y
β jα X X tα − sα,β p ~ ~sn , ~hn ) hα,β , lα,v xv = Fn ([ξ], exp iλ t − t jα +1 α α−1 B α=1 v=1 β=1 jα +1 ∼ X tα−1 − sα,β p + lα,v xv dmjα+1 (~xα ) t α − tα−1 v=β+1 jα +1 k Z Y Xp − 21 ∼ jα +1 ~ ~ ~ = Fn ([ξ], ~sn , hn ) exp iλ lα,v (Av (τ, ~sα , hα ), xv ) dm (~xα ) α=1
− 21
Bjα +1
v=1
~ ~sn , ~hn )Γn (λ, ~sn , ~hn ) = Fn ([ξ], where Γn and Av are given by (22) and (24), respectively. By Morera’s theorem, we have the result for λ ∈ C+ . Now, letting λ → −iq1 through C+ , we have the lemma. Theorem 4.4. Let the assumptions and notations be given as in Lemma 4.3. Let 1 ≤ p ≤ ∞ and let q2 be a non-zero real number. Then we have ~ = E anfq2 [Fn (·, ~sn , ~hn )]E anfq1 [Fn (·, ~sn , ~hn )|Xτ ](ξ) ~ (25) E anfq2 [Tq(p) [Fn (·, ~sn , ~hn )|Xτ ](·, ξ)] 1 ~ are given by for a.e. ξ~ ∈ Bk , where E anfq2 [Fn (·, ~sn , ~hn )] and E anfq1 [Fn (·, ~sn , ~hn )|Xτ ](ξ) (11) and (21), respectively. Proof. For λ > 0 and for s-a.e. y ∈ C0 (B), we have ~ + y, ~sn , ~hn )] ~ = E[Fn (λ− 21 (x − [x]) + [ξ] Tλ [Fn (·, ~sn , ~hn )|Xτ ](y, ξ) 1 ~ ~sn , ~hn )] = Fn (y, ~sn , ~hn )E[Fn (λ− 2 (x − [x]) + [ξ], ~ = Fn (y, ~sn , ~hn )E anwλ [Fn (·, ~sn , ~hn )|Xτ ](ξ)
for a.e. ξ~ ∈ Bk . By Lemma 4.3, we have ~ = Fn (y, ~sn, ~hn )E anfq1 [Fn (·, ~sn , ~hn )|Xτ ](ξ) ~ Tq(p) [Fn (·, ~sn , ~hn )|Xτ ](y, ξ) 1
(26)
~ is given by (21). Now, by Lemma 3.4, the result follows. where E anfq1 [Fn (·, ~sn , ~hn )|Xτ ](ξ) Corollary 4.5. Let the assumptions and notations be given as in Theorem 4.4. Then,
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for s-a.e. w ∈ C0 (B), we have
~ E anfq2 [Tq(p) [δw Fn (·, ~sn , ~hn )|Xτ ](·, ξ)] 1
= iE
anfq2
~ [Tq(p) [Fn (·, ~sn , ~hn )|Xτ ](·, ξ)] 1
jα k X X (hα,β , w(sα,β ))∼ α=1 β=1
=E
anfq2
~ [δw (Tq(p) [Fn (·, ~sn , ~hn )|Xτ ](·, ξ))] 1
(p) ~ is given by (25). for a.e. ξ~ ∈ Bk , where E anfq2 [Tq1 [Fn (·, ~sn , ~hn )|Xτ ](·, ξ)]
Proof. By (10) and (26), it is not difficult to show that, for s-a.e. y ∈ C0 (B), we have ~ Tq(p) [δw Fn (·, ~sn , ~hn )|Xτ ](y, ξ) 1 ~ = iFn (y, ~sn , ~hn )E anfq1 [Fn (·, ~sn , ~hn )|Xτ ](ξ)
(27) jα k X X
(hα,β , w(sα,β ))∼
α=1 β=1
for a.e. ξ~ ∈ Bk . Thus, by Theorem 4.4, we have ~ E anfq2 [Tq(p) [δw Fn (·, ~sn , ~hn )|Xτ ](·, ξ)] 1
= iE
anfq2
~ [Fn (·, ~sn , ~hn )]E anfq1 [Fn (·, ~sn , ~hn )|Xτ ](ξ)
jα k X X
(hα,β , w(sα,β ))∼ .
α=1 β=1
By [12, Corollary 23] and for s-a.e. w, y ∈ C0 (B), we have
~ = δw (T (p) [Fn (·, ~sn , ~hn )|Xτ ](·, ξ))(y) ~ Tq(p) [δw Fn (·, ~sn , ~hn )|Xτ ](y, ξ) q1 1
for a.e. ξ~ ∈ Bk and hence the results follow from (25). Corollary 4.6. Let the assumptions and notations be given as in Theroem 4.4. Then, for s-a.e. w ∈ C0 (B) and for a.e. ξ~ ∈ Bk , we have ~ E anfq2 [Tq(p) (δw E[Fn (·, ~sn , ~hn )|Xτ ](·, ξ))] 1
= iE
anfq2
~ ~sn , ~hn )Γn (1, ~sn , ~hn ) [Tq(p) (Fn (·, ~sn , ~hn ))]Fn ([ξ], 1
jα k X X
(hα,β , w(sα,β ))∼
α=1 β=1
=E
anfq2
~ [δw E[Tq(p) (Fn (·, ~sn , ~hn ))|Xτ ](·, ξ)] 1
(p) where E anfq2 [Tq1 (Fn (·, ~sn , ~hn ))] is given as in Theorem 3.5 and Γn is given by (22).
Proof. By Lemma 3.3 and [12, Lemma 10], we have, for s-a.e. x ∈ C0 (B) and for a.e. ξ~ ∈ Bk , ~ δw E[Fn (·, ~sn , ~hn )|Xτ ](x, ξ) ~ ~sn , ~hn )Γn (1, ~sn , ~hn ) = δw Fn (x + [ξ],
~ ~sn , ~hn )Γn (1, ~sn , ~hn ) = iFn (x, ~sn , ~hn )Fn ([ξ],
(28) jα k X X α=1 β=1
(hα,β , w(sα,β ))∼
M.J. Kim / Central European Journal of Mathematics 3(3) 2005 475–495
491
and we also have ~ E anfq2 [Tq(p) (δw E[Fn (·, ~sn , ~hn )|Xτ ](·, ξ))] 1 = iE
anfq2
~ ~sn , ~hn )Γn (1, ~sn , ~hn ) [Tq(p) (Fn (·, ~sn , ~hn ))]Fn ([ξ], 1
jα k X X
(hα,β , w(sα,β ))∼
α=1 β=1
by Theorem 3.5. The second equality in the corollary follows from [12, Corollary 24]. Theorem 4.7. Let Hn and H be given by (6) and (7), respectively. Let the assumptions and notations be given as in Theorem 4.4. Then, we have ~ E anfq2 [Tq(p) [Hn |Xτ ](·, ξ)] 1 X Z ~ E anfq2 [Tq(p) = [Fn (·, ~sn , ~hn )|Xτ ](·, ξ)]dµ sn , ~hn ) n (~ 1 ∆′n;j
j1 +···+jk =n
1 ,··· ,jk
×Hn
and E
anfq2
~ [Tq(p) [H|Xτ ](·, ξ)] 1
∞ X
=
n=1
~ E anfq2 [Tq(p) [Hn |Xτ ](·, ξ)] 1
(p) ~ are given by (15) for a.e. ξ~ ∈ Bk , where ∆′n;j1 ,··· ,jk and E anfq2 [Tq1 [Fn (·, ~sn , ~hn )|Xτ ](·, ξ)] and (25), respectively.
Proof. From [9, Theorems 14, 15] and for s-a.e. y ∈ C0 (B), we have X Z (p) ~ ~ n (~sn , ~hn ) (29) Tq1 [Hn |Xτ ](y, ξ) = Tq(p) [Fn (·, ~sn , ~hn )|Xτ ](y, ξ)dµ 1 j1 +···+jk =n
∆′n;j
1 ,··· ,jk
×Hn
and ~ Tq(p) [H|Xτ ](y, ξ) 1
∞ X
=
n=1
for a.e. ξ~ ∈ Bk . Then for λ > 0, 1 ~ E[Tq(p) [Hn |Xτ ](λ− 2 ·, ξ)] 1
=
X
j1 +···+jk =n
Z
~ Tq(p) [Hn |Xτ ](y, ξ) 1
∆′n;j
1 ,··· ,jk
×Hn
Z
C0 (B)
(30)
1 ~ Tq(p) [Fn (·, ~sn , ~hn )|Xτ ](λ− 2 y, ξ) 1
dmB (y)dµn(~sn , ~hn ) by Fubini’s theorem. By Morera’s theorem, we have the result for λ ∈ C+ . Now, letting λ → −iq2 through C+ , we have the first result in the theorem by the dominated convergence theorem. Again, by Morera’s and the dominated convergence theorem, we have the second result in the theorem.
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M.J. Kim / Central European Journal of Mathematics 3(3) 2005 475–495
Corollary 4.8. Let the assumptions and notations be given as in Theorem 4.7. In addition, for s-a.e. w ∈ C0 (B), suppose that (16) holds. Then, for s-a.e. w ∈ C0 (B), we have ~ E anfq2 [Tq(p) [δw Hn |Xτ ](·, ξ)] 1 Z X ~ = E anfq2 [Tq(p) [δw Fn (·, ~sn , ~hn )|Xτ ](·, ξ)]dµ sn , ~hn ) n (~ 1 j1 +···+jk =n
∆′n;j
1 ,··· ,jk
×Hn
~ = E anfq2 [δw (Tq(p) [Hn |Xτ ](·, ξ))] 1 and E
anfq2
~ [Tq(p) [δw H|Xτ ](·, ξ)] 1
=
∞ X
~ E anfq2 [Tq(p) [δw Hn |Xτ ](·, ξ)] 1
n=1 anfq2
=E
~ [δw (Tq(p) [H|Xτ ](·, ξ))] 1
(p) ~ is given as in Corollary 4.5. for a.e. ξ~ ∈ Bk , where E anfq2 [Tq1 [δw Fn (·, ~sn , ~hn )|Xτ ](·, ξ)]
Proof. From (17) and an application of (29), it follows that for s-a.e. w, y ∈ C0 (B),
=
~ Tq(p) [δw Hn |Xτ ](y, ξ) 1 Z X j1 +···+jk =n
∆′n;j
1 ,··· ,jk
×Hn
~ n (~sn , ~hn ) Tq(p) [δw Fn (·, ~sn , ~hn )|Xτ ](y, ξ)dµ 1
(p) ~ is given by (27). Now, by Corollary for a.e. ξ~ ∈ Bk , where Tq1 [δw Fn (·, ~sn , ~hn )|Xτ ](y, ξ) 4.5 and an application of Theorem 4.7, we have the first result in the corollary. The second result in the corollary follows from (18), an application of (30), Morera’s and the dominated convergence theorem.
Corollary 4.9. Let the assumptions and notations be given as in Corollary 4.8. Then, for s-a.e. w ∈ C0 (B) and for a.e. ξ~ ∈ Bk , we have ~ E anfq2 [Tq(p) (δw E[Hn |Xτ ](·, ξ))] 1 Z X ~ = E anfq2 [Tq(p) (δw E[Fn (·, ~sn , ~hn )|Xτ ](·, ξ))]dµ sn , ~hn ) n (~ 1 j1 +···+jk =n
∆′n;j
1 ,··· ,jk
×Hn
~ = E anfq2 [δw E[Tq(p) (Hn )|Xτ ](·, ξ)] 1 and ~ = E anfq2 [Tq(p) (δw E[H|Xτ ](·, ξ))] 1
∞ X
~ E anfq2 [Tq(p) (δw E[Hn |Xτ ](·, ξ))] 1
n=1 anfq2
=E
~ [δw E[Tq(p) (H)|Xτ ](·, ξ)] 1
(p) ~ is given as in Corollary 4.6. where E anfq2 [Tq1 (δw E[Fn (·, ~sn , ~hn )|Xτ ](·, ξ))]
M.J. Kim / Central European Journal of Mathematics 3(3) 2005 475–495
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Proof. By [12, Theorem 11], it is not difficult to show, for s-a.e. x ∈ C0 (B),
=
~ δw E[Hn |Xτ ](x, ξ) Z X j1 +···+jk =n
∆′n;j
1 ,··· ,jk
~ n (~sn , ~hn ) δw E[Fn (·, ~sn , ~hn )|Xτ ](x, ξ)dµ ×Hn
~ is given by (28), so that we have the for a.e. ξ~ ∈ Bk , where δw E[Fn (·, ~sn , ~hn )|Xτ ](x, ξ) first result in the corollary by an application of Theorem 3.7. The second result in the corollary follows from [12, Theorem 12], an application of Theorem 3.7 and the first result in the corollary. Remark 4.10. For the first results in Corollaries 4.8 and 4.9, the condition (16) is surplus. In fact, it can be replaced by (19). Theorem 4.11. Let the assumptions and notations be given as in Theorem 3.10. Moreover, let Xτ be given as in Lemma 2.4. Then, s-a.e. w ∈ C0 (B), we have ~ E anfq2 [Tq(p) [Gn (·, w, ~sn, ~hn )|Xτ ](·, ξ)] 1 X jα k X k jα +1 1 XX ∼ anfq2 (p) ~ ~ ~ (w(sα,β ), [ξ](sα,β )) − =E [Tq1 [Fn (·, ~sn , hn )|Xτ ](·, ξ)] q1 α=1 v=1 α=1 β=1 ju k 1 XX ~ ~ lα,v hAv (τ, ~sα , hα ), Av (τ, ~sα , w ~ α )i − γu,v hBu,v (hn ), Bu,v (w ~ n )i q2 u=1 v=1
(p) ~ are given by (23), (24), for a.e. ξ~ ∈ Bk , where lα,v , Av , E anfq2 [Tq1 [Fn (·, ~sn , ~hn )|Xτ ](·, ξ)] (25), respectively, and for α = 1, · · · , k
w ~ α = (w(sα,1 ), · · · , w(sα,jα )). Proof. By [12, Theorem 29] and for s-a.e w, y ∈ C0 (B), we have ~ Tq(p) [Gn (·, w, ~sn , ~hn )|Xτ ](y, ξ) 1 X jα k X (p) ~ ~ ~ α,β ))∼ = Tq1 [Fn (·, ~sn , hn )|Xτ ](y, ξ) (w(sα,β ), y(sα,β ) + [ξ](s α=1 β=1
k 1 XX ~ lα,v hAv (τ, ~sα , hα ), Av (τ, ~sα , w ~ α )i − q1 α=1 v=1 jα +1
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M.J. Kim / Central European Journal of Mathematics 3(3) 2005 475–495
for a.e. ξ~ ∈ Bk . By (26), we have ~ Tq(p) [Gn (·, w, ~sn, ~hn )|Xτ ](y, ξ) 1 ~ = Fn (y, ~sn , ~hn )E anfq1 [Fn (·, ~sn , ~hn )|Xτ ](ξ)
X jα k X
(w(sα,β ), y(sα,β ) k X X 1 ∼ ~ α,β )) − +[ξ](s lα,v hAv (τ, ~sα , ~hα ), Av (τ, ~sα , w ~ α )i q1 α=1 v=1 α=1 β=1
jα +1
~ + T (p) [Fn (·, ~sn , ~hn )|Xτ ](y, ξ) ~ = Gn (y, w, ~sn, ~hn )E anfq1 [Fn (·, ~sn , ~hn )|Xτ ](ξ) q1 X jα k X k jα +1 1 XX ∼ ~ ~ × (w(sα,β ), [ξ](sα,β )) − lα,v hAv (τ, ~sα , hα ), Av (τ, ~sα , w ~ α )i . q 1 α=1 v=1 α=1 β=1 Now, by Morera’s theorem, we have ~ E anfq2 [Tq(p) [Gn (·, w, ~sn , ~hn )|Xτ ](·, ξ)] 1 ~ + E anfq2 [T (p) [Fn (·, ~sn , ~hn ) = E anfq2 [Gn (·, w, ~sn, ~hn )]E anfq1 [Fn (·, ~sn , ~hn )|Xτ ](ξ) q1 X jα jα +1 k X k XX 1 ∼ ~ ~ α,β )) − |Xτ ](·, ξ)] (w(sα,β ), [ξ](s lα,v hAv (τ, ~sα , ~hα ), Av (τ, ~sα , w ~ α )i . q 1 α=1 β=1 α=1 v=1 By (20) and (25), we have ~ E anfq2 [Tq(p) [Gn (·, w, ~sn, ~hn )|Xτ ](·, ξ)] 1 ju k X X 1 anfq anf q ~ ~ ~ = − E 2 [Fn (·, ~sn , hn )]E 1 [Fn (·, ~sn , hn )|Xτ ](ξ) γu,v hBu,v (~hn ), q2 u=1 v=1 X jα k X anfq2 (p) ~ ~ ~ α,β ))∼ Bu,v (w ~ n )i + E [Tq1 [Fn (·, ~sn , hn )|Xτ ](·, ξ)] (w(sα,β ), [ξ](s α=1 β=1
− =E
1 q1
k jX α +1 X α=1 v=1
anfq2
lα,v hAv (τ, ~sα , ~hα ), Av (τ, ~sα , w ~ α )i X jα k X
~ [Tq(p) [Fn (·, ~sn , ~hn )|Xτ ](·, ξ)] 1
~ α,β ))∼ (w(sα,β ), [ξ](s
α=1 β=1
1 − q1
k jX α +1 X α=1 v=1
ju k 1 XX ~ ~ lα,v hAv (τ, ~sα , hα ), Av (τ, ~sα , w ~ α )i − γu,v hBu,v (hn ), Bu,v (w ~ n )i q2 u=1 v=1
which is the desired result.
Acknowledgment The author would like to express his sincere thanks to the referees for their valuable comments and suggestions.
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References [1] M.D. Brue: A functional transform for Feynman integrals similar to the Fourier transform, Thesis, Univ. of Minnesota, Minneapolis, 1972. [2] R.H. Cameron: The first variation of an indefinite Wiener intergal, Proc. Amer. Math. Soc., Vol. 2, (1951), pp. 914–924. [3] R.H. Cameron and D.A. Storvick: “An L2 analytic Fourier-Feynman transform”, Michigan Math. J., Vol. 23, (1976), pp. 1–30. [4] R.H. Cameron and D.A. Storvick: Some Banach algebras of analytic Feynman integrable functionals, An analytic functions, Lecture Notes in Math., Vol. 798, Springer, 1980, pp. 18–27. [5] K.S. Chang, D.H. Cho, B.S. Kim, T.S. Song and I. Yoo: “Conditional FourierFeynman transform and convolution product over Wiener paths in abstract Wiener space”, Integral Transform. Spec. Funct., Vol. 14(3), (2003), pp. 217–235. [6] K.S. Chang, D.H. Cho and I. Yoo: “A conditional analytic Feynman integral over Wiener paths in abstract Wiener space”, Intern. Math. J., Vol. 2(9), (2002), pp. 855–870. [7] K.S. Chang, D.H. Cho and I. Yoo: “Evaluation formulas for a conditional Feynman integral over Wiener paths in abstract Wiener space”, Czechoslovak Math. J., Vol. 54(129), (2004), pp. 161–180. [8] K.S. Chang, T.S. Song and I. Yoo: “Analytic Fourier-Feynman transform and first variation on abstract Wiener space”, J. Korean Math. Soc., Vol. 38(2), (2001), pp. 485–501. [9] D.H. Cho: “Conditional Fourier-Feynman transform and convolution product over Wiener paths in abstract Wiener space : an Lp theory”, J. Korean Math. Soc., Vol. 42(2), (2004), pp. 265–294. [10] D.H. Cho: “Conditional first variation over Wiener paths in abstract Wiener space”, J. Korean Math. Soc., (2004), to appear. [11] D.H. Cho: “Fourier-Feynman transform and first variation of cylinder type functions over Wiener paths in abstract Wiener space”, Intern. Math. J., (2004), submitted. [12] D.H. Cho: “Conditional Fourier-Feynman transforms of variations over Wiener paths in abstract Wiener space”, J. Korean Math. Soc., (2005), to apear. [13] G.W. Johnson and D.L. Skoug: “An Lp analytic Fourier-Feynman transform”, Michigan Math. J., Vol. 26, (1979), pp. 103–127. [14] G. Kallianpur and C. Bromley: “Generalized Feynman integrals using an analytic continuation in several complex variables”, In: Stochastic analysis and applications, Adv. Probab.Related Topics, Vol. 7, Dekker, New York, 1984, pp. 217–267. [15] H.H. Kuo: Gaussian measures in Banach spaces, Lecture Notes in Math., Vol. 463, Springer, 1975. [16] C. Park, D.L. Skoug and D.A. Storvick: “Fourier-Feynman transfroms and the frist variation”, Rend. Circ. Mat. Palermo II, Vol. 47(2), (1998), pp. 277–292. [17] K.S. Ryu: “The Wiener integral over paths in abstract Wiener space”, J. Korean Math. Soc., Vol. 29(2), (1992), pp. 317–331. [18] I. Yoo: “The analytic Feynman integral over paths in abstract Wiener space”, Comm. Korean Math. Soc., Vol. 10(1), (1995), pp. 93–107.
CEJM 3(3) 2005 496–507
On some properties of the functions from Sobolev-Morrey type spaces Alik M. Najafov∗ Department of Supreme Mathematics, Azerbaijan Architectural and Civil Engineering University, 5, T.Shahbazi str., AZ1073, Baku, Azerbaijan
Received 3 January 2005; accepted 23 May 2005 l Abstract: In this paper the spaces of type Sobolev-Morrey- Wp,a,Γ,τ (Q, G)-are constructed, the differential properties are studied and it is proved that the functions from these spaces satisfy Holder’s condition, in the case, if the domain G ⊂ Rn satisfies the flexible λ−horn condition. c Central European Science Journals. All rights reserved.
Keywords: Sobolev-Morrey space, flexible λ-horn, mixed derivatives, the imbedding theorems, generalized derivatives, integral representation MSC (2000): 26A33, 46E30, 42B35, 46E35
Let the domain G ⊂ Rn satisfy the condition of flexible λ−horn introduced by O.V. Besov [1] and let en = {1, 2, . . . , n}; e0n = en ∪ {0}; Q be a fixed subset of the set en ; Ø6= e ⊂ Q; p ∈ [1, ∞); a ∈ [0, 1]n ; τ ∈ [1, ∞]; l ∈ N n ; Γ∈ (0, ∞)n ; T = (T1 , T2 , . . . , Tn ), where Ti = Tj for i, j ∈ en \Q; λ = (λ1 , λ2 , , . . . , λn ), λj = 1 for j ∈ Q; λj ∈ (0, ∞) for j ∈ en \Q; le∨i = (l1e∨i , l2e∨i . . . lne∨i ), lje∨i = lj for j ∈ e ∨ i; lje∨i = 0 for j ∈ en \(e ∨ i); j ∈ e ∨ i denotes, that, either j ∈ e ⊂ Q, or j = i ∈ en \Q. Let also bj Zbe∨i Z e∨i Y f (x)dx = dxj f (x), ae∨i
j∈e∨i a
j
that is, the integral involves only the variable xj , and indices belong to set e ∨ i. Proceeding from the fact that some mixed derivatives D ν f may not be estimated by derivative functions contained in the norm of the space Wpl (G) and on the other hand from undesireable higher order derivatives from the function f , there arose a necessity to consider other types of Sobolev spaces Wpl (Q, G), that are introduced and studied in [2] ∗
E-mail:
[email protected]
A.M. Najafov / Central European Journal of Mathematics 3(3) 2005 496–507
with the finite norm: kf kWpl (Q,G) where
kf kp,G
X X
le∨i =
D f e⊂Q i∈e0n \Q
p,G
497
,
p pn p1n p3 p2 n−1 p 2 p1 Z Z Z p1 = · · · |χ (y)|f (y)| dy dy , dy · · · G 1 2 n R R R
with χ− the characteristic function of the set G.Note that the considered spaces also e∨i e∪i were defined by A.D.Djabrailov [3] but with the norm (1) replacing D l f by D l f ( note, that in case [3] dominate mixed derivatives ). Unlike in the paper [3] here either dominant unmixed derivatives, or mixed derivatives, or mixed derivatives and unmixed derivatives are equal. At |Q| = 1 (|Q| −the number of the set Q) the space Wpl (Q, G) coincides with the space of Sobolev-Wpl (G), at Q ≡ en the space Wpl (Q, G) coincides with the space of Sobolev with dominant mixed derivatives Spl W (G), introduced and studied by S.M.Nikolskii [4] with finite norm X e
D l f , kf kSpl W (G) = p,G e⊆en
where l = (l1 , l2 , · · · , ln ), lj ∈ N for j ∈ en ; le = (l1 e , l2 e , · · · , ln e ), lj e = lj for j ∈ e,lj e = 0 for j ∈ en \e. For example, in equation u + u′x + u′y + u′′xy + u′z = 0
the norm u′′xy may not be estimated by the norm of the space W (1,1,1) , but may be estimated by the norm of the Sobolev space with dominating mixed derivative S (1,1,1) W, then we require additional derivatives from the function u(x, y, z). To study partial differential equations it is necessary to study the space of functions l of many variables with parameters Wp,a,Γ (G), for some partial values of indices studied in papers of Morrey [5-7], and later on developed in papers of Greco [8], Nirenberg [9], Campanato [10,11], Barossi [12], Il’yin [13], Netrusov [14], author [15-18] and others. For all x ∈ G and t ∈ (0, ∞)n ; ti = tj for i, j ∈ en \Q assuming 1 Γj ItΓ (x) = y : |yj − xj | < tj , j ∈ en , 2 GtΓ (x) = G ∩ ItΓ (x). Let’s consider for x ∈ G the vector-function ρ(tλ , x) = (ρ1 (tλ1 1 , x), ρ2 (tλ2 2 , x), . . . , ρn (tλnn , x)), 0 ≤ tj ≤ Tj , j ∈ en where for all j ∈ en , ρj (0, x) = 0, the functions ρj (uj , x) are absolutely continuous λj λ ′ with respect to uj on [0, Tj ] and ρj (uj , x) ≤ 1 for almost all uj ∈ [0, Tj j ], where
498
A.M. Najafov / Central European Journal of Mathematics 3(3) 2005 496–507
ρ′j (uj , x) = ∂u∂ j ρj (uj , x). For θ ∈ (0, 1]n ; θi = θj for i, j ∈ en \Q. We’ll call each of the S sets V (λ, x, θ) = 0 ≤ tj ≤ Tj , j ∈ en ρ(tλ , x) + tλ θλ I and x+ V (λ, x, θ)−as flexible horn−λ and point x as a top x + V (λ, x, θ). We’ll suppose that x+ V (λ, x, θ) ⊂ G. In the case of, t1 = t2 = . . . = tn = t, θ1 = θ2 = . . . = θn = θ ∈ (0, 1], V (λ, x, θ)−is flexible λ − horn, found in O.V. Besov in [1]. l Definition. The Sobolev-Morrey-Wp,a,Γ,τ (Q, G) type space is called the Banach space of locally summable on G functions f with finite norm:
X X
le∨i kf kW l = D f (1)
p,a,Γ,τ (Q,G) p,a,Γ,τ ;G
e⊂Q i∈e0n \Q
here kf kp,a,Γ,τ ;G = kf kLp,a,Γ,τ (G) = sup
kf kp,GtΓ (x) =
Z
x∈G
GtΓn (xn )
t " Z 0 Y
Z
· · · G Γ
t 2
[tj ]1
0
Z
kf kp,G Γ (x) t
j∈en
(x2 )
Γ a − jp j
pp2 1
|f (y)| dy1 dy2 p1
GtΓ1 (x1 )
#τ
1 τ Y dtj , t j∈Q∨i j
p3 p2
(2)
p pn p1n n−1 , (3) · · · dyn
where [tj ]1 = min{1, tj }, j ∈ en ; t0 = (t01 , t02 , . . . , t0n )−a fixed positive vector, t0i = t0j l l for i, j ∈ en \Q, l0 = 0. At τ = ∞ the Wp,a,Γ,∞ (Q, G) = Wp,a,Γ (Q, G)−space studied in l [18] . At τ = ∞ and a = 0 the space Wp,a,Γ,τ (Q, G) coincides with the space of type Sobolev Wpl (Q, G). In the case τ = ∞, a = (a, · · · a) , p = (p, · · · , p), t = (t, · · · , t) Sobolev-Morrey space l ≡ Wp,a,Γ (G) were introduced and studied by V.P.Il’yin [13], more precisely,
l Wp,a,Γ,∞ (G)
kf kW l
p,a,Γ (G)
= kf kp,a,Γ;G
where
−
kf kp,a,Γ;G = sup [t]1 x∈G,t>0
n X
l
D i f + i
p,a,Γ;G
,
i=1
n P
j=1
Γj aj pj
kf kp,G Γ (x) , t
l l Besov-Morrey spaces Bp,θ,a,Γ,∞ (G) ≡ Bp,θ,a,Γ (G) with finite norm:
kf kBl
p,θ,a,Γ (G)
"
#θ 1/θ n Zh0 mi X △i (h, G)Diki f p,a,Γ dh = kf kp,a,Γ,G + , hli −ki h i=1 0
i ∆m i (h)f (x)
=
mi X i=0
j (−1)mi −j Cm f (x + jhei ), i
A.M. Najafov / Central European Journal of Mathematics 3(3) 2005 496–507
499
were introdused and studied by Yu.V.Netrusov [14]. In the papers [15-17] introdused and l studied Besov-Morrey type spaces Bp,θ,a,Γ,τ (G, λ) with finite norm (mi > li − ki > 0): kf kBl
p,θ,a,Γ,τ (G,λ)
"
#θ 1/θ n Zh0 △mi (h, G, λ)D ki f X dh i i p,a,Γ,τ = kf kp,a,Γ,τ ;G + hli −ki h i=1 0
mi l i where △m i (h, G, λ) = △i (h, Ghλ ), Lizorkin-Triebel-Morrey type spaces Fp,θ,a,Γ,τ (G) with finite norm (mi > li > ki ≥ 0):
1
Zt0 θ n
X (k −l )λ mi −ki λ ki θ dt
kf kF l = kf kp,a,Γ,τ ;G + , t i i i δi (t )Di f (·)
p,θ,a,Γ,τ (G)
t i=1 0
p,a,Γ,τ
where
δimi (tλ )f (x)
=
Z1
m λ △ i (t i u, Gtλ )f (x) du, i
−1
h0 and t0 fixed positive numbers and Besov-Morrey type spaces with dominating mixed l derivative Sp,θ,a,Γ,τ B(Gh ) with finite norm (mj > lj − kj > 0):
kf kSp,θ,a,Γ,τ l B(Gh )
e 1θ θ
h e e 0 X Z ∆m (h, Gh )D k f p,a,Γ,τ Y dhj = Q lj −kj hj hj j∈e e⊆en e 0
where
me
∆
j∈e
(h)f (x) =
Y
m ∆j j (hj )
!
f (x),
j∈e
kf kp,a,Γ,τ ;G = sup
t0 Z 1
x∈G
0
G
···
Z Γ (x1 ) t1 1
0 Ztn
0
Z
Z a n −Γj j Y [tj ]1 pj · · · j=1 G Γn (xn ) G Γ
t 2 2
tn
p2 /p1
|f | dy1 p1
dy2
p3 /p2
pn /pn−1
···
(x2 )
1/τ 1/pn τ n Y dtj dyn . j=1 tj
l Note some properties of the spaces Lp,a,Γ,τ (G) and Wp,a,Γ,τ (Q, G). 1) The embedding theorems are valid: l l Lp,a,Γ,τ (G) ֒→ Lp,a,Γ(G), Wp,a,Γ,τ (Q, G) ֒→ Wp,a,Γ (Q, G) i.e.
kf kp,a,Γ;G ≤ C kf kp,a,Γ,τ ;G
(4)
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A.M. Najafov / Central European Journal of Mathematics 3(3) 2005 496–507
and kf kW l
p,a,Γ (Q,G)
≤ C kf kW l
p,a,Γ,τ (Q,G)
.
(5)
2) For all the real c > 0 it kf kp,a,cΓ,τ ;G = and
1 1
cτ
kf kp,a,Γ,τ ;G
(6)
1
; (7) 1 kf kW l p,a,Γ,τ (G) cτ are valid. 3) a) kf kp,0,Γ,∞;G = kf kp;G and kf kW l = kf kWpl (Q,G) ; p,0,Γ,∞ (Q,G) b) kf k∞;G ≤ kf kp,1,Γ,τ ;G and kf kWpl (Q,G) ≤ kf kW l . p,1,Γ,τ (Q,G) ∞ n Let Me,i (·, y, z) ∈ C0 (R ) and support of the function Me,i (·, y, z) be such that 1 S(Me,i ) = suppMe,i ⊂ I1 = x : |xj | < , j ∈ en 2 kf kW l
p,a,cΓ,τ (G)
Denote
[
V =
0
=
o n y y : λ ∈ S(Me,i ) . t
Obviously V ⊂ IT λ , U is a open set, contained in G. Further, suppose that U + V ⊂ G. Let GT Γ (U) =
[
x∈G
GT Γ (x) = (U + IT Γ (x)) ∩ G,
if 0 < Tj ≤ 1, 0 < Γj ≤ λj , j ∈ en , then IT λ ⊂ IT Γ , it follows that U + V ⊂ GT Γ (U) = Z. Lemma 1. Let 1 ≤ p ≤ q ≤ r ≤ ∞, 0 <Γj ≤ λj , 0 < tj ≤ Tj ≤ 1, j ∈ en , 1 ≤ τ ≤ ∞, η = (η1 , η2 , . . . , ηn ), 0 < ηj ≤ Tj , j ∈ en , ηi = ηj , for i, j ∈ en \Q; ν = (ν1 , ν2 , . . . , νn ), νj ≥ 0− be integer, j ∈ en , φ ∈ Lp,a,Γ,τ (G) and let k ∈ Q X 1 1 εj = λj lj − λj νj + (λj − Γj aj ) − , p q j=k∨j=i∈en \Q e∨i
Ae,i η (x) =
Y
T −1−νj
j∈Q\e
η Z
ϕe,i (x, t)dte∨i
0e∨i
P
1−λj lj +
Q
j=k∈e∨j=i
(λj +λj νj )
,
(8)
,
(9)
j=k∨j=i∈en \Q
tj
e∨i
Ae,i ηT (x) =
Y
T −1−νj
j∈Q\e
T Z
ηe∨i
ϕe,i(x, t)dte∨i Q
j=k∈e∨j=i
where ϕe,i (x, t) =
Z Rn
φ(x + y)Me,i
P
1−λj lj +
tj
(λj +λj νj )
j=k∨j=i∈en \Q
y , + T λQ\e
λen \(Q\e)
t
A.M. Najafov / Central European Journal of Mathematics 3(3) 2005 496–507
501
! en \(Q\e) Q\e ρ(tλ + T λ , x) ′ λen \(Q\e) Q\e , ρ (t + T λ , x) dy. tλen \(Q\e) + T λQ\e
(10)
Then the following inequalities hold:
sup Ae,i η q,U x∈G
γΓ
≤ C1 kφkp,a,Γ,τ ;Z × ×
sup Ae,i ηT q,U x∈G
(x)
γΓ
(x)
Y
Γj aj q
[γj ]1
j∈en
Y
Y
j∈en
Y
−νj −(1−Γj aj )( 1p − q1 )
Tj
ε
ηj j (εj > 0),
(11)
j∈e∨i
j∈Q\e
≤ C2 kφkp,a,Γ,τ ;Z
×
Y
Γj aj
[γj ]1 q ×
−νj −(1−Γj aj )( p1 − q1 )
Tj
j∈Q\e
Q
ε
j=k∈e∨j=i
Q
Tj j , T
ln ηjj ,
j=k∈e∨j=i Q ε ηj j ,
εj > 0, εj = 0,
(12)
εj < 0,
j=k∈e∨j=i
where γ ∈ (0, ∞)n , γi = γj for i, j ∈ en \Q, C1 and C2 are constants independent of φ, γ, η and T. Proof of Lemma 1. Applying the generalized Minkowskii inequalities for all x ∈ U, we have Y −1−ν
j
≤ Tj × sup Ae,i η q,U (x) x∈G
γΓ
j∈Q\e
ηe∨i
×
Z
0e∨i
1 Q
1−λj lj +
j=k∈e∨j=i
tj
P
(λj +λj νj )
j=k∨j=i∈en \Q
kϕe,i (·, t)kq,U
γΓ
(x)
dte∨i (13)
We estimate the norm kϕe,i(·, t)kq,U Γ (x) . By virtue of Holder’s inequalities (q ≤ r) we γ have kϕe,i(·, t)kq,U
(x) γΓ
≤ kϕe,i (·, t)kr,U
(x) γΓ
Y
Γj ( 1q − r1 )
γj
(14)
j∈en
Let χ− function of the set S(Me,i ). Observing that 1 ≤ p ≤ r ≤ ∞, be the characteristic 1 1 1 s ≤ r s = 1 − p + r we represent the right hand side of equality (10) as 1
1
|φMe,i | = (|φ|p |Me,i |s ) r (|φ|p χ) p
− 1r
1
(|Me,i |s ) s
− r1
and applying to |ϕe,i (x, t)| Holder’s inequality with the exponent again 1 1 1 1 + p − r + s − 1r = 1 , we have r
(15)
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A.M. Najafov / Central European Journal of Mathematics 3(3) 2005 496–507
kϕe,i(·, t)kr,U
(x) γΓ
≤
Z p sup |φ(x + y)| χ
x∈Uγ Γ (x)
× sup y∈V
Rn
Z
Uγ Γ (x)
y + T λQ\e
tλen \(Q\e)
1r Z p |φ(x + y)| dx
Rn
en \(Q\e)
Me,i
p1 − 1r
dy
×
y , + T λQ\e
λen \(Q\e)
t
Q\e
ρ(tλ + T λ , x) ′ λen \(Q\e) λQ\e , ρ (t + T , x) e \(Q\e) Q\e n tλ + Tλ
! s ! 1s . dy
(16)
Obviously Ztλen \(Q\e) +T λQ\e (x) ⊂ ZtΓen \(Q\e) +T ΓQ\e (x) for 0 < tj ≤ Tj ≤ 1, Γj ≤ λj , j ∈ en and for any x ∈ U Z Rn
p
|φ(x + y)| χ
≤
y en \(Q\e) λ t + T λQ\e Z Z
Z
dy ≤ p
|φ(x + y)| χ
e \(Q\e) Q\e (x) tΓ n +T Γ
≤ kφkpp,a,Γ;Z For y ∈ V
Γ aj
tj j
j∈en \(Q\e)
Z
p
|φ(x + y)| dx ≤
Uγ Γ (x)
Y
Y
y en \(Q\e) λ t + T λQ\e
Γ a
(17)
j∈Q\e
|φ(x)|p dx ≤ kφkpp,a,Γ;Z en \(Q\e)
en \(Q\e)
Y
Γ a
(18)
λ
(19)
[γj ]1 j j ,
j∈en
Q\e
ρ(tλ + T λ , x) y , , tλen \(Q\e) + T λQ\e tλen \(Q\e) + T λQ\e
Rn
ρ′ (tλ
dy ≤
Tj j j .
Zγ Γ (x+y)
Z Me,i
Q\e
+ Tλ
s , x) dy = kMe,i kss
Y
λ
tj j
j∈en \(Q\e)
Y
Tj j .
j∈Q\e
From the inequalities (14) and (16)-(19) it follows that kϕe,i (·, t)kq,U
(x) γΓ
≤ kφkp,a,Γ;Z ×
Y
j∈en
Y
1−(1−Γj aj )( p1 − 1r )
Tj
j∈Q\e
Γ a [γj ]1 j j
Y
λj lj +
tj
Y
j∈en P
Γj ( q1 − 1r )
γj
×
[λj −(λj −Γj aj )( p1 − 1r )]
j=k∨j=i∈en \Q
. (20)
j=k∈e∨j=i
Taking into account inequality (4) and substituting the inequality (19) in (13) for r = q we obtain the inequality (11). Analogously the inequality (12) is proved.
A.M. Najafov / Central European Journal of Mathematics 3(3) 2005 496–507
503
Lemma 2. Let 1 ≤ p ≤ q < ∞, 0 <Γj ≤ λj , 0 < Tj ≤ 1, j ∈ en , 1 ≤ τ1 ≤ τ2 ≤ ∞, ν = (ν1 , ν2 , . . . , νn ), νj ≥ 0 be integer, j ∈ en , and εj > 0, X 1 0 εj = λj lj − λj νj + (λj − Γj aj ) , k ∈ Q, p j=k∨j=i∈en \Q
then for the function Ae,i T (x), defined by the equality (8) the following estimate is valid:
e,i
A η q,b,Γ,τ
2 ;U
≤ C kφkp,a,Γ,τ1 ;Z ,
(21)
where b = (b1 , b2 , . . . bn ), bj is any number, satisfying the inequalities: if ε0j > 0,
0 ≤ bj ≤ 1,
0 ≤ bj < 1,
ε0j
j ∈ en \(Q\e),
= 0, j ∈ en \(Q\e); 0 ≤ bj ≤ aj , j ∈ Q\e ε0j q(1 − aj ) 0 ≤ bj ≤ 1 + , if ε0j < 0, j ∈ en \(Q\e), |λ|(en \Q)∨k − |Γ, a|(en \Q)∨k if
k ∈ Q and C1 is a constant independent of φ. Theorem 1. Let the open set G ⊂ Rn satisfy the condition of flexible λ−horn, 1 ≤ p ≤ q ≤ ∞, 0 <Γj ≤ λj , 0 < Tj ≤ 1, j ∈ en , 1 ≤ τ1 ≤ τ2 ≤ ∞, ν = (ν1 , ν2 , . . . , νn ), νj ≥ 0 l be integer, j ∈ en , f ∈ Wp,a,Γ,τ (Q, G) and let εj > 0, j ∈ en (k ∈ Q) then 1 l D ν : Wp,a,Γ,τ (Q, G) ֒→ Lq,b,Γ,τ2 (G), 1
i.e., for the function f the generalized deriatives D ν f exist and the inequalities are valid: kD ν f kq,G ≤ C1
X X e⊂Q
i∈e0n \Q
Y
j∈Q∨i
kD ν f kq,b,Γ,τ2 ;G ≤ C2 kf kW l
e∨i s Tj j D l f
p,a,Γ,τ1 (Q,G)
where sj =
εj ,
−νj − (1 − Γj aj )
1 p
(22)
p,a,Γ,τ1 ;G
−
1 q
, (p ≤ q < ∞)
(23)
j ∈ e ∨ i, , j ∈ Q\e.
In particular, if ε0j > 0, j ∈ en (k ∈ Q), then D ν f is continous on G and the inequalitiy is valid: ν
sup |D f | ≤ C1 x∈G
where s0j =
X X
Y
s0 Tj j
e⊂Q i∈e0n \Q j∈Q∨i
ε0j ,
le∨i
D f
p,a,Γ,τ1 ;G
j ∈ e ∨ i,
−νj − (1 − Γj aj ) 1 , p
j ∈ Q\e,
,
(24)
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A.M. Najafov / Central European Journal of Mathematics 3(3) 2005 496–507
here Tj ∈ (0, min(1, t0j )], j ∈ en , C1 and C2 are constants independent of f, and a constant C1 doesn’t depend on T also. Proof of Theorem 1. First of all note, that as far as Γ = cΓ, c > 0, we can consider l f ∈ Wp,a,Γ,τ (Q, G) and we can replace everywhere in inequalities (22) -(24), εj and ε0j , 1 Γj into Γj , j ∈ en . We’ll prove exactly such inequalities (the more Γj , the more εj ). From the conditions of our theorem it follows from theorem 2 of [2] that on G there exist generalized derivatives D ν f and D ν f ∈ Lp (G). Really, if εj > 0, j ∈ en (k ∈ Q), P then λj lj − λj νj > 0, j ∈ en (k ∈ Q), since 1 ≤ p ≤ q ≤ ∞, 0 < aj ≤ 1, j=k∨j=i∈en \Q
l l 0 <Γj ≤ λj , j ∈ en . Since f ∈ Wp,a,Γ,τ (Q, G) ֒→ Wp,a,Γ (Q, G) ֒→ Wpl (Q, G), then by 1 virtue of theorem 2 of [2] on G the generalized derivatives exist D ν f ∈ Lp (G). Then for almost every point x ∈ G the integral identity holds: X X Y −1−ν e∨i j D ν f (x) = (−1)|ν | Tj × e⊂Q i∈e0n \Q
×
T Ze∨iZ
j∈Q\e
dte∨i Q
0e∨i Rn
j=k∈e∨j=i
1−λj lj +
tj
(ν) L (λj +λj νj ) e,i
P
j=k∨j=i∈en \Q
y , + T λQ\e
tλen \(Q\e)
! Q\e ρ(t + T λ , x) ′ λen \(Q\e) Q\e e∨i , ρ (t + T λ , x) D l f (x + y)dy en \(Q\e) Q\e λ λ t +T λen \(Q\e)
(25)
the functions Le,i ∈ C0∞ (Rn ), have support contained in in I1 , because the support of representation (25) is contained in flexible λ−horn x+ +V (λ, x, θ) ⊂ G. Here by virtue of Minkowskii inequalities, we have X X e,i
A . kD ν f (x)kq,G ≤ (26) T q,G e⊂Q i∈e0n \Q
With the help of inequality (10) for U = G, D l
e,i le∨i
A ≤ C D f
T q,G
e∨i
(ν)
f = φ, Me,i = Le,i , γ → ∞ we have
p,a,Γ,τ ;G
Y
s
Tj j ,
(27)
j∈Q∨i
and, here the inequality (22) follows. Analogously by virtue of inequality (21) the inequality (23) follows. Let now ε0j > 0, j ∈ en (k ∈ Q). Show that D ν f is continous on G. By virtue of identity (25) and inequality (22) for q = ∞, εj = ε0j > 0, j ∈ en (k ∈ Q), we have ν
ν
kD f − D fT λ k∞,G ≤ C1
X X
Y
e⊂Q i∈e0n \Q j∈Q∨i
s0 Tj j
le∨i
D f
p,a,Γ,τ ;G
.
(28)
Here it follows that the left hand side of inequality (28) converges to zero for Tj → 0 , j ∈ en . Since D ν fT λ is continuous on G then convergence on L∞ (G) in this case coincides with the uniform and consequently D ν f is continuous in G. Theorem 1 is proved.
A.M. Najafov / Central European Journal of Mathematics 3(3) 2005 496–507
505
Theorem 2. Let the domain G, the parameters p, q, τ1 , τ2 and vectors Γ, Γ, ν satisfiy the condition of theorem 1. If εj > 0, j ∈ en , k ∈ Q, then the derivatives D ν f satisfiy the Holder’s condition in metric Lq with exponent β 1 , more exactly, Y 1 1 k∆(ξ, G)D ν f kq,G ≤ C kf kW l |ξj |βj |ξ|βen \Q , (29) (Q,G) p,a,Γ,τ
j∈Q
where β 1 = (β11 , β21, . . . , βn1 ), βi1 = βj1 for i, j ∈ en \Q and βj1 − for any number satisfying the following inequalities: 0 ≤ βj1 ≤ 1, 0 ≤ βj1 < 1, 0 ≤ βj1 ≤ 1
0 ≤ βj1 ≤ εj , 0 ≤ β 1 ≤ 1,
0 ≤ β 1 < 1,
ε0 , λ0
0 ≤ β1 ≤
if
εj > 1
for j ∈ e,
if
εj = 1
for j ∈ e;
for j ∈ Q\e; if
εj < 1
if
ε0 λ0
> 1,
if
ε0 λ0
= 1,
if
ε0 λ0
< 1,
where λ0 = max λj , ε0 = min εQ,i, εQ,i = λi li − j∈en \Q
i ∈ en \Q. If ε0j > 0, j ∈ en , k ∈ Q, then
for j ∈ e,
i P h λj νj + (λj − Γj aj ) 1p − 1q ,
j∈en \Q
sup |∆(ξ, G)D ν f | ≤ C kf kW l
p,a,Γ,τ (Q,G)
x∈G
Y
j∈Q
1,0
1,0
|ξj |βj |ξ|βen \Q ,
(30)
where βj1,0 (j ∈ en ) satisfies the same conditions, that βj1 with the substitution of εj by ε0j . Proof of Theorem 2. As in proof of theorem 1, we can replace vector Γ by Γ. Let ξ be an n− dimensional vector. According to lemma 8.6 in [1] there exists domain Gσ ⊂ G, σ = (σ1 , σ2 , . . . , σn ), σj = ςj rλλ for j ∈ en ; σi = σj for i, j ∈ en \Q; rλ = distλ (x, ∂G) , x ∈ G. We’ll suppose that ξj < σj for j ∈ Q, |ξ|en \Q < σ, then for every x ∈ Gσ , segment [x, x+ ξ] is contained in G. For any point of the segment [x, x+ ξ] integral representation (25) is valid with the same kernel. After some transformation we have |∆(ξ, G)D ν f | ≤ C1
X X
Y
−1−νj
Tj
e⊂Q i∈en \Q j∈Q\e 1 λ n
|ξ e |∨|ξ|e 0\Q
×
Z
0e∨i
× dte∨i
Q
1−λj lj +
tj
P
(λj +λj νj )
j=k∨j=i∈en \Q
×
j=k∈e∨j=i Z X X e∨i (ν) × Le,i ∆(ξ, G)D l f (x + y) dy + C2 Rn
Y
e⊂Q i∈en \Q j∈Q\e
−1−νj
Tj
× (31)
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A.M. Najafov / Central European Journal of Mathematics 3(3) 2005 496–507
×
Y
j∈Q
|ξj | |ξ|en \Q
T Ze∨i
dte∨i
(λj +λj νj )
j=k∨j=i∈en \Q
t λ |ξ e |∨|ξ|e 0\Q j=k∈e∨j=i j n 1
P
1−λj lj +λj +
Q
Z Z1 (ν+1) le∨i × Le,i D f (x + y + ξ1 ς1 + · · · + ξn ςn ) dςdy = Rn
= C1
X X0
Ae,i 1 (x, ξ) + C2
e⊂Q i∈en \Q
X X
Ae,i 2 (x, ξ),
×
(32)
e⊂Q i∈en \Q
Here |ξ e | = (|ξ1e | , |ξ2e | , . . . , |ξne |), ξje = |ξj | for j ∈ e; ξje = 0 for j ∈ Q\e and we can consider, that |ξj | < Tj for j ∈ Q; |ξ|en \Q < T, 0 < Tj ≤ min(1, tj,0) for j ∈ en , consequently |ξj | < min(σj , Tj ) for j ∈ Q; |ξ|en \Q < min(σ λ0 , T λ0 ). If x ∈ G\Gσ , then by definition ∆(ξ, G)D ν f = 0. Then k∆(ξ, G)D ν f kq,G = k∆(ξ, G)D ν f kq,Gσ X X e,i X X e,i
A (·, ξ)
A (·, ξ) ≤ C1 + C . (33) 2 1 2 q,Gσ q,Gσ e⊂Q i∈en \Q
e⊂Q i∈en \Q
With the help of inequality (11) for U = G, D l
e∨i
(ν)
f = φ, Me,i = Le,i , γ → ∞ we have
εQ,i
Y
e,i
le∨i λ0 εj
A1 (·, ξ) ≤ C3 |ξj | |ξ|en \Q D f q,Gσ
p,a,Γ,τ ;G
j∈e
and, with the help of inequality (12) for U = G, D l have
e∨i
,
(34) (ν)
f = φ, Me,i = Le,i , γ → ∞ we
εQ,i
Y Y
e,i
le∨i λ0 εj
A2 (·, ξ) ≤ C |ξ | |ξ | |ξ| D f
4 j j en \Q q,Gσ
j∈e
p,a,Γ,∞;G
j∈Q\e
Y
le∨i βj β ≤ C5 |ξj | |ξ|en \Q D f j∈Q
= C6
Y
j∈Q
p,a,Γ,∞;G
1 1
e∨i |ξj |βj |ξ|βen \Q D l f
p,a,Γ,τ ;G
,
(35)
From here with the help of inequality (32) we have (29), here βj1 > βj , j ∈ en . Assume now, that |ξj | ≥ min(σj , Tj ) for j ∈ Q; |ξ|en \Q ≥ min(σ λ0 , T λ0 ). Then k∆(ξ, G)D ν f kq,G ≤ 2 kD ν f kq,G ≤ C(σ, T ) kD ν f kq,G
Y
j∈Q
1
1
|ξj |βj |ξ|βen \Q .
(36)
We estimate kD ν f kq,G with the help of inequality (22) again we have inequality (29). Theorem 2 is proved.
A.M. Najafov / Central European Journal of Mathematics 3(3) 2005 496–507
507
References [1] O.V. Besov, V.P. Il’yin and S.M. Nikolskii: Integral representations of functions and imbedding theorems, Nauka, Moscow, 1996, p. 480 (in Russian). [2] A.M. Najafov: “Some families functional spaces and imbedding theorems”, Proceeing of IMM of NAS Azerbaijan, Vol. 16(24), (2002), pp. 114–121. [3] A.D. Djabrailov: “On ones integral representation of smooth functions and some families of function spaces”, DAN SSSR, Vol. 166(6), (1966), pp. 1280–1283 (in Russian). [4] S.M. Nikolskii: “The functions with dominating mixed derivative satisfying the multiple Holder condition”, Sib. Math. Zh., Vol. 4(6), (1963), pp. 1342–1364. [5] C.B. Morrey: Multiple integral problems in the calculus of variations and related topics, Univ. California Publ.1, 1943. [6] C.B. Morrey: Second order elliptic systems of differential equations, Ann. Math. Studies, Vol. 3, Princefon Umv. Press, 1954. [7] C.B. Morrey: “Second order elliptic equations in several variables and Holder continuity”, Math. Zelt., Vol. 72(2), (1959), pp. 146–164. [8] D. Greco: “Criteri di compatteza per insieme di funzioni in ”n” variabli indeependenti”, Ricerche di Mat. Napoli, Vol. 1, (1952), pp. 124–144. [9] L. Nirenbrg: “Estimates and existence of solutions of elliptic equations”, Comm. Pure Appl. Math., Vol. 9(3), (1956), pp. 509–530. [10] S. Campanato: “Caratterizzazione delle tracce di funzioni appatenenti ad una classe di Morrey iniseme con le loro derivate prime”, Ann. Scuola Norm. Sup. di Pisa, ser. III, Vol. 15, Fasc. III, (1961), pp. 263–281. [11] S. Campanato: “Proprieta di inclusione per spazi di Morrey”, Ricerche di Mat., Vol. 12(1), (1963), pp. 67–86. [12] G.C. Barozzi: “Su una generalizzazione degli spazi L(q,λ) di Morrey”, Ann. Scuola Norm. Sup. di Pisa, ser. III, Vol. 19, Fasc. IV, (1965), pp. 609–626. l [13] V.P. Il’yin: “On some properties of the functions of spaces Wp,a,Γ (G)”, Zap. Nauchn. sem. LOMI AN SSSR, Vol. 23, (1971), pp. 33–40 (in Russian).
[14] Yu.V. Netrusov: “On some imbedding theorems of Besov-Morrey type spaces”, Zap. Nauchn. sem. LOMI AN SSSR, Vol. 139, (1984), pp. 139–147 (in Russian). [15] A.M. Najafov: “The interpolation theorems on the Besov-Morrey and TriebelLizorkin-Morrey type spaces”, In: Mater. of scientific confer. ”The questions on functional analysis and and mathematical physics” dedicated to 80-year of Baku State University named after M.A.Rasulzadeh, 1999, pp. 363–366 (in Russian). [16] V.S. Guliev, A.M. Najafov: “The imbedding theorems on the Lizorkin-Triebel-Morrey type spaces”, In: Progress in Analysis. Proceedings of the 3rd International ISAAC Congress. Berlin, Vol. 1, 2001, pp. 23–30. [17] A.M. Najafov: “The imbedding theorems for the space of Besov-Morrey type with dominant mixed derivatives”, Proceedings of Institute of mathematics and mechanics, Vol. 12(20), (2000), pp. 97–104. [18] A.M. Najafov: The spaces with parameters of functions with dominant mixed derivatives, Thesis(PhD), Baku, 1996, (in Russian).
CEJM 3(3) 2005 508–515
Radial-type complete solutions for a class of partial differential equations 1∗ 2† ¨ Ay¸segu ¸ , Nuri Ozalp ¨l Cetinkaya 1
Department of Mathematics, Faculty of Arts and Sciences, Gazi University, 40100, Kirsehir, Turkey 2 Department of Mathematics, Faculty of Sciences, Ankara University, Be¸sevler, 06100 Ankara, Turkey
Received 17 November 2004; accepted 1 June 2005 Abstract: We give some fundamental solutions of a class of iterated elliptic equations including Laplace equation and its iterates. c Central European Science Journals. All rights reserved.
Keywords: Complete solutions, iterated equations, Almansi’s expansion, Kelvin principle MSC (2000): 35A08, 35C05, 35G99
1
Introduction
Much of the physical problems are solved in spherical or cylindrical domains. It means, most of the time, that the solutions are symmetric functions with respect to a point or with respect to an axis. That is why, when investigating the solutions, we see a frequent use of the type of solutions in terms of a variable r defining a distance to a point. For some of the research have been done for various type of problems, we refer to the references [1-9]. Here in this study, we apply the idea to a class of linear partial differential equations of second order and its iterates. Let us consider the class of equations p n 2 X 1 ∂u γu 2∂ u Lu = xi 2 + αi xi + p =0 xi ∂xi ∂xi r i=1 ∗ †
E-mail:
[email protected] E-mail:
[email protected]
(1)
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509
where γ, αi (i = 1, 2, ..., n) are real parameters, p (> 0) is a real constant and r is defined by r p = xp1 + xp2 + ... + xpn . (2) The domain of the operator L is the set of all real valued functions u(x) of the class C (D), where x = (x1 , x2 , ..., xn ) denotes points in Rn and D is a regularity domain of u in Rn . 2
Almansi gave an expansion formula for the solutions of Laplace equation [1]. Lyakhov and Ryzhkov, [5], obtained Almansi’s expansions for B- polyharmonic equation i.e. obtained the solutions of the equation ∆m B f = 0, where ∆B =
n X J=1
Bj +
N X ∂2 , 2 ∂x i i=n+1
Bj = ∂ 2 /∂x2j +
γj ∂ . xj ∂x
Altın generalized the idea to a class of singular partial differential equations and obtained a Lord Kelvin principle and radial type solutions for this class of equations [2,3,4,9]. In ¨ [7], Ozalp and Cetinkaya ¸ gave an expansion formula and Kelvin principle for the class of equations given by (1). More precisely, they proved that if ui (x), i = 1, ..., k − 1 are any k solutions of the equation (1), then the functions w=
k−1 X
r jpui (x)
i,j=0
and w=r
−φ
k−1 X i=0
r ip ui
x
1 , 2 r
x2 xn , ..., r2 r2
are solutions of the iterated equation Lk u = 0 . In [6], it is shown that, if u = f (r m ), f ∈ C 2 , then r p L(u) = ′′
m2 v 2 f (v) + m (m − p + n(p − 1) +
Pn
i=1
′
αi ) vf (v) + γf (v) = 0
with v = r m . Since, this is an Euler equation, and the solutions of Euler equation are in the form f (r m ) = r cm ,( where c is a root of the characteristic equation), we conclude that equation (1) have solutions depending on powers of r m . We call these type of solutions as r m − type solutions. Here, in this study we first give r m − type solutions for the iterates of the equation (1) (i.e. for the equation Lk u = 0 ) and we generalize the idea to the equations of the form Lkq q ...Lk22 Lk11 u = 0
(3)
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510
where q, k1 , ..., kq are positive integers, p n 2 X ∂ γv 1 (v) 2 ∂ Lv = xi 2 + αi xi + p. xi ∂xi ∂xi r i=1
(4)
(v)
Here, γv , αi (v = 1, 2, .., q; i = 1, 2, ...n) are real constants, and the operator Lkvv denotes, as usual, the successive applications of the operator Lv onto itself, that is Lkvv u = Lv (Lkvv −1 u).
2
rm type solutions for Lk u = 0
We borrow the following result from [7]. Lemma 2.1. For any real or complex parameter m, Lk (r m ) =
k−1 Y j=0
[(m − pj)(m − pj + 2ψ) + γ] r m−pk
(5)
where the integer k is the iteration number and 2ψ = −p + n(p − 1) +
n X
αi
i=1
Now, we are ready to generate the r m type solutions of the iterated equation Lk u = 0. Let Φ(m) = m(m + 2ψ) + γ. Then, equation (5) becomes Lk (r m ) =
k−1 Y j=0
Φ(m − pj)r m−pk .
On the other hand, for each j(= 0, 1, ...k − 1), the roots of the quadratic equations Φ(m − pj) = 0 are p (1) m = pj − ψ + ψ2 − γ j (6) p m(2) = pj − ψ − ψ 2 − γ. j
Thus, we can rewrite (5) as k
m
L (r ) =
k−1 Yh j=0
i (1) (2) (m − mj )(m − mj ) r m−pk .
(7)
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511
From (7), we conclude that (1)
u = r mj ,
(2)
u = r mj
(j = 1, 2, ..., k − 1),
are solutions of Lk u = 0. Since the operator L is linear, by the superposition principle, the function k−1 X (1) (2) (1) (2) u= Aj r mj + Aj r mj j=0
(1)
(2)
is a complete solution of the iterated equation Lk u = 0. Here Aj and Aj are arbitrary constants. To generate the real valued solutions, we have three cases for the roots of the quadratic equations Φ(m − pj) = 0 for each fixed j. (1)
(i) (2)
ψ 2 − γ > 0, i.e. mj
(2)
and mj
(1)
are two different real roots. In this case, r mj and
r mj are real valued linearly independent solutions of Lk u = 0. (1)
(2)
(ii) ψ 2 − γ = 0, i.e. mj = mj = pj − ψ is a multiple real root. In this case, from the theory of elementary differential equations we know that r pj−ψ and r pj−ψ ln r are real valued linearly independent solutions of Lk u = 0. (iii) ψ 2 − γ < 0, i.e. the roots are complex conjugates. Once p again from the pj−ψ theory of p elementary differential equations we know that r cos( γ − ψ 2 ln r) and r pj−ψ cos( γ − ψ 2 ln r) are real valued linearly independent solutions of Lk u = 0. Now let us define three auxiliary functions as 1, if ψ 2 − γ > 0 Λ1 = , 0, else 1, if ψ 2 − γ = 0 Λ2 = , 0, else 1, if ψ 2 − γ < 0 Λ3 = . 0, else Hence, we can give the following result:
Theorem 2.2. Real valued r m type solutions of the iterated equation Lk u = 0 is given
¨ A. Cetinkaya, ¸ N. Ozalp / Central European Journal of Mathematics 3(3) 2005 508–515
512
by the formula u=
k−1 X j=0
(1) (2) (1) (2) Λ1 Aj r mj + Aj r mj
+
k−1 X
+
k−1 X
j=0
(1) (2) Λ2 r pj−ψ Bj + Bj ln r Λ3 r
pj−ψ
j=0
(i)
(i)
(i)
where Aj , Bj , Cj
3
r
m
(8)
hp i hp i (1) (2) 2 2 Cj cos γ − ψ ln r + Cj sin γ − ψ ln r
(i = 1, 2) are arbitrary constants.
kq k2 k1 type solutions for Lq ...L2 L1 u = 0
Now let v and q be positive integers, 1 ≤ v ≤ q, 2ψv = −p + n(p − 1) +
n X
(v)
αi
(9)
Φv (m) = m(m + 2ψv ) + γv
(10)
i=1
and Then, for any positive integer kv , from (5) we have Lkvv (r m )
=
kY v −1
Φv (m − pj)r m−pkv
=
kY v −1
[(m − pj)(m − pj + 2ψv ) + γv ] r m−pkv
j=0
j=0
Lemma 3.1. For any positive integer q, the following equality holds. " q k −1 !# v−1 v YY X Pq m k2 k1 kq Φv m − p(j + kl ) r m−p( l=1 kl ) . Lq ...L2 L1 (r ) = v=1 j=0
Here, we assume that
P0
l=1
l=1
kl = 0.
Proof. We use induction argument on q. For q = 1, (12) is reduced to Lk11 (r m )
=
kY 1 −1 j=0
Φ1 (m − pj)r m−pk1
(11)
(12)
¨ A. Cetinkaya, ¸ N. Ozalp / Central European Journal of Mathematics 3(3) 2005 508–515
513
which gives (11) for v = 1. Now assume that (12) holds for q − 1. Now, for q, we get kq−1 Lkq q ...Lk22 Lk11 (r m ) = Lkq q Lq−1 ...Lk22 Lk11 (r m ) = Lkq q
"q−1 k −1 v YY v=1 j=0
=
"q−1 k −1 v YY
Φv
v=1 j=0
Φv
m − p(j +
m − p(j +
v−1 X
v−1 X
!
kl ) r
P m−p( q−1 l=1 kl )
#
l=1
kl )
!#
P q−1
Lkq q (r m−p(
l=1
kl )
)
kl )
)
l=1
On the other hand, from (11) we have P q−1 Lkq q (r m−p( l=1 kl ) )
=
kq −1
Y j=0
Φq (m − p(
q−1 X l=1
P q−1
kl ) − pj)r m−p(
kl )−pkq
l=1
.
Thus, "q−1 k −1 v YY Φv Lkq q ...Lk22 Lk11 (r m ) = v=1 j=0
=
"q−1 k −1 v YY
Φv
v=1 j=0
× =
kq −1
Y
Φq
j=0
" q k −1 v YY
m − p(j +
v−1 X
kl )
!#
m − p(j +
v−1 X
kl )
!#
m − p( Φv
v=1 j=0
q−1 X l=1
l=1
l=1
kl ) − pj
m − p(j +
v−1 X
!
kl )
P q−1
Lkq q (r m−p(
P q−1
r m−p(
!#
l=1
kl )−pkq
Pq
r m−p(
l=1
l=1
kl )
.
l=1
Hence the proof is completed.
Since for any fixed v, the roots of the quadratic equation ! v−1 X Φv m − p(j + kl ) = 0 l=1
are
p Pv−1 (1) m = p(j + k ) − ψ + ψv 2 − γv l v j,v l=1
we can rewrite (12) as
p m(2) = p(j + Pv−1 kl ) − ψv − ψv 2 − γv j,v l=1
" q k −1 # v YY Pq (1) (2) Lkq q ...Lk22 Lk11 (r m ) = (m − mj,v )((m − mj,v ) r m−p( l=1 kl ) . v=1 j=0
(13)
(14)
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514
Hence, we see that for each integer v, the functions (1)
(2)
u = r mj,v , u = r mj,v (j = 1, 2, ..., kv − 1), kq k2 k1 are solutions of the iterated equations Lq ...L2 L1 u = 0. Since the operator Lv is linear for each v, by the superposition principle, the function u=
kX v −1
(1)
(1)
(2)
(2)
Aj,v r mj,v + Aj,v r mj,v
j=0
k (1) (2) is a complete solution of the iterated equation Lq q ...Lk22 Lk11 u = 0. Here Aj,v and Aj,v are arbitrary constants. Now let us define three auxiliary functions as 1, if ψ 2 − γv > 0 v Λ1,v = , 0, else 1, if ψ 2 − γv = 0 v Λ2,v = , 0, else 1, if ψ 2 − γv < 0 v Λ3,v = . 0, else
Hence, by generating the real valued solutions with similar arguments of the case Lk u = 0, we can give the following result: k Theorem 3.2. Real valued r m type solutions of the iterated equation Lq q ...Lk22 Lk11 u = 0 is given by the formula u=
q k−1 X X
Λ1,v
v=1 j=0
+
(2) (1) (2) (1) Aj r mj,v + Aj r mj,v
q k−1 X X
Λ2,v r p(j+
P v−1 l=1
kl )−ψv
Bj + Bj ln r
q k−1 X X
Λ3,v r p(j+
P v−1
kl )−ψv
Cj cos
v=1 j=0
+
v=1 j=0 (i)
(i)
(i)
where Aj , Bj , Cj
4
l=1
(1)
(1)
(2)
(15)
hp i hp i (2) γv − ψv2 ln r + Cj sin γv − ψv2 ln r
(i = 1, 2) are arbitrary constants.
Conclusion
Theorem 3.2 gives the complete solutions of the iterated equations (3) which includes a wide range of elliptic type equations with equidimensional equations. Thus, we can compute the analytical (symbolic) solutions of (3) by using (15) in a suitable algorithm.
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References [1] E. Almansi: “Sulle integrazione dell’equazione differenziale ∆2n = 0”, Ann. Math. Pura Appl., Vol. 2, (1898), pp. 1–51. [2] A. Altin: “Some expansion formulas for a class of singular partial differential equations”, Proc. Am. Mat. Soc., Vol. 85(1), (1982), pp. 42–46. [3] A. Altin: “Solutions of type r m for a class of singular equations”, Inter. J. Math. and Math. Sci., Vol. 5, (1982), pp. 613–619. [4] A. Altin: “Radial type solutions for a class of third order equations and their iterates”, Math. Slovaca, Vol. 49(2), (1999), pp. 183–187. [5] L.N. Lyakhov and A.V. Ryzhkov: “Solutions of the B - polyharmonic equation”, Differential Equations, Vol. 36(10), (2000), pp. 1507–1511; Translated from: Differetsial’nye Uravneniya, Vol. 36(10), (2000), pp. 1365–1368. ¨ [6] N. Ozalp: “r m − type solutions for a class of partial differential equations”, Commun. Fac. Sci. Univ. Ank. Series A1, Vol. 49, (2000), pp. 95–100. ¨ [7] N. Ozalp and A. Cetinkaya: ¸ “Expansion formulas and Kelvin principle for a class of partial differential equations”, Mathematica Balkanica, New Series, Vol. 15, (2001), pp. 220–226. ¨ [8] N Ozalp and A. Cetinkaya: ¸ “Radial solutions of a class of iterated partial differential equations”, Czechoslovak Mathematical Journal, Vol. 55(2), (2005), pp. 531–541. [9] A. Altin - A. Eren¸cin: “Some solutions for a class of singular equations”, Czechoslovak Mathematical Journal, Vol. 54(4), (2004), pp. 969–979.
CEJM 3(3) 2005 516–528
Survey article
On the first homology of automorphism groups of manifolds with geometric structures K¯ojun Abe1∗† , Kazuhiko Fukui2‡§ 1
Department of Mathematical Sciences, Shinshu University, Matsumoto 390-8621, Japan 2 Department of Mathematics, Kyoto Sangyo University, Kyoto 603-8555, Japan
Received 30 October 2004; accepted 1 May 2005 Abstract: Hermann and Thurston proved that the group of diffeomorphisms with compact support of a smooth manifold M which are isotopic to the identity is a perfect group. We consider the case where M has a geometric structure. In this paper we shall survey on the recent results of the first homology of the diffeomorphism groups which preserve a smooth Gaction or a foliated structure on M . We also work in Lipschitz category. c Central European Science Journals. All rights reserved.
Keywords: Automorphism group, first homology group; G-manifold, Foliated manifold MSC (2000): 57S05, 58D05, 58H10
Introduction In this paper we shall survey on the first homology of the group of automorphisms of a manifold with geometric structure, especially with respect to a manifold with G-action and foliated structure. Here the first homology H1 (K) of a group K is the quotient group of K by the commutator subgroup of K. ∗
E-mail:
[email protected] This research was partially supported by Grant-in-Aid for Scientific Research (No. 16540058), Japan Society for the Promotion of Science. ‡ E-mail:
[email protected] § This research was partially supported by Grant-in-Aid for Scientific Research (No. 14540093), Japan Society for the Promotion of Science. †
K. Abe, K. Fukui / Central European Journal of Mathematics 3(3) 2005 516–528
517
Let Dif f ∞ (M) denote the group of C ∞ -diffeomorphisms of a smooth connected manifold M with compact support which are isotopic to the identity. Because Dif f ∞ (M) is a huge group even from the group structure point of view, we see that the diffeomorphism group gives fruitful results on the structure of the manifold. Hermann [22] and Thurston [34] proved that Dif f ∞ (M) is a perfect group, which means Dif f ∞ (M) coincides with the commutator subgroup. Epstein [12] proved that the commutator subgroup of Dif f ∞ (M) is simple. Then Dif f ∞ (M) is a simple group. It is known that the result is relevant to foliation theory. We consider the case where M has a foliated structure or a smooth G-action with G a compact Lie group and treat the group of diffeomorphisms, Lipschitz homeomorphisms or homeomorphisms preserving the geometric structure. In those cases the corresponding automorphism groups are not necessarily perfect, and we calculate the first homologies of those groups. The results reflect the geometric structure on M and the difference among those categories. In §1 we give a review of the structure of the commutator subgroup of diffeomorphisms of a smooth manifold and Lipschitz homeomorphisms of a manifold. Those results are fundamental to the investigation of this paper. In §2 we treat the case of the equivariant diffeomorphisms of smooth G-manifolds. The first homology of the equivariant diffeomorphism group Dif fG∞ (M) of M depends on the orbit types of a smooth G-manifold M. When M has one orbit type, then the group Dif fG∞ (M) is perfect. If M has two orbit types, Dif fG∞ (M) is not always perfect. We can calculate H1 (Dif fG∞ (M)) when M has codimension one orbit and codimension two orbit with the orbit space 2-disk. Those results are based on the calculation of H1 (Dif f ∞ (M)) when M is a smooth manifold with boundary. For a representation space V of a finite group G, we can completely determine the structure of H1 (Dif fG∞ (V )). In §3 we consider the groups of equivariant Lipschitz homeomorphisms LG (M) of smooth G-manifolds with compact support. Let LG (M) and HLIP,G(M) denote the identity component of LG (M) when we give the compact open topology and the compact open Lipschitz topology on LG (M), respectively. If M has one orbit type, then the groups LG (M) and HLIP,G(M) are perfect. Generally those two groups have quite different properties. For a finite group G, HLIP,G (M) is a perfect group. But it is not easy to calculate H1 (LG (M)). We calculate H1 (LU (n) (Cn )) when Cn has the canonical U(n)action, and we see that H1 (LU (n) (Cn )) admits continuous moduli. From this result we can prove that H1 (L(C, 0)) also admits continuous moduli, where L(C, 0) is the subgroup of L(C) whose elements are isotopic to the identity through Lipschitz isotopies fixing the origin with compact support. In §4 we treat the case of the group of foliation preserving CAT-homeomor-phisms of a codimension q CAT-foliation (M, F ). Here CAT denotes one of the categories T OP, LIP, C ∞ and CAT-homeomorphism means homeomorphism, Lipschitz homeomorphism or C ∞ -diffeomorphism of the corresponding CAT-manifold. Let GCAT (M, F ) (resp. GCAT (M, F )) denote the group of all foliation preserving (resp. leaf preserving ) L CAT-homeomorphisms of M which are isotopic to the identity by a foliation preserving
518
K. Abe, K. Fukui / Central European Journal of Mathematics 3(3) 2005 516–528
(resp. leaf preserving ) CAT-isotopy. We can prove that the group GCAT (M, F ) is always L perfect. In 4.2 we consider the case of codimension one foliations. In this case we can decompose M into components of the following three types : Type P, Type R and Type D (see Theorem 4.2.3). If F has no components of type D and has only a finite number of components of type R, GCAT (M, F ) is perfect for CAT=TOP, LIP. We have a different phenomenon for CAT= C ∞ . If F has a type D component, GCAT (M, F ) is not perfect in general. But if F satisfies a certain condition, GCAT (M, F ) becomes a perfect group (see Remark 4.2.9). In 4.3 we consider the case of foliations of codimension ≥ 2. In this case it is difficult to determine H1 (GCAT (M, F )) in general. But we can compute it for compact Hausdorff foliations and CAT=LIP, C ∞ .
1
The first homology of the diffeomorphism groups and the Lipschitz homeomorphism groups of manifolds
In this section we give a review of the structure of the commutator subgroup of diffeomorphisms of a smooth manifold and Lipschitz homeomorphisms of a manifold.
1.1 The diffeomorphism groups of manifolds without boundary Let M be a connected smooth manifold without boundary. The following well known result is important and basic to the results in this paper. Theorem 1.1. (Hermann [22], Thurston [34], Epstein [14] ) If M is a connected smooth manifold without boundary, then the group Dif f ∞ (M) is perfect.
1.2 The diffeomorphism groups of manifolds with boundary In order to investigate our problems, it is required calculating the first homology of Dif f ∞ (M) when M is a smooth manifold with boundary. Theorem 1.2 (Fukui [15]). H1 (Dif f ∞ ([0, 1])) ∼ = R ⊕ R. Theorem 1.3. If M is a smooth manifold and dim M ≥ 2, then Dif f ∞ (M) is perfect. We proved Theorem 1.3 combining the result of Tsuboi [35], Rybicki [27] for the group of leaf preserving diffeomorphisms of a foliated manifold and the method given by Sternberg [32, 33]. Rybicki [28] also showed the result. By the parallel method to Theorem 1.3, we have the relative version. Let M be a smooth m-manifold without boundary and N be a proper n-dimensional submanifold. Let Dif f ∞ (M, N) denote the group of diffeomorphisms of M preserving N which are isotopic to the identity through an isotopy preserving N with compact support.
K. Abe, K. Fukui / Central European Journal of Mathematics 3(3) 2005 516–528
519
Theorem 1.4. (1) If N = {p1 , · · · , pk }, then H1 (Dif f ∞ (M, N)) ∼ = R × · · · × R (ktimes). (2) If n ≥ 1, then Dif f ∞ (M, N) is perfect.
1.3 The Lipschitz homeomorphism groups of manifolds Let L(M) denote the group of Lipschitz homeomorphisms of a manifold M with compact support. Let L(M) denote the identity component of L(M) when we give L(M) the compact-open topology. Theorem 1.5 ([3]). If M is a PL-manifold, then L(M) is perfect. Let HLIP (M) denote the identity component of L(M) when we give L(M) the compactopen Lipschitz topology (see Appendix). Theorem 1.6 ([3]).
2
If M is a Lipschitz manifold, then HLIP (M) is perfect.
The first homology of the equivariant diffeomorphism groups of smooth G-manifolds
In this section we discuss the structure of equivariant diffeomorphism groups of smooth G-manifolds. Let G be a compact Lie group and M be a smooth connected G-manifold. Let Dif fG∞ (M) denote the equivariant diffeomorphism group of a G-manifold M whose elements are G-isotopic to the identity through an equivariant isotopy with compact support.
2.1 The equivariant diffeomorphism groups of G-manifolds with one orbit type The following result was first proved by Banyaga [8] for the case G a torus group and we generalized it when G is any compact Lie group ([1]). Theorem 2.1. Let G be a compact Lie group. If G acts freely on M with dim M/G > 0, then Dif fG∞ (M) is a perfect group. In order to prove Theorem 2.1, Theorem 1.1 is basic and the key point is finding some functions describing each orbit preserving equivariant diffeomorphism of M by using the fragmentation lemma and the canonical coordinate of G of the second kind. From Theorem 2.1 we have Corollary 2.2. Let G be a compact Lie group. If M is a G-manifold with one orbit type and dim M/G > 0, then Dif fG∞ (M) is a perfect group.
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2.2 The equivariant diffeomorphism groups of G-manifolds with codimension one orbit If M has more than one orbit type, then the group Dif fG∞ (M) is not perfect in general. Let M be a smooth connected closed G-manifold with codimension one orbit. Then the orbit space M/G of M is homeomorphic to the circle S 1 or the unit interval [0, 1]. If M/G is homeomorphic to S 1 , then M has one orbit type and it follows from Corollary 2.2 that Dif fG∞ (M) is perfect. If M/G is homeomorphic to [0, 1], then M has two or three orbit types. Let (H) be the principal orbit type of M and (K0 ), (K1 ) be the singular orbit types of M. Let N(H) be the normalizer of the group H in G. Set W (M) =
N(H) ∩ N(K0 ) N(H) ∩ N(K1 ) × H H
0
Theorem 2.3 ([2]). H1 (Dif fG∞ (M)) ∼ = R2 × H1 (W (M)). The proof of Theorem 2.3 is based on so called smooth structure of the orbit space M/G of the G-manifold M which was investigated by Bredon [11] Bierstone [10] and Schwarz [29]. With the smooth structure of M/G and Baker-Campbell-Hausdorff formula, we can concretely describe the behavior of equivariant diffeomorphisms around the singular orbits. Then we can prove Theorem 2.3 from Theorem 1.2.
2.3 The equivariant diffeomorphism groups of G-manifolds with codimension two orbit Now consider the case of smooth connected G-manifolds with codimension two orbits. Let W 2n−1 (d) (n ≥ 2, d ≥ 0) be the Brieskorn manifold given by z0d + z12 + · · · + zn2 = 0
|z0 |2 + |z1 |2 · · · + |zn |2 = 2 for (z0 , z1 , · · · , zn ) ∈ Cn+1 . W 2n−1 (d) has the canonical O(n) action given by A · (z0 , z1 , · · · , zn ) = (z0 , A · (z1 , · · · , zn )) for A ∈ O(n), (z0 , z1 , · · · , zn ) ∈ Cn+1 . The O(n)-manifold W 2n−1 (d) was investigated by Hirzebruch and Mayer [23]. Then the orbit space W 2n−1 (d)/O(n) can be identified with D2. ∞ Theorem 2.4 ([6]). The group Dif fO(n) (W 2n−1 (d)) is perfect.
We can prove Theorem 2.4 by analyzing the structure around the singular orbits and applying Theorem 1.3.
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2.4 The equivariant diffeomorphism groups of the representation spaces of a finite group ˜ n of a finite group of In the sequel we consider the n-dimensional representation space R O(n). Set Z(G) = {A ∈ GL+ (n, R) | g · A = A · g (∀g ∈ G)}, which is the centralizer of ˜ n )G denote the fixed point set of R ˜ n. G in GL+ (n, R). Let (R ˜ n )G = 0, then Theorem 2.5 ([5]). (1) If dim(R ˜ n )) ∼ H1 (Dif fG∞ (R = H1 (Z(G)0 ). ˜ n )G > 0, then the group Dif f ∞ (R ˜ n ) is perfect. (2) If dim(R G Let M be an n-dimensional smooth orbifold. For each point p of M, there exists an open neighborhood U of p which can be identified with the orbit space of an open set U˜ in a representation space of a finite subgroup Gp in O(n). By Strub [31], the group Gp is unique up to conjugate in O(n). Applying Theorem 2.5 we have Theorem 2.6 ([5]). Let M be a smooth orbifold which has only isolated singular points p1 , ..., pk . Then H1 (Dif f ∞ (M)) ∼ = H1 (Z(Gp1 )0 ) ⊕ · · · H1 (Z(Gpk )0 ). Combining Theorem 2.5 and Theorem 2.6 we have ˜ be the non-trivial one dimensional representation space of Corollary 2.7 ([5]). Let R ˜ n /Z2 )) ∼ ˜ n )) ∼ Z2 . Then H1 (Dif fZ∞2 (R = R. = H1 (Dif f ∞ (R Let S 3 = {(w1 , w2 ) ∈ C2 | |w1 |2 + |w2 |2 = 1} with U(1)-action given by z · (w1 , w2 ) = (zw1 , z 2 w2 ),
z ∈ U(1).
Then it has two orbit types {(1), Z2 } and the orbit space S 3 /U(1) is homeomorphic to the space known as the tear drop which is the two dimensional sphere with one isolated singular point. Applying Corollary 2.7 we have Theorem 2.8.
3
H1 (Dif fU∞(1) (S 3 )) ∼ = R × U(1).
The first homology of the equivariant Lipschitz homeomorphism group of G-manifolds
In this section we shall study the structure of the group of equivariant Lipschitz homeomorphisms of a smooth G-manifold M.
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3.1 The equivariant Lipschitz homeomorphisms of G-manifolds Let G be a compact Lie group and M be a G-manifold. Let LG (M) denote the topological subgroup of L(M) consisting of equivariant Lipschitz homeomorphisms of M which are isotopic to the identity through equivariant Lipschitz homeomorphisms. Let HLIP,G(M) denote the subgroup of LG (M) whose elements are isotopic to the identity with respect to the compact open Lipschitz topology. Theorem 3.1 ([3, 4]). (1) If M is a principal smooth G-manifold with dim M/G > 0, then HLIP,G(M) is perfect. (2) Let G be a finite group. If M is a smooth G-manifold, then HLIP,G(M) is perfect. (3) If M is a smooth orbifold, then HLIP,G (M) is perfect. If h : M → M is a Lipschitz map sufficiently close to the identity in compact open Lipschitz topology, then h ∈ HLIP,G(M)(cf. [19]). This property is one of the points of the proof in Theorem 3.1. But it is not valid in the compact-open topology. Thus we have to treat the group LG (M) in a completely different way. Theorem 3.2 ([7]). If M is a principal G-manifold with dim M/G > 0, then LG (M) is perfect. More generally if M is a smooth G-manifold with one orbit type and dim M/G > 0, then LG (M) is perfect.
3.2 The equivariant Lipschitz homeomorphisms of the canonical U (n)action on Cn In the next we consider the case of Cn with the canonical U(n)-action. Let C((0, 1]) be the set of real valued functions f on (0, 1] such that there exists a positive number K satisfying K |f (x) − f (y)| ≤ (y − x) for 0 < x ≤ y ≤ 1. x Then C((0, 1]) is a vector space over R. Let C0 ((0, 1]) denote the subspace of those f ∈ C((0, 1]) with f bounded on (0, 1]. Then we have Theorem 3.3 ([7]). H1 (LU (n) (Cn )) is isomorphic to C(R)/C0 (R). Remark 3.4. Let vc (0 < c ≤ 1) be real valued smooth functions on (0, 1] such that (− log x)c (0 < x ≤ 1/2), vc (x) = 0 (3/4 ≤ x ≤ 1).
Then vc ∈ C((0, 1]). Thus the group C((0, 1])/C0 ((0, 1]) contains a linearly independent family {vc mod C0 ((0, 1]) ; 0 < c ≤ 1}.
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Let S(Cn ⊕ R) be the unit sphere in Cn ⊕ R with the canonical U(n)-action. Combining Theorem 3.3 with Theorem 3.2 we have Corollary 3.5 ([7]). H1 (LU (n) (S(Cn ⊕ R))) ∼ = C((0, 1])/C0 ((0, 1]) × C((0, 1])/C0 ((0, 1]). Here we focus on the case n = 1. Set L(C, 0) = {h ∈ L(C); h(0) = 0}. Let L(C, 0) denote the identity component of L(C, 0). >From Theorem 3.3 we have Theorem 3.6 ([7]). H1 (L(C, 0)) admits continuous moduli.
4
The first homology of the group of foliation preserving automorphisms
In this section, we survey on the first homology of the group of foliation preserving automorphisms. Let M be a compact m-dimensional CAT-manifold (CAT ≡ T OP, LIP C ∞ ). Let GCAT (M) denote the identity component of the space of all CAT-homeomorphisms of M with the suitable topology, that is, the topology is the compact open topology, the compact open Lipschitz topology, and the compact open C ∞ topology respectively in the case of CAT≡ T OP, LIP, C ∞ .
4.1 Leaf preserving CAT-homeomorphisms Let M be an m-dimensional compact connected CAT-manifold without boundary and F a codimension q CAT-foliation of M. A CAT-homeomorphism f : M → M is called a foliation preserving CAT-homeomorphism (resp. a leaf preserving CAT-homeomorphism) if for each point x of M, the leaf through x is mapped into the leaf through f (x) (resp. x), that is, f (Lx ) = Lf (x) (resp. f (Lx ) = Lx ), where Lx is the leaf of F which contains x. By GCAT (M, F )(resp. GCAT (M, F )) we denote the group of all foliation preserving L (resp. leaf preserving ) CAT-homeomorphisms of M which are isotopic to the identity by a foliation preserving (resp. leaf preserving ) CAT-isotopy. Fukui-Imanishi [17, 18] ,Rybicki [27] and Tsuboi [35] proved the following. Theorem 4.1. GCAT (M, F ) is perfect, that is, H1 (GCAT (M, F )) = 0. L L
4.2 The group of foliation preserving CAT-homeomorphisms: case of codimension one In this part we discuss the case of the group of foliation preserving CAT-homeomorphisms for codimension one foliations. Let M be a compact CAT-manifold without boundary and
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F a codimension one CAT-foliation of M. Hereafter we simply write GCAT (F ), GCAT (F ) L CAT CAT instead of G (M, F ), GL (M, F ) respectively. There exists a one dimensional CAT-foliation T of M transverse to F . Then we have the following : Let f be an element of GCAT (F ) sufficiently close to the identity. Then f is uniquely decomposed as f = g ◦ h, where h (resp. g) is an element of GCAT (F ) ∩ GCAT (T ) ( resp. L CAT GL (F )) and h and g are also close to the identity. Lemma 4.2 (Lemma 4.2 of [17]). Let f be an element of GCAT (F ) and L a leaf of F . L If f (L) 6= L, then the holonomy group of L is trivial. We define the subset S0 of M by S0 = {x ∈ M | there exists an element f of GCAT (F ) such that f (Lx ) 6= Lx }. By definition, S0 is an open F -saturated set and by Lemma 4.2, all leaves in S0 have trivial holonomy. Theorem 4.3 (see Theorem 4.3 of [17]). Let S be a connected component of S0 . Then clearly S is invariant under the action of GCAT (F ) and S is one of the following three types: Type P : S is homeomorphic to L × (0, 1) and the foliations F |S and T |S correspond to the product structure of L × (0, 1). Type R : There exists a closed transverse curve C in S such that (1) C meets each leaf of F |S at exactly one point, (2) the natural map p : S → C, p(x) = Lx ∩ C is a fibration, (3) T |S is a connection of the fibration p. Type D : All leaves of F in S are dense in S and there exists a topological flow {ϕt } on S which preserves F |S and whose orbits are leaves of T |S . Then we have the following ([17] and [18]). Theorem 4.4. Let F be a codimension one TOP-(resp. C 2 -)foliation of a compact TOP(resp. C 2 -)manifold M. Suppose that F has no components of type D and has only a finite number of components of type R. Then GTOP (F ) (resp.GLIP(F )) is perfect. Remark 4.5. From Theorem 4.4, we see that GCAT (S 3 , FR ) is perfect for CAT = TOP, LIP and the Reeb foliation FR of S 3 . We have a different phenomenon for CAT ≡ C ∞ . Indeed we have H1 (GCAT (S 3 , FR )) ∼ = SO(2) × SO(2) ([20] and Theorem 4.1). For a type D component S, the flow {ϕt } is defined as follows (see [24]). Let C be a closed transversal curve of F |S and we suppose that C is a T -orbit. Then, for a leaf L
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of F |S , G = C ∩ L has a structure of abelian group and G acts on C as the holonomy transformation group. Since all G-orbits are dense, there exists a homeomorphism h of C such that G is included in h−1 ◦ SO(2) ◦ h and is unique up to rotations of C. We call h the linearization map of the holonomy transformations. Let {ϕt } be a flow on C defined by {h−1 ◦ Rt ◦ h}, where Rt is the rotation of C of angle 2πt, and we extend {ϕt } to a flow on S by using holonomy maps. We define a submodule P er(S) of R by P er(S) = {t ∈ R | ϕt (L) = L for one and all leaves L in S}. Then we have the following([18]). Theorem 4.6. Let S be a type D component and suppose the linearization map h is not absolutely continuous, then GLIP (F |S ) coincides with GLIP L (F |S ). For CAT ≡ TOP, LIP, we have the following([17], [18]). Theorem 4.7. Let S be a type D component and suppose the linearization map h is a TOP-(resp.C 1 -)homeomorphism, then there exists a surjection π of H1 (GTOP (F ))(resp. H1 (GLIP (F ))) to R/P er(S). P Theorem 4.8. Let F be a foliation of a torus T m defined by a 1-form ω = ai dxi . If CAT one of ai /aj is irrational, then H1 (G (F )) is isomorphic to R/a1 Z + · · · + am Z. Remark 4.9. By Theorem (3.6) of [22] (CHAP. XII), there exists a C ∞ -foliation F ′ which is topologically equivalent to F of Theorem 4.8 for a suitable {ai } with non absolutely continuous linearization map, therefore by Theorem 4.1 and 4.6, we have H1 (GLIP (F ′ )) = 0.
4.3 he group of foliation preserving CAT-homeomorphisms: case of codimension greater than one In this part we discuss the case of the group of foliation preserving CAT-homeomorphisms for foliations of codimension ≥ 2. Let M be a compact CAT-manifold without boundary and F a codimension q CAT-foliation of M. General problem .
Calculate H1 (GCAT (F )).
It is difficult to determine H1 (GCAT (F )) in general. We discuss here the case where F is a compact Hausdorff foliation, that is, all leaves of F are compact and the leaf space is Hausdorff. Then we have the following([4]). Theorem 4.10. Let M be a compact C 1 -manifold without boundary and F a compact Hausdorff codimension q C 1 -foliation of M . Then GLIP (F ) is perfect.
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Remark 4.11. For CAT ≡ TOP, we have a certain result for a special case ([16]). For CAT ≡ C ∞ , we have the following for a codimension one compact C ∞ -foliation. Theorem 4.12 ([5]). Let F be a codimension one compact C ∞ -foliation of M and CAT ≡ C ∞. (i) If F is transversely orientable, then GCAT (F ) is perfect. (ii) If F is not transversely orientable, then H1 (GCAT (F )) is isomorphic to R × R. Remark 4.13. We can compute the first homology of the group of foliation preserving C ∞ -homeomorphisms for compact Hausdorff C ∞ -foliations of higher codimension ([5]).
References [1] K. Abe and K. Fukui: “On commutators of equivariant diffeomorphisms”, Proc. Japan Acad., Vol. 54, (1978), pp. 52–54. [2] K. Abe and K. Fukui: “On the structure of the group of equivariant diffeomorphisms of G-manifolds with codimension one orbit”, Topology, Vol. 40, (2001), pp. 1325–1337. [3] K. Abe and K. Fukui: “On the structure of the group of Lipschitz homeomorphisms and its subgroups”, J. Math. Soc. Japan, Vol. 53(3), (2001), pp. 501–511. [4] K. Abe and K. Fukui: “On the structure of the group of Lipschitz homeomorphisms and its subgroups II”, J. Math. Soc. Japan, Vol. 55(4), (2003), pp. 947–956. [5] K. Abe and K. Fukui: “On the first homology of the group of equivariant diffeomorphisms and its applications”, preprint. [6] K. Abe and K. Fukui: “On the structure of the group of diffeomorphisms of manifolds with boundary and its applications”, preprint. [7] K. Abe, K. Fukui and T. Miura: “ On the first homology of the group of equivariant Lipschitz homeomorphisms”, preprint. [8] A. Banyaga: “On the structure of the group of equivariant diffeomorphisms”, Topology, Vol. 16, (1977), pp. 279–283. [9] A. Banyaga: The structure of classical diffeomorphism groups, Mathematics and its Applications, Vol. 400, Kluwer Academic Publishers, Dordrecht, 1997. [10] E. Bierstone: “The Structure of Orbit Spaces and the Singularities of Equivariant Mappings”, Instituo de Mathematica Pura e Aplicada, Rio de Janeiro, 1980. [11] B. Bredon: Introduction to Compact Transformation Groups, Academic Presss, New York-London, 1972. [12] D.B.A. Epstein: “The simplicity of certain groups of homeomorphisms”, Compocio Math., Vol. 22, (1970), pp. 165–173. [13] D.B.A. Epstein: “Foliations with all leaves compact”, Ann. Inst. Fourier, Grenoble, Vol. 26, (1976), pp. 265–282. [14] D.B.A. Epstein: “Commutators of C ∞ -diffeomorphisms”, Appendix to: John N. Mather: “gA curious remark concerning the geometric transfer map”, Comment. Math. Helv., Vol. 59, (1984), pp. 111–122.
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[15] K. Fukui: “Homologies of Dif f ∞ (Rn , 0) ant its subgroups”, J. Math. Kyoto Univ., Vol. 20, (1980), pp. 457–487. [16] K. Fukui: “Commutators of foliation preserving homeomorphisms for certain compact foliations”, Publ. RIMS Kyoto Univ., Vol. 34(1), (1998), pp. 65–73. [17] K. Fukui and H. Imanishi: “On commutators of foliation preserving homeomorphisms”, J. Math. Soc. Japan, Vol. 51(1), (1999), pp. 227–236. [18] K. Fukui and H. Imanishi: “On commutators of foliation preserving Lipschitz homeomorphisms”, J. Math. Kyoto Univ., Vol. 41(3), (2001), pp. 507–515. [19] K. Fukui and T. Nakamura: “A topological property of Lipschitz mappings”, Topol. Appl., Vol. 148, (2005), pp. 143–152. [20] K. Fukui and S. Ushiki: “On the homotopy type of F Dif f (S 3 , FR )”, J. Math. Kyoto Univ., Vol. 15(1), (1975), pp. 201–210. [21] M. Hermann: “Simplicit´e du groupe des diff´eomorphismes de class C ∞ , isotopes ´a l’identit´e, du tore de dimension n”, C. R. Acad. Sci. Pari S´er. A-B, Vol. 273, (1971), pp. 232–234. [22] M. Hermann: “Sur la conjugasion diff´erentiable des diff´eomorphismes du cercle ´a des rotations”, Publ. I.H.E.S., Vol. 49, (1979), pp. 5–233. [23] F. Hirzebruch and K.H. Mayer: O(n)-Manigfaltigkeiten, exotische Sph¨ aren und Singularit¨aten, Springer Lecture Notes, Vol. 57, 1968. [24] H. Imanishi: “On the theorem of Denjoy-Sacksteder for codimension one foliations without holonomy”, J. Math. Kyoto Univ., Vol. 14, (1974), pp. 607–634. [25] J. Luukkainen and J. V¨ais¨al¨a: “Elements of Lipschitz topology”, Ann. Acad. Sci. Fennicae, Ser. A.I. Math., Vol. 3, (1977), pp. 85–122. [26] J.N. Mather: “Commutators of diffeomorphisms I and II”, Comment. Math. Helv., Vol. 49, (1992), pp. 512–528; Vol. 50, (1975), pp. 33–40. [27] T. Rybicki: “The identity component of the leaf preserving diffeomorphism group is perfect”, Monatsh. Math., Vol. 120, (1995), pp. 289–305. [28] T. Rybicki: “Commutators of diffeomorphisms of a manifold with boundary”, Ann. Polon. Math., Vol. 68(3), (1998), pp. 199–210. [29] G.W. Schwarz: “Smooth invariant functions under the action of a compact Lie group”, Topology, Vol. 14, (1975), pp. 63–68. [30] G.W. Schwarz: “Lifting smooth homotopies of orbit spaces”, Inst. Hautes Etudes Sci. Publ. Math., Vol. 51, (1980), pp. 37–135. [31] R. Strub: “Local classification of quotients of smooth manifolds by discontinuous groups”, Math. Zeitschrift, Vol. 179, (1982), pp. 43–57. [32] S. Sternberg: “Local contractions and a theorem of Poincar´e”, Amer. Jour. of Math., Vol. 79, (1957), pp. 809–823. [33] S. Sternberg: “The structure of local homeomorphisms, II”, Amer. Jour. of Math., Vol. 80, (1958), pp. 623–632. [34] W. Thurston: “Foliations and group of diffeomorphisms”, Bull. Amer. Math. Soc., Vol. 80, (1974), pp. 304–307.
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[35] T. Tsuboi: “On the group of foliation preserving diffeomorphisms”, preprint. [36] J.H.C. Whitehead: “Manifolds with transverse fields in euclidean space”, Ann. of Math., Vol. 73(2), (1961), pp. 154–212.
CEJM 3(3) 2005 529–557
The generalized de Rham-Hodge theory aspects of Delsarte-Darboux type transformations in multidimension Anatoliy Mykhaylovich Samoilenko1,2∗ , Yarema Anatoliyovych Prykarpatsky1,2† , Anatoliy Karolevych Prykarpatsky2‡ 1
The Institute of Mathematics, National Academy of Sciences, Kiev 01601, Ukraine 2 Department of Applied Mathematics, The AGH University of Science and Technology, Krakow 30059 Poland
Received 18 November 2004; accepted 23 April 2005 Abstract: The differential-geometric and topological structure of Delsarte transmutation operators and their associated Gelfand-Levitan-Marchenko type eqautions are studied along with classical Dirac type operator and its multidimensional affine extension, related with selfdual Yang-Mills eqautions. The construction of soliton-like solutions to the related set of nonlinear dynamical system is discussed. c Central European Science Journals. All rights reserved.
Keywords: Delsarte transmutation operators, parametric functional spaces, Darboux transformations, inverse spectral transform problem, soliton equations, generalized de RhamHodge differential complex, Zakharov-Shabat equations, Laplace and Dirac type operators MSC (2000): 34A30, 34B05, 34B15
∗ † ‡
E-mail:
[email protected] E-mail:
[email protected] E-mail:
[email protected],
[email protected]
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Generalized de Rham-Hodge theory aspects and related Delsarte-Darboux type binary transformations
1.1 A differential-geometric analysis of Delsarte-Darboux type transformations that was done in [23] for differential operator expressions acting in a functional space H = L2 (T;H), where T = R2 and H := L2 (R2 ; C2 ), appears to have a deep relationship with a classical generalized de Rham-Hodge theory devised around the middle of the past century [3, 4, 5, 6] for a set of commuting differential operators defined, in general on a smooth compact m-dimensional metric space M. Concerning our problem of describing the differential-geometric and spectral structure of Delsarte-Darboux type transmutaions acting in H, we preliminarily consider following [23] to obtain insight into a generalized de Rham-Hodge differential complex theory devised for studying these transformations of differential operators. Consider a smooth metric space M being a suitably compactified form of the space Rm , m ∈ Z+ . Then one can define on MT := T × M the standard Grassmann algebra Λ(MT ; H) of differential forms on T×M and consider a generalized by I.V. Skrypnik [3, 4] exterior anti-differentiation operator dL : Λ(M T ; H) → Λ(MT ; H) acting as follows: for any β (k) ∈ Λk (MT ; H), k = 0, m, dL β (k) :=
2 X j=1
dtj ∧ Lj (t; x|∂)β (k) +
m X i=1
dxi ∧ Ai (t; x; ∂)β (k) ∈ Λk+1 (MT ; H),
(1)
where Ai ∈ C 2 (T;L(H)), i = 1, m, are some mappings and Lj (t; x|∂) := ∂/∂tj − Lj (t; x|∂)
(2)
j = 1, 2, are suitably defined linear differential operators in H, commuting to each other, that is [L1 , L2 ] = 0 and [Lj , Ai ] = 0 (3) for all j = 1, 2 and i = 1, m. We will put, in general, that differential expressions nj (L)
Lj (t; x|∂) :=
X
a(j) α (t; x)
|α|=0
∂ |α| , ∂xα
(4)
(j)
with coefficients aα ∈ C 1 (T; C ∞ (M; EndCN )), |α| = 0, nj (L) nαj ∈ Z+ , j = 0, 1, are some closed normal densely defined operators in the Hilbert space H for any t ∈ T. It is easy to observe that the anti-differentiation of dL defined by (1) is a generalization of the usual exterior anti-differentiation m X
2
X ∂ ∂ d= dxj ∧ + dts ∧ ∂xj ∂ts s=1 j=1
(5)
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for which, evidently, commutation conditions [
∂ ∂ ∂ ∂ ∂ ∂ ; ] = 0, [ ; ] = 0, [ ; ]=0 ∂xj ∂xk ∂ts ∂tl ∂xj ∂ts
(6)
hold for all j, k = 1, m and s, l = 1, 2. Substituting ∂/∂xj −→ Aj , ∂/∂ts −→ Ls , j = 1, m, s = 1, 2, in (5) one gets the anti-differentiation dA :=
m X j=1
dxj ∧ Aj (t; x|∂) +
2 X j=1
dts ∧ Ls (t; x|∂),
(7)
where the differential expressions Aj , LS : H −→ H for all j, k = 1, m and s, l = 1, 2, satisfy the commutation conditions [Aj , Ak ] = 0, [Ls , Ls ] = 0, [Aj , Ls ] = 0, then then operation (7) defines on Λ(M T ; H) an anti-differential with respect to which the co-chain complex. d
d
d
d
A A A A H −→ Λ0 (M T ; H) −→ Λ1 (M T ; H) −→ ... −→ Λm+2 (M T ; H) −→ 0
(8)
is evidently closed, that is dA dA ≡ 0. As the anti-differential (1) is a particular soliton-like of (7), we obtain that the corresponding with it co-chain complex (8) is closed too.
1.2 Below we will follow ideas developed [3, 4, 5, 6, 32].A differential form β ∈ Λ(M T ; H) will be called dA -closed if dA β = 0 and a form γ ∈ Λ(M T ; H) will be called exact or dA -homological to zero if there exists on MT such a form ω ∈ Λ(M T ; H) that γ = dA ω. Consider now the standard [31, 32, 8, 34] algebraic Hodge star-operation ∗ : Λk (M T ; H) −→ Λm+2−k (M T ; H),
(9)
k = 0, m + 2, as follows: if β ∈ Λk (M T ; H), then the form ∗β ∈ Λm+2−k (M T ; H) is such that: • (m − k + 2) - dimensional volume | ∗ β| of the form ∗β equals k-dimensional volume |β| of the form β; ⊺ • the (m + 2) -dimensional measure β¯ ∧ ∗β > 0 under the fixed orientation on MT . Define also on the space Λ(M T ; H) the following natural scalar product: for any β, γ ∈ Λk (M T ; H), k = 0, m, Z (β, γ) :=
MT
⊺ β¯ ∗ γ.
(10)
Subject to the scalar product (10) one can naturally construct the corresponding Hilbert space m+2
HΛ (MT ) := ⊕ HΛk (MT ) k=0
(11)
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well suitable for our further consideration. Notice also here, that the Hodge star ∗operation satisfies the following easily checkable property: for any β, γ ∈ HΛk (MT ), k = 0, m, (β, γ) = (∗β, ∗γ), (12) that is the Hodge operation ∗ : HΛ (MT ) → HΛ (MT ) is unitary and its standard adjoint with respect to the scalar product (10) operation satisfies the condition (∗)′ = (∗)−1 . ′ Denote by dL the formally adjoint expression to the weak differential operation (1). By means of the operations d′L and dL in the HΛ (MT ) one can naturally define [8, 31, 32, 3, 34] the generalized Laplace-Hodge operator ∆L : HΛ (MT ) −→ HΛ (MT ) as ∆L = d′L dL + d′L dL .
(13)
Take a form β ∈ HΛ (MT ) satisfying the equality ∆L β = 0.
(14)
Such a form is called [3, 32, 34, 8] harmonic. One can also verify that a harmonic form β ∈ HΛ (MT ) satisfies simultaneously the following two adjoint conditions: d′L β = 0,
dL β = 0
(15)
easily obtained from (13) and (14). It is easy to check that the following differential operators in HΛ (MT ) d∗L := ∗d′L (∗)−1
(16)
defines also a new exterior anti-differential operation in HΛ (MT ). Lemma 1.1. The corresponding dual to (8) co-chain complex d∗
d∗
d∗
d∗
L L L L H −→ Λ0 (MT ; H) −→ Λ1 (MT ; H) −→ ... −→ Λm+2 (MT ; H) −→ 0
(17)
is exact. Proof. A proof follows from the property d∗L d∗L = 0, which is a direct consequence of the definition (16).
1.3 k Denote further by HΛ(L) (MT ), k = 0, m + 2, the cohomology groups of dL -closed and by k HΛ(L∗ ) (MT ), k = 0, m + 2, k = 0, m + 2, the cohomology groups of d∗L -closed differential k forms, respectively, and by HΛ(L ∗ L) (MT ), k = 0, m + 2, the abelian groups of harmonic differential forms from the Hilbert subspaces HΛk (MT ), k = 0, m + 2. Before formulating the next results, define the standard Hilbert-Schmidt rigged chain [14, 15] of positive and negative Hilbert spaces of differential forms k k HΛ,+ (MT ) ⊂ HΛk (MT ) ⊂ HΛ,− (MT ),
(18)
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the corresponding hereditary rigged chains of harmonic forms: k k k HΛ(L ∗ L),+ (MT ) ⊂ HΛ(L∗ L) (MT ) ⊂ HΛ(L∗ L),− (MT )
(19)
and chains of cohomology groups: k k k HΛ(L),+ (MT ) ⊂ HΛ(L) (MT ) ⊂ HΛ(L),− (MT ),
k HΛ(L ∗ ),+ (MT )
⊂
k HΛ(L ∗ ) (MT )
⊂
(20)
k HΛ(L ∗ ),− (MT )
for all k = 0, m + 2. Assume also that the Laplace-Hodge operator (13) is reduced upon the space HΛ0 (M). Now by reasoning similar to that in [8, 32, 34] one can formulate a little generalized [4, 5, 6, 32] de Rham-Hodge theorem. k The groups of harmonic forms HΛ,+ (MT ), k = 0, m + 2, are, respectively, isomorphic k to the homology groups (H (MT ; C))|Σ| , k = 0, m + 2, where H k (MT ; C) is the k-th cohomology group of the manifold MT with complex coefficients, a set Σ ⊂ Cp , p ∈ Z+ , is the set of suitable ”spectral” parameters marking the linear space of independent d∗L -closed 0 0-form from HΛ(L),− (MT ) and, moreover, the following direct sum decompositions k k k HΛ,+ (MT ) = HΛ(L ∗ L),+ (MT ) ⊕ ∆L HΛ,+ (MT )
(21)
k k−1 ′ k+1 = HΛ(L ∗ L),+ (M T ) ⊕ dL HΛ,+ (M T ) ⊕ dL HΛ,+ (MT )
hold for any k = 0, m + 2. Another variant of the statement similar to that above was formulated in [3, 4] and reads as the following generalized de Rham-Hodge theorem. k The generalized cohomology groups HΛ(L),+ (MT ), k = 0, m + 2, are isomorphic, rek spectively, to the cohomology groups (H (MT ; C))|Σ|, k = 0, m + 2. A proof of this theorem is based on some special sequence [3, 4, 5, 6, 7] of differential Lagrange type identities.⊲ Define the following closed subspace 0 ∗ (0) H0∗ := {ϕ(0) (η) ∈ HΛ(L (η) = 0, ϕ(0) (η)|Γ, η ∈ Σ} ∗ ),− (MT ) : dL ϕ
(22)
for some smooth (m+1)-dimensional hypersurface Γ ⊂ MT and Σ ⊂ (σ(L)∩ σ ¯ (L))×Σσ ⊂ p 0 C , where HΛ(L∗ ),− (MT ) is, as above, a suitable Hilbert-Schmidt rigged[14, 15] zero-order cohomology group Hilbert space from the co-chain given by (20), σ(L) and σ(L∗ ) are, respectively, mutual generalized spectra of the sets of differential operators L and L∗ in H at t = 0 ∈ T. Thereby, the dimension dimH0∗ = card Σ := |Σ| is assumed to be known. The next lemma first stated by I.V. Skrypnik [3, 4] is of fundamental importance for a proof of Theorem 1.2. Lemma 1.2. There exists a set of differential (k + 1)-forms Z (k+1) [ϕ(0) (η), dL ψ (k) ] ∈ Λk+1 (MT ; C), k = 0, m + 2, and a set of k-forms Z (k) [ϕ(0) (η), ψ (k) ] ∈ Λk (MT ; C), k =
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0, m + 2, parametrized by the set Σ ∋ η, being semilinear in (ϕ(0) (η), ψ (k) ) ∈ H0∗ × k HΛ,+ (MT ), such that Z (k+1) [ϕ(0) (η), dLψ (k) ] = dZ k [φ(0) (η ), ψ (k) ]
(23)
for all k = 0, m + 2 and η ∈ Σ. Proof. A proof is based on the following Lagrange type identity generalizing that of [3, 4] k and holding for any pair (ϕ0 (η), ψ ( k)) ∈ H0∗ × HΛ,+ (MT ) : 0 = < d∗L φ(0) (η), ∗(ψ (k) ∧ γ) >=< ∗d′L (∗)−1 ϕ(0) (η), ∗(ψ (k) ∧ γ) > =<
=< =<
∗d′L (∗)−1 φ(0) (x), ψ (k) ∧ γ) >= (∗)−1 ϕ(0) (η), dL ψ (k) ∧ γ > +Z (k+1) [ψ (0) (η), dL ψ(k)] ∧ (∗)−1 ϕ(0) (η), dL ψ (k) ∧ γ > +dZ (k) [ϕ(0) (η), ψ (k) ] ∧ γ,
(24)
γ >=
where Z (k+1) [ϕ(0) (η), dL ψ (k) ] ∈ Λk+1 (MT ; C), k = 0, m + 2, and Z (k) [ϕ(0) (η), ψ (k) ] ∈ Λk (MT ; C), k = 0, m + 2, are some semilinear differential forms on MT parametrized by a parameter λ ∈ Σ, and γ ∈ Λm+1−k (MT ; C) is arbitrary constant (m + 1 − k)-form. Thereby, the semilinear differential (k + 1)-forms Z (k+1) [ϕ(0) (η), dLψ (k) ] ∈ Λk+1 (MT ; C) and k-forms Z (k) [ϕ(0) (η), ψ (k) ] ∈ Λk (MT ; C), k = 0, m + 2, λ ∈ Σ, constructed above exactly constitute those searched for the Lemma.
1.4 Based now on Lemma 1.3 one can construct the cohomology group isomorphism claimed in Theorem 1.2 formulated above. Namely, following [3, 4], let us take some singular simplicial [31, 32, 33, 34] complex K(MT ) of the compact metric space MT and introduce (k) k a set of linear mappings Bλ : HΛ,+ MT −→ Ck (MT ; C), k = 0, m + 2, λ ∈ Σ, where Ck (MT ; C), k = 0, m + 2, are free abelian groups over the field C generated, respectively, by all k-chains of singular simplexes S (k) ⊂ MT , k = 0, m + 2, from the simplicial complex K(MT ), as follows: Z X (k) (k) (k) Bλ (ψ ) := S Z (k) [ϕ(0) (λ), ψ (k) ] (25) S (k) ∈Ck (MT ;C))
S (k)
k with ψ (k) ∈ HΛ,+ (MT ), k = 0, m + 2. The following theorem [3, 4] based on mappings (25) holds.
Theorem 1.3. The set of operators (25) parametrized by λ ∈ Σ realizes the cohomology group isomorphism formulated in Theorem 1.2 Proof. One obtains a proof of this theorem by passing over in (25 ) to the corresponding k cohomology HΛ(L),+ (MT ) and homology Hk (MT ; C) groups of MT for every k = 0, m + 2.
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k If one takes an element ψ (k) := ψ (k) (µ) ∈ HΛ(L),+ (MT ), k = 0, m + 2, solving the equation (k) dL ψ (µ) = 0 with µ ∈ Σk being some set of the related ”spectral” parameters marking k elements of the subspace HΛ(L),− (MT ), then one finds easily from (25) and identity (23) (k) (0) (k) that dZ [ϕ (λ), ψ (µ)] = 0 for all (λ, µ) ∈ Σ × Σk , k = 0, m + 2. This, in particular, means that due to the Poincar´e lemma [30, 31, 32] there exist differential (k − 1)-forms Ω(k−1) [ϕ(0) (λ), ψ (k) (µ] ∈ Λk−1(M; C), k = 0, m + 2, such that
Z (k) [ϕ(0) (λ), ψ (k) (µ)] = dΩ(k−1) [ϕ(0) (λ), ψ (k) (µ)]
(26)
k for all pairs (ϕ(0) (λ), ψ (k) (µ)) ∈ H0∗ × HΛ(L),+ (MT ) parametrized by (λ, µ) ∈ Σ × Σk , k = 0, m + 2. As a result of passing on the right hand-side of (25) to the homology groups Hk (MT ; C), k = 0, m + 2, one obtains from the standard Stokes theorem [30, 32, 31] that the mappings (k) k Bλ : HΛ(L),+ (MT ) −→ Hk (MT ; C) (27)
are isomorphisms for every k = 0, m + 2 and λ ∈ Σ. Making further use of the Poincar´e duality [8, 31, 32] between the homology groups Hk (MT ; C), k = 0, m + 2, and the cohomology groups H k (M; C), k = 0, m + 2, respectively, one obtains finally the statement claimed in Theorem 1.4.
2
The spectral structure of Delsarte-Darboux type transmutation operators in multidimension
2.1 Take now into account that our differential operators Lj : H → H, j = 1, 2, are of the special form (2). Assume also that differential expressions (4) are normal closed operators defined on dense subspace D(L) ⊂ L2 (M; CN ). k Then due to Theorem 1.4 one can find such a pair (ϕ(0) (λ), ψ (k) (µ)) ∈ H0∗ ×HΛ(L),+ (MT ) parametrized by elements (λ, µ) ∈ Σ × Σk , for which the equality Z (m) (m) (0) Bλ (ψ (µ)dx) = S(t;x) Ω(m−1) [ϕ(0) (λ), ψ (0) (µ)dx] (28) (m)
∂S(t;x)
(m)
holds, where S(t;x) ∈ Hm (MT ; C) is some arbitrary but fixed element of parametrized by (m)
an arbitrarily chosen point (t; x) ∈ MT ∩ ∂S(t;x) . Consider the next integral expressions Z Ω(t;x) (λ, µ) : = Ω(m−1) [ϕ(0) (λ), ψ (0) (µ)dx], (29) (m−1)
σ(t;x)
Ω(t0 ;x0 ) (λ, µ) : = (m)
Z
(m−1) σ(t ;x ) 0 0
Ω(m−1) [φ(0) (λ), psi(0) (µ)dx], (m−1)
with a point (t0 ; x0 ) ∈ MT ∩ ∂S(t0 ;x0) being taken fixed at the boundaries σ(t;x) (m−1)
(m)
(m)
:= ∂St;x ,
σ(t0 ;x0 ) := ∂St0 ;x0 , which are assumed to be homological to each other as (t; x0 ) −→ (t; x) ∈
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MT , (λ, µ) ∈ Σ × Σk , and interpret them as the kernels [14, 15, 35] of the correspond(ρ) ing invertible integral operators of Hilbert-Schmidt type Ω(t;x) , Ω(t0 ;x0 ) : L2 (Σ; C) −→ (ρ) L2 (Σ; C), where ρ is some finite Borel measure on the parameter set Σ. Define now the invertible operators expressions ˜ (0) (µ) Ω± : ψ (0) (µ) −→ ψ
(30)
m ˜ (0) (µ)dx ∈ Hm for ψ (0) (µ)dx ∈ HΛ(L),+ (MT ) and ψ Λ(L),+ (MT ), µ ∈ Σ, where, by definition, for any η ∈ Σ
˜ (0) (η) : = ψ (0) (η) · Ω−1 · Ω(t ;x ) ψ 0 0 Z Z (t;x) = dρ(µ) dρ(ξ)ψ (0) (µ)Ω−1 (t;x) (µ, ξ)Ω(t0 ;x0 ) (ξ, η), Σ
(31)
Σ
being motivated by the expression (28). Namely, consider the following diagram Ω±
m HΛ(L),+ (MT )
−→
(m)
˜ (m) ւB λ
Bλ
↓
m HΛ( (MT ), ˜ L),+
(32)
Hm (MT ; C)
which is assumed to be commutative for some another co-chain complex d˜
d˜
d˜
d˜
L L L L H −→ Λ0 (MT ; H) −→ Λ1 (MT ; H) −→ ... −→ Λm+2 (MT ; H) −→ 0.
(33)
Here, by definition, the generalized anti-differentiation is dL˜ :=
2 X j=1
˜ j (t; x|∂) dtj ∧ L
(34)
with ˜ j = ∂/∂tj − L ˜ j (t; x|∂), L ˜ nj (L)
˜ j (t; x|∂) : = L
X
|α|=0
a ˜(j) α (t; x)
(35)
∂ |α| , ∂xα
(j) ˜ nj (L) ˜ := nj (L) ∈ Z+ , where coefficients a ˜α ∈ C 1 (T; C ∞ (M; EndCN ), |α| = 0, nj (L), ˜ (m) : Hm j = 1, 2. The corresponding isomorphisms B λ Λ(L),+ (MT ) −→ Hm (MT ; C), λ ∈ Σ, act, by definition, as follows: Z (m) ˜ (0) (m) ˜ (0) (µ)dx], ˜ ˜ (m−1) [˜ Bλ (ψ (µ)dx) = S(t;x) Ω ϕ(0) (λ), ψ (36) (m)
∂S(t;x)
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537
˜ ∗ ⊂ H0 ∗ (MT ), λ ∈ (σ(L) ˜ ∩σ ˜ ∗ )) × Σσ , where ϕ ˜ (0) (λ) ∈ H ¯ (L 0 Λ(L ),− m ∗ (0) ˜ ∗ := {˜ H ϕ(0) (λ) ∈ HΛ(L ˜ (x) = 0, ϕ ˜ (0) (λ)|Γ˜ = 0, λ ∈ Σ} ∗ ),− (MT ) : d ˜ϕ 0 L
(37)
˜ ⊂ MT . One defines the following closed subspace for some hypersurface Γ ˜ (0) (µ) ∈ H0 ∗ (MT ) : d∗˜ψ ˜ (0) (λ) = 0, ψ ˜ (0) (µ)| ˜ = 0, µ ∈ Σ} ˜ 0 := {ψ H Λ(L ),− Γ L
(38)
˜ ⊂ MT , introduced above. for the hyperspace Γ Suppose now that the elements (31) belong to the closed subspace (38), that is (0)
˜ (µ) = 0 dL˜ψ
(39)
˜ ∗ ⊂ Hm ∗ (MT ) as follows: Define similarly to (38) a closed subspace H 0 Λ(L ),− 0 (0) H0 := {ψ (0) (λ) ∈ HΛ(L (λ) = 0, ψ (0) (λ)|Γ = 0, λ ∈ Σ} ∗ ),− (MT ) : dL ψ
(40)
for all µ ∈ Σ. Then due to the commutativity of the diagram (32) there exist the corresponding two invertible mappings ˜0, Ω± : H0 → H
(41)
m depending on how they are extened over the whole Hilbert space HΛ,− (MT ). Extend m now operators (41) upon the whole Hilbert space HΛ,− (MT ) by means of the standard method [24, 26] of variation of constants, taking into account that for kernels (p) (p) Ω(t;x) (λ, µ), Ω(t0 ;x0 ) (λ, µ) ∈ L2 (Σ; C) ⊗ L2 (Σ; C), λ, µ ∈ Σ, one can write down the following relationships:
Ω (λ, µ) − Ω(t0 ;x0 ) (λ, µ) Z (t;x) Z (m−1) (0) (0) = Ω [ϕ (x), ψ (µ)dx] − = =
Z Z
(m) ∂S(t;x)
(m) (m−1) (m−1) S± (σ(t;x) ,σ(t ;x ) ) 0 0
(m) (m−1) (m−1) S± (σ(t;x) ,σ(t ;x ) ) 0 0
(m) ∂S(t ;x ) 0 0
(42) Ω(m−1) [ϕ(0) (λ), ψ (0) (µ)dx]
dΩ(m−1) [ϕ(0) (λ), ψ (0) (µ)dx] Z (m) [ϕ(0) (λ), ψ (0) (µ)dx], (m)
(m−1)
(m−1)
where, by the definition, m-dimensional open surfaces S± (σ(t;x) , σ(t0 ;x0 ) ) ⊂ MT are (m−1)
spanned smoothly without self-intersection between two homological cycles σ(t;x)
(m) (m−1) (m) ∂S(t;x) and σ(t0 ;x0 ) = ∂S(t0 ;x0 ) ∈ (m−1) (m) (m−1) (m−1) σ(t0 ;x0 ) ) ∪ S− (σ(t;x) , σ(t0 ;x0 ) ))
Cm−1 (MT ; C) in such a way that the boundary
=
(m) (m−1) ∂(S+ (σ(t0 ;x0 ) ,
= ∅. Making use of the relationship (42), one can thereby find easily the following integral operator expressions in H− : Z ˜ (0) (ξ)Ω−1 (ξ, η) Ω± = 1 − dρ(η)ψ (43) (t0 ;x0 ) Z Σ × Z (m) [ϕ(0) (η), (·)dx] (m)
(m−1)
(m−1) ) 0 ;x0 )
S± (σ(t;x) ,σ(t
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˜ (0) (µ)) ∈ H ˜∗ × H ˜ 0, defined for fixed pairs (ϕ(0) (ξ), ψ (0) (η)) ∈ H0∗ × H0 and (˜ ϕ(0) (ξ), ψ 0 λ, µ ∈ Σ, being bounded invertible operators of Volterra type [20, 21, 16, 35] on the ˜ j : H −→ H, j = 1, 2, whole Hilbert space H. Moreover, for the differential operators L one can get easily the following expressions ˜ j = Ω± Lj Ω−1 L ±
(44)
for j = 1, 2, where the left hand-side of (44) does not depend on sings ”±” of the right-hand sides. Thereby, the Volterra integral operators (43) are the Delsarte-Darboux transmutation operators, mapping a given set L of differential operators into a new set L˜ of differential operators transformed via the Delsarte expressions (44).
2.2 Suppose now that all of differential operators Lj (t; x|∂), j = 1, 2, considered above do not depend one the variable t ∈ T. Then, evidently, one can take H0 := {ψµ(0) (ξ) ∈ L2.− (M; CN ) : Lj ψµ(0) (ξ) = µj ψµ(0) (ξ), ˜ ∩ σ(L∗ ), ξ ∈ Σσ } j = 1, 2, ψµ(0) (ξ)|Γ˜ = 0, µ = (µ1 , µ2 ) ∈ σ(L) ˜ (0) (ξ) ∈ L2.− (M; CN ) : L ˜ (0) (ξ) = µj ψ ˜ (0) (ξ), ˜ 0 := {ψ ˜jψ H µ µ µ
˜ (0) (ξ)| ˜ = 0, µ = (µ1 , µ2 ) ∈ σ(L) ˜ ∩ σ(L∗ ), ξ ∈ Σσ } j = 1, 2, ψ µ Γ (0) (0) ¯ j ϕ(0) (η), j = 1, 2, H0∗ := {ϕλ (η) ∈ L2.− (M; CN ) : L∗j ϕλ (η) = λ λ (0) ∗ ˜ ϕλ (η)|Γ˜ = 0, λ = (λ1 , λ2 ) ∈ σ(L) ∩ σ(L ), η ∈ Σσ } (0) (0) ¯ j ϕ(0) (η), ˜ ∗ := {˜ ˜ ∗ϕ H ϕ (η) ∈ L2.− (M; CN ) : L ˜ (η) = λ 0
j
λ
j = 1, 2,
(0) ϕ ˜ λ (η)|Γ˜
λ
(45)
λ
˜ ∩ σ(L∗ ), η ∈ Σσ } = 0, λ = (λ1 , λ2 ) ∈ σ(L)
and construct the corresponding Delsarte-Darboux transmutation operators Z Z Ω± = 1 − dρσ (λ) dρΣσ (ξ)dρΣσ (η) ∗) ˜ σ(L)∩σ(L Σσ ×Σσ Z (0),⊺ ˜ (0) (ξ)Ω−1 (λ; ξ; η)¯ × dxψ ϕλ (η)(·) λ x0
(46)
(m) (m−1) (m−1) ,σ 0 ;x0 ) (t0 ;x0 )
S± σ(t
˜ ∩ σ(L∗ ) × Σ2 acting yet in Hilbert space L2,+ (M; CN ), where for any (λ; ξ, η) ∈ (σ(L) σ kernels Z (0) (0) Ω(x0 ) (λ; ξ, η) := Ω(m−1) [ϕλ (ξ), ψλ (η)dx] (47) (m−1)
σx0
(ρ)
(ρ)
˜ ∩ σ(L∗ ) belong to L2 (Σσ ; C) ⊗ L2 (Σσ ; C). Moreover, for (ξ, η) ∈ Σ2σ and every λ ∈ σ(L) as ∂Ω± /∂tj = 0, j = 1, 2, one gets easily the set of differential expressions ˜ j (x|∂) := Ω± Lj (x|∂)Ω−1 L ± j = 1, 2, also commuting, evidently, to each other.
(48)
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The Volterra operators (46) possess some additional properties. Namely, define the following Fredholm type integral operator in H : Ω := Ω−1 + Ω− ,
(49)
Ω=1+Φ(Ω),
(50)
which can be written in the form
where the operator Φ(Ω) ∈ B∞ (H) is compact. Moreover, due to the relationships (48) one gets easily that the following commutator conditions [Ω, Lj ] = 0
(51)
hold for j = 1, 2. ˆ ˆ + (Ω), K ˆ − (Ω) ∈ H− ⊗ H− the kernels correDenote now by Φ(Ω) ∈ H− ⊗ H− and K sponding [14, 15] to operators Φ(Ω) ∈ B∞ (H) and Ω± − 1 ∈ B∞ (H). Then due to the (m−1) (m−1) fact that the supports supp K+ (Ω) ∩ suppK− (Ω) = σx ∪ σx0 , one gets from (49) and (50) the well known Gelfand-Levitan-Marchenko linear integral equation ˆ +K ˆ + (Ω) + Φ(Ω) ˆ + (Ω)+ ∗ Φ(Ω) ˆ ˆ − (Ω), K =K
(52)
ˆ + (Ω)(x; y) ∈ H− ⊗ H− for enabling the factorization of Fredholm operator (49) kernel K all y ∈ suppK+ (Ω). The conditions (51) can be rewritten suitably as follows: ˆ ˆ (Lj,ext ⊗ 1)Φ(Ω) = (1 ⊗ L∗j,ext )Φ(Ω)
(53)
for j = 1, 2, where Lj,ext ∈ L(H− ), j = 1, 2, and their adjoints L∗j,ext ∈ L(H− ), j = 1, 2, are the corresponding extensions [14, 27, 15] of the differential operators Lj and L∗j ∈ L(H), j = 1, 2. Concerning the relationships (48) one can write down [14, 27] kernel conditions similar to (53): ˜ j,ext ⊗ 1)K ˆ ± (Ω) = (1 ⊗ L∗ )K ˆ ± (Ω), (L (54) j,ext ˜ j,ext ∈ L(H− ), j = 1, 2 are the corresponding rigging extensions of the where as above, L ˜ j ∈ L(H), j = 1, 2. differential operators L
2.3 Proceed now to analyzing the question about the general differential and spectral structure of transformed operator expression (44). It is evident that the found above conditions ˆ ± (Ω) ∈ H− ⊗ H− of Delsarte- Darboux transmutation op(52) and (53) on the kernels K erators are necessary for the operator expressions (44) to exist and be differential. The question now is whether these conditions are also sufficient?
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For studying this question let us consider Volterra operators (43) and (46) with kernels satisfying the conditions (52) and (53), assuming that suitable oriented surfaces (m) S± (σ(t;x)(m−1) , σ(t0 ;x0 )(m−1) ) ∈ Cm (MT ; C) are given as follows: (m)
S+ (σ(t;x)(m−1) , σ(t0 ;x0 )(m−1) ) = {(t′ ; x′ ) ∈ MT :
t′ = P (t; x|x′ ), t ∈ T},
(m)
S− (σ(t;x)(m−1) , σ(t0 ;x0 )(m−1) ) = {(t′ ; x′ ) ∈ MT :
t′ = P (t; x|x′ ) ∈ T\[t0 , t]},
(55) (m)
where a mapping P ∈ C ∞ (MT × M; T) is smooth and such that the boundaries ∂S± (m−1) (m−1) (m−1) (m−1) (m−1) (m−1) (σ(t;x) , σ(t0 ;x0 ) ) = ±(σ(t;x) − σ(t0 ;x0 ) ) with cycles σ(t;x) and σ(t0 ;x0) ∈ K(MT ) being homological to each other for any choice of points (t0 ; x0 ) and (t; x) ∈ MT . Then one can see by means of some simple but cumbersome calculations, based on considerations from [37] and [9], that the resulting expressions on the right hand-sides of ˜ = L + [K± (Ω), L] · Ω−1 L ±
(56)
are exactly equal to each other differential ones if such there is such an expression for an operator L ∈ L(H). Concerning the inverse operators Ω−1 ± ∈ B(H) present in (56) one can notice here ˜0 ⊂ H ˜ − , the that due to the functional symmetry between closed subspaces H0 and H defining relationships (41) and (31) are reversible, that is there exist the inverse operator ˜ mappings Ω−1 ± : H0 → H0 , such that (0)
(0)
(0) ˜ ˜ ˜ −1 ˜ Ω−1 ± : ψ (λ) −→ ψ (λ) := ψ (λ) · Ω(t;x) Ω(t;x)
(57)
˜ (t;x) (λ, µ) and Ω ˜ (t ;x ) (λ, µ) ∈ L(ρ) (Σ; C) ⊗ L(ρ) (Σ; C), related for some suitable kernels Ω 0 0 2 2 ˜ ∈ L(H). Thereby, due to the exnaturally with the transformed differential expression L pressions (57) one can write down similar to (46) the following inverse integral operators: Ω−1 ±
= 1− Z ×
Z
Σ
dρ(ξ)
Z
Σ
˜ −1 (ξ, η) dρ(η)ψ (0) (ξ)Ω t0 ;x0
(m) (m−1) (m−1) S± (σ(t;x) ,σ(t ;x ) ) 0 0
(58)
Z˜ (m) [˜ ϕ(0) (η), (·)dx]
˜ (0) (η)) ∈ H ˜ 0∗ × H ˜ 0 and (ϕ(0) (ξ), ψ (0) (η)) ∈ H0∗ × H0 , defined for fixed pairs (˜ ϕ(0) (ξ), ψ ξ, η ∈ Σ, and being bounded invertible operators of Volterra type in the whole Hilbert −1 space H. In particular, the compatibility conditions Ω± Ω−1 ± = 1 = Ω± Ω± must be fulfilled identically in H, involving some restrictions identifying measures ρ and Σ and possible asymptotic conditions of coefficient functions of the differential expression L ∈ L. Such kinds of restrictions were already mentioned before in [40, 41, 42], where in particular the relationships with the local and nonlocal Riemann problems were discussed.
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2.4 Within the framework of the general construction presented above one can give a natural interpretation of so called B¨acklund transformations for coefficient functions of a given differential operator expression L ∈ L(H). Namely, following the symbolic considerations in [44], we reinterpret the approach devised there for constructing the B¨acklund transformations making use of the techniques based on the theory of Delsarte transmutation operators. Let us define two different Delsarte-Darboux transformed differential operator expressions L1 = Ω1,± LΩ−1 L2 = Ω2,± LΩ−1 (59) 1,± , 2,± , where Ω1,+ , Ω2,− ∈ B(H) are some Delsarte transmutation Volterra operators in H with Borel spectral measures ρ1 and ρ2 on Σ, such that the following conditions −1 Ω−1 1,+ Ω1,− = Ω = Ω2,+ Ω2,−
(60)
hold. Making use now of the conditions (59) and relationships (60) one finds easily that the operator B := Ω2,− Ω−1 1,+ ∈ B(H) satisfies the following operator equations: L2 B = BL1 ,
Ω2,± B = BΩ1,± ,
(61)
which motivate the next definition. Definition 2.1. An invertible symbolic mapping B : L(H) −→ L (H) will be called a Darboux-B¨acklund transformation of an operator L1 ∈ L(H) into the operator L2 ∈ L(H) if there holds the condition [QB, L1 ] = 0 (62) for some linear differential expression Q ∈ L(H). The condition (62) can be realized as follows. Take any differential expression q ∈ L(H) satisfying the symbolic equation [qB, L] = 0.
(63)
Then, making use of the transformations like (59), from (60) one finds that [QB, L1 ] = 0,
(64)
−1 QB := Ω1,+ qBΩ−1 1,+ = Ω1,+ qΩ2,+ B.
(65)
where owing to (61) Therefore, the expression Q = Ω1,+ qΩ−1 2,+ appears to be differential one too owing to the conditions (61). The consideration above related with the symbolic mapping B :L(H) → L(H) gives rise to an effective tool of constructing self-B¨acklund transformations for coefficients of differential operator expressions L1 , L2 ∈ L(H) having many applications [17, 38, 29, 35, 26] in spectral and soliton theories.
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2.5 Return now to studying the structure Delsarte-Darboux transformations for a polynomial differential operators pencil n(L)
L(λ; x|∂) :=
X
Lj (x|∂)λj ,
(66)
j=0
where n(L) ∈ Z+ and λ ∈ C is a complex-valued parameter. It is asked to find the corresponding to (66) Delsarte-Darboux transformations Ωλ,± ∈ B(H), λ ∈ C, such that ˜ x|∂) ∈ L(H) the following Delsartefor some polynomial differential operators pencil L(λ; Lions [2] transmutation condition ˜ λ,± = Ωλ,± L LΩ
(67)
holds for almost all λ ∈ C. For such transformations Ωλ± ∈ B(H) to be found, let us consider a parameter τ ∈ R dependent differential operator Lτ (x|∂) ∈ L(Hτ ), where n(L)
Lτ (x|∂) :=
X
Lj (x|∂)∂ j /∂τ j ,
(68)
j=0
acts in the functional space Hτ = C q(L) (Rτ ; H) for some q(L) ∈ Z+ . Then one can easily construct the corresponding Delsarte-Darboux transformations Ωτ,± ∈ B(Hτ ) of Volterra type for some differential operator expression n(L)
˜ τ (x|∂) := L
X
˜ j (x|∂)∂ j /∂τ j , L
(69)
j=0
if the following Delsarte-Lions [2] transmutation conditions ˜ τ Ωτ,± = Ωτ,± Lτ L
(70)
hold in Hτ . Thus, making use of the results obtained above, one can write down that Z Z ˜ (0) (λ; ξ)Ω−1 (λ; ξ, η) Ωτ,± = 1 − dρΣ (ξ) dρΣ (η)ψ (71) τ (τ0 ;x0 ) Σ Σ Z × Z (m) [ϕ(0) τ (λ; η), (·)dx] (m)
(m−1)
(m−1) ) 0 ;x0 )
S± (σ(τ ;x) ,σ(τ
∗ ∗ defined by means of the following closed subspaces Hτ,0 ⊂ Hτ,− and Hτ,0 ⊂ Hτ,− :
Hτ,0 : = {ψτ(0) (λ; ξ) ∈ Hτ,− : Lτ ψτ(0) (λ; ξ) = 0,
ψτ(0) (λ; ξ)|τ =0 = ψ (0) (λ; ξ) ∈ H, Lψ (0) (λ; ξ) = 0, ψ (0) (λ; ξ)|Γ = 0, λ ∈ C, ξ ∈ Σ},
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∗ ∗ (0) Hτ,0 : = {ϕ(0) τ (λ; η) ∈ Hτ,− : Lτ ϕτ (λ; η) = 0,
543
(72)
(0) ∗ (0) ϕ(0) τ (λ; η)|τ =0 = ϕ (λ; η) ∈ H , Lϕ (λ; η) = 0,
ϕ(0) (λ; η)|Γ = 0, λ ∈ C, η ∈ Σ}.
Recalling now that our operators Lj ∈ L(H), j = 0, r(L), do not depend on the parameter τ ∈ R, one can derive easily from (71) Z Z ˜ (0) (λ; ξ)Ω−1 (λ; ξ, η) Ω± = 1 − dρΣ (ξ) dρΣ (η)ψ (73) (x0 ) Σ Σ Z (m) × Z0 [ϕ(0) (λ; η), (·)dx], (m)
(m−1)
S± (σ(x)
(m−1)
where we put σx
(m−1)
(m−1)
:= σ(τ0 ;x) , σx0
(m−1) ) 0)
,σ(x
(m−1)
:= σ(τ0 ;x0 ) ∈ Cm−1 (Rm ; C) and
(m)
(0) Z0 [ϕ(0) (λ; η), ψ (0) dx] := Z (m) [ϕ(0) τ (λ; η), ψτ dx]|dτ =0 .
(74)
∗ The closed subspaces H0 ∈ H− and H0∗ ∈ H− corresponding to (73) are given as follows:
H0 := {ψ (0) (λ; ξ) ∈ H− : Lψ (0) (λ; ξ) = 0, ψ (0) (λ; ξ)|Γ = 0, λ ∈ C, ξ ∈ Σ},
(75)
∗ ∗ Hτ,0 := {ϕ(0) (λ; η) ∈ H− : Lϕ(0) (λ; η) = 0, ϕ(0) (λ; η)|Γ = 0, λ ∈ C, η ∈ Σ}.
Thereby, making use of the expressions (73) one can construct the Delsarte-Darboux ˜ ∈ L(H), whose coefficients are related with those transformed linear differential pencil L for the pencil L ∈ L(H) via some B¨acklund type relationships useful for applications (see [26, 22, 45, 46, 41]) in the soliton theory.
3
Delsarte-Darboux type transmutation operators for special multi-dimensional expressions and their applications
3.1 A perturbed self-adjoint Laplace operator in Rn Consider the Laplace operator −∆m in H := L(Rm ; C) perturbed by the multiplication operator on a function q ∈ W22 (Rm ; C), that is the operator L(x|∂) := −∆m + q(x),
(76)
where x ∈ Rm . The operator (76) is self-adjoint in H. Applying the results from the Section 1 to the differential expression (76) in the Hilbert space H, one can write down the following invertible Delsarte-Darboux transmutation operators: Z Z Z Z Ω± = 1 − dρσ (ξ) dρσ (ξ) dρΣσ (ξ) dρΣσ (η) (77) σ(L)
(0)
σ(L)
˜ (λ; ξ)Ω−1 (λ; ξ, η) ×ψ (x0 )
Z
Σσ
Σσ
(0)
(m) (m−1) (m−1) S± (σ(x) ,σ(x ) ) 0
dy ϕ ¯ (0)⊺(λ; η), (·),
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where σx ∈ K(Rm ) is some closed maybe non-compact simplicial hyper-surface in Rm (m−1) (m−1) parametrized by a running point x ∈ σx , and σx0 ∈ K(Rm ) is a suitable homolog(m−1)
(m−1)
ical to σx simplicial hypersurface in Rm parametrized by a point x0 ∈ σx0 . There (m) (m−1) (m−1) exist exactly two m-dimensional subspaces spanning them, say S± (σx , σx0 ) ∈ (m) (m−1) (m−1) (m) (m−1) (m−1) K(Rm ), such that S+ (σx , σx0 )∪ S− (σx , σx0 ) = Rm . Taking into account these subspaces, one can rewrite the Delsarte-Darboux transmutation operators (77) for (76) more concisely as: Z ˆ ± (Ω)(x; y)(·), Ω± = 1+ dy K (78) (m)
(m−1)
S± (σx
(m−1)
where, as before, x ∈ σx or equivalently,
(m−1)
,σx0
)
ˆ ± (Ω) ∈ H− ⊗ H− satisfy the equations (54), and kernels K
ˆ ± (Ω)(x; y) + ∆m (y; ∂)K ˆ ± (Ω)(x; y) −∆m (x; ∂)K ˆ ± (Ω)(x; y) = (q(y) − q˜(x))K
(79)
ˆ ± (Ω). Take for simplicity, a non-compact closed simplicial hypersurfor all x, y ∈ suppK (m−1)
(m−1)
face σx = σx,γ := {y ∈ Rm :< x − y, ±γ >= 0} and the degenerate simplicial cycle (m−1) σx0 := x0 = ∞ ∈ Rm , where γ ∈ Sm−1 is an arbitrary versor with ||γ|| = 1. Then, evidently, (m)
(m)
(m−1)
(m−1) ) := S±γ,x = {y ∈ Rm :< x − y, ±γ > ≥ 0} S± (σx,γ) , σ∞
and our transmutation operators (78) take the form Z ˆ ±γ (Ω)(x; y)(·), Ω±γ = 1+ dy K
(80)
(81)
(m)
S±γ,x
(m)
(m)
(m)
(m−1)
(m−1)
(m)
(m)
ˆ ±γ (Ω) = S±γ,x , S+γ,x ∩ S−γ,x = σx,γ ∪ σ∞ where suppK and S+γ,x ∩ S−γ,x = Rm for any direction γ ∈ Sm−1 . The invertible transmutation Volterra operators like (81) were constructed before by L.D. Faddeev [9] and R.G. Newton [11, 12] for the self-adjoint perturbed Laplace operator (76) in R3 . They called them [9] transformation operators with a Volterra direction γ ∈ Sm−1 . It is easy to see that Faddeev’s expressions (81) are very special cases of the general expressions (78) obtained above. Define now making use of (78) the following Fredholmian operator in the Hilbert space H: Ω := (1+K+ (Ω))−1 (1+K− (Ω)) = 1+Φ(Ω) (82) with the compact part Φ(Ω) ∈B∞ (H). Then the commutation equality [L, Φ(Ω)] = 0
(83)
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together with the Gelfand-Levitan-Marchenko equation ˆ ˆ + (Ω)+Φ(Ω) ˆ ˆ − (Ω) K+ (Ω)+Φ(Ω)+ K =K
(84)
ˆ ± (Ω) and Φ(Ω) ˆ for the corresponding kernels K ∈H− ⊗ H− holds. ˆ ± (Ω) ∈H− ⊗ In [9, 12] there was thoroughly analyzed the spectral structure of kernels K H− in (81) making use of the analytical properties of the corresponding Green’s functions of the operator (76). As one can see from (77), these properties depend strongly both on the structure of the spectral measures ρσ on σ(L) and ρΣσ on Σσ and on analytical behav(ρ) (ρ) ior of the kernel Ω∞ (λ; ξ, η) ∈ L2 ( Σσ ; C)⊗L2 ( Σσ ; C), ξ, η ∈ Σσ , for all λ ∈ σ(L). In [9] ˆ ± (Ω) ∈H− ⊗ H− there was stated for any direction γ ∈ Sm−1 the dependence of kernels K on the regularized determinant of the resolvent Rµ (L) ∈ B(H), µ ∈ C/σ(L) is a regular point, for the operator (76). This dependence can be also clarified if one makes use of the approach from Section 2.
3.2 A two-dimensional Dirac type operator Let us define in H := L2 (R2 ; C2 ) a two-dimensional Dirac type operator
∂/∂x1 u˜1 (x) ˜ ∂) := L(x; , u˜2 (x) ∂/∂x2
(85)
where x := (x1 , x2 ) ∈ R2 , and coefficients u˜j ∈ W21 ( R2 ; C), j = 1, 2. The transformation properties of the operator (85) were studied thoroughly by L.P. Nizhnik [18]. In particular, he constructed some special class of the Delsarte-Darboux transmutation operators in the form Z ˆ ±(Ω)(x; y)(·), Ω± = 1+ dy K (86) (2)
(1)
(1)
S± (σx ,σ∞ )
where for two orthonormal versors γ1 and γ2 ∈ S1 , ||γ1|| = 1 = ||γ2||, (2)
(1) S+ (σx(1) , σ∞ ) : = {y ∈ R2 :< x − y, γ1 >≥ 0} (2)
(87)
∩{y ∈ R2 :< x − y, γ2 >≥ 0},
(1) S− (σx(1) , σ∞ ) : = {y ∈ R2 :< x − y, γ1 >≤ 0}
∪{y ∈ R2 :< x − y, γ2 >≤ 0}.
In case when < x, γj >= xj ∈ R, j = 1, 2, the corresponding kernel (1) (0) (1) (0) K+,11 δ
+ K+,11 (x; y) K+,12 δ + K+,12 ) ˆ + (Ω) = K (1) (0) (1) (0) K+,21 δ + K+,21 (x; y) K+,22 δ + K+,22
(88)
is Dirac delta-function singular, being, in part, localized on half-lines < y−x, γ2 >= 0 and (l) < y − x, γ1 >= 0, with all regular coefficients K+,ij ∈ C 1 (R2 × R2 ; C) for all i, j = 1, 2
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and l = 0, 1. Such a property for the transmutation kernels of the perturbed Laplace operator (76) was also observed in [9], where it was motivated by the necessary condtion ˜ ∂) ∈ L(H) to be differential. As one can check, the for the transformed operator L(x; same reason of the existence of singularities holds in (88). Let us now consider the general expression like (78) for the corresponding hypersur(2) (1) (1) (1) faces S± (σx , σ∞ ) spanning a closed non-compact smooth cycle σx ∈ K(R2 ) and the (1) (1) infinite point σ∞ := ∞ ∈ K(R2 ). A running point x ∈ σx is taken arbitrary but, as usual, fixed. ˆ ± (Ω) ∈ H− × H− in (86) satisfy the standard conditions (53) and (54), The kernels K that is ˜ 1,ext ⊗ 1)K ˆ ± (Ω) = (1 ⊗ L∗ )K ˆ ± (Ω), (L 1,ext
(89)
[L1 , Φ(Ω)] = 0
for some matrix differential Dirac type operator L1 ∈ L(H) of the form (76). Together with this Dirac operator the following matrix second order differential operator ∂u ˜1 ∂2 ∂2 ± ∂x2 − v˜2 −2 ∂x2 ∂x2 2 ˜ 2 := 1 ∂ + L (90) 1 2 2 ∂t ∂u ˜2 ∂ ∂ −2 ∂x1 ± ∂x2 − v˜1 ∂x2 1
2
in the parametric space H := C 1 (R; H) was studied in [18, 19] for which there was developed scattering theory that was used for constructing soliton-like exact solutions to the so called Devey-Stewartson nonlinear dynamical system in partial derivatives. The ˜ 1 and L ˜ 2 ∈ L(H) commute. latter was based on the fact that the two operators L Namely, consider the Volterra operators Ω± ∈ B(H) realizing the following DelsarteDarboux transmutations: ˜ 1 Ω± = Ω± L 1 , L
˜ 2 Ω± = Ω± L 2 . L
(91)
Here we put
∂/∂x1 0 L1 (x; ∂) : = , 0 ∂/∂x2 ∂2 ∂2 ∂ ∂x21 ± ∂x22 − α2 (x2 ) L2 : = 1 + ∂t 0
(92)
0 ∂2
∂x21
±
∂2
∂x22
− α1 (x1 )
,
where αj ∈ W21 (R; C), j = 1, 2, are some given functions. It is evident that operators (92) are commuting to each other. then, if the operators Ω± ∈ B(H) exist and satisfy (91), the following commutation condition ˜ 1, L ˜ 2] = 0 [L
(93)
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547
holds, as was exactly claimed above and effectively exploited before in [18, 19]. Recall now that for the operators Ω± ∈ B(H) to exist they must satisfy additionally the kernel conditions (89) and ˜ 2,ext ⊗ 1)K ˆ ± (Ω) = (1 ⊗ L∗ )K ˆ ± (Ω), (L 2,ext
(94)
[L2 , Φ(Ω)] = 0,
where, as before, the operator Φ(Ω) ∈B∞ (H) is defined by (82) as Ω := 1+Φ(Ω).
(95)
Owing to the evident commutation condition (93) the set of equations (89) and (94) is ˆ + (Ω) ∈ H− ⊗ H− compatible giving rise to the expression like (86), where the kernel K satisfies the set of differential equations generalizing those from [18, 19]: ∂K+,12 ∂K+,12 ∂K+,11 ∂K+,11 + + u˜1 K+,21 = 0, + + u˜1 K+,22 = 0, ∂x1 ∂y1 ∂x1 ∂y1 ∂K+,21 ∂K+,21 ∂K+,22 ∂K+,22 + + u˜2 K+,11 = 0, + + u˜2 K+,12 = 0, ∂x2 ∂x1 ∂x2 ∂y2 ∂K+,11 ∂2 ∂2 ∂2 ∂ u˜1 K+,21 = + [( 2 − 2 ) ± ( 2 ∂x2 ∂t ∂x1 ∂y1 ∂x2 +(α2 (x2 ) − v˜2 (x))K+,11 ∂ u˜1 ∂K+,22 ∂2 ∂2 ∂2 ± K+,21 = + [( 2 − 2 ) ± ( 2 ∂x2 ∂t ∂x1 ∂y1 ∂x2 +(α1 (x1 ) − v˜1 (x))K+,22 , ∂ u˜1 ∂K+,12 ∂2 ∂2 ∂2 ∓2 K+,22 = + [( 2 − 2 ) ± ( 2 ∂x2 ∂t ∂x1 ∂y1 ∂x2 +(α1 (x1 ) − v˜2 (x))K+,22 . ∂ u˜2 ∂K+,21 ∂2 ∂2 ∂2 2 K+,22 = + [( 2 − 2 ) ± ( 2 ∂x1 ∂t ∂x1 ∂y1 ∂x2 +(α2 (x2 ) − v˜1 (x))K+,11 . ±
−
∂2 )]K+,11 ∂y22
−
∂2 )]K+,22 ∂y22
−
∂2 )]K+,12 ∂y22
−
∂2 )]K+,21 ∂y22 (96)
Moreover, the following conditions (0)
u˜1 (x) = −K+,12 |y=x ,
(0)
u˜2 (x) = −K+,21 |y=x ,
(97)
v˜2 (x)|x1 =−∞ = α2 (x2 ), v˜1 (x)|x2 =−∞ = α1 (x1 )
ˆ + (Ω), where we take into account the singular series hold for all x ∈ R2 and y ∈suppK expansion p(K+ ) X (s) (s−1) ˆ K+ δ (1) (98) K+ (Ω) = s=0
σx
for some finite integer p(K+ ) ∈ Z+ with respect to the Dirac function δσx(1) : W2q (R2 ; C) → R, q ∈ Z+ , and its derivatives, having the support (see [37], Chapter 3) coinciding (1) with the closed cycle σx ∈ K(R2 ).
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Remark 3.1. Concerning the special case (88) discussed before in [18, 19], one gets easily (1) ˆ + (Ω). It was also that p(K+ ) = 1 and σx = ∂(∩j=1,2 {y ∈ R2 :< y −x, γj >= 0}) ⊂suppK shown before that equations like (??) and (97) possess solutions if the Gelfand-LevitanMarchenko equation (52) does. Making use also of the exact forms of operators L1 and L2 ∈ L(H), one obtains easily from (89) and (94) the corresponding set of differential equations for components of the ˆ kernel Φ(Ω) ∈ H− ⊗ H− : ∂Φ12 ∂Φ12 ∂Φ11 ∂Φ11 + = 0, + = 0, ∂x1 ∂y1 ∂x1 ∂y1 ∂Φ21 ∂Φ21 ∂Φ22 ∂Φ22 + = 0, + = 0, ∂x2 ∂y2 ∂x2 ∂y2 ∂Φ11 ∂t ∂Φ12 ∂t ∂Φ21 ∂t ∂Φ22 ∂t
∂2 ±( 2 ∂x2 ∂2 ±( 2 ∂x2 ∂2 +( 2 ∂x1 ∂2 +( 2 ∂x1
− − − −
∂2 )Φ11 + (α2 (y2 ) − α2 (x2 ))Φ11 ∂y22 ∂2 )Φ12 + (α1 (y1 ) − α2 (x2 ))Φ12 ∂y22 ∂2 )Φ21 + (α2 (y2 ) − α1 (x1 ))Φ21 ∂y12 ∂2 )Φ22 + (α1 (y1 ) − α1 (x1 ))Φ22 ∂y12
(99)
= 0, = 0, = 0, =0
for all (x, y) ∈ R2 × R2 . The obtained above equations (99) generalize those found in [18, 19] and used for exact integration of the well known Devey-Stewartson differential equation [40, 38, 10] and finding so called soliton-like solutions. Concerning our generalized case, the kernel (98) is a solution to the following Gelfand-Levitan-Marchenko type equations: Z (0)
K+ (x; y) + Φ(0) (x; y) + +
Z
(1)
(2)
(1)
K+ (x; ξ)Φ(0) (ξ; y)dσx(1) = 0,
(1)
+ Φ (x; y) + +
K+ (x; ξ)Φ(0) (ξ; y)dξ
(1)
σx
(1) K+ (x; y)
(0)
(2) (1) (1) S+ (σx ,σ∞ )
Z
Z
(2) (1) (1) S+ (σx ,σ∞ )
(100)
(0)
K+ (x; ξ)Φ(1) (ξ; y)dξ
(1)
(1) σx
K+ (x; ξ)Φ(1) (ξ; y)dσx(1) = 0,
(1)
where y ∈ S+ (σx , σ∞ ) for all x ∈ R2 and, by definition, ˆ Φ(Ω) := Φ(0) + Φ(1) δσx(1)
(101)
is the kernel expansion corresponding to (98). Since the kernel (101) is singular, the differential equations (99) must be treated naturally in the distributional sense [37].
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Taking into account the exact forms of ”dressed” differential operators Lj ∈ L(H), j = 1, 2, given by (85) and (90) one gets easily that the commutativity condition (93) ˜ j ∈ L(H), j = 1, 2, being equivalent to the mentioned before gives rise to that of L Devey-Stewartson dynamical system d˜ u1 /dt = −(˜ u1,xx + u˜1,yy ) + 2(˜ v1 − v˜2 ),
(102)
d˜ u2 /dt = u˜2,xx + u˜2,yy + 2(˜ v2 − v˜1 ), v˜1,x = (˜ u1u˜2 )y , v˜2,x = (˜ u1u˜2 )x
on a functional infinite-dimensional manifold Mu ⊂ S(R2 ; C). The exact soliton-like solu(1) tions to (102) are given by expressions (97), where the kernel K+ (Ω) solves the second linear integral equation of (100). On the other hand, there exists the exact expression (31) which solves the set of ”dressed” equations ˜ (0) (η) = 0, L ˜ (0) (η) = 0. ˜ 1ψ ˜ 2ψ L (ρ)
(103)
(ρ)
(2)
(1)
(1)
Since the kernels Ω(λ, µ) ∈ L2 (Σ; C)⊗L2 (Σ; C), for λ, µ ∈ Σ, (t; x) ∈ MT ∩ S+ (σx , σ∞ ) are given by means of exact expressions (29), one can find via simple calculations the corresponding analytical expression for the functions (˜ u1 , u ˜2) ∈ Mu , solving the dynamical system (102). This procedure is often called the Darboux type transformation and was recently extensively used as a particular case of the construction above in [26] for finding soliton-like solutions to the Devey-Stewartson (102) and the related two-dimensional modified Korteweg-de Vries flows on Mu . Moreover, as it can be observed from the technique used for constructing the Delsarte-Darboux transmutation operators Ω± ∈ B(H), the set of solutions to (102) obtained by means of Darboux type transformations coincides completely with the corresponding set of solutions obtained by means of solving the related set of Gelfand-Levitan-Marchenko integral equations (99) and (100).
3.3
An affine de Rham-Hodge differential complex and related generalized self-dual Yang-Mills flows
Consider the following set of affine differential expressions in H := C 1 (Rm+1 ; H), H := L2 (Rm ; CN ) : ∂ ∂ Li (λ) := 1 −λ + Ai (x; p|t), (104) ∂pi ∂xi where x ∈ Rm , (t, p) ∈ Rm+1 , matrices Ai ∈ C 1 (Rm+1 ; S(Rm ; EndCN )), i = 1, m, and a parameter λ ∈ C. One can easily now construct an exact affine de Rham-Hodge differential complex on MT := Rm+1 ×Rm as dL(λ)
dL(λ)
H → Λ(MT ; H) → Λ1 (MT ; H) → dL(λ) ... → Λ2m+1 (MT ; H) → 0,
(105)
where, by definition, the differentiation dL(λ) := dt ∧ B(λ) +
m X i=1
dpi ∧ Li (λ)
(106)
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and the affine matrix n(B)+q
B(λ) := ∂/∂t −
X
Bs (x; p|t)λn(B)−s
(107)
s=0
with matrices Bs ∈ C 1 (Rm+1 ; S(Rm ; EndCN )), s = 0, n(B) + q, n(B), q ∈ Z+ . The affine complex (105) will be exact for all λ ∈ C iff the following generalized self-dual Yang-Mills equations [46] ∂Ai /∂pj − ∂Aj /∂pi − [Ai , Aj ] = 0, ∂Ai /∂xj − ∂Aj /∂xi = 0, ∂B0 /∂xi = 0, ∂Bn(B)+q /∂pi = 0, ∂Bs /∂xi = ∂Bs−1 /∂pi + [Ai , Bs−1 ] = 0, ∂Ai /∂t + ∂Bn(B) /∂pi − ∂Bn(B)+1 /∂xi + [Ai , Bn(B) ] = 0
(108)
hold for all i, j = 1, m and s = 0, n(B) ∨ n(B) + q, n(B) + 2. Assume now that the conditions (108) are satisfied on MT . Then, making the change C ∋λ → ∂/∂τ : H → H, τ ∈ R, one finds the following set of pure differential expressions ∂ ∂2 − + Ai (x; p|t), ∂pi ∂τ ∂xi n(B)+q X ∂ : = ∂/∂t − Bs (x; p|t)( )n(B)−s , ∂τ s=0
Li(τ ) : = 1 B(τ )
(109)
where matrices Ai , i = 1, m, and Bs , s = 0, n(B) + q, do not depend on the variable τ ∈ R. By means of the operator expressions (109) one can now naturally construct a new differential complex related with that of (105): d
d
L L H(τ ) → Λ(MT,τ ; H(τ ) ) → Λ1 (MT,τ ; H(τ ) ) → d...L → Λ2m+2 (MT,τ ; H(τ ) ) → 0,
(110)
where, by definition, H(τ ) := C 1 (Rm+1 ; H(τ ) ), H(τ ) := L2 (Rm × Rτ ; CN ) and dL := dt ∧ B(τ ) +
m X
dpi ∧ Li(τ ) .
i=1
(111)
Owing to the condition (108) the following lemma holds. Lemma 3.2. The differential complex (110) is exact. Therefore, one can construct the standard de Rham-Hodge type Hilbert space decomposition HΛ (MT,τ ) :=
k=2m+2
⊕
k−0
HΛk (MT,τ )
(112)
as well as the corresponding Hilbert-Schmidt rigging HΛ,+ (MT,τ ) ⊂ HΛ (MT,τ ) ⊂ HΛ,− (MT,τ ).
(113)
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551
Making use now of the results obtained in Subsection 1.5, one can define the Delsarte ˜ 0(τ ) ⊂ H(τ )− , related with the exact complex (110): closed subspaces H0(τ ) and H (0)
(0)
0 H0(τ ) : = {ψ(τ ) (ξ) ∈ HΛ,− (MT,τ ) : Lj(τ ) ψ(τ ) (ξ) = 0, (0)
(0)
(0)
(114)
(0)
0 B(τ ) ψ(τ ) (ξ) = 0, ψ(τ ) (ξ)|Γ = 0, ψ(τ ) (ξ)|t=0 = eλτ ψλ (η) ∈ HΛ,− (MRm ,τ ), (0)
(m)
Lj (λ)ψλ (η) = 0, ξ = (λ; η) ∈ Σ : = C × ΣC }, ˜ (0) (ξ) ∈ H0 (MT,τ ) : L ˜ (0) ˜ 0(τ ) : = {ψ ˜ (0) ψ H (τ ) Λ,− j(τ ) (τ ) (ξ) = 0,
˜ (0) (ξ) = 0, ψ ˜ (0) (ξ)| ˜ = 0, ψ ˜ (0) (ξ)|t=0 = eλτ ψ ˜ (0) (η) ∈ H0 (MRm ,τ ), ˜ (τ ) ψ B (τ ) (τ ) (τ ) λ Λ,− Γ
˜ (0) (η) = 0, ξ = (λ; η) ∈ Σ : = C × Σ(m) }, ˜ j (λ)ψ L λ C
˜ ⊂ MT,τ are some smooth hyper-surfaces. Similar expressions correspond where Γ and Γ ∗ ∗ ˜∗ to the adjoint closed subspaces H0(τ ) and H0(τ ) ⊂ Hτ,− : ˜ 0(τ ) : = {ϕ(0) (ξ) ∈ H0 (MT,τ ) : L∗ ϕ(0) (ξ) = 0, H Λ,− j(τ ) (τ ) (τ )
(0) B(τ ) ϕ(τ ) (ξ) (0) L∗j (λ)ϕλ (η)
= 0,
(0) ϕ(τ ) (ξ)|Γ
= 0,
(0) ϕ(τ ) (ξ)|t=0
¯ −λτ
=e (m)
(0) ϕλ (η)
(115)
0 ∈ HΛ,− (MRm ,τ ),
= 0, ξ = (λ; η) ∈ Σ : = C × ΣC },
(0) (0) 0 ˜ 0(τ ) : = {˜ ˜∗ ϕ H ϕ(τ ) (ξ) ∈ HΛ,− (MT,τ ) : L j(τ ) ˜ (τ ) (ξ) = 0,
¯ (0) (0) (0) (0) 0 ˜∗ ϕ ˜ λ (η) ∈ HΛ,− (MRm ,τ ), ˜ (τ ) (ξ)|t=0 = e−λτ ϕ ˜ (τ ) (ξ)|Γ˜ = 0, ϕ B (τ ) ˜ (τ ) (ξ) = 0, ϕ
(0) (m) ˜ ∗j (λ)˜ L ϕλ (η) = 0, ξ = (λ; η) ∈ Σ : = C × ΣC }.
Based on the closed subspaces (115) and (114), one can suitably define the Darboux ˜ (t,x;τ ) (η, ξ) ∈ L(ρ) (Σ(m) ; C) ⊗ L(ρ) (Σ(m) ; C), η, ξ ∈ Σ(m) , and further, the type kernel Ω 2 2 C C C corresponding Delsarte transmutation mappings Ω± ∈ B(H(τ ) ). Namely, assume that the following conditions (0) ˜ (0) (ξ) · Ω ˜ −1 ˜ (t0 ,p0, x0 ;τ ) Ω (116) ψ (ξ) := ψ (τ ) (t,p;x;τ )
(τ )
(m)
for any ξ ∈ C×ΣC
hold, where
˜ (t,x;τ ) (µ, ξ) := Ω
Z
σ(t;x;τ )
¯ ˜ (0) (η)dx ∧ dp ∧ dt], ˜ (2m+1) [e−λτ Ω ϕ ˜ (0) (µ), eλτ ψ (τ )
¯ (2m+1) −λτ Z˜(τ ) [e ϕ ˜ (0) (µ),
m X
(0)
m
˜ (ξ(i) ) ∧ dτ ∧ dx ∧ dpj ] eλτ ψ
i=1 m X
¯ ˜ (2m) [e−λτ : = dΩ ϕ ˜ (0) (µ), (τ )
i=1
j6=i
(0)
m
˜ (ξ(i) ) ∧ dτ ∧ dx ∧ dpj ], eλτ ψ j6=i
(117)
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and, similarly to (24), there holds the relationship ¯
< d∗L˜ϕ ˜ (0) (µ)e−λτ , ∗ −1
m X i=1
¯ −λτ
(0)
= < (∗) ϕ ˜ (µ)e (2m+1)
+dZ˜(τ )
m
(0)
˜ (ξ(i) )dt ∧ dτ ∧ dx ∧ dpj > eλτ ψ
(118)
j6=i
, dL˜(
m X
¯
[˜ ϕ(0) (µ)e−λτ ,
i=1 m X i=1
m
(0)
˜ (ξ(i) )dt ∧ dτ ∧ dx ∧ dpj ) > eλτ ψ j6=i m
(0)
˜ (ξ(i) )dt ∧ dτ ∧ dx ∧ dpj ], eλτ ψ j6=i
(2m+1) defining the exact (2m + 1)-form Z˜(τ ) ∈ Λ2m+1 (MT,τ ; C). Compute now the Delsarte transformed differential expressions
˜ ˆ ˆ −1 L ˆ −1 ˜ ˆ Lj(τ ) := Ω (τ )± j(τ ) Ω(τ )± , B(τ ) := Ω(τ )± B(τ ) Ω(τ )±
(119)
for any j = 1, m, where, by definition, 2 ˜ j(τ ) : = 1 ∂ − ∂ + A¯j , L ∂pj ∂τ ∂xj n(B)+q
X
B(τ ) : = ∂/∂t −
s=0
(120)
¯s ( ∂ )n(B)−s B ∂τ
¯s ∈ EndCm , s = 0, n(B) + q, being with all matrices A¯j ∈ EndCm , j = 1, m, and B constant. This implies that, in particular, the commuting relationships ˜ j(τ ) , L ˜ i(τ ) ] = 0, [L ˜ j(τ ) , B ˜ (τ ) ] = 0 [L
(121)
hold for all i, j = 1, m. Owing to the expressions (119) the induced commuting relationships [Lj(τ ) , Li(τ ) ] = 0, [Lj(τ ) , B(τ ) ] = 0 (122) evidently hold, coinciding exactly with relationships (108). Moreover, reducing our differential expressions (119) upon functional subspaces H(λ) := eλτ H, λ ∈ C, one easily obtains the set of affine differential expressions (104) and (107). Write down now the respective reduced Delsarte transmutation operators Z Z ˜ −1 ˆ Ω± = 1− dρΣ(m) (ν) dρΣ(m) (η)ψ (0) (λ; ν)Ω (t0 ,p0 ;x0 ) (λ; ν, η) (m)
× (2m)
Z
ΣC
(2m+1)
S±
(m)
C
(2m)
ΣC
(2m) ) 0 ,p0 ;x0 )
C
Z˜ (2m+1) [˜ ϕ(0) (λ; ν), (·)
(σ(t,p;x) ,σ(tt
m X i=1
m
dt ∧ dx ∧ dpj ], j6=i
(2m)
(123)
where σ(t,p;x) and σ(tt0 ,p0 ;x0) ∈ K(MT ) are some 2m-dimensional closed singular simplexes, and by definition, Z˜ (2m+1) [˜ ϕ(0) (λ; ν),
m X i=1
m ˜ (0) (λ; η(i) )dt ∧ dx ∧ ψ dpj ]
¯ (2m+1) −λτ : = Z˜(τ ) [e ϕ ˜ (0) (λ; ν),
j6=i
m X i=1
(0)
m
˜ (λ; η(i) )dτ ∧ dt ∧ dx ∧ dpj ]|dτ =0 , eλτ ψ j6=i
A.M. Samoilenko et al. / Central European Journal of Mathematics 3(3) 2005 529–557
˜ (t,p;x) (λ; ν, η) := Z˜ (2m+1) [˜ dΩ ϕ(0) (λ; ν),
m X i=1
m ˜ (0) (λ; η(i) )dt ∧ dx ∧ ψ dpj ], j6=i
553
(124) (m)
since the (2m + 1)-form (124) is owing to (118) also exact for any (λ; ν, η) ∈ C × (ΣC × (m) ΣC ). Thus, the operator expression (123) if applied to the operators (120) reduced upon the functional subspace H(λ) ≃ H, λ ∈ C, gives rise to the differential expressions ˜ ˆ −1 ˆ ˆ −1 ˜ ˆ Lj (λ) := Ω ± Lj (λ)Ω± B(λ) := Ω± B(λ)Ω± ,
(125)
where Lj (λ)H(λ) = Lj(τ ) H(λ) , B(λ)H(λ) = B(τ ) (λ)H(λ) , j = 1, m, coinciding with affine differential expressions (104) and (107). Concerning application of these results to finding exact soliton-like solutions to self-dual Yang-Mills equations (108), it is enough to mention that the relationship (116) reduced upon the subspace H(λ) ≃ H, λ ∈ C, gives rise to the following mapping: ˜ (0) (λ; η) · Ω ˜ −1 Ω ˜ ψ (0) (λ; η) := ψ (126) (t,p;x) (t0 ,p0 ;x0 ) , ˜ (t,p;x;τ ) (λ; η, ξ) ∈ L(ρ) (Σ(m) ; C)⊗L(ρ) (Σ(m) ; C), η, ξ ∈ Σ(m) , for all (t, p; x) ∈ where kernels Ω 2 2 C C C (m) MT and λ ∈ C. Since the element ψ (0) (λ; η) ∈ H− for any (λ; ξ) ∈ C×ΣC satisfies the set of differential equations Li (λ)ψ (0) (λ; η) = 0, B(λ)ψ (0) (λ; η) = 0,
(127)
for all i = 1, m, from (126) and (127) one finds easily exact expressions for the corresponding matrices Aj and Bs ∈ C 1 (R×Rm+1 ; S(Rm ; EndCN )), j = 1, m, s = 0, n(B) + q, satisfying the self-dual Yang-Mills equations (108). Thereby, the following theorem is stated. Theorem 3.3. The integral expressions (123) in H are the Delsarte transmutation operators corresponding to the affine differential expressions (104), (108) and constant operators n(B)+q X ∂ ∂ ˜ ¯ ˜ ¯s λn(B)−s Li (λ) := 1 −λ + A, B(λ) := ∂/∂t − B (128) ∂pi ∂xi s=0 for any λ ∈ C. The mapping (126) realizes the isomorphisms between the closed subspaces (0) (0) H0 : = {ψ (0) (λ; η) ∈ H− : dL(λ) ˜ ψ (λ; η) = 0, ψ (λ; η)|t=0
(129)
˜ (0) (λ; η) ∈ H− : d(0) ψ(λ; ˜ η) = 0, ψ ˜ (0) (λ; η)|t=0 ˜ 0 : = {ψ H ˜ L(λ)
(130)
=
(0) ψλ (η)
∈ H− , ψ (0) (λ; η)|Γ = 0, (λ; η) ∈ C ×
(m) ΣC }
and
˜ (0) (η) ∈ H− , ψ ˜ (0) (λ; η)| ˜ = 0, (λ; η) ∈ C×Σ(m) } =ψ λ Γ C
for any parameter λ ∈ C. Moreover, the expressions (126) generate the standard Darboux type transformations for the set of operators (128) and (104), (107) via the corresponding
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set of linear equations (127), thereby producing exact soliton-like solutions to the self-dual Yang-Mills equations (108). As a simple partial consequence from Theorem 3.2 one retrieves all of the results obtained before in [46], where the Delsarte-Darboux mapping (126) was chosen completely a priori without any proof and motivation in the form of some affine gauge transformation. Results similar to the above can be with a minor change applied also to the affine differential de Rham-Hodge complex (105) with the exterior differentiation (106), where ni (L) ni (L) X X ∂ k+1 ∂ Li (λ) : = 1 −( aik λ ) + Aik λk , ∂pi ∂x i k=0 k=0 n(B)+q
˜ B(λ) : = ∂/∂t −
X
¯s λn(B)−s , B
(131)
s=0
or ni (L) ni (L) X (j) X ∂ k+1 ∂ −( aik λ ) + Aik λk , Li (λ) : = 1 ∂pi ∂x j k=0 k=0 n(B)+q
˜ B(λ) : = ∂/∂t −
X
¯s λn(B)−s , B
(132)
s=0
for i = 1, m, λ ∈ C. The case (131) was analyzed recently in [45] by means of the same affine gauge transformation that there was used before in [46]. Unfortunately, the results obtained there are too complicated and unwieldy, thus one needs to use more mathematically motivated, clear and less cumbersome techniques for finding DelsarteDarboux transformations and related soliton-like exact solutions.
Acknowledgment The authors cordially thank the Organizing Committee of the ”Geometry and Toplogy” conference held in Krynice, 2003, Poland, for the invitation to deliver a report on the generalized de Rham-Hodge theory aspects of the Delsarte type transmutations of differential operators, and useful discussions of results obtained. The third author thanks AGH for the patial support of a local grant. The last but not least our deep gratitude is addressed to the anonimous Referees whose important comments helped to correct some statements and favored in improvement of the final draft of the article.
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CEJM 3(3) 2005 558–577
General spectral flow formula for fixed maximal domain ∗ Bernhelm Booss–Bavnbek1† , Chaofeng Zhu2‡ 1
Institut for Matematik og Fysik, Roskilde University, 4000 Roskilde, Denmark 2 Nankai Institute of Mathematics, Key Lab of Pure Mathematics and Combinatorics of Ministry of Education, Nankai University, Tianjin 300071, People’s Republic of China
Received 30 November 2004; accepted 26 April 2005 Abstract: We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory. c Central European Science Journals. All rights reserved.
Keywords: Spectral flow, Maslov index, elliptic boundary value problems MSC (2000): 58J30, 53D12
∗
This work was supported in part by The Danish Science Research Council, SNF grant 21-02-0446. The second author is partially supported by FANEDD 200215, 973 Program of MOST, Fok Ying Tung Edu. Funds 91002, LPMC of MOE of China, and Nankai University. † E-mail: [email protected] ‡ E-mail: [email protected]
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1
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Statement of the problem and main result
1.1 Statement of the problem Roughly speaking, the spectral flow counts the net number of eigenvalues changing from the negative real half axis to the non-negative one. The definition goes back to a famous paper by M. Atiyah, V. Patodi, and I. Singer [3], and was made rigorous by J. Phillips [23] for continuous paths of bounded self-adjoint Fredholm operators, by K.P. Wojciechowski [30] and C. Zhu and Y. Long [34] in various non-self-adjoint cases, and by B. BoossBavnbek, M. Lesch, and J. Phillips [7] in the unbounded self-adjoint case. We shall give a rigorous definition of spectral flow, most suitable for our purpose, below in Subsection 2.1 together with a review of its basic properties. For a definition of spectral flow admitting zero in the continuous spectrum, we refer to A. Carey and J. Phillips [13]. In various branches of mathematics one is interested in the calculation of the spectral flow of a continuous family of closed densely defined (not necessarily bounded) self-adjoint Fredholm operators in a fixed Hilbert space. We consider the following typical problem of this kind. Assumption 1.1. Let {As : C ∞ (M; E) → C ∞ (M; E)}s∈[0,1] be a family of formally selfadjoint linear elliptic differential operators of first order with continuously varying smooth coefficients over a smooth compact Riemannian manifold M with boundary Σ, acting on sections of a Hermitian vector bundle E over M. Let {Ps } be a continuous family of orthogonal pseudodifferential projections in L2 (Σ; E|Σ ). Define As,Ps to be the unbounded operator in L2 (M; E) with domain Ds := {x ∈ H 1 (M; E) | Ps (γ(x)) = 0},
(1)
where 1
γ : H 1 (M; E) → H 2 (Σ; E|Σ )
(2)
denotes the (continuous) trace map from the first Sobolev space over the whole manifold to the 12 Sobolev space over the boundary. (Note that in this paper the symbols x and y do not denote points of the underlying manifolds M or Σ, but points in Hilbert spaces, sections of vector bundles, etc., following the conventions of functional analysis and dynamical systems.) Assume that each Ps defines a self-adjoint elliptic boundary condition for As , i.e., As,Ps is a self-adjoint Fredholm operator for each s ∈ [0, 1]. Then the spectral flow sf{As,Ps ; s ∈ [0, 1]} or, shortly, sf{As,Ps } is well defined. As a spectral invariant it is essentially a quantum variable which one may not always be able to determine directly by eigenvalue calculations. As an alternative, one is looking for a classical method of calculating the spectral flow. There are two different approaches. One setting expresses the spectral flow (of a loop of Dirac operators on a closed manifold) as an integral over a 1-form induced by the heat kernel (for a review see [13]). The other setting is reduction to the boundary, i.e., one expresses the spectral flow (of a path of
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self-adjoint boundary value problems on a compact manifold with boundary) in terms of the intersection geometry of the solution spaces of the homogeneous differential equations and the boundary conditions. That is the approach we shall follow in this paper. Problem 1.2. Give a classical method of calculating the spectral flow of the family {As,Ps } by reduction to the boundary, i.e., a method not involving the determination of the spectrum near 0 and yielding an expression on Σ. The preceding spectral flow calculation problem is formulated for families by analogy with Bojarski’s Theorem for single operators which expresses the index (which is the difference between the multiplicities of the 0-eigenvalue of the original and the formally adjoint problem and so a priori a quantum or spectral invariant) of an elliptic operator over a closed partitioned manifold M = M− ∪Σ M+ by the index of the Fredholm pair of Cauchy data spaces from two sides along the hypersurface Σ (which are classical objects, see Bojarski [4] and Booss and Wojciechowski [10, Chapter 24]).
1.2 General functional analytic setting and announcement of the General Spectral Flow Formula Now we translate our problem into a functional analytic setting. For any such family there are three geometrically defined relevant Hilbert spaces of global sections which remain fixed under variation of the coefficients of the operators and under variation of the boundary conditions: L2 (M; E),
H01 (M; E),
and H 1 (M; E).
(3)
Here H01 (M; E) denotes the closure of C0∞ (M \Σ; E) in the first Sobolev space H 1 (M; E), where C0∞ (M \ Σ; E) denotes the smooth sections with support in the interior of M \ Σ. 1 Since the trace map γ : H 1 (M; E) → H 2 (Σ; E|Σ ) is continuous, we have H01 (M; E) = ker γ, i.e., the space H01 (M; E) consists exactly of the elements of H 1 (M; E) which vanish on the boundary Σ . For each s ∈ [0, 1], we shall denote the unbounded operator As acting in L2 (M; E) with domain H01 (M; E) also by As . Since the differential operator As is elliptic, the unbounded operator As is closed by G˚ arding’s inequality kxkH 1 (M ;E) ≤ C kxkL2 (M ;E) + kAs xkL2 (M ;E) for x ∈ H01 (M; E) . (4)
Denote by dom(A) the domain of an operator A, by A∗ the adjoint operator of A, and Dmax (A) := dom(A∗ ). (5) Since A is closed and symmetric, it follows that Dmax (A) = {x ∈ L2 (M; E) | Ax ∈ L2 (M; E)} with Ax taken in the distributional sense. For As formally self-adjoint, it follows immediately that H 1 (M; E) ⊂ Dmax (As ) and that As (with domain H01 (M; E)) is symmetric.
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In local coordinates, we view each coefficient of {As } as a continuous map which assigns to s ∈ [0, 1] a smooth section. Then the continuity of the curve {As }s∈[0,1] in the sense of continuously varying coefficients implies the continuity of the curve [0, 1] ∋ s 7→ A∗s |H 1 (M ;E) ∈ B(H 1 (M; E), L2 (M; E)) ,
(6)
as a curve of bounded operators from H 1 (M; E) to L2 (M; E). We denote by Qs : L2 (Σ; E|Σ ) → L2 (Σ; E|Σ ) the Calder´ on projection. It is a projection ∗ onto the Cauchy data space of As which is defined as the L2 -closure of γ(ker(A∗s |H 1 (M ;E) ). It can be described as a pseudodifferential operator, e.g., when continuing As to an elliptic f ⊃ M, see R.T. Seeley [29, Sections 4 and 8] and [10, operator on a closed manifold M Chapter 12]. For an alternative canonical construction based on a natural boundary value problem and avoiding the choices of closing the manifold and continuing the operator, see B. Himpel, P. Kirk, and M. Lesch [16, Section 3] and recent joint work of the authors with M. Lesch [8]. For each s ∈ [0, 1], there is a natural (strong) symplectic form ωs on the quotient space Dmax (As )/H01(M; E) induced by Green’s form of As as ωs (γ(x), γ(y)) := hA∗s x, yi − hx, A∗s yi ,
x, y ∈ Dmax (As ).
(7)
Here γ denotes the natural projection Dmax (As ) → Dmax (As )/H01(M; E) . Identifying the quotient space Dmax (As )/H01 (M; E) with a subspace of the Sobolev (distribution) space H −1/2 (Σ; E|Σ ), we obtain that this γ extends the Sobolev trace map of (2). A rigorous definition of symplectic structures and Lagrangian subspaces will be given below in Subsection 2.2. For our formally self-adjoint differential operators of first order, we have an explicit description of the form in (7), restricted to H 1 (M; E) , by Stokes’ Theorem Z ωs (γ(x), γ(y)) = − hσ1 (As )(·, dt) x|Σ , y|ΣidvolΣ , (8) Σ
where σ1 (As )(·, dt) denotes the principal symbol of As at the boundary, taken in inner (co-)normal direction dt. Notice that we do not require that the manifold M is orientable: for our application of Stokes’ Theorem it suffices that any collar neighborhood of Σ in M is oriented by the normal structure. Then the form ωs |H 1 (M ;E) of (8) extends to a (strong) symplectic structure ω s on L2 (Σ; E|Σ ). One can show that ωs |H 1 (M ;E)/H01 (M ;E) is a weak (but not strong) symplectic form on the Hilbert space H 1 (M; E)/H01 (M; E) ∼ = 1 H 2 (Σ; E|Σ ) (cf. Booss and Zhu [11, Remark 1.6b]). We have H 1 (M; E) = Dmax (As ) if and only if dim M = 1. For higher dimensional case, the strict inclusion H 1 (M; E) ⊂ Dmax (As ) and the weakness of ωs |H 1 (M ;E) causes technical difficulties. However, we still have the following theorem (cf. Theorem 0.1 of [11]).
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Theorem 1.3 (General Spectral Flow Formula). Let {As }s∈[0,1] and {Ps }s∈[0,1] be operator families like in Assumption 1.1. We assume that {ker Ps }s∈[0,1] is a continuous family of Lagrangian subspaces in (H, ω s ). If As satisfies weak inner UCP, i.e., ker As = {0} for each s ∈ [0, 1], we have: (a) The family {As,Ps }s∈[0,1] of closed self-adjoint Fredholm operators on X is a continuous family (in the gap norm, or equivalently, in the projection norm). (b) The Cauchy data spaces im Qs are Lagrangian subspaces in the weak symplectic Hilbert 1 1 space (H 2 (Σ; E|Σ ), ω s ) and form a continuous family in H 2 (Σ; E|Σ ) for s ∈ [0, 1]. (c) Finally, the following formula holds: sf{As,Ps } = − Mas{ker Ps , im Qs },
(9)
where the spectral flow sf and the Maslov index Mas are defined by Definitions 2.1 and 2.11 below respectively. Remark 1.4. (a) The General Spectral Flow Formula contains and generalizes all previously known spectral flow formulae, as given by M. Morse [21], W. Ambrose [1], J.J. Duistermaat [14], A. Floer [15], P. Piccione and D.V. Tausk [24] and [25], and C. Zhu [32] and [33] for the 1-dimensional setting of the study of geodesics, and for the higher dimensional setting the formulae given by T. Yoshida [31], L. Nicolaescu [22], S.E. Cappell, R. Lee, and E.Y. Miller [12], B. Booss, K. Furutani, and N. Otsuki [5] and [6], and P. Kirk and M. Lesch [18]. (b) The main difference to [5] and [6] is that we admit varying maximal domain and varying Fredholm domain. The main difference to [18] is that we admit more general operators than Dirac type operators with constant coefficients in normal direction close to the boundary. (c) The proof of the above theorem is rather technical and complicated. In this review article, we only prove the following fixed maximal domain case which completely covers all above cited one-dimensional cases (cf. Corollary 2.14 in [11]). Moreover, it contains [5] and [6] and generalizes it to varying Fredholm domains, and contains [18] for the case of fixed maximal domain and generalizes it under that restriction to more general operator families.
1.3 Statement of the result for fixed maximal domain Let X be a Hilbert space, and Dm ⊂ Dmax be two dense linear subspaces of X. Let {As }s∈[0,1] be a family of symmetric densely defined operators in C(X) with domain dom(As ) = Dm . Here we denote by C(X) all closed operators in X. Assume that dom(A∗s ) = Dmax , i.e., the domain of the maximal symmetric extension A∗s of As is independent of s. ´ We recall from [5] (see also B. Lawruk, J. Sniatycki, and W.M. Tulczyjew [19] for early investigation of symplectic structures and boundary value problems) for each s ∈ [0, 1]:
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(1) The space Dmax is a Hilbert space with the graph inner product hx, yiGs := hx, yiX + hA∗s x, A∗s yiX
for x, y ∈ Dmax .
(10)
(2) The space Dm is a closed subspace in the graph norm and the quotient space Dmax /Dm is a strong symplectic Hilbert space with the (bounded) symplectic form induced by Green’s form ωs (x + Dm , y + Dm ) := hA∗s x, yiX − hx, A∗s yiX
for x, y ∈ Dmax .
(11)
(3) If As admits a self-adjoint Fredholm extension As,Ds := A∗s |Ds with domain Ds , then the natural Cauchy data space (ker A∗s + Dm )/Dm is a Lagrangian subspace of (Dmax /Dm , ωs ) . (4) Moreover, self-adjoint Fredholm extensions are characterized by the property of the domain Ds that (Ds + Dm )/Dm is a Lagrangian subspace of (Dmax /Dm , ωs ) and forms a Fredholm pair with (ker A∗s + Dm )/Dm . (5) We denote the natural projection (which is independent of s) by γ : Dmax −→ Dmax /Dm . The main result of this paper is the following theorem which reproves parts of the preceding list. Theorem 1.5 (General Spectral Flow Formula for fixed maximal domain). We assume that on Dmax the graph norms induced by As , 0 ≤ s ≤ 1 are mutually equivalent. Then we fix a graph norm G on Dmax induced by A0 . Assume that {A∗s : Dmax → X} is a continuous family of bounded operators and each As is injective. Let {Ds /Dm } be a continuous family of Lagrangian subspaces of (Dmax /Dm , ωs ), such that each As,Ds is a Fredholm operator. Then: (a) Each Ds /Dm , γ(ker(A∗s ) is a Fredholm pair in Dmax /Dm . (b) Each Cauchy data space γ(ker A∗s ) is a Lagrangian subspace of (Dmax /Dm , ωs ) . (c) The family {γ(ker A∗s )} is a continuous family in Dmax /Dm . (d) The family As,Ds is a continuous family of self-adjoint Fredholm operators in C(X). (e) Finally, we have sf{As,Ds } = − Mas{γ(Ds ), γ(ker A∗s )}. (12)
2
Definition of spectral flow and Maslov index
2.1 Spectral flow, revisited and generalized Let X be a Hilbert space. For a self-adjoint Fredholm operator A ∈ C(X), there exists a unique orthogonal decomposition X = X + (A) ⊕ X 0 (A) ⊕ X − (A)
(13)
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such that X + (A), X 0 (A) and X − (A) are invariant subspaces associated to A, and A|X + (A) , A|X 0 (A) and A|X − (A) are positive definite, zero and negative definite respectively. We introduce vanishing, natural, or infinite numbers m+ (A) := dim X + (A), m0 (A) := dim X 0 (A), m− (A) := dim X − (A), and call them Morse positive index, nullity and Morse index of A respectively. For finitedimensional X, the signature of A is defined by sign(A) = m+ (A)−m− (A) which yields an integer. The APS projection QA (where APS stands for Atiyah-Patodi-Singer) is defined by QA (x+ + x0 + x− ) := x+ + x0 , for all x+ ∈ X + (A), x0 ∈ X 0 (A), x− ∈ X − (A). Let {As }, 0 ≤ s ≤ 1 be a continuous family of self-adjoint Fredholm operators. The spectral flow sf{As } of the family should be equal to m− (A0 ) − m− (A1 ) if dim X < +∞. We will generalize this definition to general X. For each t ∈ [0, 1], there exists a bounded open neighborhood Nt of 0 such that ∂Nt is of class C 1 , σ(At ) ∩ ∂Nt = ∅, and P (At , Nt ) is a finite rank projection. Here we denote the spectrum of a closed operator A by σ(A), and the spectral projection by Z 1 P (A, N) := − √ (A − zI)−1 dz 2π −1 ∂N if N is a bounded open subset of C with C 1 boundary and ∂N ∩σ(A) = ∅. The orientation of ∂N is chosen to make N stay on the left side of ∂N. Since the family {As }, 0 ≤ s ≤ 1 is continuous, there exists a δ(t) > 0 for each t ∈ [0, 1] such that σ(As ) ∩ ∂Nt = ∅, for all s ∈ (t − δ(t), t + δ(t)) ∩ [0, 1]. Then
P (As , Nt ) s∈(t−δ(t),t+δ(t))∩[0,1]
for fixed t ∈ [0, 1],
is a continuous family of orthogonal projections. By Lemma I.4.10 in Kato [17], they have the same rank. We denote by A(s, t) the operator As acting on the finite-dimensional space im P (As , Nt ). Since [0, 1] is compact, there exists a partition 0 = s0 < . . . < sn = 1 and tk ∈ [sk , sk+1 ], k = 0, . . . , n − 1 such that [sk , sk+1] ⊂ (tk − δ(tk ), tk + δ(tk )) for each k = 0, . . . , n − 1. Definition 2.1. The spectral flow sf{As } of the family {As }, 0 ≤ s ≤ 1 is defined by n−1 X sf{As } := m− A(sk , tk ) − m− A(sk+1 , tk ) .
(14)
k=0
After carefully examining the above definition, inspired by [23], we find that the necessary data for defining any spectral flow are the following:
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• a co-oriented bounded real 1-dimensional regular C 1 submanifold ℓ of C without boundary (we call such an ℓ admissible, and denote by ℓ ∈ A(C)); • a Banach space X; • and a continuous family of admissible operators As , 0 ≤ s ≤ 1 in Aℓ (X). Here we define A ∈ C(X) to be admissible with respect to ℓ, if there exists a bounded open neighborhood N of ℓ in C with C 1 boundary ∂N such that (i) ∂N ∩ σ(A) = ∅; (ii) N ∩ σ(A) ⊂ ℓ is a finite set; and (iii) Pℓ0 (A) := P (A, N) is a finite rank projection. Note that Pℓ0 (A) does not depend on the specific choice of N. We call νh,ℓ (A) := dim im Pℓ0 (A) the hyperbolic nullity of A with respect to ℓ. We denote by Aℓ (X) the set of closed admissible operators with respect to ℓ. It is an open subset of C(X). Similarly as before, we can define the spectral flow sf ℓ {As }. It counts the number of spectral lines of As coming from the negative side of ℓ to the non-negative side of ℓ. For the details, see [34]. √ Example 2.2. a) In the above self-adjoint case, ℓ = −1(−ǫ, ǫ) (ǫ > 0) with coorientation from left to right. Then a self-adjoint operator A is admissible with respect to ℓ if and only if A is Fredholm. b) Another important case is that ℓ = (1 − ǫ, 1 + ǫ) (ǫ ∈ (0, 1)) with co-orientation from downward to upward, and all As unitary. A unitary operator A is admissible with respect to ℓ if and only if A − I is Fredholm. The spectral flow has the following properties (cf. [23] and Lemma 2.6 and Proposition 2.2 in [34]). Proposition 2.3. Let ℓ ∈ A(C) be admissible and let {As }, 0 ≤ s ≤ 1 be a curve in Aℓ (X). Then the spectral flow sf ℓ {As } is well defined, and the following holds: (1) Catenation. Assume t ∈ [0, 1]. Then we have sf ℓ {As ; 0 ≤ s ≤ t} + sf ℓ {As ; t ≤ s ≤ 1} = sf ℓ {As ; 0 ≤ s ≤ 1}.
(15)
(2) Homotopy invariance. Let A(s, t), (s, t) ∈ [0, 1] × [0, 1] be a continuous family in Aℓ (X). Then we have sf ℓ {A(s, t); (s, t) ∈ ∂([0, 1] × [0, 1])} = 0.
(16)
(3) Endpoint dependence for Riesz continuity. Let Bsa (X), respectively C sa (X) denote the spaces of bounded, respectively closed self-adjoint operators in X. Let R : C sa →
Bsa (X) 1
A 7→ A(A2 + I)− 2
denote the Riesz transformation. Let As ∈ C sa (X) for s ∈ [0, 1]. Assume that {R(As )}, 0 ≤ s ≤ 1 is a continuous family. If m− (A0 ) < +∞, then m− (A1 ) < +∞ and we have sf{As } = m− (A0 ) − m− (A1 ). (17)
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(4) Product. Let {Ps } be a curve of projections on X such that Ps As ⊂ As Ps for all s ∈ [0, 1]. Set Qs = I − Ps . Then we have Ps As Ps ∈ Aℓ (im Ps ) ⊂ C(im Ps ), Qs As Qs ∈ Aℓ (im Qs ) ⊂ C(im Qs ), and sf ℓ {As } = sf ℓ {Ps As Ps } + sf ℓ {Qs As Qs }.
(18)
(5) Bound. For A ∈ Aℓ (X), there exists a neighborhood N of A in C(X) such that N ⊂ Aℓ (X), and for curves {As } in N with endpoints A0 =: A and A1 =: B, the relative Morse index Iℓ (A, B) := − sf ℓ {As , 0 ; ≤ s ≤ 1} is well defined and satisfies 0 ≤ Iℓ (A, B) ≤ νh,ℓ (A) − νh,ℓ (B).
(19)
(6) Reverse orientation. Let ℓˆ denote the curve ℓ with opposite co-orientation. Then we have sf ℓ {As } + sf ℓˆ{As } = νh,ℓ (A1 ) − νh,ℓ (A0 ). (20)
(7) Zero. Suppose that νh,ℓ (As ) is constant for s ∈ [0, 1]. Then sf ℓ {As } = 0. (8) Invariance. Let {Ts }s∈[0,1] be a curve of bounded invertible operators. Then we have sf ℓ {Ts−1 As Ts } = sf ℓ {As }.
(21)
Now we give a method of calculating the spectral flow of differentiable curves, inspired among others by J.J. Duistermaat [14] and J. Robbin and D. Salamon [28]. Definition 2.4. Let ℓ ∈ A(C) be admissible and {As }s∈[0,1] be a curve in Aℓ (X). (1) A crossing for As is a number t ∈ [0, 1] such that νh,ℓ (At ) 6= 0. (2) Set Ps = Pℓ0 As . A crossing t is called regular if dom(As ) = D fixed for s near t, As x is differentiable at s = t for all x ∈ D, and Pt A˙ t Pt is hyperbolic, i.e. νh,ℓ (Pt A˙ t Pt ) = 0, where A˙ s is the unbounded operator with domain D defined by d A˙ s x = As x ds for all x ∈ D. (3) A crossing t is called simple if it is regular and νh,ℓ (At ) = 1. Proposition 2.5 (cf. Theorem 4.1 of [34]). Let X be a Banach space and ℓ = √ −1(−ǫ, ǫ) (ǫ > 0) with co-orientation from left to right. Let As , −ǫ ≤ s ≤ ǫ (ǫ > 0), be a curve in Aℓ (X). Suppose that 0 is a regular crossing of As . Set P = Pℓ0 (A0 ), A = A0 and B = A˙ s |s=0. Assume that P (AB − BA)P = 0.
(22)
Then there is a δ ∈ (0, ǫ) such that νh,ℓ (As ) = 0 for all s ∈ [−δ, 0) ∪ (0, δ] and sf ℓ {As ; 0 ≤ s ≤ δ} = −m− (P BP ), +
sf ℓ {As ; −δ ≤ s ≤ 0} = m (P BP ).
(23) (24)
Here we denote by m+ (P BP ) (m− (P BP )) the total algebraic multiplicity of eigenvalues of P BP with positive (negative) imaginary part respectively.
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2.2 Symplectic functional analysis and Maslov index A main feature of symplectic analysis is the study of the Maslov index. It is an intersection index between a path of Lagrangian subspaces with the Maslov cycle, or, more generally, with another path of Lagrangian subspaces. The Maslov index assigns an integer to each continuous path of Fredholm pairs of Lagrangian subspaces of a fixed Hilbert space with continuously varying symplectic structures. Firstly we define symplectic Hilbert spaces and Lagrangian subspaces. Definition 2.6. Let H be a complex vector space. A mapping ω : H × H −→ C is called a (weak) symplectic form on H, if it is sesquilinear, skew-hermitian, and nondegenerate, i.e., (i) ω(x, y) is linear in x and conjugate linear in y; (ii) ω(y, x) = −ω(y, x); (iii) H ω := {x ∈ H | ω(x, y) = 0 for all y ∈ H} = {0}. Then we call (H, ω) a complex symplectic vector space. Definition 2.7. Let (H, ω) be a complex symplectic vector space. (a) The annihilator of a subspace λ of H is defined by λω := {y ∈ H | ω(x, y) = 0 for all x ∈ λ}. (b) A subspace λ is called isotropic, co-isotropic, or Lagrangian if λ ⊂ λω ,
λ ⊃ λω ,
λ = λω ,
respectively. (c) The Lagrangian Grassmannian L(H, ω) consists of all Lagrangian subspaces of (H, ω). Definition 2.8. Let H be a complex Hilbert space. A mapping ω : H × H → C is called a (strong) symplectic form on H, if ω(x, y) = hJx, yiH for some bounded invertible skew-adjoint operator J. (H, ω) is called a (strong) symplectic Hilbert space. Before giving a rigorous definition of the Maslov index, we fix the terminology and give a simple lemma. We recall: Definition 2.9. (a) The space of (algebraic) Fredholm pairs of linear subspaces of a vector space H is defined by 2 Falg (H) := {(λ, µ) | dim (λ ∩ µ) < +∞ and dim H/(λ + µ) < +∞} (25) with
index (λ, µ) := dim(λ ∩ µ) − dim(H/(λ + µ)).
(26)
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(b) In a Banach space H, the space of (topological) Fredholm pairs is defined by 2 F 2 (H) := {(λ, µ) ∈ Falg (H) | λ, µ, and λ + µ ⊂ H closed}.
(27)
We need the following well-known lemma (see, e.g., [11, Lemma 1.7]). Lemma 2.10. Let (H, ω) be a (strong) symplectic Hilbert space. Then (1) there is a 1-1 correspondence between the space U J = {U ∈ B H + , H − | U ∗ J|H − U = −J|H + } and L(H, ω) under the mapping U → L := G(U) (= graph of U), where H ± = √ H ∓ ( −1J) in the sense of the decomposition (13); (2) if U, V ∈ U J and λ := G(U), µ := G(V ), then (λ, µ) is a Fredholm pair if and only if U − V , or, equivalently, UV −1 − I is Fredholm. Moreover, we have a natural isomorphism ker(UV −1 − I) ≃ λ ∩ µ .
(28)
Definition 2.11. Let (H, h·, ·is), s ∈ [0, 1] be a continuous family of Hilbert spaces, and ωs (x, y) = hJs x, yis be a continuous family of symplectic forms on H, i.e., {As,0 } and {Js } are two continuous families of bounded invertible operators, where As,0 is defined by hx, yis = hAs,0x, yi0
for all x, y ∈ H.
Let {(λs , µs )} be a continuous family of Fredholm pairs of Lagrangian subspaces of (H, h·, ·is, ωs ). Then there is a continuous splitting √ √ H = Hs− ( −1Js ) ⊕ Hs+ ( −1Js )
(29)
√ associated to the self-adjoint operator −1Js ∈ B(H, h·, ·is) for each s ∈ [0, 1]. By Lemma 2.10, λs = Gs (Us ) and µs = Gs (Vs ) with Us , Vs ∈ U Js , where Gs denotes the graph associated to the splitting (29). We define the Maslov index Mas{λs , µs } by −1
Mas{λs , µs } = − sf ℓ {Us Vs },
(30)
where ℓ := (1 − ǫ, 1 + ǫ) with, ǫ ∈ (0, 1) and with upward co-orientation. Remark 2.12. For finite-dimensional H, constant µs = µ0 , and a loop {λs }, i.e., for λ0 = λ1 , we notice that Mas{λs , µs } is the winding number of the closed curve {det(Us−1 V0 )}s∈[0,1] . This is the original definition of the Maslov index as explained in Arnol’d, [2]. Lemma 2.13. The Maslov index is independent of the choice of the complete inner product of H.
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Proof. Let h·, ·is,k , s ∈ [0, 1] with k = 0, 1 be two continuous families of complete inner products of H. We define h·, ·is,t = (1 − t)h·, ·is,0 + th·, ·is,1 for each (s, t) ∈ [0, 1] × [0, 1]. Let (λs , µs ) be a continuous family of Fredholm pairs of Lagrangian subspaces of (H, ωs ). For each inner product h·, ·is,t, we denote by Us,t and Vs,t the associated generated ”unitary” operators of λs and µs respectively. We also denote by Mast the Maslov index defined with h·, ·is,t for each t ∈ [0, 1]. By Proposition 2.3 we have Mas0 {λs , µs } − Mas1 {λs , µs }
−1
−1
= − sf ℓ {Us,0Vs,0 } + sf ℓ {Us,1 Vs,1 }
−1 = − sf ℓ {Us,t Vs,t ; (s, t) ∈ ∂ [0, 1] × [0, 1] }
= 0.
Now we give a method of using the crossing form to calculate Maslov indices (cf. [14], [28], [5, Theorem 2.1]; for a full proof of the following Proposition see [33, Corollary 3.1]). Let λ = {λs }s∈[0,1] be a C 1 curve of Lagrangian subspaces of H. Let W be a fixed Lagrangian complement of λt . For v ∈ λt and |s−t| small, define w(s) ∈ W by v +w(s) ∈ λs . The form d Q(λ, t) := Q(λ, W, t)(u, v) = |s=t ω(u, w(s)), ∀u, v ∈ λt (31) ds is independent of the choice of W . Let {(λs , µs )}, 0 ≤ s ≤ 1 be a curve of Fredholm pairs of Lagrangian subspaces of H. For t ∈ [0, 1], the crossing form Γ(λ, µ, t) is a quadratic form on λt ∩ µt defined by Γ(λ, µ, t)(u, v) = Q(λ, t)(u, v) − Q(µ, t)(u, v),
∀u, v ∈ λt ∩ µt .
(32)
A crossing is a time t ∈ [0, 1] such that λt ∩ µt 6= {0}. A crossing is called regular if Γ(λ, µ, t) is nondegenerate. It is called simple if it is regular and λt ∩µt is one-dimensional. Proposition 2.14. Let (H, ω) be a symplectic Hilbert space and {(λs , µs )}, 0 ≤ s ≤ 1 be a C 1 curve of Fredholm pairs of Lagrangian subspaces of H with only regular crossings. Then we have X Mas{λ, µ} = m+ (Γ(λ, µ, 0)) − m− (Γ(λ, µ, 1)) + sign(Γ(λ, µ, t)). (33) 0
3
Symplectic analysis of symmetric operators
3.1 Local stability of weak inner UCP Let X be a complex Hilbert space and A ∈ C(X) a linear, closed, densely defined operator in X. We assume that A is symmetric, i.e., A∗ ⊃ A where A∗ denotes the adjoint operator.
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We denote the domains of A by Dm (the minimal domain) and of A∗ by Dmax (the maximal domain). Definition 3.1. Let X be a Hilbert space and A ∈ C(X) with dom A = Dm and A∗ ⊃ A. We shall say that the operator A satisfies the weak inner Unique Continuation Property (UCP) if ker A = {0}. It is well known that weak UCP and weak inner UCP can be established for a large class of Dirac type operators, see the first author with Wojciechowski [10, Chapter 8], and the first author with M. Marcolli and B.-L. Wang [9]. However, it is not valid for all linear elliptic differential operators of first order as shown by one of the Pli´s counterexamples [26]. Moreover, one has various quite elementary examples of linear and nonlinear perturbations which invalidate weak inner UCP for Dirac operators. Two such examples are listed in [9]. In the same paper, however, it was shown that weak UCP is preserved under certain ‘small’ perturbations of Dirac type operators. Here we show an elementary result, namely the local stability of weak inner UCP. Lemma 3.2. Let X be a Hilbert space. Let As ∈ C(X), 0 ≤ s ≤ 1 be a family of symmetric operators with dom As = Dm and dom A∗s = Dmax independent of s. Assume that {A∗s : Dmax → X} is a continuous curve of bounded operators, where the norm on Dmax is the graph norm induced by A∗0 . If A0 satisfies weak inner UCP and there exists a self-adjoint Fredholm extension A∗0 |D of A0 , then for all s ≪ 1 the operators A∗s are surjective and the operators As satisfy weak inner UCP. Proof. By our assumptions, im A∗0 |D is closed and is of finite codimension. Since im A∗0 |D ⊂ im A∗0 ⊂ X, the full range im A∗0 is closed. Since A0 satisfies weak inner UCP, im A∗0 = X. Then A∗0 is semi-Fredholm. By Theorem IV.5.17 of Kato [17] we have im A∗s = X for s ≪ 1. Since As are symmetric, As satisfy weak inner UCP for s ≪ 1.
3.2 Continuity of the family {As,Ds } Let X be a complex Hilbert space, and M, N ⊂ X be two closed linear subspaces. Let PM , PN be the orthogonal projections onto M, N respectively. Then the distance d(M, N) is defined by d(M, N) = kPM − PN k and called the gap between M and N. For any two closed operators A, B on X, we define d(A, B) as the distance between their graphs. Let A ∈ C(X) be a linear, closed, densely defined operator in X. By Footnote 1 (page 198), Theorems IV.1.1 and IV.2.14 in [17], it is easy to verify the following Lemma 3.3. Let B ∈ B(dom(A), X) be a bounded operator, where the norm on dom(A) is the graph norm GA induced by A. Let d := kB − AkGA < 12 . Then we have (1) B ∈ C(X), and it holds that (1 − 2d)hx, xiGA ≤ hx, xiGB ≤ (1 + d)2 hx, xiGA for x ∈ D.
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(2) d(B, A) ≤
571
√
2d . (1−d)−1
Lemma 3.4. Let X be a Hilbert space, and Y be a closed linear subspace of H. Then there exists a bijection between the space of closed linear subspaces of X containing Y and that of closed linear subspaces of X/Y which preserves the metric. Proof. We view X/Y as Y ⊥ . Let M, N ⊂ Y ⊥ be two closed subspaces and PM , PN be the orthogonal projections onto M, N respectively. Then we have d(M + Y, N + Y ) = kPM +Y − PN +Y k = kPM − PN k = d(M, N). From the definition of the gap norm and by some computations we have Lemma 3.5. Let Dm ⊂ Dmax ⊂ X be three Hilbert spaces such that Dm is a closed sub space of Dmax and a dense subspace of X. Let As ∈ C(X) s∈[0,1] be a family of densely de fined symmetric operators with domain Dm , and Ds s∈[0,1] be a family of closed subspaces of Dmax containing Dm . We assume that dom(A∗s ) = Dmax , each graph norm Gs of Dmax induced by A∗s is equivalent to the original norm G of Dmax , and A∗s ∈ B(Dmax , X) , Ds /Dm ⊂ Dmax /Dm are two continuous families. Then As,Ds ∈ C(X) s∈[0,1] is a continuous family of closed operators.
3.3 Continuity of natural Cauchy data spaces In this subsection we generalize the proof of the continuity of Cauchy data spaces given in [5, Section 3.3]. We need the following Proposition 3.6 (Proposition 3.5 of [5]). Let X be a Hilbert space, and A ∈ C(X) be a symmetric operator. Set Dm = dom(A) and Dmax = dom(A∗ ). If A admits a selfadjoint Fredholm extension with domain D , then the quotient space D/Dm and the natural Cauchy data space (ker A∗ + Dm )/Dm form a Fredholm pair of Lagrangian subspaces of the (strong) symplectic Hilbert space Dmax /Dm (introduced above in Subsection 1.3, Item (ii)). Remark 3.7. From the arguments in Ralston [27] one can deduce (see [8]) that all linear formally self-adjoint elliptic differential operators over a compact smooth Riemannian manifold with smooth boundary admit a self-adjoint Fredholm extension. Now we can prove Proposition 3.8. Let X be a Hilbert space, and Dm ⊂ Dmax be two dense linear subspaces of X. Let {As : Dm → X}s∈[0,1] be a family of closed symmetric densely defined operators in X. We assume that
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(1) each As admits a self-adjoint Fredholm extension with domain Ds ; (2) dom(A∗s ) = Dmax is independent of s and that all graph norms Gs of Dmax induced by A∗s are mutually equivalent; (3) each As satisfies weak inner UCP relative to Dm ; and (4) {A∗s : Dmax → X} forms a continuous family of bounded operators, where the norm on Dmax is the graph norm G induced by A0 . Then the natural Cauchy data spaces Dm + ker A∗s /Dm are continuously varying in Dmax /Dm . Proof. We denote the projection of Dmax onto Dmax /Dm by γ . Note that ker A∗s is closed in Dmax . To prove the continuity, we need only to consider the local situation at s = 0. First we show that {ker A∗s }s∈[0,1] is a continuous family of subspaces of Dmax ; then we show that γ(ker A∗s ) is a continuous family in Dmax /Dm . We consider the bounded operator Fs : Dmax −→ X ⊕ ker A∗0 x
7→
,
(A∗s x, P0 x)
where P0 : Dmax → ker A∗0 denotes the orthogonal projection of the Hilbert space Dmax onto the closed subspace ker A∗0 . By definition, the family {Fs } is a continuous family of bounded operators. Clearly, F0 is injective. Since im A∗0 |D0 ⊂ im A∗0 ⊂ X and A∗0 |D0 is Fredholm, im A∗0 is closed. From weak inner UCP we get im A∗0 = X. So the operator F0 is also surjective. This proves that F0 is invertible with bounded inverse. Then all operators Fs are invertible for small s ≥ 0, since Fs is a continuous family of operators. Note that Fs (ker A∗s ) ⊂ {0} ⊕ ker A∗0 ,
(Fs )−1 ({0} ⊕ ker A∗0 ) ⊂ ker A∗s .
Since Fs are invertible for small s ≥ 0, we have Fs (ker A∗s ) = {0} ⊕ ker A∗0 .
(34)
We define −1 ∼ ϕs := Fs−1 ◦ F0 : Dmax ∼ = Dmax and ϕ−1 s = F0 ◦ Fs : Dmax = Dmax
for s small. Since Fs are invertible for small s ≥ 0, from (34) we obtain that ϕs (ker A∗0 ) = ker A∗s . From (35) we get that ∗ {Ps := ϕs P0 ϕ−1 s : Dmax −→ ker As }
(35)
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is a continuous family of projections onto the solution spaces ker A∗s . The projections are not necessarily orthogonal, but can be orthogonalized and remain continuous in s like in [10, Lemma 12.8]. This proves the continuity of the family {ker A∗s } in Dmax . Now we must show that {γ(ker A∗s )} is a continuous family in the quotient space Dmax /Dm . This is not proved by the formula γ(ker A∗s ) = γ(ϕs (ker A∗0 )) alone. We must modify the endomorphism ϕs of Dmax in such a way that it keeps the subspace Dm invariant. By Proposition (3.6), the Cauchy data space γ(Dm + ker A∗0 ) is closed in Dmax /Dm . So Dm + ker A∗0 is closed in Dmax . We define a continuous family of mappings by ψs : Dmax = Dm + ker A∗0 + (Dm + ker A∗0 )⊥ −→ x+y
+
z
Dmax
7→ x + ϕs (y) + z
with ψ0 = id. Hence all ψs are invertible for s ≪ 1, and ψs (Dm ) = Dm for such small s. Hence we obtain a continuous family of mappings {ψes : Dmax /Dm → Dmax /Dm } with ψes (γ(ker A∗0 )) = γ(ker A∗s ). From that we obtain a continuous family of projections as above. Remark 3.9. From the preceding arguments it also follows that the Cauchy data spaces form a differentiable family, if {A∗s } is a differentiable family.
3.4 Proof of the spectral flow formula We begin with a simple case. Lemma 3.10. Let X be a Hilbert space, and A ∈ C(X) be a symmetric operator with dom(A) = Dm and dom(A∗ ) = Dmax . Let AD := A∗ |D be a self-adjoint Fredholm extension of A. We assume that A satisfies weak inner UCP. Then there exists an ǫ > 0 such that AD + aI is Fredholm and satisfies weak inner UCP for each a ∈ [0, ǫ]. Let γ : Dmax → Dmax /Dm denote the natural projection. Then we have sf{AD + aI; a ∈ [0, ǫ]} = − Mas{γ(D), γ(ker(A∗ + aI)); a ∈ [0, ǫ]}. Proof. By the definition of the spectral flow we have X sf{AD + aI; a ∈ [0, ǫ]} = dim ker(AD + aI).
(36)
a∈(0,ǫ]
Let ω denote the Green form on Dmax induced by A∗ . Let W ∈ L(Dmax /Dm ) be a Lagrangian complement of γ(ker(A∗ + a0 I)). By Proposition 3.8, γ(ker(A∗ + aI)) and ker(A∗ + aI) are two differentiable families. For each y(a0 ) ∈ ker(AD + a0 I), there exists a continuous family w(a) ∈ W + Dm , |a − a0 | small, such that w(a0 ) = 0 and y(a) := y(a0 )+w(a) ∈ ker(A∗ +aI). Since A∗ (y(a)) = −ay(a) and the family {y(a)} is continuous
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in Dmax , the family {y(a)} is also continuous in X. For all x(a0 ) ∈ ker(AD + a0 I), we have ω(γ(x(a0 )), γ(w(a)) = hA∗ (x(a0 )), y(a) − y(a0)i − hx(a0 ), A∗ (w(a))i
= h−a0 x(a0 ), y(a) − y(a0 )i − hx(a0 ), A∗ (y(a)) − A∗ (y(a0 ))i = h−a0 x(a0 ), y(a) − y(a0 )i − hx( a0 ), −ay(a) + a0 y(a0 )i = (a − a0 )hx(a0 ), y(a)i
Let the crossing forms Q and Γ be defined by (31) and (32) respectively. Then we have Q(γ(ker(A∗ + aI)), a0 )(γ(x(a0 )), γ(y(a0)) = hx(a0 ), y(a0)i and Γ(γ(D), γ(ker(A∗ + aI)), a0 )(γ(x(a0 )), γ(y(a0)) = −hx(a0 ), y(a0)i. By Proposition 2.14 we have Mas{γ(D), γ(ker(A∗ + aI)); a ∈ [0, ǫ]} = −
X
a∈(0,ǫ]
dim ker AD + aI .
Combine equations (36), (37), and our lemma follows.
(37)
Now our main result follows at once. Proof of Theorem 1.5. By Lemma 3.2, for each s0 there exists an ǫ(s0 ) > 0 such that the operators As + aI satisfy weak inner UCP for all s, a with |s − s0 |, |a| < ǫ(s0 ). Here we use the continuity of the family A∗s } as bounded operators from Dmax to X. Since [0, 1] is compact and As,Ds are Fredholm operators for all s ∈ [0, 1], there exists an ǫ > 0 such that the operators As + aI satisfy weak inner UCP and As,Ds + aI are Fredholm operators for all s ∈ [0, 1] and |a| < ǫ. We only need to prove the formula (12) in a small interval [s0 , s1 ]. We consider the two-parameter families {As,Ds + aI} and {γ(Ds ), γ(ker A∗s + aI)} for s ∈ [s0 , s1 ] and a ∈ [0, ǫ]. Because of the homotopy invariance of spectral flow and Maslov index, both integers must vanish for the boundary loop going counter clockwise around the rectangular domain from the corner point (s0 , 0) via the corner points (s1 , 0), (s1 , ǫ), and (s0 , ǫ) back to (s0 , 0). Moreover, for s1 sufficiently close to s0 we can choose ǫ sufficiently small so that ker(As,Ds + ǫI) = {0} for all s ∈ [s0 , s1 ]. Hence, spectral flow and Maslov index must vanish on the top segment of our box. Finally, by the preceding lemma, the left and the right side segments of our curves yield vanishing sum of spectral flow and Maslov index. So, by additivity under catenation, our assertion follows.
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Acknowledgment The first author thanks the organizers Jan Kubarski, Tomasz Rybicki, and Robert Wolak of the 6th Conference on Geometry and Topology of Manifolds (Krynica, Poland, May 2-8, 2004) for the opportunity to present the main ideas and various ramifications of this paper in a mini-course of four hours. We both thank the referee for corrections, thoughtful comments, and helpful suggestions which led to many improvements. The referee clearly went beyond the call of duty, and we are indebted.
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CEJM 3(3) 2005 578–579
Corrections for “The closure diagram for nilpotent orbits of the split real form of E8” ˇ Dragomir Z.
okovi´c∗
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
There is one serious error in Table 2 of [2], which lists the children of each node of the closure diagram, and several minor errors. Let us first deal with the minor ones. The formula displayed at the bottom of p. 578 defines the set Ii by giving two expressions for it. The first of these expressions is incorrect and should be omitted. The types E8 (a7 ) in Figure 6 (p. 640) and E8 (b4 ) in Figure 8 (p. 642) should be deleted. The nilpotent elements E that they represent do not belong to the orbits 67 and 107, respectively, but to some orbits of lower dimension. Consequently, the following corrections to Table 1 are required: Delete the representatives of the orbits 67 and 68 of type E8 (a7 ) (p. 593) and the representative of the orbit 107 of type E8 (b4 ) on p. 597. We point out that these particular representatives were not used in any of the proofs in the paper. The diagrams of type Dn (ak ) and Dn (ak ) given in Figure 5 and the corresponding representatives E and sl2 -triples are studied in detail in [1]. There are a few misprints or inaccuracies that we have noticed: 1) In the second column of Table 3 on p. 601, when i = 98 there should be two possible values, 94 and 92, for j. 2) In Table 4 on p. 603 in the row for i = 40 and j = 37, the index −88 for the representative E should be replaced by −82. And on p. 606 in the row for i = 86 and j = 84 the index −83 occurs twice; the second one should be replaced by 103. 3) In Table 4, the type of the representative E is incorrectly given in the following cases: On p. 604 in the row for i = 66 and j = 65 the type should be D6 (a2 ) + A1 . On p. 605 in the row for i = 73 and j = 69 it should be D6 (a2 ) + 2A1 and in the row for i = 78 and j = 76 it should be E7 (a4 ) . On p. 607 in the row for i = 107 and j = 105 the type should be D8 (a1 ). In all these cases, the representative E itself is correct. ∗
E-mail: [email protected]
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okovi´c / Central European Journal of Mathematics 3(3) 2005 578–579
579
4) In the 8-th row of Table 9 on p. 634, where k takes the values 89 and 91, only the pairs 93, 95 and 93, 99 should occur as possibilities for i, j. Let us now address the error in Table 2: The entry 63 → 52, 55, 58 should be replaced by 63 → 55, 58, 59 As mentioned in the paper, each node, i, was studied separately and various computations were performed in order to identify the children of the node. The method uses two prehomogeneous vector spaces. The first of these is (KH i , gH i (1, 2)), with KH i reductive. We denote by l the length of this module and by r the number of its basic relative invariants. The inequality r ≤ l is always valid and l can be easily computed. It is more complicated to determine r as it requires the knowledge of the generic isotropy subgroup of this prehomogeneous vector space (see [3, Proposition 2.12]). We decided to re-examine all nodes where we took for granted that r < l and found that in three cases this is not so and that r = l holds. One of these cases led to the above error. There remain now 14 nodes for which r < l, and in all these cases r = l − 1. The fact that the orbit O159 is contained in the closure of O163 follows from the observation that the representative E 59 of O159 of type D5 (a1 ) + A2 (see Table 1, p. 592) is contained in the subspace p2 (H 63 ) (see Table 7, p. 617). As a consequence of the correction in Table 2, some further adjustments are in order. In Table 3 (p. 600) remove the value j = 59 for the entry with i = 63. Table 4 should be modified to incorporate the above change in Table 2: On p. 604 remove the value i = 63 from the row with j = 52 and insert it in the row with i = 61 (and j = 59). The two pairs (59, 48) and (63, 59) that appear in Table 5, are no longer critical and should be deleted from that table. Finally, two lines in Table 9, p. 632 should be modified. One should delete the pair (i, j) = (59, 63) from the fifth last line and replace “52 57” in column k with “59”. In the next line the values of k should be “58 59” instead of “52 58”.
References ˇ okovi´c: “On semiregular nilpotent orbits in simple Lie algebras [1] B. Arbour and D.Z. of type D”, J. Algebra Appl., Vol. 3, (2002), pp. 341–356. ˇ okovi´c: “The closure diagram for nilpotent orbits of the split real form of E8 ”, [2] D.Z. CEJM, Vol. 4, (2003), pp. 573–643. [3] T. Kimura: Introduction to prehomogeneous vector spaces, Translation of Mathematical Monographs, Vol. 215, Amer. Math. Soc., Providence, 2003.
