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(u) = +00.
8. Introduction
33
the level of sequences (that are not necessarily realized by orbits). I found it a convenient bridge between the study of the dynamics of circle homeomorphisms (which appears in the appendix to this chapter) and that of Aubry-Mather sets.
Aubry-Mather Theorem as Topological Stability. There are important notions in the theory of dynamical systems that help to compare different systems. We refer to Hasselblat & Katok (1995) for more details. Suppose / : M —> M and g : N —> N are two Cr, r > 0 maps on manifolds. We say that / and g are topologically conjugate if there is a homeomorphism h : M —> N such that ho f = g oh. Orbits of conjugate maps are in 1-1, continuous correspondence (given by the map h). If the map h is continuous but only surjective (and not necessarily injective), we say that g is a factor of / and we call h a semiconjugacy. Finally, if / is a diffeomorphism and if it is a factor of any homeomorphisms in a C° neighborhood of it, we say that / is topologically stable. In light of this terminology, we can say that the Aubry Mather theorem is a "weak" stability statement: All maps in a C 1 neighborhood of the completely integrable map have the completely integrable map restricted to irrational rotation invariant circles as a factor.
B. From the Annuius to the Cylinder We precede our study by a Lemma, which implies that we can reduce our study to twist maps of the cylinder. Lemma 8.2 Let f be a Ck,k
> 2, twist map of a compact annuius A- Then f can
be extended to a Ck twist map of the cylinder C, in such a way that it coincides with the shear map (x, y) i-+ (a; + cy, y) outside a compact set. In particular, generating function lim
of any lift of the extended map satisfies the growth
the
condition
S(x, X ) —• +oo.
[X— x\—>oo
To prove this lemma, one extends the generating function S from ip(A) to H 2 by interpolating it to the quadratic § (X — x)2 outside of some appropriate compact set. See Forni & Mather (1994) or Moser (1986a) . As a corollary of this lemma, we obtain the following version of the Aubry-Mather theorem:
34
2: AUBRY-MATHER THEOREM
Theorem 8.3 (Aubry-Mather on the compact annulus) Let F be the lift of a twist map of the bounded annulus and suppose that the rotation numbers of the restriction of F to the lower and upper boundaries are p^, and p+ respectively. Then F has orbits of all rotation numbers in [p_, p+]. These orbits are minimizers,
recurrent,
cycli-
cally ordered and they lie on compact invariant sets that form (uniformly)
Lipschitz
graphs over their projections.
invariant
These sets may either be periodic orbits,
circles or invariant Cantor sets on which the map is semi-conjugate
to lifts of circle
rotations.
9. Cyclically Ordered Sequences and Orbits If a map G : IR —> H is the lift of a circle homeomorphism which preserves the orientation, it is necessarily strictly increasing and must satisfy G(x+l)
= G(x) +1. Hence, if
{xk}kez
is an orbit of G, it must satisfy:
(9.1)
xk <Xj+p=>
We will say that a sequence {xk}kei
xk+i < xj+1 +p, V k,j,p
€ Z.
in IR Z is Cyclically Ordered, (or CO in short) if it
satisfies (9.1). Clearly the CO sequences form a closed set for the topology of pointwise convergence in M z : x^
—» x whenever x?k —* xk for all k. Note that this topology is
the same as the product topology on the space of sequences. Using the partial order on sequences (it comes with three degrees of strictness): x < y *> {\fk, xk < yk} x < y o {Vk, xk < yk x
and
x ^ y}
< yk}
we let the reader check that an equivalent definition of CO sequences is: (9.2) where
Vm, n G 2 ,
9. CO Sequences and Orbits
35
We say that the orbit {(xfc, yk)}k£Z of a twist map is a Cyclically Ordered orbit or CO orbit if {xk}ke~z
ls
CO. These orbits come with various other names in the literature: Well
Ordered (does not evoque the cyclic ordering), Monotone (is used in too many contexts), BirkhofF(this order was apparently never mentioned by Birkhoff).'7' The following lemma, whose proof is in great part due to Poincare (1885), is central to our use of CO sequences.
Lemma 9.1 Let {xkjkez
be a CO sequence then p(x) = limfc^ooZfc/fc exists and:
(9.3)
\xk - xQ - kp{x)\ < 1.
Moreover x —> p(x) is a continuous function on CO sequences, with the topology of pointwise
convergence.
Define: CO[aM
=
{x€CO\p(x)e[a,b}}.
The following lemma shows that it is easy to find limits of CO sequences, as long as their rotation numbers are bounded. Lemma 9.2 The sets C O ^ / T ^ O and CO[a^ f l { i 6 M z | x0 e [0,1]} are compact for the topology of pointwise
convergence.
We give the proofs of both these lemmas in the appendix to this chapter. The fact, given by these lemmas, that the rotation number behaves well under limits of CO-sequences is one of the essential points in the theory of twist maps that does not generalize to higher dimensional maps: to our knowledge, there is no dynamically natural definition of CO sequences in 1R™, n > 2 which ensures the existence of rotation vectors which behave well under limits. Note that there is, however, a natural generalization of CO sequences in the context of maps 2 d —> M, see Chapter 9. There is a visual way to describe CO sequences, which we now come to. A sequence x in IR is a function 7L —> H . One can interpolate this function linearly and obtain a piecewise affine function IR —> IR that we denote by 11—> xt. The graph of this function is sometimes This is not an indictment of the authors who have used these terminologies: the author of this book has himself used them all in various publications...
36
2: AUBRY-MATHER THEOREM
called the Aubry diagram of the sequence. We say that two sequences x and w cross if their corresponding Aubry diagrams cross. There are two types of crossing: at an integer k, in which case (xk-i
- Wk-\){xk+i
— w/t+i) < 0 or at a non integer t 6 (k,k + 1), in
which case (xk — Wk){xk+\ — W/H-I) < 0. These inequalities can be taken as a definition of crossings. Non-crossing of two sequences can be put in terms of the partial order on sequence: x, y do not cross if and only if x < y . In particular a sequence x is CO if and only if it has no crossing with any of its translates
Tm^nx.
F i g . 9.0. Aubry diagrams of sequences and their crossings: in this example the sequences x and w have crossings at the integer fc and between the integers j and j + 1.
10. Minimizing Orbits Throughout the rest of this chapter, we consider a lift F of a given twist map / of the cylinder, and its corresponding generating function S, action function W, periodic action function Wmn
and change of variable ip. A sequence segment (xk,. • •, xm) is (action)
minimizing if W(xk,•••,xm)
< W(yk,•
••,ym)
for any other sequence segment (yk, • • •, ym) with same endpoints: xk = yk,xm
= ym.
Since minimizing segments are necessarily critical for W, they correspond to orbit segments called (action) minimizing orbit segment. A bi-infinite sequence is called a (global action) minimizer if any of its segments is minimizing. The orbit it corresponds to is a minimizing orbit, or simply minimizer, when the context is clear. Note that the set of minimizers is closed under the topology of pointwise limit (see Exercise 10.5). Finally a is a periodic sequence in Xm^n that minimizes the function
Wmn.
Wmn-minimizer
10. Minimizing Orbits
37
A recurrent theme in the Calculus of Variation is that minimizers have regimented crossings. In the case of geodesies on a Riemannian manifold, geodesies that (locally) minimize length cannot have conjugate points, i.e. small variations with fixed endpoints of a minimizing geodesic only intersect that geodesic at the endpoints ( Milnor (1969)), and geodesies that minimize length globally cannot have self intersections (Mane (1991), page 102 ). We will see, in the present theory, that minimizers satisfy a non-crossing condition, which implies that Wm n -minimizers (and more generally, recurrent minimizers) are CO.
Lemma 10.1 (crossing) Suppose that (x — w)(X — W) < 0. Then: S(x, X) + S(w, W) - S{x, W) - S(w, X) < 0, and equality occurs iff {x — w)(X — W) = 0 Proof.
We can write: S(x,X)-S{x,W)=
f d2S{x,Xs){X Jo
-W)ds,
where Xs — (1 — s)W + sX. Applying the same process to h(x) = S(x, X) — S(x, W), we get: S(x, X) + S{w, W) - S(x, W) - S{w, X) = h{x) - h(w) =
-L
f di2S(xr, Jo
XS){X - W)(x - w)dsdr = \{X - W){x - w)
for some strictly negative A, by the positive twist condition and for xr = (1 — r)w + rx.
D
The following is a watered down version of the Fundamental Lemma in Aubry & Le Daeron (1983). We follow Meiss (1992): Lemma 10.2 (Aubry's Fundamental Lemma) Two distinct minimizers
cross at most
once. Proof.
Suppose that x and w are two distinct minimizers who cross twice. We perform
some surgery on finite segments of x and w to get two new sequences x' and w' with at least one of them of lesser action, contradicting minimality. There are three cases to consider: (i)
38
2: AUBRY-MATHER THEOREM
both crossings are at non integers, (ii) one crossing is at an integer, (iii) both crossings are at integers.
Fig. 10.2. A crossing of Case (ii) Case (i):Letti € (i — l,i)andt 2 € (j,j + 1) be the crossing times. Define: x, = k
(wk if k e [i,j] \ Xk otherwise
fc
w,
=
(xk if ke [i,j] \ Wk otherwise
Letting W denote the action over an interval [N, M] containing [j — 1, k + 1], we easily compute that:
W(x') + W(w')-W(x)
- W(w) =
S(Xi-l,Wi)
+S{xj,wj+i)
+ S(Wi-!,Xi)
+ S(wj,xj+i)
- S(Xi-i,Xi)
- S(xj,xj+1)
-
S(Wi-i,Wi)
-
S(wj,wj+i).
The Crossing Lemma 10.1 shows that this difference of actions is negative, contradicting the minimality of x and w. Case (ii): In this case, only one crossing will contribute negatively to the difference of action of new and old sequences. We still get a contradiction. Case (iii) Let i — 1 and j + 1 be the crossing times of x and w, and construct x' and w' as before. In this case the difference in action between old and new segments is null. The sequences x',w' must be minimizing, and hence correspond to orbits. But we have Xj_2 = u^_2,
Xi-i = w'i_1. Hence the points ip~1(xi^2,Xi-i)
and i>~1(w'i_2,w'i_1)
2
of IR are the same and thus generate the same orbit under F. This in turn implies that x = w, a contradiction to our assumption.
D
11. Minimizing Orbits Corollary 10.3 Wmn-minimizing
39
sequences are CO and their set is completely ordered
for the partial order on sequences. Proof.
Since the proof of Aubry's Lemma deals with finite segments of sequences only,
it also applies to show that two Wmn-minimizers in X m n , may not cross twice within one period n. But two m, n-periodic sequences that cross once must necessarily cross twice within one period. Hence two Wmn-minimizers cannot cross at all. It is easy to check that Wmn is invariant under Tjj for all integers i, j . Thus, if a: is a Wmn minimizer, ritjX is also a Wmn-minimizer. Since they do not cross, one must have either x < Ti^x or n^x
< x,
for all i, j e Z , i. e. x is CO.
•
We end this section by a proposition which we will need in Chapter 3. Proposition 10.4 Any Wmn-minimizer Proof.
is a
minimizer.
We will show that if m is a W mn -minimizer, it is also a Wkmkn minimizer for
any k. This implies that x is a minimizer on segments of arbitrary length: if x is a Wkmkn minimizer, any segment of x of length less than kn is minimizing. Hence x is a minimizer. Now, take a Wfcmfcn-minimizer w. Ifw is not m, n-periodic, then w and Tm,nw are distinct. By Corollary 10.3, they cannot cross. Suppose, say, that rm
and its action over intervals of any
length multiple of n cannot be less than that of x. Hence x is also a Wkmkn minimizer.
•
Exercise 10.5 Show t h a t the set of minimizers (either sequences or orbits) is closed under pointwise limits. Exercise 10.6 a) Show t h a t the set of recurrent minimizers of rotation number u> is completely ordered. {Hint. Mimic the proof of Proposition 10.4: if an appropriate inequality is not satisfied, there must be a crossing. By recurrence, there is another one, a contradiction to Aubry's Lemma). b) Show t h a t a minimizer corresponding to a recurrent (not necessarily periodic) orbit of the twist map is CO. (Remember t h a t the orbit zk of a dynamical system is called recurrent if zo is the limit of a subsequence Zkj • Equivalently, zo is in its own u;-limit set).
40
2: AUBRY-MATHER THEOREM
11. CO Orbits of All Rotation Numbers A. Existence of CO Periodic Orbits We prove that the set of Wmn-minimizers is not empty. By Corollary 10.3 this will show the existence of CO orbits of all rational rotation numbers. Proposition 11.1 Let F be the lift of a twist map with a generating function satisfies the coercion condition Ymi\x-x\^>oo S{x,X) Wmn Proof.
has a minimum
on
which
—> +oo. Then, for all m,n,
Xmn.
Note that, by periodicity of S, the ranges of Wmn on X m n and on its subset
Xm,n n {xi G [0,1]} are the same: we can translate any sequence of Xm^n by an integer to bring it to that sub>> i without changing its action. Now, if S satisfies the coercion condition, then for x € Xm,n D \x\ G [0,1]}, limn-,.^^ Wmn(x)
—> +oo: if ||x|| —> oo and xi
remains bounded, at least one \xk — Xk-\\ must tend to +oo. In particular, for any large enough K G H , W^(—oo, K] is bounded and not empty. Since, by continuity, this set is also closed, it must be compact. Thus Wmn attains its minimum on that set.
D
An interesting sufficient condition for S to satisfy the coercion condition is that the "twist" of the map be uniformly bounded below (see MacKay & al. (1989)): Proposition 11.2 Let the twist condition for the lift of a twist map F be uniform: ^ p y l > dy
a >
0
V(x,y)GlR 2 .
Then there is a constant a, and two strictly positive constants (3 and 7 such that : S(x, X) > a - 0 \X - x\ + 7 \X - x\2 .
Proof.
We can write:
S(x,X) = S(x,x)+
f Jo
d2S(x,Xs)(X-x)ds,
where Xs = (1 — s)x + sX. Applying the same process to 92-5, we get:
12. Aubry-Mather Sets
S(x,X) = S(x,x) + f Jo
41
d2S(Xs,Xs)(X-x)ds d12S(Xr,Xs)(X~x)2.dr
- f ds [ Jo Jo
We can conclude the proof of the lemma by taking a = mmS(x,x),
f3 = max \d2S(x, x)\
x£lR
ieIR
(a, /? exist by periodicity of S) and 7 = a/2.
•
B. Existence of CO Orbits of Irrational Rotation Numbers
The existence of CO orbits of irrational rotation numbers is a simple consequence of the existence of CO periodic orbits: pick a sequence x^> of VFmfc]nfc-minimizers, with rrik/nk —> u ask —» 00. By using appropriate translations ofthe type rm>o on x^ (which neither change their rotation numbers, nor the fact that they are minimizers) we can assume
that 4*° e [0,1]. The sequence mk/rik is bounded and hence, by Corollary 10.3 the sequences x^ are in CO[atb] n {a: e IRZ | xo G [0,1]} for some a, b G IR. Lemma 9.2 guarantees the existence of a converging subsequence in CO[a^] and Lemma 9.1 shows that the limit of this subsequence has rotation number LJ. Finally, note that the periods nk go to infinity as k goes to infinity. In particular, any finite segment of a limit x of x^ is the limit of minimizing segments, hence minimizing itself (Exercise 10.5). •
12. Aubry-Mather Sets We have proven Part (1) and (3) of the Aubry-Mather theorem: existence of cyclically ordered, minimizing orbits of all rotation numbers. We now prove Part (2): the cyclically ordered orbits that we found in the previous section lie on Aubry-Mather sets, which we describe in this section. We say that a set M in IR2 is F-ordered if, for z, z' in M, n(z) < ir{z') => w(F{z)) < n(F{z')), where 7r is the a;-projection. A set is F-ordered invariant if it is JP-ordered and invariant under both F and F _ 1 . On such a set, the sequences x, x' of ^-coordinates of z and z'
42
2: AUBRY-MATHER THEOREM
must satisfy x -< x'. An example of F-ordered invariant set is the set of points in a CO orbit and all their integer translates. In fact, this can be used to give an alternative definition of CO orbits: an orbit is CO if and only if its points form an F-ordered invariant set. Note that an invariant circle which is a graph is F-ordered invariant (we will see in Chapter 6 that all invariant circles are graphs). We now want to explore the properties of F-ordered invariant sets. Crucial to the properties of these sets is the following ratchet phenomenon (I owe this terminology to G.R. Hall), which is a somewhat quantitative expression of the twist condition. This phenomenon, or condition is best described by the following picture: a
='•
G F ^
Ft 72)
.
n.,
•"*>
F i g . 1 2 . 0 . T h e ratchet phenomenon for the lift of a positive twist map F: there are two cones (shaded in this picture) <9„ and OH in IR2 centered around t h e y and a;-axes respectively, such that, if z, z' are two points of R 2 with z' £ z + 6>„, then F(z') e F(z) + 0/,. More precisely, for a positive twist map z' e z + Gi => F(z') e F(z) + 6>+, where the half cones 0^,0+ have the obvious meaning. The same holds for the half cones 0^ and 0^ • If g is negative twist (eg. F _ 1 ) , then the signs are reversed. T h e same cones can be used for F'1 as for F.
Lemma (Ratchet) 12.1 Let F be the lift of a twist map satisfying ^ > a > 0 in some region. Then, in that region, F satisfies the ratchet phenomenon for some cones 0v,&h whose angles only depend on a. Proof. See Exercise 12.9. Proposition 12.2 The closure of an F-ordered invariant set is F-ordered and invariant. Proof. The invariance is by continuity of F. Let M be an F-ordered invariant set. We let the reader prove that the uniform twist condition^ > a > 0 is automatically satisfied on an
12. Aubry-Mather Sets
43
F-ordered invariant set (essentially, such a set is necessarily bounded in the y direction, see Exercise 12.9). Suppose that, in the closure M of M there are z, z' in M , with -K{Z) < -ir(z') but n(F(z))
= 7r(F(z')) (the worst case scenario). By the ratchet phenomenon for F ~ \
F(z) must be above F(z') and TT(F2(Z'))
< TT(F2(Z)),
i.e. the x orbits of z and z' switched
order. This is impossible since in M the (strict) order is preserved by F.
Proposition 12.3 If M is an F-ordered invariant set, then it is a Lipschitz
graph
over its projection: there exists a constant K depending only on F such that, if (x, y) and (x1 ,y') are two points of M,
then:
\y' - y\ < K\x' - x\ with K only depending on the twist constant a = inf M ^ •
Note that a, and hence K could also be chosen the same for all F-ordered sets in a compact region.
Proof.
The proof of Lemma 12.2 shows that if M is F-ordered, we cannot have z, z'
in M and ir(z) = TT(Z') unless z = z'. Hence IT is injective on M , and M is a graph. To show that M forms a Lipschitz graph over its projection, let z and z' be two points of M and x and x' the corresponding sequences of x-coordinates of their orbits. Assuming 7r(z) < ir(z'), we must have x -< x'. If z' € z + 6>+, the ratchet phenomenon implies that F~1(z')
€ F~l{z)
+ O^, i.e. x'_x > x~\, a contradiction. Likewise z' cannot be in
the cone z + 0~, and hence it must be in the cone complementary to Ov at z. This cone condition is easily transcribed into a uniform Lipschitz condition \y' — y\ < K\x' — x\.
O
Remark 12.4 Applied to the special case of invariant circles, Proposition 12.3 shows that any invariant circle for a twist map which is a graph is Lipschitz. This is a theorem originally due to Birkhoff, who also proved (see Chapter 6) that all non-homotopically trivial invariant circles for twist maps must be graphs.
Lemma 12.5 All points in an F'-ordered set have the same rotation
number.
44
2: AUBRY-MATHER THEOREM
Proof.
This is a consequence of the simple fact (Lemma 13.3 in the appendix) that if
x < x' are two CO sequences, they must have the same rotation number.
D
Definition 12.6 An Aubry-Mather set M for the lift F of a twist map / of the cylinder is a closed, F-ordered invariant set which is also invariant under the integer translation T.
Note that some authors call Aubry-Mather sets the projections of the above sets to the annulus. Exercise 12.9 shows that these projections are necessarily compact. Taking the closure of all the integer translates of the points in the CO orbits found in the previous section, we immediately get:
Theorem 12.7 Let F be the lift of a twist map of the cylinder. Then F has AubryMather sets of all rotation numbers in IR. Any CO orbit is in an
Aubry-Mather
set.
Note that this theorem gives part (b) of the Aubry-Mather theorem.
Theorem 12.8 (Properties of Aubry-Mather sets) Let M be an Aubry-Mather
set for
a lift F of a twist map of the cylinder. (a) M forms a graph over its projection TV(M),
which is Lipschitz
with
Lipschitz
constant only depending on the twist constant a = infM ^~ • (b) All the orbits in M are cyclically ordered and they all have the same
rotation
number, which is called the rotation number of M. (c) The projection ir(M) is a closed invariant set for the lift of a circle homeomorphism, and hence F restricted to M is conjugated to the lift of a circle
homeomor-
phism via -n.
Proof of Theorem 12.8. We have shown in Lemmas 12.5 and 12.6 that (a) and (b) are in fact properties of F-ordered invariant sets. As for Property (c), since ir is one to one on M, F induces a continuous (Lipschitz, in fact) increasing map G on TT{M), defined by G(TT(Z))
= TT(F(Z).
Since M and thus 7r(M) are invariant under integer translation, we
have G(x + 1) = G(x) + 1. The set ir(M) is closed and invariant under integer translation
12. Aubry-Mather Sets since M is. If 7r(M) = IR, then G is the lift of a circle homeomorphism. If n(M)
45 ^ IR,
then its complement is made of open intervals. Extend G by linear interpolation on each interval in the complement of n(M). Since G is increasing on n(M), its extension to IR (call it G) is increasing as well, continuous and G(x + 1) = G(x) + 1, hence the lift of a circle homeomorphism. By construction G(TT(Z)) = n(F(z)), and 7r | M is a continuous, 1 -1 map on the compact set M, hence a homeomorphism M —> 7r(M). Thus 7r is a conjugacy between i*1 on M and G on 7r(M), which is a closed and invariant set under G and G _ 1 .
•
Recapitulation on the Dynamics of Aubry-Mather Sets. If G is the lift of a circle homeomorphism constructed in the proof of Theorem 12.7, the possible dynamics for invariant sets of circle maps described in the appendix become, under the conjugacy, possible dynamics on Aubry-Mather sets M for F. Hence an Aubry-Mather set M is either: (i) an ordered collection of periodic orbits with (possibly) heteroclinic orbits joining them, or (ii) the lift of an /-invariant circle, or (iii) an F-invariant Cantor set with (possibly) homoclinic orbits in its gaps. The rotation number of M is necessarily rational in Case (i), and necessarily irrational in Case (iii). In Case (ii), M may either have a rational or irrational rotation number, as the example of the shear map shows. However, maps with rational invariant circles are non generic. Indeed, as a circle map, the restriction of the twist map to the invariant circle must have a periodic orbit. For generic twist maps, periodic orbits must be hyperbolic and the circle must be made of stable and unstable manifolds of such orbits, that coincide. But generically, such manifolds intersect transversally. See Herman (1983) and Robinson (1970) for more details. As for homoclinic and heteroclinic orbits as in (i) and (iii), they have been shown to exist each time there are no invariant circles of the corresponding rotation numbers, see Hasselblat & Katok (1995), Mather (1986). The feature that is striking in the Aubry-Mather theorem is the possible occurrence of Aubry-Mather sets as in (iii). The F-invariant Cantor sets have been called Canton by Percival (1979)who constructed them for the discontinuous sawtooth map (a standard map with sawtooth shaped potential). This type of dynamics does occur in twist map, since it can be shown that many maps have no invariant circles, and hence the irrational Aubry-Mather sets must be of type (iii), i.e. contain Cantori.
46
2: AUBRY-MATHER THEOREM Although one can construct many Aubry-Mather sets that are not made of minimizers
(Mather (1985)), the name "Aubry Mather set" is often reserved to the action minimizing Cantori Mw as defined below:
Proposition 12.9 For each irrational rotation number ui there is a unique Mu made of recurrent minimizing CO minimizing
Proof.
Cantorus
orbits of rotation number w. The closure of any
orbit of rotation number w is contained in
Mu.
A CO minimizing orbit forms an F-ordered set, contained in an Aubry-Mather set,
and hence conjugated to an orbit of a circle homeomorphism. The closure of the irrational CO minimizing orbit is therefore in a Cantorus, conjugated to the w-limit set of the circle homeomorphism. As limit of minimizers, this Cantorus is made up of minimizers. We now prove that this Cantorus is unique: suppose there are two of them. Exercise 10.6 implies that the (disjoint) union of these two Cantori forms an F-ordered set, hence conjugated to a closed invariant set of a circle homeomorphism. Each Cantorus is the w-limit set of its points. This is a contradiction to the uniqueness of ui limit sets of circle homeomorphisms proven in Theorem 13.4.
•
Exercise 12.9 a) Prove the Ratchet Lemma 12.1. b) Prove that if F is an .F-ordered invariant set, then the projection proj(M) of M to the cylinder is compact, /-invariant. Deduce from this that M satisfies the uniform twist condition dX/dy > a > 0. [Hint. Use Lemma 9.2]. Exercise 12.10 Show that a twist map / restricted to a Cantorus (irrational Aubry-Mather set) is semiconjugate to a rotation of the same rotation number.
13. Appendix: Cyclically Ordered Sequences and Circle Maps
In this section, we prove Lemma 9.1, and Lemma 9.2. We then recover important facts about circle homeomorphisms and their invariant sets using the language of CO sequences.
13. Appendix: CO Sequences
47
A. Proofs of Lemmas 9.1 and 9.2 We recall the statements of each lemma before proving it. Part of the proof below is classical, due to Poincare in his study of circle homeomorphisms. Lemma 9.1 Let {xk}kzz
be a CO sequence then p(x) = limfc_,oo Xk/k exists and:
(13.1)
\xk - x0 - kp(x)\ < 1.
Moreover x —> p(x) is a continuous function
on CO sequences, when the set of
sequences has been given the topology of pointwise
convergence.
Proof . Let x be a CO sequence. We first prove that the sequence { X r t ~ x o } r a ez is Cauchy as n —> ±oo. We do the case n —• +oo first. Given n 6 IN, let an be the integer such that: (13.2)
xo + a „ < xn < x0 + an + 1.
We prove by induction that (13.3)
xo + kan < Xkn < ^o + kan + k,
Vfc e IN.
Indeed, step 1 in the induction is just (13.2), and if we assume step k, i.e. (13.3) then, since x is CO, we get xn + kan < X(fc+i)n < xn + kan + k. Using (13.2) this gives x0 + (k + l)an < a; (fc+1)n <x0 + (k + l ) a „ + (k + 1), which is the step k + 1 and finishes the induction. Dividing (13.3) by k we get Xkn
(13.4)
an <
Xg
< an + 1.
Since this is true for all k > 0, we must have, for all n ^ 0, the two equivalent inequalities (13.5) Writing zn =
*£fcn
*^0
3-n
2-0
l Xn XQ n ,
< 14*
•Efcn
kn
^0
~^n
*^0
<
1
and assuming m > 0, n > 0, the triangular inequality gives:
48
2: AUBRY-MATHER THEOREM
(13.0)
\Zn — Zm\
< \Zn — Zmn\
+ \zmn
~
z
m\
SL
I
j
and hence {-zn}neiN, is a Cauchy sequence whose limit we call p(x). Let m —> oo in (13.6), and multiply by n: (13.7)
\xn - xo - np(x)\ < 1.
To see how the case n —> — oo follows, note that in all the above we could have replaced xo by an arbitrary xm,m S 2 and obtained: (13.8)
\xn — xm — (n — m)p(x)\
<1
\lm,n^7L.
We let the reader check that this last inequality implies that liirin^-oo zn = p{x). The continuity of p is also a consequence of (13.7). Indeed, suppose the CO sequences x^fi tend to x pointwise as j —> oo. Constructing sequences z^
as above, and denoting
p(xW>) = Wj, (13.7) yields a)
(13.9)
1
14 -^I
. . 1 \zk-P(x)\<-. \zk-p{x)\ < -.
Since z^~> —> z, for all A; and e > 0, |W, -
Wi|
< | W , - s£>| + \z? - 4 4 >| + | z « -
Wi|
< | +e
whenever z, j are big enough. Hence {u)k}kei. is a Cauchy sequence, whose limit we denote by u>. Letting j —> oo in (13.9) yields w = /J(:E)-
•
Lemma 13.1 T/ie sets COfa.&j/n.o a«d C,0[Ojt] PI {x € 1RZ | xo € [0,1]} are compact for the topology of pointwise Proof.
convergence.
We have already remarked that, trivially, CO is closed for pointwise convergence,
i.e. the product topology on sequences. Lemma 9.1 implies that CO [„_(,] fl{ x | XQ € [0,1]} is a closed subset of the set: {x £ E
Z
| xk = x0 + ku) + yk,(x0,u>,y)
e [0,1] x [a,b] x [ - 1 , l ] 2 , withj/o = 0 }
which is compact for the product topology, by Tychonov's theorem. We let the reader derive a similar proof for C O ^ / T ^ O .
•
13. Appendix: CO Sequences
49
B. Dynamics of Circle Homeomorphisms
Rotation Numbers and Circle Homeomorphisms. The orbits of an orientation preserving circle homeomorphism are by definition Cyclically Ordered. From Lemma 9.1, we can deduce the following theorem, due to Poincare (1885): Theorem 13.2 All the orbits of the lift F of an orientation preserving circle homeomorphism f have the same rotation number, denoted by p(F). The rotation number p is a continuous function of F, where the set of lifts of homeomorphisms of the circle is given the C° topology. Proof. We start by a simple but useful lemma. Lemma 13.3 If two CO sequences x,x' satisfy x < x' then p{x) = p(x'). Proof. The rotation numbers are the respective asymptotic slopes of the Aubry diagram of x and x'. Thus, if p{x) / p(x'), the Aubry diagrams of x and x' must cross. In this case, there must be a ko and a fci such that Xk0 > x'ko and xkl < x'ki. This contradicts x < x'. • Continuing with the proof of Theorem 13.2, since F is increasing, two CO sequences x and w corresponding to distinct orbits of F must satisfy x -< w or w -< x. From the previous lemma x and w have same rotation number. Finally, if /„ —> / in the C° topology, then the /„ orbit of a point x (a CO sequence) tends pointwise to the / orbit of x. By Lemma 9.1, ]imp(fn) = limp({/*(:r)}fc6z) = p({fk(x)}keZ) = p(f). D Dynamical classification of circle homeomorphisms. We now review the classification of circle homeomorphisms by Poincare (1885). Recall some general terminology from dynamical systems. The Omega limit set u>(x) of a point x under a dynamical system / is the set of limit points of the forward orbit, i. e. the set of limit points of all subsequences {xkj} where xk = fk(x) and kj —> +oo as j —> +oo. Likewise, the Alpha limit set a{x) is the set of limit points of the backward orbit. A minimal invariant set for a dynamical system is a closed, (forward and backward) invariant set which contains no closed invariant proper subset. A heteroclinic orbit between two invariant sets A and B is the orbit of a point x such that ot(x) C A and ui(x) C B. The term homoclinic is used when A = B.
50
2: AUBRY-MATHER THEOREM
Theorem 13.4 Let f be a circle homeomorphism then, for any x € $ , ui{x) and a(x) periodic (in which case x € u(x)
and F a lift of f. If p(F) is rational,
are periodic orbits. The orbit of x is either
= ot{x)) or it is heteroclinic between a(x)
and
If p(F) is irrational, then, for any x,x' € S 1 , a{x) = a{x') = OJ(X) = w(x').
Call
OJ(X).
this set O(f).
Then f2{f) is either the full circle, or a minimal invariant set which
is a Cantor set. In the first case any orbit is dense in the circle, and f is conjugated to a rotation by p(F). In the second case, a point x of S 1 is either in Q{f)
and
recurrent, or it is homoclinic to fi{f),
to a
a "gap orbit", and f is semi-conjugate
rotation by p(F).
We remind the reader that a Cantor set K is a closed, perfect, and totally disconnected topological set. Perfect means that each point in K is the limit of some (not eventually constant) sequence in K, and totally disconnected means that, given any two points a and b in K, one can find disjoint closed sets A and B with a G A,b 6 B and A U B = K. In the real line or the circle, a closed set is totally disconnected if and only if it is nowhere dense. A set X is nowhere dense if Interior(Closure{X))
Proof of Theorem
= 0.
13.4.
Rational rotation number. Suppose p{F) = m/n. Then Fn(-) — m must have a fixed point, otherwise for all x G H , Fn(x)
— x ^ m and we can assume Fn{x)
— x > m. By
1
compactness of S , p(F) > m/n, a contradiction. Hence F has an m, n-periodic orbit. By continuity, on any interval / where Fn — Id — m is non zero, it must stay of a constant sign. This sign describes the direction of progress of points inside the orbit of / towards its endpoints: they must be heteroclinic to the endpoint orbits. Conversely, if F has an m, n-periodic orbit, its rotation number and thus that of F must be m/n. Irrational rotation number. Suppose p(F) is irrational. Let x e S {xk}kel
its
1
and denote by x =
1
orbit under / (with x = x0). Suppose u(x) = S . We show that LU(X') = S 1
for any other x' e S 1 . Suppose not, and there is an interval (a, b) which contains no x'k = fk(x').
But (a, b) must contain some [ x n , i m ] by density of x. Again by density,
the intervals f-^™--™)[xn,xm]
must cover S 1 and hence / i ( m -™)x / £ (a, b) for some i, a
13. Appendix: CO Sequences
51
contradiction. We guide the reader through the proof that / is conjugated to a rotation by p(f) in Exercise 13.6. Suppose OJ(X) •£ S 1 . Then, since u>(x) is closed, its complement is the union of open intervals. Take another point x'. We want to show that ui(x') = u){x). We will prove that UJ(X') C u>(x): by symmetry UJ(X) C ui(x'). This is obvious if x' G w(x). Suppose not. Then x' is in an open interval I in the complement of UJ(X) whose endpoints are in ui(x). The orbit of I is made of open intervals in the complement of UJ (x) whose endpoints are orbits in w(x).
Since there is no periodic orbit, these intervals are disjoint: by the intermediate value
theorem fk(I)
C I would imply the existence of a fixed point for fk, hence a periodic
orbit. The length of these intervals must tend toward 0 under iteration. Thus the orbit of x' approaches the endpoint orbit of I arbitrarily i.e. the orbit of x' is asymptotic to UJ(X). Hence UJ(X') C u)(x). In particular UJ(X) = (2(f) is a minimal invariant set: any closed invariant subset of (2(f) must contain the w-limit set of any of its point, hence (2(f) itself. We now show that (2(f) is a Cantor set. That it is closed is a property of w-limit sets. It is perfect since x 6 (2(f) means that x e u(x) and hence fnk (x) —> x for some nk / nk
the f
oo and
(x)'s are in w(x), and are all distinct. To prove that (2(f) is nowhere dense, first note
that the topological boundary d(2(f) = (2\Interior((2(f)) or df2(f)
must satisfy d(2(f)
= (2(f)
= 0: dQ(f) is closed, invariant under / and included in fi(f) which is a minimal
set. But dil(f)
= 0 means fi(f)
= Interior(il(f))
is open, and because it is also closed,
1
it must be all of S , which we have ruled out. The alternative is dfl(f) means Interior(f2(f))
= O(f), which
= 0, which is what we wanted to prove. Exercise 13.6 walks the
reader through the proof that / is semi-conjugate to a rotation in this case.
•
Remark 13.5 A circle homeomorphism with an invariant Cantor set cannot be too smooth: Denjoy (see Hasselblat & Katok (1995), Robinson (1994)) proved that if / is a C 1 diffeomorphism of S 1 with irrational rotation number and derivative of bounded variation, then / has a dense orbit (i.e. fi(f) p(F).
= S 1 ) and is therefore conjugated to a rotation of angle
On the other hand, Denjoy did construct a C 1 diffeomorphism with O(f) a Cantor
set. The idea is simple: take a rotation by irrational angle a. Cut the circle at some point x and at all its iterate fk(x).
Glue in at these cuts intervals Ik of length going to 0 as k -+ oo,
in such a way that the new space you obtain is again a circle. Extend the map / by linear interpolation on the Ik. You get a circle homeomorphism with rotation number a. With
52
2: AUBRY-MATHER THEOREM
some care, one can make this homeomorphism differentiable, but only up to a point ( C 1 with Holder derivative). T h e complement of the Ik's in the new circle is a Cantor set, which is minimal for the extended map.
Exercise 13.6 In this exercise, we prove t h a t all orientation preserving circle homeomorphism with irrational rotation number ui has the rotation of angle u a s a factor. This is sometimes called Poincare's Classification Theorem (see Hasselblat & Katok (1995)). a) Prove t h a t a; is a CO sequence with irrational p(x) iff Vn,m,p£"E,
xn < xm + p •*=> np(x) < mp{x) + p
(Hint. Use Formula (13.8) for multiples of m and n). W h a t is t h e proper corresponding statement for CO sequences of rational rotation number? b) Suppose the circle homeomorphism / has a dense orbit, which lifts t o an orbit x of some F. Build a monotone map h : 1R —• IR by first defining it on x by: i t + m i - > kp(x) + m,
Vm, k £ 7L.
Use a) to show t h a t h is order preserving and show t h a t its extension by continuity is well defined, has continuous inverse and preserves orbits of F, and it commutes with the translation T (Hint, density of the orbit in S 1 means density of t h e set {x/c + m}k.mez in IR). Hence in this case / is conjugate to a rotation. c) Suppose now t h a t O(f) ^ S 1 . Following the steps in b), take a dense orbit x in Q(f) and build a map h : fi(F) —> ]R as before (0(F) denotes the lift of P(f) here). Check t h a t this m a p is onto, non decreasing and extend it to a m a p IR —* IR by mapping each the gap of the Cantor set t o a single point. d) Conclude that, in both cases, h provides a (semi)-conjugacy between / and a rotation by u.
3 GHOST CIRCLES In Chapter 2, we saw how traces of the invariant circles of the completely integrable map persist, either as invariant circles, as periodic orbits or as invariant Cantor sets, in any twist map. The main result of this chapter, Theorem 18.1, provides a vertical ordering of these Aubry-Mather sets in the cylinder for each given map. Indeed, we show that each Aubry-Mather set is a subset of a circle in a family of disjoint, homotopically nontrivial circles that are graph over the circle {y = 0}. The circles in this family are ordered according to the rotation number of the Aubry-Mather sets. To prove this, we establish important properties of the gradient flow of the action functional in the space of sequences. The central property, given by the Sturmian Lemma, is that the intersection index of two sequences cannot increase under the gradient flow of the action. One consequence is that the flow is monotone: it preserves the natural partial order between sequences. This fact yields a new proof of the Aubry-Mather Theorem. It also enables us to define special invariant sets for the gradient flow that we called ghost circles, which we study in some detail here. The family of circles that neatly arranges the Aubry-Mather sets are projections of ghost circles in the cylinder. The results of this chapter come from three sources: Gole (1992 a), in which properties of ghost circles were systematically investigated; Gole (1992 b), where gradient flow techniques were used to give a proof of the Aubry-Mather theorem. There was a gap in that last paper, pointed out to me by Sinisa Slijepcevic which isfixedhere thanks to a lemma from Koch & al. (1994). Finally, the bulk of this chapter comes from Angenent & Gole (1991), in which we gave a proof of the ordering of Aubry-Mather sets via ghost circles. I am deeply indebted to Sigurd Angenent for letting me publish this work here for the first time. The notion of ghost circles originated in my thesis, in which I was looking for regularity properties for
54
3: GHOST CIRCLES
ghost tori, their higher dimensional counterparts. In Chapter 5, a link is made between ghost tori and Floer Homology.
14. Gradient Flow of the Action A. Definition of the Flow Throughout this chapter, we consider a twist map / of the cylinder and its lift F whose generating function S is C2. For simplicity, we will also assume that the second derivative of S is bounded. This mild assumption is satisfied for twist maps of the bounded annulus which are extended to maps of the cylinder as in Lemma 8.2, as well as for standard maps. In this section we investigate the property of the "gradient" flow of the action associated with the generating function S of F solution to: (14.1)
xk = -VW(x)k
= -[d1S(xk,xk+1)
+ d2S(xk-i,Xk)},
fceZ
Since this is an infinite system of ODEs, we need to set up the proper spaces to talk about such aflow.We endow 1RZ with the norm :
—oo
We let X be the subspace of IR of elements of bounded norm, which is a Banach space. On bounded subsets of X, the topology given by the above norm is equivalent to the product topology, itself equivalent to the topology of pointwise convergence. Remember from Chapter 2 that Z 2 acts on IR by:
The map
T0>I
which we also denote by T has the effect of translating each term of the
sequence by 1. The map r^o which we denote also by a is called the shift map, as it shifts the indices of a sequences by 1. We define X/~Z := X/T and we can choose as a representative of a sequence x one such that xo € [0,1). More generally, in this chapter, the quotient of any subset of IRZ by Z will be with respect to the action of the translation T = T 0> I.
Proposition 14.1 Suppose that the generating function S is C2 with bounded second derivative. The infinite system of O.D.E's
14. Gradient Flow of the Action
(14.2)
xk = -VW(x)k
= -[dxS{xk, xk+i) + d2S(xk-i,
55
xk)]
defines a C1 local flow £* on X as well as on XjlL, for the topology of pointwise convergence. The rest points of £' on X correspond to orbits of the map F. Proof. We prove that the vector field — VW is C 1 by exhibiting its differential. The proposition follows from general theorems on existence and uniqueness of solutions of ODEs in Banach spaces (Lang (1983), Theorems 3.1 and 4.3). The following map is the derivative of x — i > — WW(x): L • {vk}keJ. >-> {PkVk-i + akVk + Pk+iVk+i}keZ Oik = -d22S{xk-i,xk)
-dnS{xk,xk+1),
(3k =
~di2S{xk-i,xk)
Indeed, this map is linear with (uniformly) bounded coefficients, hence a continuous linear operator. Clearly: -VW{x
+ v) + VW{x) - L{v) = \\v\\ ip(v)
with lim„_o i>{v) = 0.
•
B. Order Properties of the Flow
Angenent (1988) was the first author, to my knowledge, to notice the similarity between the ODE (14.1) and the heat flow of parabolic PDEs. Indeed, when we consider the standard map with generating function S{x, X) = \(X - a:)2 + V(x), the ODE (14.1) becomes xk = (-Ac)fc
-V'(xk)
where A(x)k = Ixk — Xk-i — xk+\ is the discretized Laplacian. It is not too surprising therefore, that the gradient flow solution of (14.1) inherits analogous order properties to those of heatflows{eg. , the comparison principle). In a nice reversal of roles, de la Llave (1999) has now proven Aubry-Mather type theorems for certain PDEs, using order properties (see Chapter 9). To explore these properties in twist maps, we come back to the notion of order introduced in Chapter 2. IRZ is partially ordered by: i
xk
56
3: GHOST CIRCLES
We also define x < y to mean x < y, but x ^ y; and we write x -< y to denote the condition Zj < %jj for all j e 2 . The order interval [x, y] is defined by: [x,y] =
{zeMx\x
The positive order cone at a;
y+(aO = {y e X I a; < y} with a similar definition for V~(x). These cones are closed for the topology of pointwise convergence. The following statement was observed by Angenent (1988). It is related to the comparison principle for parabolic PDEs (In the case of the standard map. Theorem 14.2 (Strict Monotonicity of £*) For x, y e X with x < y one has Cix) ~< C*(y) forallt>0. We will give a simple proof of this theorem in Section 22. It is also a consequence of the Sturmian Lemma (see below), which was stated in Angenent (1988), and written in Angenent & Gole (1991). Both proofs were communicated to the author by Sigurd Angenent. In Chapter 2, we defined the notion of crossing of two sequences a;, y in H z in terms of their Aubry diagrams. We remind the reader that such a crossing occurs when there is a k € Z at which either xk - yk and xk+i - yk+i have opposite signs, or xk = Vk and Xk-i - j/fc-i and xk+i — yk+i have opposite signs. We say that two sequences are transverse if they have no tangency, i.e. there is no k € Z at which xk = yk and xk-i — yk-i and xk+\ — yk+i have same sign. We denote the transversality of x and y by x f|) y. We now define the intersection index I(x, y) to be the number of crossings of transverse sequences. Lemma 14.3 (Sturmian Lemma) Let x, y e X have different rotation numbers. Ifx,y are not transverse, then for all sufficiently small e > 0 (,±ex, C,±ey are transverse and: i(Cex,Cey)>I(Cex,Cey). Otherwise, as long as (fx and C^y stay transverse, their intersection index does not change.
15. The Gradient Flow and the Aubry-Mather Theorem Proof.
57
See Section 22.
As in Section 5.C, Xp
Corollary 14.4 The sets CO, C O u , and XP:q are all invariant under the flow £', and so are their quotients by the action ofT = TO,I.
Proof.
The inequalities of the type x < r m n x , which define the sets CO and CO w are
all preserved under £'. The invariance of Xp>q comes from the periodicity of the generating function S and its derivatives: when x £ XPtQ the infinite dimensional vector field — VW for the ODE (14.1) is a sequence of period n (made of subsequences of length n equal to VWpq).
D
15. The Gradient Flow and the Aubry-Mather Theorem In this section, we show how the existence of CO orbits of all rotation numbers can be recovered from the monotonicity of the gradient flow £*. From Lemma 9.2 and Corollary 14.4, we know that the set C O w / Z is compact and invariant under the flow £'. Rest points of the flow in this set lift to CO orbits of rotation number u>. It turns out that, even though C' is not the gradient flow of any function, we can still make it gradient like when restricted to the appropriate subsets. Denote by XK = {x e X | sup f c e Z \xk — Xk-i\ < K}. Theorem 15.1 Let C C X fZ.
be a compact invariant set under a and forward
invariant under the flow £'. Then C must contain a rest point for the flow. In particular C O w / Z contains a restpoint and thus the map has a CO orbit of rotation number ui.
Proof.
Assume, by contradiction, that there are no rest points in C. We show that, for
some large enough N, the truncated energy function WN = 5Z-AT S(xk,xk+i)
is a strict
Lyapunov function for the flow £' on C. More precisely, we find a real a > 0 such that 2IWM{X)
< —a for all x in C. This immediately yields a contradiction since on one hand
58
3: GHOST CIRCLES
WN decreases to — oo on any orbit in C, on the other hand, the continuous WN is bounded on the compact K. To show that WN is a Lyapunov function for some N, we start with: Lemma 15.2 Let C be as in Theorem 15.1. Suppose that there are no rest points in C. Then, there exist a real so > 0, a positive integer No such that, for all x € C j+N
N>N0=>VjeZ,
J2 (VW{x)kf
> e0.
Proof. Suppose by contradiction that there exist sequences jn,Nn oo such that (15.1)
J2
(vW(* ( n ) )k)
and x^n~> with Nn —•
-+0.
In
Let m(n) = —jn~ [Nn/2] where [•] is the integer part function, and let x'^ This new sequence cc'(n' is still in C, and satisfies:
=
am^x^n\
Nn~[Nn/2]
rVW{x'(n))k)
Y2
->0
asn-^co.
k=~[Nn/2]
By compactness of C, it has a subsequence that converges pointwise to some x°° in C. Since S is C 2 , VW(x°°)k
= l i m ^ ^ \/W{x'^)k
= 0 for all k and thus x°° is a rest
point, a contradiction.
•
We now show that WN is a strict Lyapunov function on C. By chain rule: ,
-WN{x) d t
N
=-J2
[9iS(xk,xk+1)VW{x)k N
(15.2)
+
d2S{xk,xk+l)VW{x)k+1]
-N
= ~YldlS(xk,xk+1)VW(x)k_N = - diS(x-N, x-N+i)VW(x)^N
- £ (VW(x)kf -N+l
N+l
£ -N+i -
d2S(xk^,xk)VW{x)k d2S(xN,xN+1)\7W(x)N+1
16. Ghost Circles
59
For all x in XK, we have \xk - Xk-i\ < K and hence, by periodicity, S(xk-i,Xk), its partial derivatives as wellas VWk are bounded on XK. In particular, we can find some M depending only on K such that \-dxS(x-N)x-N+x)VW(x)^N
- d2S(xN,xN+1)VW(x)N+1\
<M
for all x in XK and all integer k. Let p = [M/2e0] and AT > (p + 1)JV0, where N0, s0 are as in Lemma 15.2. We claim that for such an N, Wjv is a Lyapunov function. Indeed, we can split the sum 2 - J V + I ( V f ( i ) k ) into 2p + 2 sums of length greater than N0- By Lemma 15.2, each of these subsums must be greater than eo, and thus the total sum must be greater than M + 2EQ, making the expression in (15.2) less than —2eotH Remark 15.3 As in Chapter 2, we can derive from Theorem 15.1 the existence of AubryMather sets of all rotation numbers. This proof does not yield the fact that the orbits found are minimizers. This apparent weakness may be an asset in considering possible generalizations of this theorem to higher dimensions (see Chapter 9). This proof is a variation of the one given in Gole (1992 b). We are very grateful to Sinisa Slijepcevic, who pointed to a gap in Section 3 of that paper. The above is essentially a rewriting of that section. It was inspired by arguments found in Koch & al. (1994), who prove an interesting generalization of the Aubry-Mather Theorem for functions on lattices of any dimensions (see Chapter 9).
16. Ghost Circles The set of critical sequences corresponding to the orbits of an invariant circle of the twist map / , is itself a circle in IR Z /Z. Trivially, this circle is invariant under £*, as it is made of rest points of theflow.This circle is one instance of a ghost circle. In general, we think of ghost circles as £' -invariant sets that are the surviving traces in the sequence space H z of such critical circles. Definition 16.1 A subset T C M 2 is a Ghost Circle, hereafter GC, if it is 1. strictly ordered: x,y e F => x ~< y or y *!, x. 2. invariant under the Z 2 action (by r m n ), as well as under theflow£*,
60
3: GHOST CIRCLES
3. closed and connected. We will see in the Section 17 that GC's can be constructed by bridging the gaps of the Aubry-Mather sets (identified to their corresponding subsets of rest points in IR Z ) with connecting orbits of the gradient flow £'. Any sequence x in a ghost circle J 1 is CO: since r m n x must also lie in r, which is ordered, we must have x -< r m n a ; or Tm>nx -< x. Moreover, the fact that r is ordered implies, by Lemma 13.3, that all sequences in r have same rotation number. We will call this number p(F), the rotation number of the ghost circle.
Proposition 16.3 Let r be a ghost circle. a) The coordinate projection map IR
— t > IR defined by x i—t xo induces a homeo-
morphism of r to IR. The corresponding projection map IR / 2 — i > I R / 2 induces a homeomorphism
between rfS.
and the circle.
b) The set of ghost circles is closed in the Hausdorff topology of closed sets ofJR
,
and it is compact in COra w/2Z. The rotation number on GCs is continuous in this topology.
Proposition 20.2 improves on part b) of this proposition by giving a sufficient condition for convergence of sequences of GCs
Proof of Proposition 16.3. We show that, for any x, y in r, the projection 5 : x i-> XQ defines a homeomorphism from [x, y] n r to the interval [xo, yo] in IR. As before, we give IR the product topology. The projection map S is continuous and the set [x, y] is compact, by Tychonov Theorem, as a product of closed intervals. Clearly 6 preserves the strict order: x -< y =>- XQ < yo and hence it is one to one on F. Take any two points x -< y in T. As a continuous injection, the map 5 defines a homeomorphism between the compact set F D [x, y] and its image. We show that 6(r n [a:, y]) = [S(x), 5(y)). For this, it suffices to show that r D [x, y] is connected. Suppose not and FC\[x,y]
= A U B where A and B are
closed and disjoint in r f~l [x, y]. There are two possibilities: either both x and y belong to the same set, say A or else x € A, y e B. In the first case, we could write J 1 as the union of two disjoint closed sets:
I 16. Ghost Circles
61
r = [(v_(x) n r) u A u (v+(y) n r)] |J s, a contradiction since r is connected. The other case yields the same contradiction. Since r is ordered, any bounded open ball for the product topology intersects r inside an interval [x,y\. Hence what we have shown above implies in particular that <5 is a local homeomorphism on r. To show that it is a global homeomorphism, it remains to show that it is onto. Since r is r-invariant, if x is a point of Z\ then r m j 0 x is as well, and hence the set {XQ + m | m £ 2 } is in S(r). By what we proved above, all the points in between are also in 6(r) and hence 5 is onto IR. This proves a). To prove b), note that if i~fc —> i-1 (in the Hausdorff topology) as k —> oo then any point x € r is limit (in the product topology of 1RZ) of points x(fc) e /&. Since Tm>n and the flow £* are continuous, r must be invariant under these maps. "Close" and "connected" are adjectives that also behave well under Hausdorff limits. Finally, to see that F is strictly ordered, note that if x ^ y are in J1, we can find sequences x^, y^> € i~fc with x = limx(fc),y = limy(k\ If Xj < y-j, we can assume xV*> -< y^> for all k sufficiently large. Since I*, is strictly ordered and C'-invariant, we must have £~'x(fc) -< £~*y(fc) and hence C - 4 ^ < C-*!/- The strict monotonicityofthe flow now implies: x -< y. The continuity of the rotation number is a direct consequences of the continuity of the rotation number on CO sequences, given by Lemma 9.1. • It follows from this proposition that any GC has a parameterization £ G IR i-> x(£) 6 f of the form (16-1)
x(0
= (• • • , z - i ( 0 , W O . S 2 K ) , • • •) •
where the a; j (£) are strictly increasing andcontinuous functions of £. In particular^ H-> XI(£) is a homeomorphism of IR. Invariance of r under the Z 2 action T implies that Xj (£ + 1) = Xj(£) + 1, so that the Xj define homeomorphisms of the circle as well; r-invariance also implies that z2(£) = ^ I C ^ I C O ) ' and more generally that the x n are alliterates of a; 1. Any GC projects naturally to a circle wT in the annulus, where the projection •K : 1RZ —> A is defined by ?r(x) = (x0,-diS{x0,xx)) Proposition 16.3 Let r be a GC for the twist map f. Then 71T and / ( ^ r ) are periodic graphs of periodic functions
62
3: GHOST CIRCLES
L < oo, depending only on the map, and, where the derivatives are defined,
p'(0 > -L,
V'(0 < L.
Proof. If one parameterizes F as in (16.1), then TTF is the graph of (16-2)
y = -ihS&X! ( 0 ) d = ¥>(£)•
Likewise, the image /(nF) is the graph of y = #2,S(:r_i(f), £) = V(0- We now give a proof of the Lipschitz estimate. Using the parameterization of the projection of our GC as in (16.2), it is enough to prove that the derivative of tp is bounded below. The same proof would hold for the estimate for the image f{irr) of our circle. Applying the chain rule to (16.2), we find:
- faS • ?£•
>-dnS.
This last term is bounded below by our assumption on the second derivative of S. A similar argument proves the estimate for ip' (£). D Remark 16.4 As mentioned before (see also Exercise 16.6), the set of critical sequences corresponding to an invariant circle of / is a GC, call it F. In this case TTT = f{irr), and Proposition 16.3 provides a proof that invariant circles are Lipschitz, a result of Birkhoff (see also Proposition 12.3). We end this section by giving a condition that insures that GCs do not intersect. We can define a partial ordering on GCs as follows. Let J i , r2 be GCs. We say that 7"i -< r2 if (i) for all x € Ti,x' £ T2 one has x ffl x' and I(x, x') = 1; (ii) p(A) < p{r2), i.e. p(x) < p(x'). Lemma 16.5 (Graph Ordering Lemma) If T\ -< F2 then the circle -KF\ lies below -nF2. Proof. Leta;^(£) be parameterizations of the form (16.1) for .T, (j = 1,2). Then -KF3is the graph of <^(£) = -diS(€, 4 % ) ) . We claim that x?\£) < x f } (f) for all f. Indeed, for a given £ the sequences x^(£)
and x^(£)
transverse, we must have x^\$,) ^ xf\$)\
intersect at site n = 0. Since they are
by comparing rotation numbers we then get
17. Construction of Ghost Circles x[ (£) < x[ (£). By combining this inequality with the twist condition d^S conclude that
63
< 0 we then O
Exercise 16.6 Prove that the set of x sequences corresponding to orbits of an nontrivial invariant circle for the map is a GC. [If the map has a transitive invariant circle of rotation number ui, then its associated GC is the only GC with rotation number u> (Gole (1992 a), Lemma 4.22. We conjecture that this remains true when the invariant circle is not transitive (i.e., of Denjoy type).
17. Construction of Ghost Circles This section will show that GCs are plentiful. In the first subsection we construct GCs whose projection passes through any given Aubry-Mather set. The next subsection will specialize to GCs with rational rotation numbers. For generic twist maps, we construct smooth GCs containing periodic minimizers. In Section 18 we will refine this construction to obtain ordered sets of GCs, whose projections do not intersect.
A*. Ghost Circles Through Any Aubry-Mather Sets Let Mu the minimal, recurrent Aubry-Mather set of rotation number u>, as defined in Proposition 12.9. It corresponds bijectively to the set, call it Ew of x sequences of orbits in Mu. By Aubry's Fundamental Lemma 10.2, Ew is a completely ordered subset of CO w . If x is a recurrent minimizer, than so is r m > n a; for any m, n e Z , so Su is invariant under T. Each point of Eu corresponds to an orbit of F, and thus is a rest point of £'. In Gole (1992 a), we proved the following theorem:
Theorem 17.1 The set Eu is included in a ghost circle r, and hence the AubryMather set Mu is included in the projection ITF of a ghost circle. Proof (Sketch).
Sw is a Cantor set whose complementary gaps are included in order
intervals of the type ]x, y[ where x,y
e Eu. A theorem of Dancer and Hess (1991) on
monotone flows implies that, in conditions that are satisfied in the present case, if x -< y are two rest points for the strictly monotone flow C* and there is no other restpoint in [x, y] then there must be a monotone orbit (i.e. completely ordered) of £' joining x and y. Hence we
64
3: GHOST CIRCLES
can bridge all the gaps of Eu with ordered orbits of £', taking care to do so in an equivariant way with respect to the r action. The resulting set is a GC. • B. Smooth, Rational Ghost Circles
We now build rational Ghost Circles by piecing together the unstable manifolds of mountain pass points for Wpq in Xpq. This construction will be crucial when we build disjoint GCs in Section 18. Let u> = p/q be given. Beginning here and throughout Sections 18 and 19 , we shall assume the following: For any p/q € Q,
Wpq is a Morse-function on Xp
(17.1)
This is a generic condition on twist maps, as will be proven in Proposition 29.6. Since a GC consists of CO sequences we may assume that p and q have no common divisor (see the proof of Proposition 10.4). Let x G Xp
(17.2)
2
-V Wpq(x)
Oil
Pi
Pi
Q2
0 Pi
0
Pi
e*3
LP*
0
0
P< 9-1 Pq-
where a, = —d2iS(xj-i,Xj) — duS(xj,Xj+i), and Pj = —d\2S{xj-i,Xj) > 0. Due to the Perron-Frbbenius theorem, the largest eigenvalue Ao of — V2Wpq{x) is simple, and the eigenvectors = ( VJ) corresponding to Ao can be chosen to be positive: Vj > 0,j = l,...,q. Moreover all other eigenvectors are in different orthants (See Angenent (1988), Proposition 3.2andLemma3.4).Ifo;isacriticalpointofindex 1, there exist two orbits a±(x;t),t € IR of the gradient flow £' of Wpq with a±(x; t) —> x as t —> —oo, and with a ± (x; t)=x±
eA°'£ + o (eXot).
These two orbits, together with x itself, form the unstable manifold of x. The orbits a± (x; i) are monotone, a+ being increasing, and a_ decreasing; since
T±I$X
= x ± 1 are also
critical points, we have x — 1 < a± (x; t) < x + 1 so that a± (x; t) is bounded. Hence the limits
17. Construction of Ghost Circles
65
u)±(x) = lim a±(x;t) t—>oo
exist and they are critical points of Wpq. Since £' is monotone, there are no other critical points y witho>_(x) < y < x o r x < y < ui+(x). If y > x is another critical point, then y > u+(x). Moreover, since the Morse index must decrease along the negative gradient flow, the points ui±(x) have index 0, i.e. they are local minima of Wpq. We now show that the orbits ce± (x; t) converge to these points along their "slow stable manifold", tangent to the largest eigenvalue of — V2Wpq(w±(x)).
Indeed, since LJ±{X) are minima, all the
eigenvalues are negative, and thus the largest one has the smallest modulus. All orbits in the stable manifold of u>± (x) except for a finite number that are tangent to the eigenspaces of the other eigenvalues, are tangent to this "slow stable manifold". But the other eigenvectors are in different orthants than the positive or negative ones. Hence a±(x;i),
which are
in the positive or negative orthant of w±(x), must converge to u>±(x) tangentially to the eigenvector of largest eigenvalue. To construct a GC in Wpq we first consider the set of critical points such a GC must contain. Definition 17.2 A subset A C XPiq is a skeleton if the following hold. Si A consists of critical points of Wpq with Morse index < 1, 52 A is invariant under the Z 2 action r, 53 A is completely ordered. A skeleton A is maximal if the only skeleton A' with A C A' C XPiq is A itself. Lemma 17.3 A maximal skeleton A for Wpq Proof.
exists.
Chooser, s with rp+qs = landdefineT = r r>s . By Aubry's fundamental lemma
the set Ao of absolute minimisers of Wpq is a skeleton. We fix some element x G Ao- Any skeleton A D Ao is completely determined by
B = An[x,T(x)]
= {zeA:x
Indeed, given B we can reconstruct A as follows: 00
(17-3)
A=
|J j=—oo
T*{B).
T(x)}.
66
3: GHOST CIRCLES
Conversely, any ordered set B C [x, T(x)} of critical points determines a skeleton A D Ao by (17.3). The closed order interval [x, T(x)} is compact and Wpq is a Morse function, so there are only finitely many critical points in [x, T{x)]. We can therefore choose a maximal ordered set of critical points B C [x, T(x)] and be sure that the corresponding A is a maximal skeleton.
•
Lemma 17.4 (Mountain Pass Lemma) / / the skeleton A is maximal, other point (according to the order) is a local minimum;
then every
the remaining points are
minimaxes. Proof.
If x < y are consecutive elements of a maximal skeleton A then we must show
that exactly one of x and y is a local minimum. S t e p 1. If x and y are both local minima then the following standard minimax argument shows that there is a third critical point of index 1 between x and y. Define Q = [x,y] and consider Q~> = {zeQ:
Wpq{z) < 7}
Each Q1 is compact, and if 7 > max Wpq | Q then Q 7 = Q is connected. On the other hand, Q 70 with 70 = max (Wpq(x),
Wpq(y)) is not connected, since x and y are local minima
of Wpq. Consider 71 = inf{7 > 70 : x and y are in the same connected component of Q 7 }. By compactness, x and y are connected in Q 7 1 , and hence 71 > 70. Suppose there is no critical point of Wpq in ]x,y[. Note that, by order preservation, Q = [x,y] is forward invariant under the gradient flow: C'(Q)
c
n 7 > 7 l Q 7 there is an e > 0 such that C x (2 7 1 ) 7l
£
Q for t > 0. By compactness of Q 7 1 = c
2 7 l ~ £ > which implies that x and y are
connected in Q ~ , a contradiction. Hence there is at least one critical point z
e]x,y[,
with Wpq (z) — 71. If the Morse index of all such z were 2 or more, then the Morse Lemma 61.1 would show that Q 7 with 7 slightly less than 71 would still connects x and y, so the index of at least one such z is 1. But now we have a contradiction: if x and y are both local minima, then there is a minimax point z €]x,i/[and«4u{T m > n z : m , n g 2 } is a skeleton; this cannot be since A is maximal.
18. Construction of Disjoint Ghost Circles
67
S t e p 2. Next we show that either x or y is a local minimum.If x is not a local minimum, then w+(x) = lim^oo a + ( x ; £) is a local minimum. But OJ+{X) < y, so w+(x) = y, and we find that y must be a local minimum. Likewise, if y is not a local minimum, then x = w_ (y) must be one.
•
We have all the ingredients necessary to show the following, which was proven in a slightly different form in Gole (1992 a), Theorem 3.6. Theorem 17.5 Assume Wpq is a Morse function. If A is a maximal skeleton,
then
r_A — {a±(x; t) :t G H , x e A is a minimax} U A is a C 1 ghost circle. Proof.
It is simple to check that, by maximality, r^ is connected, and a ghost circle.
As a union of unstable manifolds, i~U is smooth except perhaps where different unstable manifold meet, at the minima. But we showed above how the orbits a± (x; t) must converge tangentially to the one dimensional eigenspace in the positive-negative cone of the minima. Hence the GC constructed is also smooth at the minima.
•
Exercise 17.6 Check that rU is indeed a GC.
18. Construction of Disjoint Ghost Circles We now arrive at the main result of this chapter, which provides a vertical ordering of Aubry-Mather sets: Theorem 18.1 (Ordering of Aubry-Mather Sets) Given any interval [a, b] in IR there is a family of nontrivial circles C „ , w £ [a, b] in the cylinder such that: (a) Each Cu is the projection of a GC ru and hence is a graph over {y = 0} (as is
ncu)j. (b) The Cw are mutually disjoint and if' w > w1, Cu is above Cu*. (c) Each Cu contains the Aubry-Mather number ui.
set Mu of recurrent minimizer
of rotation
68
3: GHOST CIRCLES
This section and the next two are devoted to the proof of this theorem. We will first construct, in this section and the next one,finitefamilies of rational ghost circles. In Section 20, we will take limits of such families and conclude the proof of the theorem. Let w i , . . . , Wfc be distinct rational numbers. The construction of the preceding section provides us with maximal skeletons Ai,...,Ak and corresponding GCs rAl,..., rAk. It is not immediatly clear from this construction that the projections Cj = TVTA. are disjoint. In this section we show that the skeletons can be chosen so that the Cj are indeed disjoint. Definition 18.2 A family of skeletons Aj C XPjqj is minimally linked if any pair x £ Ai, y 6 Aj with i ^ j is transverse with I(x, y) = 1. Theorem 18.3 (Disjointness Theorem) If Aj c XPjqj is a minimally linked family of maximal skeletons, then the projected ghost circles Cj = TTT^ are disjoint. Proof. Order the Aj so that their rotation numbers pj = Pj/qj are increasing. Then we claim that
(i8.i)
rM
Disjointness of the projected GCs then follows directly from the Graph Ordering Lemma 16.5. To see why (18.1) holds, we consider any pair i ' 1 ' e rAi, x^ e i"^. and assume that they are not transverse. Since p(Clx^) ^ p(C'x^') we must always have J (C*xW, C*a;(j)) > 1 when defined. By the Sturmian Lemma 14.3, (18.2)
/(C'a;W,C*a; W ) > 1
for all those t < 0 at which C*x(i) fl C ' ^ - B u t l i r n t ^ - o o C * ^ = y(l) for some y® € Ai (I = i,j). Since the Ai are minimally linked we must have I{y^\y^) = 1, which contradicts (18.2). D Theorem 18.4 For any k-tuple u>i,..., u>k of rational numbers there exists a minimally linked family of skeletons Ai,...,
Ak such that each Aj is a maximal skeleton.
This theorem, combined with the Disjointness Theorem, provides us with a disjoint family of circles Cj = ^TAj in the annulus. The construction of the Aj's will be such that
18. Construction of Disjoint Ghost Circles
69
they automatically contain the absolute minimizers of WPiQi, which by Proposition 10.4 are the minimal energy orbits of Aubry-Mather. In our proof of Theorem 18.4 we begin with constructing a maximal fe-tuple of skeletons, and then show that each skeleton in this fc-tuple is maximal. Proof of Theorem
18.4- Let Mj be the set of absolute minimizers of WPjqj on
Aubry's fundamental lemma implies that M\,...,Mk
XVjqj.
is a minimally linked family of
skeletons. As in the proof of Lemma 17.3 one easily finds a maximal fc-tuple of minimally linked skeletons A\,...,Ak
with Mj C Aj, by observing that there are only finitely many
such extensions. We shall now show that each Aj is a maximal skeleton. Assume that one of the Aj, say Ai is not maximal. Then there is a critical point z € WPiqi with index 0 or 1, such that Ai U {z} is completely ordered. In particular, there must exist a couple of adjacent critical points x < y in A\ with z e]x, y[. We must deal with two different cases: A . Both x and y are local minima of WPiqi. B . At least one of the critical points x or y has index 1. Case A. By a minimax argument we will show that there is a critical point between x and y which allows us to extend Ai to a larger skeleton A[ for which (A'x,... minimally linked. This would then contradict maximality of (A\,...
,Ak) is still
,Ak), and thereby
show that Case A cannot occur. To carry out the minimax argument we consider H = {w e WPiqi : x < w < y, Vj > 2, Vv € Aj, v ffl w and I(v, w) = 1}. and its closure J7. The Sturmian Lemma implies that Q, and hence ft are forward invariant under the gradient flow f*. Also, as in Mountain Pass Lemma 17.4, Q is compact, as are the sublevel sets ffl = {w G Q : WPiqi (w) < 7 } . To obtain a critical point other than x and y in f2 we must show that not all the D^'s have the same topology. If 70 = max(W P 1 ? 1 (a;), WPiqi(y)),
then Q10 is again not connected, since x and y are local
minima. On the other hand we have
70
3: GHOST CIRCLES
Lemma 18.5 Q is connected. Postponing the proof of this statement to the next section, we can now easily complete the minimax argument. Indeed, as in the Mountain Pass Lemma, 7i = inf (7 > 70 : ff connected) is a critical value of WPiqi, so there must be a third critical point w G J?. By the Sturmian Lemma w must lie in Q, and it follows from the Morse lemma that w has index 1. Put (18.3)
A[ = Ai U {Tm,nw
:m,ne~Z.};
then (A'i,. •., Ak) is a minimally linked family of skeletons extending ( . 4 i , . . . ,Ak), and we have our contradiction. Case B. Assume that x has Morse index 1, and put w = ui+(x).
Then w is a critical
point of WPiqi and is therefore transverse to any v € Aj with j > 2, by the Sturmian Lemma. We claim that I(w, v) = 1. Indeed, for t —> —00 we have a+(x; t) —> x. Since (Ai,...
,Ak) is minimally linked, we find that for all t sufficiently large negative a+ (x; t)
and v are transverse with I(a+(x;t),v)
— 1. By the Sturmian Lemma
I(a+{x;t),v)
cannot increase, and since a+ (x; t) and v have different rotation numbers I{a+ (x;t),v) 1 for all t: hence I(a+(x;
>
t), v) = 1 for all t. Letting t —+ +00 we get I(w, v) = 1, as
claimed. Defining A'x as in ((18.3) ) we again get a larger minimally linked family of skeletons, a contradiction. If x is a local minimum then y cannot be one by Case A, and considering o>_ (y) leads to a similar contradiction.
D
19. Proof of Lemma 18.5 We must show that Q is connected. We shall do this by showing that any w e Q can be connected to x via a path 7 : [0,1] —> O U {x}. For any j € Z and any x < w € XPiqi we put A,-(x : w) = {VJ : v G -4 2 U • • • U A } l"l [xj,Wj). For simplicity we shall write as rfl ^ U ' - U A when we mean that x fjl w for every v G * 4 2 U - - - U Ak-
19. Proof of Lemma 18.5 Proposition 19.1 Given x < w in
71
Xpiqi,
(i) Aj(x : w) is finite, for each j € Z . (ii) Aj+qi(x
: w) = Aj(x
(iii) If z € XPiqi
:w)+pi.
and x < z < w, then z ffl Ai U • • • U Ak, if and only if they are
tranverse in the index range 0 < j < q\. Proof.
(i): WPjQj is a Morse function, (ii) holds because x,w€
XPiqi and the Ai are
invariant under the action of T mi „, m , n £ l (iii) is a consequence of (ii).
•
We define the height of w over x by 91-1
If the height h(x : w) vanishes then all the Aj(x : w) are empty and we can define 7(i) = tw + (1 — t)x. Since Xj < 7j(t) < Wj for all j and 0 < t < 1, it follows from part (iii) of our last proposition that j(t) ffl ^.2 U • • • U Ak for 0 < t < 1, so that 7(4) stays within /2. Call this a move of type 0. We now assume that h(x : w) > 0, and we show how to decrease it to zero. Suppose that for some / one has wi = vi > xi for some v G A2 U • • • U Ak- Then there is an e such that 0 < £ < wi — xi and (w; — e, w;) n A;(a;: w) is empty and we can define , ^
( w-j — e if j = / m o d g i , 1 uij otherwise.
As before one can connect w and « / by 7(i) = tto + (1 — i)iu' without leaving £2. Call this a move of type 1. Assuming now that Wi / v^ for all i, we will move the sequence w down by interpolating it linearly to: (i) _ j max Ai{x : w) \ u>i
1
\fi = lmo&qi, otherwise
for some judiciously chosen I. Call this a move of type 2. Clearly z^ (
£ XPiqi
and x <
(I
zW < w, zW = z ( +9) and h(x : z )) = h(x : w) - 1. We need to show that for at least one I € Z , this move does not change the intersection index of w with the elements of Ai U • • • U Ak- Consider the set of elements in A2 U • • • U Ak that are immediately below w:
72
3: GHOST CIRCLES
(si) def
. ,
.
a{ = max.Ai(x : w). Assume that, among the sequences a^ at least one has rotation number greater than that of x and pick the one, say a ^ ' which has the largest rotation number (If all a1-"^ have lower rotation number than x, pick the one that has the lowest and proceed similarly). In the following, we only worry about the possible changes of intersection index in the range 0 < j < <7i- The periodicity condition (ii) of Proposition 19.1 insures that if there are changes of index, they must occur periodically. There are three cases (see Figure 19.1) to consider:
Wj+l
Case 1
Case 2
Fig. 19.1. The two possible moves of type 2. Case 1: a^l > w J + 1 Choose I = j and move w to «W as defined above. This could only change the intersection index of w with a^s'\ But in this case this intersection index remains the same: since p(a(-3^) > p(w) = p(x), and I(a,(Si\w) = 1, we must have aj_x < afli < Wj-iSj Hence the one crossing of w and a^ ^, which occured between j and j +1 is now moved to a crossing that occurs at j , with no other crossing introduced with this or any other sequence of.4 2 U---U Ak. Case 2: a^> < afct0 Since by assumption p(a<-s^+1'>) < p(a<-3j^), we must have a^ +1' > a^' and thus a.j > Wj, by maximality of aj- . Now choose I = j + 1 and move w to z^: the one crossing of w and a^Sj+1\ which occured between j and j + 1 is now moved to a crossing that occurs at j + 1.
20. Proof of Theorem 18.1 C a s e 3 : a%\ =
73
afc^
Si
The equality a\ ' = a\Si' cannot be true for all i > j since otherwise w and a.(s^ would have same rotation number. Hence for some i > j , Case 1 or 2 must occur. Apply the procedure for these cases there. Concatenating moves alternating between type 1 and 2, we get a curve in Q between w and and a sequence which has zero height. Concatenate this with a move of type 0 to get a curve in i? between w and x.
•
20. Proof of Theorem 18.1 Let wi, W2) • • • be an enumeration of the rational numbers in the interval (a, b). Proposition 20.1 There is a family of GCs {i^ number uij, and where r>
X -T-
, . . . ,.T„ }, where r j n ) has rotation
i/ a>j < u)j. Each r>n
contains at least one
minimizing periodic orbit of rotation number u>i, and generically all of them. Proof.
If one assumes that the map / is such that the Morse property 17.1 holds, then,
according to Theorem 18.4, one can find a minimally lirked family of maximal skeletons {.4.1 , . . . ,An
} such that A™ has rotation number ujj and contains all the absolute (n)
minimizers of that rotation number. The corresponding GCs J!> ' = -T. M then satisfy the required conditions. In general, when the Morse property 17.1 is not satisfied, one can approximate / by smooth twist maps f£ which do satisfy 17.1 (since this condition is generic); One thus obtains ghost circles F- ™ , and by the compactness of the set of GCs with a fixed rotation number (Proposition 16.3) one can extract a convergent subsequence whose limit will then be a family { r 1 ( n ) , . . . , i i n ) } of GCs. But we need to make sure that limits of strictly ordered rational GCs stay strictly ordered. To see this, notice that the set r^J
x r^J is, when i / j ,
included in: (2ij = {(v,w)
£ PCO W i x PCO w . : v fjl w
and
I(v,w)
where PCO w is the set of periodic CO sequences of rotation number u>: P C O p / 9 = COp/g fl XPiq.
= 1}
74
3: GHOST CIRCLES
The set i?^ is, by the Sturmian lemma, positively invariant under the product gradient flow C* x £' corresponding to any twist map. In fact: (£* x ^(Clos £2^) C (Int fi^), as can easily be checked (i.e.Clos fi^ is an "attractor block" in the sense of Conley). As Hausdorff limit of compact sets in ntj, the set J1/"' x .T>n) is in Clos £2^ .But, since it is both positively and negatively invariant under £' x £', r> n ' x r j n ) must in fact be in Int fi^ where the intersection number is well defined and always equal to 1. In other words, we have shown that, whenever Wj < Wj one must have r\n' -< r^'. Finally, the set r>n' contains at least a minimizing periodic orbit, since the sets r>E' contain by construction all the minimizing periodic orbits of period uii for fs, and limits of minimizers are minimizers. • A. Rational CVs We now construct the CJs of Theorem 18.1, starting with all the rational a; £ [a, b]. Again, we use the compactness of the set of GCs: For each n, Proposition 20.1 provides us with GCs i f ' , ..., r „ with rotation numbers u>i, ..., wn. By compactness we can extract a subsequence {rij} for which the r^3' converge as j —» oo to a GC of rotation number (n')
u>i. Using compactness again, we can extract a further subsequence n' for which 7\ ' and (n')
F2 ' both converge; repetition of this argument and application of the diagonal trick then finally gives a subsequence n" for which all rk 3 converge to some limiting GC i")[ (of rotation number w^) as j —» oo. By the same argument as in the previous proposition, the limits r^ oo) satisfy j ; ( o o ) -< r^°°] whenever w* < uij. We then define CUk = Trr^00' and by the Graph Ordering Lemma 16.5, the CUk 's are disjoint. In the generic case, each r>n' contains all the periodic minimizers of rotation number u)i, and hence so must the limit r^00'. In the non generic case, r>°°' must contain at least one periodic minimizer of the energy. B. Irrational CJs To complete our family of rational GCs with irrational ones, we once again take a limit. We could proceed in a way similar to what we did in order to get all rational GCs, but we would have to appeal to the axiom of choice (no diagonal tricks on uncountable sets!). To avoid this, we first prove a proposition of monotone convergence of GCs. We shall write A ^ i~2 if either Ji -< T2 or p(A) = p(r2) and 7ri~i is ( not necessarily strictly) below 7iT2.This
20. Proof of Theorem 18.1
75
last condition is equivalent to xy (£) < x{' (f) in the notation of the proof of the Graph Ordering Lemma 16.5. Proposition 20.2 (Monotone Convergence for Ghost Circles) LetT^)
be an increasing
sequence of GCs, i.e. assume that
r(!) -< f (2) x _r(3) x . . •. Asswme ako £/ia£ the rotation numbers pj = p(r^') there is a unique GC r(°°) such that f ^ is the parametrization
of F^
are bounded from above. Then
- • r(°°) as j —> oo. Moreover, if
with XQ (£) = £, i/ien Wie a:*. (£) converge
ically and uniformly to x£° (£), where x^°°\^)
is the parametrization
x^($)
monoton-
of T1-00' with
Of course, the corresponding theorem for decreasing sequences of GCs also holds. We postpone the proof of this proposition till the end of this section. Assume now that we have constructed the rational GCs i ^
as above. For any number
w € {a,b), rational or otherwise, we can then define two GCs F^ as follows. Choose a sequence of rational numbers w n . which increases monotonically to UJ. The Monotone Convergence Theorem tells us that the limit of the corresponding GCs r„°° must exist. We denote this limit by r~. This procedure might produce an ambiguous definition of F~, since the result could depend on the choice of the sequence nj: If one has two such sequences, nj and n'3,, then the r „ ° ° ' and / „ ,
might have two different limits F and i~". However, one can
take the union of the two sequences to obtain a third sequence n'', i.e. {n'j} = {nj} U {n'j}. The u)n" then also increase to u, so that the r „ ' also must converge to some GC F". Since nj and n'j are subsequences of n", both sequences rij and n'j must produce the same limiting GC: hence r = F1 = r", and the definition of r~ is independent of the choice of the nj. We choose to define Cu = -KT~ (or -KF^, but with the same choice of + or — for all u) in order to avoid using the axiom of choice...). We now check that, for ui irrational, the unique Aubry-Mather set Mu of recurrent minimizers (see Proposition 12.9) is included in C w . We can take a sequence of periodic Aubry minimizing sequences xk e i ^
where u>k /
w ( \ if one chose Cw =
nT+).
Then xk —* cc, an Aubry minimizing sequence in F~. The orbit that CE corresponds to is
76
3: GHOST CIRCLES
recurrent and minimizing, as limit of recurrent and minimizing orbits. Its closure, which is also included in Cu, must be the Aubry-Mather set Mw . From our definition of J ^ , it is clear that:
wi < <* < Uj => i f 5 -< rz < rj -< rj°°\ for rational u>j,Wj and irrational w. Hence the set formed by the rational GCs i ^
and
the irrational ones ru is completely ordered according to their rotation numbers. By the Graph Ordering Lemma 16.5, the C w 's (irrational and rational) that we have constructed are mutually disjoint.
•
Remark 20.2 If u is a rational number, r~ is no longer necessarily in PCOu but is certainly in CO w . It may contain the sequences corresponding to homo(hetero)clinic orbits joining hyperbolic periodic orbits of rotation number ui. Hence we may (and, probably, generically do) have three distinct Ghost Circles r~
X ru
< Fj for each rational u> where ru is
PJf° for some k. We will call their projections C~, Cu and C+ respectively. Instead of the set {Cw}u,e[a,6] °f strictly non mutually intersecting curves that we have found in Theorem 18.1, one might prefer to consider the bigger set {Cu U C+ U Cz}u>e[a,b]- K is n o t hard
to
check that this is a closed set of GCs. Proof of Proposition
20.2. It follows from the Graph Ordering Lemma 16.5 that the
2^ (£) are monotonic in j . We have assumed that the rotation numbers of the r^) bounded, say by some integer M. Since x^ x
i
are
is CO, this bound implies for I > 0 that
(0 < £ + KM + 1), and in view of the monotonicity of the x\r (£) they converge to
some x, ( o o ) (0- For negative I one finds that x[j)(£) decrease to some i ,
(£). Clearly xj
> £ + l(M + 1), so that the
x\j)(£)
(£) is a nondecreasing function of £. We shall show
that it is strictly increasing, and continuous. x
i
(0 *s strictly increasing.
Let £ < rj be given. Then t \-> C*(x
C ' O ^ f a ) ) both are on the GC r^\
(£))
an
^ * l—•
so that they must be ordered in the same way for all
t 6 M. At t — 0 we have
Z=
e(x(i)(Oh
so this ordering must hold for all t. Upon taking the limit j —> oo we findthatC'(x(oo)(£)) < 00 C* (a^ ) (7/)) holds for all t. By the strict monotonicity of (*, we must have strict inequality
21. Proof of Theorem 18.1
77
for all t, unless we have equality for all t. Equality cannot happen of course, since XQ (£) = £ < 77 = a ^ f a ) - H e n c e w e h a v e a; (oo) (0 < a(°°)(?7); in particular a^ 0 0 ^) < x^irj). x i ( 0 *s continuous. Since the i j (f) are monotonically increasing in both jf and £, their limit is increasing and lower semicontinuous in £. Thus we only have to show that 4°°> (£) = x
D
Exercise 20.3 Complete the sketch of the following alternate conclusion to the proof of Theorem 18.1, which does not use Proposition 20.3, but uses the axiom of choice. For each p = (u>i,... ,ujk) in Q fc , and k arbitrary, consider the set, given by Theorems 18.3 and 18.4, Qp = (J g rUi, union of GC's whose projections do not intersect. Let
J[a,b) = closure{(a:, y) G (CO[o.)b])
| I{rm,nx,y)
< 1, V(m,n)
€ Z2} .
This is a compact attractor block for the flow C* x C' o n the cartesian product (CO[ a 6 ]) . Let K C J[a,b] be the maximum invariant set in J[a,b]- Then K and its projection K' on the first component are both compact. Take an increasing (for the inclusion) sequence of finite subsets 11 of Q, say { W } J 6 I N such t h a t (J W = Q n [a,b]. Since K' is compact, assume t h a t the sequence of compact sets {Q-^i }igiN converges (in t h e Hausdorff topology) to a set C in K'. Now show t h a t for all u E [a,b], £ n CO„ contains at least one GC. Show t h a t two GCs ru,rui of different rotation numbers in C must satisfy i~L/ n r „ ; = 0. To construct a partition, i.e. a family of non intersecting circles , pick (using the axiom of choice!) one GC of C for each UJ in [a, b].
78
3: GHOST CIRCLES
21 .* Remarks and Applications A*. Remarks 1) The techniques introduced in this chapter have a scope that goes beyond proving the vertical ordering of Aubry-Mather sets. Angenent (1988) introduced the flow £' and its monotonicity. He used it to prove, for instance, the existence of periodic orbits that, in the generic case, would come from "elliptic islands around elliptic islands", as well as homoclinic and heteroclinic orbits between hyperbolic points. The remarkable fact is that his results do not make any generic assumption. This is a definite advantage of the variational techniques over the hyperbolic techniques with which removing generic assumptions about transversality of unstable manifolds is often a major hurdle. As an example, it was this kind of technical hurdle that barred Tangerman & Veerman (1990a) to obtain a complete proof that the Aubry-Mather sets are vertically ordered, a fact that they conjecture in that paper. In Chapter 9, we review work by Angenent (1990), Koch & al. (1994) and Candel & de la Llave (1997) which use the monotone properties of variational problem in higher dimensional and PDE contexts. 2) Ghost circles first appeared in Gole (1992 a). They stemmed from an effort I was making in understanding the ghost tori of my thesis (C'-invariant sets for symplectic twist maps, see Chapter 5). In the realm of twist maps, I had constructed C* invariant circles within the ghost tori. My advisor G. Hall as well as R. MacKay and J. Meiss asked me if their projections were graphs. I proved that in Gole (1992 a), where I also recover a result similar to that of Mather (1986) on the existence of invariant circles. MacKay and Muldoon showed numerical evidence that well chosen ghost circles were disjoint, which led to the work of Angenent & Gole (1991) which makes the bulk of this chapter. In his work on toral and annulus homeomorphisms, LeCalvez (1997) proposes another way to construct ghost circles: take an ordered circle in CO w /Z which is 2Z2 invariant, but not necessarily £* invariant. A simple choice is the "straight" circle with #&(£) = ku) + £. Apply the flow £' to this whole circle, and take a limit as the time t —• oo. Le Calvez suggested to us that letting the flow act on non-intersecting collections of rational GCs may be a way to prove Theorem 18.4. In a way that is reminiscent to Le Calvez' construction of GCs, Fathi (1997) has obtained, in the context of convex Lagrangian systems, the generalized Aubry-Mather sets of Mather (see Chapter 9) by applying a flow in a functional analytic space of Lagrangian graphs. Finally Katznelson & Ornstein (1997) find Aubry-Mather sets
21. Remarks and Applications
79
on a collection of pseudo-graphs that are (not strictly) ordered vertically. They do this by iterating the map on circles in the annulus, trimming the image of the circles at each step so that they remain pseudo-graphs (see Chapter 6). It would be interesting to investigate the parallel between these different methods. B. Approximate Action-Angle Variables for an Arbitrary Twist Map If in some well chosen coordinate system (say (x, y)) of IR2 a twist map is completely integrable, these coordinates are called Action-Angle variables (x is the angle, y the action). Dewar & Meiss (1992) attempt the construction of approximate action-angle variables using almost-invariant circles defined through a mean square flux variational principle. We refer the reader to their paper as to the physical relevance of such coordinates. We show here that similar approximate action variables can easily be defined from our GC's. Given any finite number of minimal Aubry-Mather sets, we will produce a continuous foliation of the annulus by circles such that each of the Aubry-Mather set of our chosen collection is contained in a different circle of the foliation. Moreover, such a construction will also produce a completely integrable, albeit not necessarily differentiable map of the annulus that coincides with the original map on the collection of Aubry-Mather sets and leaves the foliation invariant. We sketch here the simple construction. Let MUl,..., MUn be an arbitrary collection of minimal Aubry-Mather sets. From Theorem 18.1, we know that we can produce a corresponding collection JTi,..., rn of GC's whose disjoint projections contain the chosen Aubry-Mather sets. Parameterize these GC's by their coordinate xo as in (16.1) and order them by increasing rotation number. Between two succesive GC's , say A and Tfc+j, construct the continuous family:
r t (o = (---,a;t-i(o.e^i(o.---) with
x\(0 =
(l-t)x[k\0+tx[k+1)(0
*5(0 = (*i) J '(0 where, since both x^ and x\k+1^ are lifts of homeomorphisms of the circle, x[ also is (it must be periodic and monotone); (x\)j represents the jth iterate of this homeomorphism. It is not hard to see that, for t jt 0 or 1, Ft has all the properties of a GC except for that of being invariant under theflow.In particular it is a circle in CO Wt /r 0 ,i on which the shift 7-1,0 acts as a circle homeomorphism with rotation number ut — (1 — t)ojk + tuk+i- Its
80
3: GHOST CIRCLES
projection irrt is a graph in the annulus. The circles 7rJi do not intersect for different t's since in the (x0, xi) coordinates, they are the linear interpolation along the xi axis of the non intersecting graphs of x[' and x[ + '. Repeating this process between each pair of adjacent Ifc's in our finite collection gives the continous foliation ixFt advertised. The completely integrable map is given by TI>0 acting on the family Tt of Ghost Circles, or alternatively by "" ° Ti,o ° TT-1 acting on the annulus, which is the topologically embedded image (by ii) of the family rt. Since for generic maps the rational GC's can be made C 1 , the above construction yields, when starting with a generic map and rational Aubry-Mather sets, a C1 foliation (after smoothing the glueing of our interpolations with suitable time reparameterizations). All the minimizing periodic orbits of the chosen rotation numbers are then embedded in the construction. One can also take a limit of this process, by adding more and more AubryMather sets. One obtains an ordered continuum of circles in 1RZ which contains our set C of the proof of Theorem 18.1. Alternatively, we could have started with the set L of GCs and filled its gaps as above, all at once (gaps will occur between the F~ and the r~ of a given rotation number). Further study of this object might be interesting in order to draw a parallel between twist maps and families of circle maps, eg. in the theory of renormalization (see MacKay (1993)). C*. Partition for Transport In the theory of transport of MacKay, Meiss & Percival (1984) and (1986), it is sought to use almost invariant circles in order to form disjoint boxes containing the "resonance zones" around the elliptic islands (or hyperbolic points with reflexion) of the periodic minimax orbits of different rational rotation numbers. It is not hard to see that the pairs Cp/q± of projections of the p/q± GC's each enclose the circle Cp/q of Theorem 18.1: they are defined as limits of circles that are respectively strictly above or strictly below Cp/q. Moreover, as in the almost invariant circles (or partial separatrices) of MacKay, Meiss & Percival (1986), Cp/q and the Cp/q± all meet at the minimum p/q orbits, at least when there are finitely many of these minima (i.e. generically). Cp/q+ (resp. Cp/q_) contains the advance (resp. retrograde) homoclinic orbits (min and minimax), by an argument of Hasselblat & Katok (1995), in their Proposition 13.2.11. We therefore hope that the boxes defined by the pairs
22. Remarks and Applications
81
Cp/q± of GC's may be used as intended for the partial separatrices in MacKay, Meiss & Percival (1986). The advantage of our boxes over those formed by partial separatrices is that their boundaries are graphs and that they are disjoint from one another (statements unproven to our knowledge for partial separatrices in the general case. See Tangerman & Veerman (1990a) for partial results). Hence the calculation of the flux through them does not rely on the hypothesis that the turnstiles of MacKay et al. always have the simple shape of afigure8. One of the advantages of their partial barriers is that they can canalise the flux through "cheminees", i.e., points exit a resonance zone through one turnstile (as opposed to infinitely many in our case).
D*. An extension of Aubry's Fundamental Lemma
As a consequence of Theorem 18.4, we get that any pairs of points in two unlinked maximal skeletons of distinct rotation numbers have intersection index 1. By Aubry's Fundamental Lemma, we knew this to be the case for minimizers, but our results shows that it is also true for the minimaxes and local minima in the skeletons. The relevance of this appears clearer in the light of LeCalvez (1991), where he shows that this intersection number is geometrically a linking number for the corresponding orbits of the suspension flow of the map. Extending an idea of Hall (1984), he shows that this linking is an obstruction to continue periodic orbits simultaneously, through paths of periodic orbits in an isotopy of the map to some completely integrable twist map. In our terminology, his result implies that the periodic orbits corresponding to critical sequences in a set of minimally linked skeletons can "continue" simultaneously through curves of periodic orbits of an isotopy of our map to a well chosen completely integrable map. In particular, LeCalvez already noted that, because of Aubry's Fundamental Lemma, any collection of minimum periodic orbits can be continued simultaneously to orbits of a completely integrable map. A consequence of Theorem 18.4, where we construct minimally linked sets that contain minimum and minimax orbits, we get, using LeCalvez' result, periodic local minimizers as well as orbits of minimax type continuing simultaneously to orbits of a completely integrable map /o, through paths of periodic orbits of a curve of maps joining / to /o.
82
3: GHOST CIRCLES
22. Proofs of Monotonicity and of the Sturmian Lemma In this section, we give the proofs of Theorem 14.2 and Lemma 14.3. Eventhough it is a consequence of the latter, we start with a simpler, direct proof of the former. Both proofs are by S. Angenent. A. Proof of Strict Monotonicity
We let the reader show that if the operator solution of the linearised equation: (22.1)
u(t) = Lu(t)
with L : {vk}keJ. *-> {PkVk-i + akvk + pk+\vk+i}keT otk = -d22S(xk-i,xk)
-duS(xk,xk+i),
Pk =
-d-L2S{xk-i,xk)
is strictly positive, then the flow £' is strictly monotone. L(x(t)) is an infinite tridiagonal matrix with positive off diagonal terms —di2S(xk, xk+i) (see Formula (17.1) for afinitedimensional version of this matrix). The diagonal terms dnS(xk, xk+i) + d22d2S(xk-i, xk) are uniformaly bounded by assumption on S. Hence, for any T > 0 for which x (t) = C* (x) is defined when 0 < t < T, we can find a positive A such that: B{t) = L(x{t)) + Md is a strictly positive matrix. If u{t) is solution of the equation (22.1) then eXtu(t) is solution of: (22.2)
v(t) = B(t)v(t),
hence the strict positivity of the solution operator for (22.1) is equivalent to that of (22.2). Looking at the integral equation: v(t) = v(0) + f B(s)v(s)ds, Jo one sees that Picard's iteration will give positive solutions for a positive vector v(0). This will imply, assuming that vk(0) > 0, vi(0) > 0, for l ^ k:
22. Proofs of Monotonicity and of the Sturmian Lemma
83
vk+i(t) > vk+l(0) + / Bklk+i{s)vk(s)ds > 0 Jo The same holding for vk-i. By induction, vk(t) > 0, Vfc 6 Z and the operator solution is strictly positive. Thisfinishesthe proof of Theorem 14.2.
•
B. Proof of the Sturmian Lemma
Lemma 22.1 (Sturmian Lemma) Let x(-),y(-) e CO be different solutions of — - = -d2S(xk~i,xk)
then I (x(t),y(i))
- diS(xk, %k+i) ;
at does not increase, and decreases whenever x(t) and y(t) are not
transverse. To prove this lemma, we will examine a more general situation. Let Xi(t) (io < i < h, —T < t < T) be a solution of dec' (22.3)
—± = ai{t)xi-i + bi(t)xi(t) + Ci{t)xi+i(t)
(i0 < i < h)
where we assume that the coefficients ai(t), 6j(t), Cj(t) are continuous and satisfy (22.4)
ai(t),Ci{t) >6;
at, bua < M
for all — T
Then the sequence {xi0(t),... t< 0.
i\.
,£^(4)} has less sign changes when t > 0 than when
We will now see how Lemma 22.2. gives us a proof of the Sturmian Lemma 22.1. Proof of Lemma 22.1. By the mean value theorem the difference z(t) = x(t) — y(t) satisfies a linear equation of the form (22.3). If x (t0) fl y (to) > then I (x(t),y(t)) is constant fort near t 0 .
84
3: GHOST CIRCLES
If x(to) and y(t0) are not transverse, then since x(to) ^ 2/(*o) one can choose i0 < i\ such that zio(t0) ^ 0, zh(t0) / 0, while z^to) = 0 for i0 < i < ix. Lemma 22.2 then implies the Sturmian Lemma. D Proof of Lemma 22.2. First a few reductions. Consider yt(t) = Bi(t)xi(t) with Bi{t) = exp{— / 0 bi(r)dr}; then —£ = Ai(i)yi-i + Ci(t)yi+i, where * »
=
% # « • « > •
« » " % # = • « > •
In other words, we may assume that the &;(£) vanish. Note that {xi(i}} and {yi(t)} have the same sign changes. The coefficients Ai, d satisfy Se~MT < Ai(t), d(t) < Me+MT
(22.5)
By integrating the differential equation for yi(t) we find that for io < i < i% one has (22.6)
Vi(t)
= f {AiWyi-^T)
+ Ci(T)yi+1(T)}dT
Jo
Proposition 22.3 For io
Vi(t) = Mif-10 + Nit*-' + o (|*l*—*° + |*|**—*)
where the constants Mi and TVj are given by Mi=M0)Ai-i(0)---Aio+1(0) Ni=Ci(p)Ci+1(0)---Cil-1(0)
(t -* 0)
Vio{0)
(t-io)!' yii(0) (ii-J)'-"
We shall prove this by induction. The relevant property of the coefficients Mi, Nt is that the Mi have the same sign as yio (0), and the Ni have the same sign as yix (0). Furthermore,
22. Proofs of Monotonicity and of the Sturmian Lemma
85
one of the two terms in (22.7) always dominates the other, unless i — io = ii — i, i.e. = M^0
unless i = *a±^; if i < *>±± then yi(t) j/j(t)=JVit Proof.
il_i
il
+ o ( t i _ i o ) , if i > ^ ^
then
i
+ o(t - ).
We may assume ii — io > 2. The j/j(i) are continuous, and hence bounded as
t -> 0. Therefore it follows from (22.6) that |j/j(t)| < C |t| for |t| < T. If ii — i 0 = 2, then the only i with io < i < ii is i = io + 1 = ii — 1, and we have j/ i o + 1 (t) = / {Alo+1{0)yio{0) +Cil-1{0)yil(0) Jo = Mio+it + Nio-it + o{t),
+
o(l)}dr
as claimed. If ii — io > 2, then yio+2(t)
= o(l), and (22.6) implies
Vio+i(t) = f {A i o + 1 (0)y i o (0) + o(l)}dr Jo = Mio+lyio(0)t + o{t). Likewise (22.6) implies yi a _i(t) = N^^y^^t
+ o(t). If i\ — io = 3 this proves the
claim; if i\ — i0 > 3, then for all io + 1 < i < ii — 1 one deduces from (22.6) and the estimate |i/i±i(t)| < C\t\ that \yi(t)\ < Ct2. The general induction step in the derivation of (22.7) is as follows. Assume that it has been shown that (22.7) holds for all i with io < i < io + k, or ii — k < i < i\\ moreover assume it has been shown that \yi(t)\ < C \t\k for i 0 + k < % < i\ — k. If i0 + k = i\ - k, then (22.7) implies 2/io+fcW = / {Aio+k(0)Mio+k^Tk-1 Jo = Mio+ktk
+ Nh-ktk
+ Ch-k(0)Nil_k+1rk'1
+o
{rk-l)}dT
+ o (i f c ),
with Mio+k = A i o + f c ( 0 ) - M i o + f c _ i , iVil_fe = Ci 1 _fc(0)-JVi 1 _ fc+ i. In this case the claim is proved. Otherwise i0 + k
and a similar computation shows
that (22.7) holds when i = i 0 + k and i = ii - k. Finally, using (22.6) again, one finds
86
3: GHOST CIRCLES
that for i0 + k < i < ix - k the estimate \yl±1{t)\ < C \t\k implies \yi(t)\ < C |£|fc+1, which completes the induction step. • Lemma 22.2 follows directly from the proposition. If yi0 (0) and y^ (0) have the same sign, say they are positive, then the expansion (22.7) implies that all yi (t) are positive for t > 0; For small negative t the sequence yi0 (t), yi0+i {t),..., y^ (t) alternates signs, except in the middle, i.e. if i\ — io is odd then yio+k{t) and yio+k+i(t) (with k = [ n ^ i p ]) will have the same sign. Indeed, (22.7) says the sequence {yi0(t),..., y^ (£)} has the signs as the sequence {co,C\t,C2t
,...,Ck-lt
,Ckt
, Cfc+i* ~ ,•••
,C2k-lt,C2k)
if i\ — io = 2k is even, and {yi0 (<),..., y^ (t)} will have the same signs as the sequence (co, cii, c2t2,..., cktk+1,ck+itk,...,
c2kt, c2fc+i)
if Zi — io = 2/c + 1 is odd; here the Cj 's are positive constants, with the possible exception of the coefficient ck of tk+1 in the second sequence. If yia (0) and yit (0) have opposite signs, then one can again use the expansion (22.7) to derive that the sequence {y^ (t)} has exactly one sign change for t > 0, and i\ — io — 1 sign changes for t < 0. If i\ — io = 2, then {yi0(t),yio+i(t),yio+2(t)} is "transverse" to the zero sequence for all small t, whatever the sign of yi0+i(t) is. Thus, if {yio (t),..., y^ (t)} is not transverse to the zero sequence at t = 0, then either ii > iQ + 2, or ii = i0 + 2, and ?/j0 (0) and yti (0) have the same sign. In either case we have shown that the number of sign changes of {yi0(t),... ,yn(i)} drops at t = 0. • Lemma 22.2 implies the following: Lemma 22.4 If {xio (t), ...,xix (£)} is a C 1 solution of (22.3) , with xio (t), xh (t) =/= 0 for all to < t < t\, then (a) the number of sign changes of {xi0(t),... (b) this number drops whenever {xi0(t),... sequence.
,x^ (t)} does not increase; ,xil(t)}
is not transverse to the zero
4 SYMPLECTIC TWIST MAPS
In this chapter, we generalize the definition of twist maps of the annulus to that of symplectic twist maps in higher dimensions. We will do that first in the natural higher dimensional analog of the cylinder, T™ x lRn = Cn, and then treat the case of general cotangent bundles. There are several reasons to favor the cotangent bundle of the torus T" x H n . This space arises naturally in many classical mechanical settings whose configuration spaces can be described by n angles. It also arises as local polar coordinate systems near elliptic fixed points of symplectic maps. Another reason to start with maps of T n x H n is that they are the most accessible: although these notions are at least implicitly present, little knowledge of manifolds, fiber bundles and differential forms is needed in the study of this case. Finally, these maps are prone to numerical studies, and for this reason have given rise to many studies by physicists and astronomers. Nonetheless, cotangent bundles of other manifolds than the torus do occur in mechanics (eg. the configuration space of the solid rigid body is SO(3)) and there too it is possible to define and make use of symplectic twist maps. We will see in later chapters that they offer a convenient handle on Hamiltonian systems on cotangent bundles. For the part of the chapter treating general symplectic twist maps, the reader should be familiar with the notion of cotangent bundle and differential forms which are reviewed in Section 58 of Appendix 1. Higher dimensional Symplectic Twist Maps have appeared in many forms, under many names in the literature. I owe the name to MacKay & al. (1989), who use a more restrictive twist condition. The work of Herman (1990), as well as my desire to find a geometric twist condition suitable for general cotangent bundles inspired the definition given here, which first appeared (to my knowledge) in Banyaga & Gole (1993) and Gole (1994).
88
4: SYMPLECTIC TWIST MAPS
23. Symplectic Twist Maps of T n xR n A. Definition Let T " = H n / Z n be the n-dimensional torus. An analog to the annulus in higher dimensions which is most natural in mechanics is the space T n x H n , which can be seen as the cartesian product of nannuli. We give T n x H n the coordinate (q, p) = (qi,...,qn,Pi,--In mechanics, q\,...,
,Pn)-
qn would be n angular configuration variables of the system, whereas
would be their conjugate momentum, and T n x M n is the cotangent bundle
Pi,...,pn n
T*T of the torus T n . The following is a generalization of the definition of twist maps of the cylinder:
Definition 23.1 F(q,p).
Let F b e a diffeomorphism of M 2 n and write (Q(q,p),P(q,p))
=
Let F satisfies:
1) F(q + m,p)
= F(q,p)
+ (m,0),
Vme2"
2) Twist Condition: the map ipp : (q, p) t-+ (q, Q(q, p)) is a diffeomorphism of H 2 n . 3) Exact Symplectic: in the coordinates (q, Q), (23.1)
PdQ - pdq = dS(q, Q)
where S is a real valued function on H 2 n satisfying: (23.2)
S(q + m,Q
+ m) = S(q,Q),
VmeZ"
Then the map / that F induces on T " x IR™ is called a Symplectic Twist Map. As for maps of the annulus, S(q, Q) is called a generating function of the map F: Equation (23.1) is equivalent to p= P =
-drS{q,Q) d2S(q,Q),
and thus F is implicitly given by S since F{q,p)
=
(23 3) with
(Qoi}F(q,p),d2S°ipF(q,p)) 1
TPF (q,Q) =
(q,-d1S(q,Q))
Note that the prescription of F through its generating function S is often more theoretical than computational: it involves the inversion of the diffeomorphism tpp1.
23. Symplectic Twist Maps of T n x M n
89
B. Comments on the Definition •The periodicity condition F(q + m,p)
= F(q, p) + (m, 0) insures that F induces a map
/ on T " x E " . It also implies that (in fact is equivalent to) / is homotopic to Id (see the Exercise 23.3, b)). The periodicity condition is also a consequence of (23.2) (Exercise 23.3 , a). We chose to include it for its importance. •The twist condition (2) of definition 23.0 implies the local twist condition often used in the literature: Local Twist Condition (2')
det dQ/dp
± 0.
We will explore in Section 25 extra assumptions under which this local twist condition implies the global twist of Condition (2). •In terms of differential forms, PdQ — pdq = F*pdq—pdq. The periodicity of S given by S(q+m,Q+m)
= S(q, Q) in the (q,Q) coordinates becomes S(q+rn,p) 1
=
S(q,p)in
the (q, p) coordinates (i.e. applying Wp ). In particular S induces a function s on T x IR n such that fpdq—pdq
n
= ds(q is seen as coordinate on T " here). This last equality expresses
the fact that / is exact symplectic. As we have seen in Chapter l(see also Appendix 1), if / is exact symplectic it is also symplectic. f* pdq — pdq = ds => d(f*pdq
- pdq) — 0 =>• / * (dq A dp) = dq A dp.
Any symplectic map of IR 2n is exact symplectic, but it is not true of maps of T n x H™: the map f(q, p) — i > (q, p + m ) , m ^ 0 is symplectic but not exact symplectic. As in the case of maps of the annulus, exact symplecticity can be interpreted as a zero flux condition, but the flux is now an n dimensional quantity.
C. The Variational Setting As in the case of monotone twist maps of the annulus, the generating function of a symplectic twist map induces a variational approach to finding orbits of the map. Proposition 23.2 (Critical Action Principle) Let fi,..., n
ofT*T ,
fN
be symplectic twist maps
and let Fk be a lift of Fk, with generating function Sk-There is a one to
one correspondence between orbits {(9fe+i,P/t+i) = Fk(qk,Pk)} Fk 's and the sequences {9fc}fcgz in ( K n ) z
satisfying:
under the successive
90
4: SYMPLECTIC TWIST MAPS
(23.4)
d1Sk(qk,qk+1)
+
The correspondence is given by: pk = Proof.
d2Sk-1(qk_1,qk)=0 —diSk(qk,qk+1).
It is identical to the case n = 1 , see Lemma 5.4.
•
Remark 23.3 As in the case n = 1, Equation (23.4) can be interpreted as: JV-l
VW = 0 With W(q0, ...,qN)=J2
Sk(qk, qk+1). 0
And, as in the proof of Corollary 5.5, (23.4) can also be written: Ffc_i-pfc=0 where we use the notation pk = -diSk(qk,qk+l),
Pk =
d2Sk{qk,qk+l).
Exercise 23.4 a) Prove that Condition 1) in Definition 23.3 is a consequence of Conditions 2) and 3). b) You will now prove that Condition 1) implies that the map / induced by F is homotopic to Id. It is known that each homeomorphism of the torus T" is homotopic to a unique torus map induced by a linear map A of Gl(n, 2 ) (the group of invertible integer nxn matrices, see Stillwell (1980)). Likewise, each homotopy classes of homeomorphisms of T" x IRn has exactly one represent of the form A x Id where A 6 Gl(n, Z). Show that any lift F of a map homotopic to A x Id satisfies: F(q,
P)
= (Q, P) => F(g + m, p) = (Q + Am, P).
Exercise 23.5 Show that if F and F1 are two lifts of the same symplectic twist map F, their corresponding generating functions S and S' satisfy: S(q,Q) = S'(Lq,Q + m), where ro 6 2 " is such that F'(-) = F(-) + (m, 0).
24. Examples
91
24. Examples A. The Generalized Standard Map The generalized standard map or standard family is the family of symplectic twist map whose lift is generated by the following type of functions: Sx(q,Q) = \\\Q-qf
+ Vx(q).
where Vx is a family of C 2 functions that are ^"-periodic, with A a (possibly multidimensional) parameter and Vb = 0. In the following, we assume that the norm is the usual Euclidean one, but that could be changed. It is trivial to see that S satisfies the periodicity condition S\(q + m, Q + m) = S\(q, Q). To find the corresponding map, we compute: P = -dxSxiq, Q) = Q~qP = d2Sx(q,Q)
=
VVx(q)
Q-q
from which we immediately get: Q = q + p + VVx(q) P
= p +
vy A (q)
In other words, the standard map is given by: (24.1)
Fx(q,p) = (q+p + VVx(q),p + VVx(q)).
In the case n = 2, the following is the most widely studied potential. It is due to Froeschle (1972) (see also Kook & Meiss (1989), Froeschle, Laskar & Celletti (1992)): V\{qi,q2) = 7^—vai-^i cos(27rgi) + K2 cos{2nq2) + hcos(2-K{qi + q2))}. In this case A = (Ki,K2, h) 6 H 3 , and the standard family attached to this potential is a three parameter family of symplectic maps of T 2 x H 2 . The picture on the book cover, gracefully provided by Eduardo Tabacman, represents the stable and unstable manifolds of a periodic orbit for this map, with parameter k0 = ki = 2, h = 1/2. The difference in colour is to suggest the fourth dimension. When A = 0, the map Fx of (24.1) becomes: F0[q,p) =
(q+P,p).
92
4: SYMPLECTIC TWIST MAPS
This is an instance of a completely integrable symplectic twist map: such maps preserve a foliation of T n x H n by tori homotopic to T n x {0}. On the covering space of each of these tori, the lift of the map is conjugate to a rigid translation. The term "completely integrable" comes from the corresponding notion in Hamiltonian systems (see Example 24.2.) The reason why the standard map has attracted so much research is that it is a computable example of a higher dimensional conservative system. Because of the relative tractability of this system, one may understand questions about persistence of invariant tori as the parameter A varies away from 0, study the various properties of its periodic orbits as well as problems of diffusion. B. Hamiltonian systems As we will see, the relationship between symplectic twist maps and Hamiltonian systems runs wide and deep. Chapter 7 is devoted to this relationship. Let us say here that there are two ways to generate a symplectic twist map from a Hamiltonian system. They occur either as Poincare return maps around elliptic orbits in Hamiltonian systems (we develop this idea in the next subsection and Section 40) or as time e maps of a Hamiltonian system, when restricted to an appropriate domain. As a basic example of the latter, the Hamiltonian flow generated by: H(q, p) = \ {Ap, p)
with A1 = A, det A / 0
is completely integrable, in that it preserves each torus {p = p0} and its time t map: 5*(g,p) = (Q + t(Ap),p) is a completely integrable symplectic twist map. If A is positive definite, gl restricted to {H = 1} is just the geodesicflowfor the flat metric \ {A~lv, v) on T n (See Section 38). More generally, if F(q, p) = (Q, P) is the lift of the time s of some Hamiltonian flow generated by the function H, then: Q = q(e) = q(0) + e.Hp + o(e2) P = p(e)=p(0)-e.Hq
+ o(e2),
and F satisfies the local twist condition det §2(z(0)) / 0 whenever Hpp is non degenerate. This remark was made by Moser (1986a) in the dimension 2 case. From this local argument we will derive in Chapter 7 conditions under which the time e of a Hamiltonian is a symplectic
24. Examples
93
twist map . We will also see that, even if the time e map of a Hamiltonian system is not twist, its time 1 map can, for large classes of Hamiltonian systems, still be decomposed into the product of twist maps (see Chapter 7). C. Elliptic Fixed Points As we will see in Section 40, the study of Hamiltonian dynamics around a periodic orbit of a time independent Hamiltonian reduces to that of a symplectic map: U : B 2 n -• H2n,
such that 11(0) = 0,
called the Poincare return map. The case where 0 is an elliptic fixed point (i. e. the differential DTZ has all its eigenvalues on the unit circle, see Sections 33and 56) is of particular interest, as the dynamics around it offers a microcosm of all the possible symplectic dynamics. Elliptic fixed points have also been a focus of attention in the discussion on stability with the KAM theory, Nekhoroshev estimates and Arnold diffusion (see Chapter 6). We now follow Moser (1977) in proving that, generically, one can find good symplectic coordinates around the elliptic fixed point which makes the map TZ a symplectic twist map . A normal form theorem for elliptic fixed points says that the map 1Z is, around 0 given by: Qk = qkcos$k(q,p) - pksin$k(q,p)
+ fk{q,p)
-Pfc = qksin$k(q,p)
+ gk(q,p)
+ pkcos$k(q,p)
n
$k(q,p) = ak + Y^Pki{qf +pl)1=1
where the error terms fk, gk are C 3 S8^ We now show that this map is, in "polar coordinates", a symplectic twist map of T*T", whenever the matrix B = {(3ki} is non singular. Let V be a punctured neighborhood of 0 defined by: 0 < ^2k(qk + pi) < e. We introduce on V new coordinates (rk, Qk) by: qk = y/2erkcos2ir6k,
pk = \Z2erksin2ir9k
where 6k is determined modulo 1. One can check that V is transformed into the "annular" set: 8
actually, one only need t h e m t o have vanishing derivatives up t o order 3 at t h e origin and be C 1 otherwise.
94
4 : SYMPLECTIC TWIST MAPS
U=l(ek,rk)eTnxmn
| T> > 0
and
^
rfc < | i
Since the symplectic form dqAdp is transformed into 2nedrAd0,TZ remains symplectic in these new coordinates, with the symplectic form dr A dO. In fact, TZ is exact symplectic. To check this, it is enough to show that (Exercise 58.6), for any simple closed curve 7: /
rdO= f rd6.
It is easy to see that 4werkd0k = Pkdlk — Qkdpk, so by Stokes' theorem: 4-ire / rdO = / J~i
pdq — qdp = —2 / dq A dp
JdD
JD
where D is a 2 manifold in V with boundary dD = 7. Since TZ preserves dq A dp in V, it must preserve the last integral, and hence the first. To see that 7Z satisfies the two other conditions for being a symplectic twist map, we write 71(0, r) = (0, R) in the new coordinates: 6>fc=0fc+#- t (r)+01(e) Rk =Tk + 01 (e) n
with ipFk - Qfc + e ^
2pkin.
1=1
where e _1 oi(e, 6, r) and its first derivatives in r, 0 tend to 0 uniformly as e —> 0. We can rewrite this as: 71(0,r) = (0 + eBr + a + oi(e),r + So for small e, the condition det 80/dr
0l(e)).
^ 0 is given by the nondegeneracy of B = {fiki},
1
one uses the fact that TZ is C close to a completely integrable symplectic twist map to show that TZ is twist in U (the twist condition is open). The fact that it is homotopic to Id derives from Exercise 23.4. Note that the set V and therefore U are not necessarily invariant under TZ. Note also that the symmetric matrix B, even though it is generically nondegenerate, is not necessarily positive definite. Herman (1992 b) has examples of Hamiltonian systems and symplectic maps arbitrarily close to completely integrable which have elliptic fixed point with B not positive definite.
25. Generating Functions
95
Exercise 24.3 Compute the expression of the lift of a symplectic twist m a p generated by: S(q, Q) = \ (A(Q - q), (Q - q)) + c.(Q - q) +
V(q),
where A is a nondegenerate n x n symmetric matrix (This is yet a further generalization of the standard map).
25. More on Generating Functions A. Homeomorphism Between Twist Maps and Generating Functions The following proposition justifies the name "generating function". Proposition 25.1
between the set of lifts F of C1
There is a homeomorphism^ n
symplectic twist maps of T*T
2
and the set of C
real valued functions
S
onTR71
satisfying the following: (a) S(q + m,Q
+ Tn)=,S(q,Q),
(b) The maps: q —> d2S(q,Q0) for any Q0 and q0
VmeT, and Q —• diS(q0,Q)
are diffeomorphisms ofTR™
respectively,
(c) 5(0,0) = 0. This homeomorphism
(25.1) Proof.
is implicitely given by:
Ft,.,)-<«.">*{K.
-^Xy
Let F be a lift of a symplectic twist map and S(q, Q) be its generating function.
For such F and 5 , we have already derived (25.1) from PdQ - pdq = dS, and (a) is part of our definition of symplectic twist maps. To show that 5 satisfies (b), first note that, by (25.1), Q —• —diS(q0, Q) is just the inverse of the map p —» Q(q0,p),
which is a
diffeomorphism since ipF '• (
(q,Q)^
(q,p)$(Q,P)
In the compact open topologies of the corresponding sets
96
4: SYMPLECTIC TWIST MAPS
which implies that the map q —* P(q,p0) = d2S(q,Q0) is a diffeomorphism, which finishes to prove that S satisfies (b). Since two generating functions of the same F only differ by a constant there is exactly one such 5(0,0) = 0. Conversely, given an S satisfying (b), we can define a C 1 map F of lR2n by: F(q,p) = (25.2)
(QoipF(q,p),d2SoipF(q,p))
where i>pl{q,Q) = (q, -SiS(g,Q)).
F is a diffeomorphism, as it is the composition of the two diffeomorphisms: (q,p)^
(q,Q) -*
(Q,d2S(q,Q)).
It is easy to check that such a pair F, S satisfies (25.1). Since S satisfies (a), F is a lift of a diffeomorphism of T*Tn : (a) also holds for diS and d2S, which implies (as the reader should check) that F(q + m,p) = (Q + m, P) whenever F(q,p) = (Q, P). Exercise 23.4 shows that F must be homotopic to the Identity. Because of (b), F satisfies the twist condition. Hence the map F (uniquely) defined from (25.1) is a symplectic twist map and it is not hard to see that the correspondence we built between the maps F and the functions 5 is continuous in the C 1 and C2 compact open topologies respectively. D B. Local vs. Global Twist
It is useful to have "local" computational criteria to determine when a function S satisfies the hypotheses of Proposition 25.1 in order to be the generating function of some map. This is the purpose of the following proposition: Proposition 25.2 Let S : M 2n —> 1R be a C2 function satisfying: (i) S(q + m,Q + m) = S(q,Q), (25.3)
(") (tit)
det
Vm € Z "
duS ^ 0 sup
(9)Q)eiR2"
||(9125(g,Q))-1|=ir
Then S is the generating function for the lift of a symplectic twist map . The next proposition gives a way to insure the "global" twist condition from a local condition on the map:
25. Generating Functions Proposition 25.3 Let F(q,p) m
,P)
= (Q,P)
be a symplectic map ofMn
97
with F(q +
m
= {Q + > P)• Suppose that
(25.4)
\\(dQ(q,p)/dp)-1\\
sup 2
( 9 l P )eiR "
Then F is the lift of a symplectic twist map . The proof of both these propositions are direct consequences of the following: Lemma 25.4 Let f : IR
—> IR
be a local diffeomorphism
at each point, such that:
1
sup I^D/^)" )! = K < oo. Tent" Then f is a global Proof.
diffeomorphism.
We first prove that / is onto. Let y0 = /(O) and take any y € IR^. Let y(t) =
(1 — t)yo + ty. By the inverse function theorem, / _ 1 is defined and differentiable on an interval j/([0, e)). Let a be the supremum of all such e in [0,1]. If we prove that / _ 1 is also defined and differentiable at a, then a = 1, otherwise, by the inverse function theorem, we get the contradiction t h a t / - 1 is defined on [O,a+a),forsomea > O.Foranyio,*i £ [0,a), we have:
\\rHv{ti))-rHv(to))\\<
suP ||£>r1(i/(t))||iii/-W)iiiti-toi te[o,a)
existence of f~ (y(a)).
is Cauchy. This proves the
By the Inverse Function Theorem, /
_ 1
is differentiable at y{a).
This finishes the proof that / is onto. Since / is onto and open, it is a covering map from H ^ to IR^. Such a covering has to be one sheeted, i.e. a diffeomorphism, since IR^ is connected and simply connected (see Appendix 2). This finishes the proof. Proof of Proposition
•
25.2. In order to apply Proposition 25.1, we need to show that
S satisfies condition (b) in that proposition. But this is an immediate consequence of Lemma 25.4 applied to the two maps q —> d2S(q,Q0) that||(ft 1 S')- 1 || = ||(a 1 2 S , )- 1 ||).
and Q -> diS(q0,Q)
(note D
98
4: SYMPLECTIC TWIST MAPS
Proof of Proposition 25.3. By Lemma 25.4, for each fixed q, the map p —• n
is a global diffeomorphism of M . This implies that ipF : (q,p)
Q(q,p)
—> (q, Q) is a global
diffeomorphism of M2n.
D
C. Differential of the Map vs. Generating Function
Proposition 25.5 The following formula relates the differential of a symplectic map F to the second derivatives of its generating function
DF
(O,P)
(
-duS.idnS)-1
\d21S
- d22S.dnS.iduS)-1
twist
S:
-(d12S)-1
\
-d22S.(d12S)~1
J
=
where all the partial derivatives are taken at the point (q,Q) = ipF(q,p)Proof.
We will show that -& = —(di2S)~1(q,Q),
where, as usual, we have set
F(<1> P) = (Q> P)- Differentiating the equality: p = —diS(q, Q) with respect to p , viewing Q as a function of q, p , one gets:
Id =
-d12S(q,Q)(^y
The computations for the other terms are similar.
•
Exercise 25.6 a) Show that if instead of Condition (1) in the definition of symplectic twist maps we ask F to be homotopic to A x Id, where a lift A of A is in Gl+(n, Z), then Proposition 25.1 remains true, replacing (a) by: S(q + m,Q +
A(m))=S(q,Q).
b) Find the map generated by
S(q,Q) = l(q-A-1Qf
+ V(q)
Note that this exercise shows, in particular, that there are plenty of examples of exact symplectic maps of T*T n that are not homotopic to Id and hence cannot be Hamiltonian maps. Exercise 25.7 Let B n denote a compact ball in ]R". Show that if / : B " -> K " is a differentiable map satisfying :
26. Twist Maps on Cotangent Bundles inf (dfxV.v) > a(v,v),
99
Vu £ IRn
then / is an embedding (diffeomorphism on its image) of B n in IR".
26. Symplectic Twist Maps on Cotangent Bundles of Compact Manifolds A. Definition Our definition of symplectic twist maps of T " x M™ is geometric enough to allow a generalization to cotangent bundles of general compact manifolds. The main difference between our general definition and the one in the case of T™ x IR™ = T*Tn is that we do not work with the universal covering space of our manifold any more, to the cost of a less global definition.^10) In this book, the main examples of symplectic twist maps on general cotangent bundles will arise in the context of Hamiltonian systems (see Chapter 7). We also present, in the next section, a generalization of the standard map in cotangent of hyperbolic manifolds. We refer the reader to Appendix 1 for a review of the concepts of cotangent bundles and their symplectic structure. In the following, U will denote an open subset of T*M such that: (26.1)
ir-1(q)nU~interior{TBn)
where -K : T*M —> M is the canonical projection, and IB™ C H™ denotes the n-ball. Hence U is a relatively compact ball bundle over M, diffeomorphic to T*M. As in Appendix 1, we denote by A the canonical 1-form on a cotangent bundle. Definition 26.1 A symplectic twist map F is a diffeomorphism of an open ball bundle U C T*M (as in (26.1)) onto itself satisfying the following: (1) F i& homotopic to Id. (2) F is exact symplectic. F*X — A = S_ for some real valued function S_ on U. (3) Twist condition: the map ipF : U -* M x M given by IPF(Z) = {n{z), -K O F{z)) is an embedding. 10
If the manifold M is not covered (topologically) by IR", problems occur when we want to make the definition of symplectic twist maps of T*M as global as in T*T": there cannot be a global diffeomorphism from a fiber of T*M to the universal cover M.
100
4: SYMPLECTIC TWIST MAPS
The function S = 5 o ipp1 on IJJF(U) is called the generating function for F. Often, the kind of neighborhood we have in mind is of the form:
U=
{(q,p)eT*M\H(q,p)
for some function H convex in p. One could use a less restrictive class of neighborhoods, to the cost of possible domain complications. If M = E " , one can take U = T*M and modify the above definition slightly to make it more global by changing (2) into: (2') If F : T*M -+ T*M is a lift of F, the map •tpp : U —> M x M given by ipF(z) = (n(z),7r o F(z)) is a diffeomorphism (of IR 2 n ). We leave the reader check that when M = Tn, Definition 26.1 with 2') replacing 2) is an obvious, coordinate free generalization of the definition of symplectic twist map of T*Tn, with the appropriate restrictions of domains. Finally, one could further relax the above definition by asking that F be only an embedding of U into T*M, letting go of the invariance of U. B. Maps vs. Functions, Revisited It is not hard to adapt the proof of Proposition 25.1 to the more general: Proposition 26.2 Given a point q* in the compact manifold M, there is a homeomorphism between the set of pairs (F, U) where F is a C1 symplectic twist map of an open ball bundle U C T*M
and the pairs (S, V) , where S is in the set of C2
real valued functions S on an open set V (diffeomorphic to U) of M x M
satisfying
the following: (i) The map q —* d2S(q,Q0) open set {(q,Q0)} (T;OM^ (ii)
(resp. Q —• d\S{q0,Q))
n V ( resp. {{q0,Q)}
is a diffeomorphism
(~l V) of M into ( l ^ A f ) !~l U (resp.
n U) for each Q0 (resp. q0).
S(q*,q*)=0. This correspondence is given by:
(26,,
of the
*,,,)-W,JT{£-_
^$»>
26. Twist Maps on Cotangent Bundles
101
Remark 26.3 1) As noted before, if M ^ IR", we can choose 0 = M x R,™ = IR 2n in the above proposition. In this case Corollaries 25.2 and 25.3 also remain valid. 2) Even though we have written Proposition 26.2 in coordinates, the relationship between symplectic twist maps and their generating function is independent of the canonical system of coordinates chosen, see Exercise 26.4. In particular, the prescription of Formula (26.2) will yield the same covectors p and — P regardless of the local coordinates q and Q are expressed in.
C. Examples Time t of Hamiltonian Systems. The examples of symplectic twist maps in general cotangent bundles will mainly come from Chapter 7, as time t of Hamiltonian systems satisfying the Legendre condition. We will also show how to decompose any Hamiltonian map of a large class into symplectic twist maps . Standard Map on Hyperbolic Manifolds. In Proposition 38.7 of Chapter 7, we will prove that the function S(q, Q) = | D i s 2 ( g , Q) + V(q), where Dis is the distance function on the hyperbolic half space H n and V : H n —» IR is some C2 function, generates a global symplectic twist map that we call generalized standard map on hyperbolic space. By using potential functions V that are equivariant under discrete groups of isometries, this type of map provides many examples of symplectic twist maps on compact hyperbolic manifolds that are covered by H n .
Exercise 26.4 Check that, with the appropriate restriction on the set of sequences, Proposition 23.2 and Remark 23.3 remain valid for symplectic twist mapson general cotangent bundles. Exercise 26.5 a) Prove Proposition 26.2. Verify that, although we have written things in local coordinates, everything in Proposition 26.2 has intrinsic meaning (e.g. diS(q,Q)is an element of TgM, which is independent of the canonical systems of coordinate chosen above either q or Q). b) Prove that if M in Proposition 26.2 is the covering space of a manifold N with fundamental group r, and if S satisfy S(jq, 7Q) = S(q, Q), V7 e r, as well as (i) and (ii), then the symplectic twist map that S generates is a lift of a symplectic twist map on N. Exercise 26.6 Show that the set of C 1 twist maps on a relatively compact, open neighborhood in the cotangent bundle of a manifold is open in the set of C 1 symplectic maps on that neighborhood (Hint: prove first that the twist condition is an open condition).
5 PERIODIC ORBITS FOR SYMPLECTIC n TWIST MAPS OF T*T
27. Presentation of the Results In this Chapter, we give some results on the existence of multiple periodic orbits of different rotation vectors for symplectic twist map of T*Tn. The introduction in Appendix 2 of new topological tools yields an improvement on the results of Gole (1989) and (1991), as well as simplifications and unification in the proofs. For the sake offluidity,some of the settings and arguments are repeated from the two dimensional case. A. Periodic Orbits and Rotation Vectors Similarly to the case n — 1, a point (q, p) € !R2n is called a m, d-periodic point for the HftFofamap/ofT*T n if Fd(q,p) = (q + m,p) where m g Z n and d e Z + . The rational vector —- is called the rotation vector of the d orbit of (q,p). We will say that m and d are relatively prime when d is relatively prime with at least one of the components of the vector m. In general, when it exists, the rotation vector of a sequence q = {qk}kez £ (M™)2 is given by the limit:
p(q)= Urn
f.
104
5: Periodic Orbits for Symplectic Maps
B. Theorems of Existence of Periodic Orbits The maps that we consider here satisfy either one of the following assumptions of coercion or asymptotic linearity. In both cases, we assume that our map is a composition of symplectic twist maps : F = F N C . O F , , where each Fk is the lift of a symplectic twist map of T*T", with generating function Sk satisfying either: Coercion (27.1)
lim
Sk(q,Q)
-+oo
||Q-g||-»oo
Asymptotic Linearity sk(q,Q)
= \(Ak(Q-q),(Q-q))
+
Rk(q,Q)
with: (27.2) (a)
Ak = A'fc, det Ak # 0 N
(27.2) (6)
d e t ^ A ^ ^ O l
(27.2) (c)
lim
^
^
=
0
.
Equivalently: Fk(q,p)
= (q + Aklp
+ e(q,p),
p+
T{q,p))
with (27.2) (a) and (b) holding for Ak and: (27-2)
(C ')
lim ^ IIPIHOO
= ||p||
lim ^ iipii-oo
= 0 ||p||
Theorem 27.1 Let F = FN ° • • • ° JF\ be a finite composition of symplectic maps Fk ofT*Tn
condition (27.2). Then, for each relatively prime (m,d) e 2 " x Z n
n + 1 periodic orbits of type m, d. It has at least 2 non-degenerate.
twist
satisfying either the convexity condition (27.1) or the asymptotic +
, F has at least
of them when they are all
27. Presentation of the Results
105
A periodic orbit is called nondegenerate when the composition of the differential of the map along the orbit has no eigenvalue equal to 1, see Section 29. Outline Of The Proof. In the coercive case, we start by finding a minimum for the discrete action function W, sum of generating functions. This minimum is given by the coercion of S and its periodicity. The multiplicity is given by Morse-Conley theory on an adequately chosen sublevel set {W < C}. The case with the asymptotically linear condition is a relatively easy consequence of Proposition 64.6 of Appendix 2 (first published here), which is really the technical heart of the proof. We only have to prove that the action function W on the appropriate quotient space of sequences is indeed quadratic at infinity in the sense required by that proposition. C. Comments on the Asymptotic Conditions Coercion vs. Convexity. As in the case of dimension 2, the convexity of the generating functions implies the coercion condition (27.1). Namely: Lemma 27.2 Let S be the generating function of a symplectic twist map satisfying the following convexity (27.3)
condition: Vq, Q,v € Et n , k £ { 1 , . . .
(d12S(q,Q)v,v)<-a\\vf,
,N}.
Then S is coercive. More precisely, there are a real number a and positive real numbers f3 and 7 such that: (27 A)
S(q, Q)>a-p\\q-Q\\+1
\\q - Q\\2 .
The proof is identical to that of Proposition 40 of Chapter 2.
The convexity con-
dition (27.3) can be seen directly on the map. Indeed, in Proposition 25.5, we derived f f (q,P) = - ( d ^ S f a . Q ) ) - 1 , by implicit differentiation of p = -<9iS(g,Q). The convexity condition (27.3) thus translates to: (27.5)
F(q,p)
= (Q,P)
and
((jp)
v,v\
> a\\vf,
Vv e H n .
106
5: Periodic Orbits for Symplectic Maps
uniformly in (q, p). Note that Condition (27.5) means that F has bounded, positive definite twist. MacKay & al. (1989) imposed this condition in their definition of symplectic twist maps, a terminology that we have taken from them. Remember that Proposition 25.3 in Chapter 4 shows that the bounded twist condition (27.5)implies the global twist condition. About Asymptotic Linearity. The most important feature of Condition (27.2) is that each Ak is not necessarily positive definite, but only a nondegenerate symmetric matrix. In particular, no coercion on S is assumed here and the function W will in general not have a minimum. This is what Herman (1990) called the indefinite case. Note that if we set Rk = 0 in Sk, we obtain a quadratic generating function for a linear symplectic twist map Lk(q,p) = (q + A^lp,p). Thus, if L = LN ° • • • ° L\, condition (27.2) implies that dN
(27.6)
L(q,p) = (q + Ap,p)
Y/^1
with A = k=\
and L is a symplectic twist map. Hence Condition (27.2) can be expressed as saying that F is asymptotically linear (and asymptotically completely integrable), in that it is close to L at oo: (27.2) (c') shows that \\F(q,p) - L(q,p)\\ _ Q Um INI-oo ||p|| We leave it to the reader to show that the generating function and map conditions in (27.2) are indeed equivalent. Example 27.3 The generalized standard map satisfies both conditions (27.4) and (27.2).
D. History The results of Theorem 27.1 have a rich history, which can be traced back to the PoincareBirkhofffixedpoint theorem. Birkhoff & Lewis (1933) gave a proof of existence of infinitely many periodic orbits around an ellipticfixedpoint of a symplectic map - which can basically be reduced to a symplectic twist map, as we saw in Section 91. An elegant proof of the Birkhoff-Lewis Theorem given by Moser (1977) follows the lines of the sketch of the Poincare-Birkhoff Theorem given in the introduction: since close to the elliptic periodic
28. Finite Dimensional Setting 107 orbit the map is, in polar coordinates, close to integrable, the set r of points that only move radially under the map Fd(-) — (m, 0) is a graph over the angular coordinates, for suitable choices of m, d. The intersection of F with its image Fd(r) — (m, 0) is a set of periodic orbits. This intersection is not empty, and can be obtained by finding critical points of an appropriate function related to the generating function of the exact symplectic map. This can be repeated over infinitely many m , d, where d —> oo. One of the main purposes of this book is to introduce the reader to the relatively simple variational methods that are adapted to systems (discrete or continuous, see Chapter 8) not necessarily close to integrable. Some methods were introduced for Hamiltonian systems in the seminal article of Conley & Zehnder (1983), in which they prove the existence of n + 1 homotopically trivial (i.e. m = 0) periodic orbits for Hamiltonian systems on T™ x IR n which are linear outside of a compact set. Theorem 27.1 generalizes Conley and Zehnder's global version of the Birkhoff-Lewis Theorem (see Theorems 42.3 and 42.2 for its Hamiltonian corollaries) in two ways: in its linearly asymptotic condition and in the variety of the rotation vectors. Theorem 27.1 appeared in several pieces: Kook & Meiss (1989) gave the proof of existence in the convex case. Their proof of multiplicity was corrected in Gole (1994), inspired by the work of Bernstein & Katok (1987), who also consider the close-to-integrable case. The asymptotically linear case is one of the main results of the author's thesis (see also Gole (1991)). Note that Josellis (1994) proves, at a considerable technical cost of analysis and topology, a slightly stronger theorem for Hamiltonian systems (he requires less smoothness on the system).
28. Finite Dimensional Variational Setting Let F = FN o ... o Fi where each Fk is the lift of a symplectic twist map of T*Tn with generating function Sk. The critical action principle in Chapter 4 tells us that finding orbits of F can be done by finding solutions of: (28.1)
d1Sk(qk,qk+1)+d2Sk-1(qk_1,qk)=0,
k € Z.
The appropriate space of sequences in which to look for solutions of (28.1) corresponding to m, d-points of F is: f
=i{qe(JRn)Z\qk+dN
= qk+m}
108
5: Periodic Orbits for Symplectic Maps
which is isomorphic to (Eln)dN:
the terms (q1,...,
qdN) determine a whole sequence in
X , and we will use them as a coordinate system for this space. Finding a sequence satisfying (28.1) in X, is equivalent to finding q = (q1,...,
qdN) which is a critical point for the
periodic action function: d.N-1 W(
$)
= Yl
Sk q
( k> qk+i) + sdN(qdN,
9i + m).
k=l
In fact, the proof of the critical action principle (see Proposition 23.2 and also Corollary 5.5) reduces in this case to the suggestive formula: dN
(28.2)
dW(q) = £ ( J V i -
Pk)dqk.
k=i
Similarly to the 2 dimensional case in Chapter 3, we will search for critical points of W by studying the gradient flow solution of
« = -™w» where t is an artificial time variable. Written in components, this equation is a system of dN differential equations: Qk = -diSk(qk,qk+1)
-
d2Sk-1(qk_1,qk)
which, for C 2 functions Sk's, defines a local flow tp1 on X. This flow is defined for all t e 1R whenever the second derivatives of the Sk 's are bounded (as is the case in the Standard Map): the vector field — VW is then globally Lipschitz (see Lang (1983)). But the existence of a local flow, garanteed by the existence of the second derivatives is enough for our purpose. We need to complicate matters some more, to take advantage of the topology induced by the periodicity of the generating functions. Formally, this can be done by remarking that W is invariant under the diagonal Z™ action: W o T„ = W, n € Z " where Tn(qi,
•••, qdN) = (qi+n,...,qdN
+ n).
Hence W induces a function on the quotient X = X / 2 " . But we go one step further. We are not satisfied with finding distinct m , d-points, but we want to make sure that different critical points of our function W correspond to different m , d-orbits of F. To this effect, we note that W is also invariant under the Nth iterate aN of the shift map:
29. Finite Dimensional Setting (o-q)fc =
109
qk+v
This is because Sk+N = Sk, and thus aN permutes circularly the terms of W. Hence we can define W successively on the quotients: X = l/r = X/Zn d
and
N
X =X/o
of X by the actions of T„, n € 2 n and aN. Since the action of aN on critical sequences corresponds to the action of F on points of T*Tn, distinct critical points of W on X correspond to distinct orbits of F. The following lemma, due to Bernstein & Katok (1987), describes the topology of the different spaces: Lemma 28.1 The quotient maps: X —> X and X —> X are covering maps , and thus so is X —> X . The space X is homeomorphic (not always trivial) fiber bundle with base T Proof.
n
to T n x (JRn)dN~1,
whereas X is a
n dN 1
and fiber (JR ) ~ .
We make the change of variables: 1
q =
dN
l N ^
q k
v
k = qk+i-Qk-rn/dN>
ke{l,...,dN
-1}
and think of q as the base coordinate and v as the fiber. In these coordinates: Tn(q,v)
= (q + n,v) dN-l
a(q, « i , . . . , Vdiv-i) = \q+—,v2,..., dN' adN(q,v)
vdN-U
-
^
\
w^
= (q + m,v)
(the reader should verify this...) From the first equality, we get: X = X/Z™ ~ T n x ( E " ) ^ 1 . The map aN induces a ef-periodic, fixed point free diffeomorphism on X, and thus taking the quotient of X by aN gives again a covering map. Finally, these coordinates show that X = X/aN
is a fiber bundle over ( I R n / Z n ) / f Z ~ T n .
D
110
5: Periodic Orbits for Symplectic Maps
29. Second Variation and Nondegenerate Periodic Orbits In this section, we show a relationship between the second derivative of W and the spectrum of the differential of the map along a periodic orbit which will help us detect nondegenerate orbits variationally. We will delve more on this relationship in Section 33. Second Variation vs. Dynamical Type. Suppose F = FN O ... o JF\ where each Fk is a symplectic twist map and let W be defined as before. Proposition 29.1 The eigenvalues of the differential DF£ at a m, d periodic point z are in 1 to 1 correspondence with the the values A £ IR such that the following matrix M(X) is degenerate: f S22 + Sn
M{\)
S12
0
"21
"22 + "11
0
Si2
0
...
0
0
...
V ASg*
...
"12
0
j,S 2 1
'
\
0
0 1 S™w
0
S™-1
S™-?+Sf»J
where each entries represents an n x n matrix, which we have abbreviated. S*j=dijSk{qk,qk+1).
Proof. Suppose that {qx, p1) = z is an m, d point for F. We want to solve the equation: (29.1)
DF*{v) = Xv
with v £ T(T*Tn)z. We follow MacKay & Meiss (1983): If q corresponds to the orbit of z under the the successive Fk's, it must satisfy: dW{q) dqk
d2Sk-i(qk_vqk)
Therefore, a "tangent orbit" 6q must satisfy:
+ diSk(qk,qk+1)
=0.
29. Second Variation and Nondegenerate Periodic Orbits (29.2)
S^Sq*-!
+ (Skn + Sk2^)5qk
+ Sku6qk+1
111
= 0
When q corresponds to a periodic point (q1, p1), Equation (29.1) translates, in terms of the 5q, to: (29.3)
5qdN+1
=
\6qi
Clearly, equations (29.2), (29.3) can be put in matrix form as M(X)Sq = 0 where is the matrix advertised in the proposition, which finishes the proof.
M{\) D
Remark 29.2 This rather physical argument can be given a more mathematical footing. Consider the following: T * H T = { ( f o i . P i ) , • • •, {qdN,pdN) = { g e (ET)
dAr+1
I VW(q)k
e (T*iRn)dN
\ Fk(qk,Pk)
= 0,k =
=
(qk+1,Pk+1)}
l,...,dN-l}
The first homeomorphism is between points in the space and their orbit segments of a given length, the second is given by the correspondence between orbit segments and critical points of the action. If one expresses a parameterization of an element of T(T*Mn)
with the first
representation, one gets the orbit of a tangent vector under the differentials of the Fk's. If one uses the second identification , one gets (29.2) .
Nondegenerate Periodic Orbits. As an immediate consequence of Proposition 29.1, we have the second variation criterium for degenerate periodic orbits: Definition 29.3 A periodic point z of period d for a symplectic twist map F is called nondegenerate if DF£ has no eigenvalue 1. A periodic orbit is called nondegenerate if one, and therefore all of its points are nondegenerate. Lemma 29.4 An m, d periodic point is nondegenerate for F if and only if the critical point of W to which it corresponds is
nondegenerate.
Lemma 29.2 proves in particular that the condition "all m, d orbits are nondegenerate" is equivalent to "W is a Morse function" (i.e. a function all of whose critical points are
112
5: Periodic Orbits for Symplectic Maps
nondegenerate, see Appendix 2). The following proposition shows that both properties are true for generic symplectic twist maps.
Proposition 29.5 For generic symplectic twist maps , all periodic orbits are nondegenerate and hence all the periodic action functions W are simultaneously Proof.
Morse.
We remind the reader that a property is generic on a topological space if it satisfied
on a residual set of that space, i.e. & countable intersection of open and dense sets. Robinson (1970), in his theorem IBi, proves that the set of Ck symplectic maps with nondegenerate periodic points is residual in the space of all Ck symplectic maps. He proceeds by induction on the period d of the points^ 11 '. We want to adapt his proof to the space of C 1 symplectic twist maps . First note that, since the twist condition is open, this space is an open set in the space of C 1 exact symplectic maps. The only thing that we have to check, therefore, is that the perturbations that Robinson uses to remove degeneracy transform exact symplectic maps into exact symplectic maps. But this is not hard to check: each of these perturbations is given by composing the original map / with the time one map of the hamiltonian flow associated to a bump function in a small neighborhood of a given periodic point. Hence the perturbed map is the composition of the original exact symplectic map with the time 1 map of a Hamiltonian, also exact symplectic by Theorem 59.7. The composition of two exact symplectic maps being exact symplectic, we are done.
•
30. The Coercive Case The standing assumption in this section is that F = FN o . . . o F\ where Ffc is a symplectic twist map with generating function Sk satisfying the coercion condition:
Corollary 30.1 Let F satisfy the coercion condition (27.1) . Then, for each relatively prime, m, d, there is a minimum for the corresponding periodic action function
W
and hence an m, d-point for F. 11
C.Robinson actually deals with higher order resonances as well, i.e, roots of unity in the spectrum of Dfz.
31. The Coercive Case
113
Proof. Identical to that of Proposition 11.1. • We have thus found at least one m, d-orbit corresponding to a minimum of W. The reader should be aware that, unlike the 1 degree of freedom case, this does not imply that the orbit is a global minimizer (see Herman (1990) and Arnaud (1989)). We now turn to the multiplicity of orbits. Remember that X is a bundle over T n . Let S = T n be its zero section. Let K > sup^ 6 £ W(q) • Trivially, we have: E C WK =f {q e X | W < K} Note that since W is proper, for almost every K, WK is a compact manifold with boundary, by Sard's Theorem. From this we get the commutative diagram in homology: (30.1)
H.(E) i,\
—•
H.(X) /U
Ht{WK) where i, j , k are all inclusion maps. But k* = Id since S and X have the same homotopy type. Hence i* must be injective. If all the m, d-points are nondegenerate, W is a Morse function (a generic situation by Proposition 29.5) and according to Morse Theory (see Section 61 in Appendix 2) WK has the homotopy type of a finite CW complex, with one cell of dimension k for each critical point of index k in WK. In particular, we have the following Morse inequalities: ^{critical points of index k} > 6fc where bk is thefcthBetti number of WK. In the present case bk > (£) since -H*(Tn) ^-> H*(WK). Hence there are at least 5Zfc=i (fc) = 2™ critical points in this nondegenerate case. If W is not a Morse function, rewrite the diagram (30.1), but in cohomology, reversing the arrows and raising the stars. Since k* — Id, j * must be injective this time. We know that the cup length cl(X) = cl(Tn) = n + 1. By definition, this means that there are n cohomology classes oti,...,an in H 1 (X) such that Qi U ... U an / 0. Since j * is injective, j*ai U ... U j*an ^ 0 and thus cl(WK) > n + 1. WK being compact, and strictly forward invariant under the gradient flow, we can apply Lyusternik-Schnirelman theory which implies that W has at least n + 1 critical points in WK (The proof of Theorem 1 in CH.2 Section 19 of Dubrovin & al. (1987) , which is for compact manifolds without boundaries can easily be adapted to this case.) D
114
5: Periodic Orbits for Symplectic Maps
31. Asymptotically Linear Systems In this section we swap the coercion condition (27.1) for asymptotic linearity of the map (27.2) In this case, the periodic action function W does not necessarily have any minimum. The topological tool we use here is Proposition 64.6. We roughly sketch the content and philosophy of this proposition in the next section. We remind the reader of our assumption (27.2): F = FN O ... o F\ is a product of lifts of symplectic twist maps of T*Tn. The generating function Sk of Fk satisfies: Sk(q, Q) = \(Ak{Q
- q), {Q ~ q)) + Rk{q, Q)
with: N
(27.2)
Ak=Al
det Ak / 0,
det £
A~k * * 0,
lim
VRk(q,Q) ^ ^
= 0
We view Rk as a global perturbation term. As before we let Lk (q, p) = (q + A^p, and L = LN O . . . o L\. Then L(q,p)
— (q + Ap,p)
p)
1
with A = J2i A^ . It is crucial to
note that, because of the conditions on the determinants of Ak and J^A^1,
L and all the
Lfc's are completely integrable symplectic twist maps . As before, we are looking for critical points of: dN
w{
dN
Sk qk q
dN
1
Ak( qk i
$) = YJ ( > ^
= Y 2(
k=l
Qk
9fc
^ + ~ >' ( +i ~ qk)}+YRk^k'
fc=l
where q G X i.e., qjN+i
=
9i +
qk
+^-
fc=l
Tn
- The first sum in the right hand side is the (quadratic)
action function for the integrable symplectic twist map L denned above. We change coordinates & : (qi,...,
qdN-i)
^ (?> v )
as in
Section 28:
dN
Vk = qk+i-qk-m/dN'
k
€{!,...,dN-l}.
In these coordinates, W is of the form: W(q, v) = Q(v) + a • v + b + R{q, v)
32. Asymptotically Linear Systems
115
where R = Y1\N R-k ° ^~x, Q is a homogeneous quadratic function and a, b are constant def
vectors. The function WL (V) = Q(v) + a-v + b comes up while replacing q^+i ~ qk by Vk + m/dN in the quadratic part of W. Thus WL is the action function of the integrable map L in our new coordinates. Postponing the proof that Q(v) is nondegenerate, we conclude the proof of the theorem. The maps T„ (n e Z n ), and a introduced in Section 28 all map fiber to fiber diffeomorphically and linearly in the trivial bundle X —> IR™ with projection (q, v) H-> q. Hence Q(q, v) = Q(v), which is quadratic nondegenerate in the fibers, induces in the quotient X of X a function Q which is also quadratic nondegenerate in the fibers of the bundle X —> T n . Finally, it is easy to see that the asymptotic condition on Rk given in (27.2) implies that:
Ml{w'Q)
=
¥\\(^
+a
)^°
as
IHH
°°
in X and hence also in its quotient X. Hence W is a gpqi, in the sense of Proposition 64.6 which provides the estimates advertised and concludes the proof of Theorem 27.1. We now turn to the proof that, given the assumption (27.2), Q{v) is nondegenerate. The reader could work the linear algebra out directly. We prefer to give a dynamical argument which might enlighten us a bit about the linear asymptotic condition. We need to show that the quadratic form Q has zero kernel. This is true if and only if the linear equation dQ(v)+a = 0 has a unique solution. Now, m, d orbits of L are in one to one correspondence with critical points of WL in X, i.e. solutions of the following linear equations: (31.1)
-8q-^=° dQ(v) + a = 0.
Since WL does not depend on q, the first equation is always satisfied. Since the second equation does not depend on q either, one solution of this equation yields exactly an ndimensional plane of solutions of the form (q, v*), for afixedv*. We now show dynamically that the set of solution of Equations (31.1) is indeed n dimensional, thereby proving that the second equation has a unique solution. The solutions of (31.1) in X are in one to one correspondence with the m, d points of the map L. Since L is a linear, completely integrable symplectic twist map , these orbits form an n dimensional plane parallel to the 0 section of
116
5: Periodic Orbits for Symplectic Maps
T*Tn. Since the generating function of L is quadratic and the above change of coordinate ty is affine, this plane corresponds 1-1 to an n-plane of critical points of W^q, v) inX. •
32. Ghost Tori On the Topological Part of the Proof of Theorem 27.1. In the proof of Theorem 27.1, we invoked a topological "black box", Proposition 64.6. This proposition says that, if a function W is asymptotically quadratic (in a precise sense) on afiberbundle over a compact manifold M (the bundle is X and the manifold is M = Tn here), then the number of critical point of the function W is regimented by the cohomology of M. We now try in a paragraph to peal the successive layers that constitute the proof of Proposition 64.6 in order to extract the gist of the ideas, and motivate the concept of ghost torus. Please see Appendix 2 for the rigorous definitions of the concepts and for the proofs of the following statements. Thefinallayer in the proof of Proposition 64.6 (in the proof of the proposition itself and in the proof of Proposition 64.1) consists, through changes of variables, in coming back to the simpler situation of Proposition 62.4. This latter proposition investigates the gradient flow of a function which has an isolating block B of the form:
X
M
x
D+x D~
Fig. 32.0. The isolating neighborhood in Proposition 62.4. +
Here, D , D~ are homeomorphic to disks of some euclidean spaces. The picture suggests that the flow leaves the boundary of the block in either positive or negative time. The proof of Proposition 62.4 uncovers (via Lemma 63.4) a relationship between the topology of the largest invariant set (12' G for the gradient flow in B and the manifold M. More 12
This set is often denoted by I instead of G in Appendix 2.
32. Ghost Tori
117
precisely, one proves the existence of an injection H*(M) •-* H*(G). This a way to say that the set G is at least as topologically complex as the manifold M. Ghost Tori. As roughly explained in the previous paragraph, the topological part of the proof of Theorem 27.1 brings about the existence of an invariant set G for the gradient flow of the action function W on X. We call this set G a ghost torus, because it lives in the nether world of sequences (vs. the "real" world T*T™ where the dynamics takes place) and has a topology at least as complex as that of the torus: H* (T n ) <-> H* (G). Looking a little more closely at the proof of Proposition 64.6, one would see that G is in fact made of all the bounded orbits of the gradientflowof W in X. As we did with ghost circles in Chapter 3, we can indeed think of the ghost tori as the ghosts of the tori of T*T™ invariant under a completely integrable symplectic twist map L. Indeed, the L-invariant torus of rotation vector m/d has a homeomorphic counterpart in the set X of sequences, namely, the critical sequences corresponding to the periodic this torus is filled with. This torus of critical point is the ghost torus of rotation vector m/d for L: it is easy to see that the gradient flow has no other bounded orbits. What are the ghost tori good for? My hope when I introduced the concept in Gole (1989) was to prove the existence of irrational ghost tori, and thus a generalization of the Aubry-Mather Theorem. Indeed, the only ghost tori whose existence we can secure for all symplectic twist maps (at this point, and to my knowledge) are those in periodic, m, dsequence spaces. This prompted my investigation of the dimension 2 case in Gole (1992 a), where the existence of ghost circles of all rotation number was indeed proven. Ghost circles turned out to be very useful in ordering Aubry-Mather sets as well, see Chapter 3. [Beware that the notion of ghost circle is actually more restrictive than that of ghost torus: a ghost circle may miss some bounded orbits for the gradient flow in a given X and be a proper subset of the ghost torus of a particular rotation number]. Unfortunately, two of the main tools that make things work in the dimension 2 case are the monotonicity of the flow with respect to the natural order on numerical sequences, and the notion of cyclic order. These two notions break down in higher dimension, and my attempt to find irrational ghost tori by other means (eg. proving some regularity of subsets of ghost tori and taking limits, or using Conley continuation in appropriate Banach spaces) have been unsuccessful. At this point, I am rather pessimistic that such a program could be carried out.
118
5: Periodic Orbits for Symplectic Maps
On the other hand, the construction and point of view ghost tori are based on are, for those who know it, very reminiscent of that of Floer Cohomology, where the set of loops of bounded action is used to construct the Floer Cohomology complex. Indeed, the variational/topological theory involved in this chapter and in Chapter 8 could be interpreted as a discrete version of Floer's theory for cotangent bundles (13'. I hope it could be put to some of the use Floer's Theory has. It is interesting to note, for instance, that Floer's theory has concentrated on homotopically trivial periodic orbits - which is not the case here.
33. Hyperbolicity vs. Action Minimizers Dynamical Types of Periodic Orbits. We now return to the connection between the dynamical type of a periodic orbit and the spectrum of Hessian of the action along that orbit, which we started investigating in Section 29. For more detail, the reader is urged to consult MacKay & Meiss (1983) and Kook & Meiss (1989). In Section 56 of Appendix 1, it is shown that the dynamics of a linear symplectic map around the origin in M 2n , which is a fixed point for such a map, are determined by the spectrum of the map. This spectrum has the special property that if A is an eigenvalue, than so are 1/A, A and 1/A. As a consequence, IR2™ can be decomposed in even dimensional eigenspaces of 3 flavors: elliptic (corresponding to pairs of conjugate eigenvalues on the unit circle), parabolic (double eigenvalues 1 or -1), and hyperbolic (either real pairs A, 1/A or complex quadruplets (A, 1/A, A, 1/A)). The origin is stable only if it is an elliptic fixed point (or parabolic, if there are as many eigenvectors as the multiplicity of the eigenvalue ±1). We can now apply this stability analysis to periodic orbits of symplectic twist maps, by considering the spectrum of DFN, where N is the period. There is obviously a loss of control when taking this step. Hyperbolicity of the differential at a periodic orbit implies that of the map itself (by the Hartmann-Grobman theorem, see Robinson (1994)), so instability persists when passing from linear to nonlinear. On the other hand stability does not necessarily survive. Indeed, in the elliptic case, stability cannot be guaranteed while perturbing a linear simplectic map, except in dimension 2, when a certain, generic, twist condition is satisified (Moser (1973) , Theorem 2.11). This is because KAM theory (see Chapter 6), 13
Floer's theory originally took place on compact manifolds. See Cielieback (1992) for a cotangent bundle version of Floer's theory.
33. Hyperbolicity vs. Action Minimizers
119
which guarantees stability in dimension 2, does not anymore in higher dimensions (an n dimensional torus does not separate T*Tn, unless n = 2. However, these invariant tori can be "sticky" and provide long term stability. At any rate, knowing the linear type of the orbit gives a good amount of information about the dynamics around this orbit. Hyperbolicity vs. Minimization: The Periodic Case. If Ai, 1 / A j , . . . , A n , 1/An are the eigenvalues of a linear symplectic map T, a convenient way to track down stability is with the traces pk = Afc + l/\k
and residues Rk = | ( 2 — pk)- Indeed, it is not hard to see that
the origin is stable only when the traces are real and fall in [—2,2], or the residues are real and in [0,1]. Moreover the characteristic polynomial can be neatly expressed in terms of the traces (use l/A(Afc - A)(l/A fc - A) = p - pk):
XnH(p-pk).
det (T-XId) =
(33.1)
We now look back at the matrix M(A) introduced in Section 29:
/stf + sh D
21
0
0 addJV-1
0
0 \
1 odN X' 5 21
0
'S'22 + ^ 1 1 C2 '-'12
0
M(A)
0
<~>12
ASff
0
0
-.dJV-1 '21
J
22
+ S?»J
We showed in Section 29 that the eigenvalues of T = DF™ are exactly the solutions of the equation det M(A) = 0. Hence we must have
det M(\) =
(33.2)
C~[[(p-pk)
fc=i for some constant C. It is not too hard to see that C is in fact the product of the determinants of (minus) the superdiagonal blocks: C = Ylk=i
det
(-£12)' which, if we assume the
convexity condition (27.4) of Lemma 27.2, or just the ordinary positive twist condition for the dimension 2 case, happens to be positive. Finally, we set A = 1 in (33.2). In this case, note that M ( l ) = HessW{qa),
p - pk = 4Rk and we obtain:
1 \ " det HessWjq*) fc=l
4
/
nf=i 1 det(-5 1 fc 2 )'
120
5: Periodic Orbits for Symplectic Maps
In the twist map case n = 1, there is only one residue Ri, which is negative if
HessW{qt)
is positive definite. This gives the following, due to MacKay & Meiss (1983): Theorem 33.1 (MacKay & Meiss) Let z* be a periodic point for a positive twist map of the cylinder, corresponding to a critical sequence qt. Then, if qt is a nondegenerate local minimum for the periodic action, the orbit of z* is hyperbolic.
The Infinite Orbit Case. It is interesting to note that Theorem 33.1 has counterexamples in higher dimensions (see Arnaud (1989), where open sets of maps with no hyperbolic fixed points are found arbitrarily close to an integrable map, even though the action does have a minimum). However there is a higher dimensional result, due to Aubry et. al. (1991) which relates hyperbolicity to a stronger form of minimization of arbitrary orbits. Given a stationary infinite configuration q for a symplectic twist map on T*Tn (i.e. the q coordinate of a (not necessarily periodic) orbit of the map), the Hessian of W as the following infinite matrix, similar to M ( l ) :
/-.
Po HessW(q)
\ CCl
=
01 Oil
02
02
«3 3q-l
0q-l
0q
\ with: ak = d22S(qk_1,qk)
+ dnS(qk,
qk+1),
0k = di2S(qk_lt
qk).
If q is a local minimizer, i. e. minimizes W locally on any finite segment, then the spectrum of HessW(x)
is positive (here HessW
is seen as an operator on "variation" sequences in
2
l , i.e. those sequences y such that Y, \\Vk\\2 < oo). We say that q has a phonon gap if moreover Spec(HessW(q))
€ [a, oo), a > 0. An invariant set has a phonon gap a if each
of the orbits it contains does, and if their phonon gaps are all greater or equal to a. Theorem 33.2 (Aubry-Baesens-MacKay) Let A be a closed invariant set for a symplectic twist map of JR 2n and A' be the associated set of critical sequences for W.
33. Hyperbolicity vs. Action Minimizers Suppose that duS{qk,qk+1),
(d12S)
1
(qk,qk+1),dnS(qk,qk+1)
121
andd22S(qk,qk+l)
are all bounded for q G A',k e Z . Then A is uniformly hyperbolic if and only if A has a phonon gap.
Concluding Remarks. 1) We have not talked about the different types of bifurcations that govern the changes of periodic orbits from one type to another. We refer to Kook & Meiss (1989) and the references therein for more information about this vast and interesting subject. 2) The above theorems are related to a general principle, first unveiled by Morse in Riemannian geometry. In that context, Morse (see Milnor (1969) ) revealed a tie between the index of the second derivative of the action of a segment of geodesic and the number of conjugate points this segment has. In terms of more general Lagrangian systems, this number can be formulated as a certain rotation index (the Maslov index) of Lagrangian subspaces under the differential of the flow along an orbit segment (see Duistermaat (1976), Conley & Zehnder (1984) ). If the orbit is hyperbolic, the Lagrangian tangent subspace can be chosen to be the unstable manifold. It would be interesting to cast the above theorems in a symplectic setting, using the invariance under symplectic reduction of the Maslov index from the manifold graph(dW)
to that of
graph(FN).
6 INVARIANT MANIFOLDS
34. The Theory of Kolmogorov-Arnold-Moser KAM theory, which proves the existence of many invariant tori for systems close to integrable, is one of the greatest achievements in Hamiltonian dynamics. It has historical roots going back to Weierstrass who, in 1878, wrote to S. Kovalevski that he had constructed formal power series for quasi-periodic solutions to the planetary problem. The denominators of the coefficients of these series involved integer combinations of the frequencies of rotation of the planets around the sun. These denominators could be close to zero and hence impede the convergence of the series. Weierstrass advised Mittag-Leffler to make this problem of convergence the subject for a prize sponsored by the king of Sweden. In the 271 pages work (Poincare (1890)) for which he won the prize, Poincare does not solve the problem completely, and his tentative answer to the convergence is negative. In Poincare (1899), he speculates on the possibility of such a convergence, given appropriate number theoretic conditions, but still deems it improbable. It was therefore a significant event when Arnold (1963) (in the analytic, Hamiltonian context) and Moser (1962) (in the differentiable twist map context) gave, following the ideas of Kolmogorov (1954), a proof of existence of quasi-periodic orbits on invariant tori filling up a set of positive measure in the phase space. We can only give here a very limited account of this complex theory, and refer to Moser (1973) and de la Llave (1993) for introductions as well as Bost (1986) for an excellent survey and bibliography. There are many KAM theorems, the most applicable ones being often the hardest ones to even state. We present here a relatively simple statement, cited in Bost (1986).
124
6: INVARIANT MANIFOLDS
Theorem 34.1 (KAM for symplectic twist maps ) Let f0 be an integrable
symplectic
twist map of Tn x ID" of the form: MQ,P)
= (q +
u(p),p)
where ID™ is a disk in IR n and u> : ID™ >—> H n is C°° (since f0 is twist, Du> is invertible). Let p0 be an interior point o/D™. Suppose that the following
condition
is satisfied: Diophantine condition: there are positive constants r and c such that: fn+1
Vfe G Z n + 1 \ { 0 } ,
(34.1)
XX'WJ'(Po) + kn+l
> C
J'=l
Then there is a neighborhood W of /o of C°° exact symplectic maps such that, for each f G W, there exists an embedded invariant torus Tf ~ T™ in the interior of Tn x E>™ such that: (i) Tf is a C°° Lagrangian graph over the zero section, (ii) f\
is C°° conjugated to the rigid translation by cj(p0),
(Hi) Tf and the conjugacy depend C°° on f. Moreover the measure of the complement of the union of the tori Ty(p 0 ) goes to 0 as | | / - / o | | goes to 0. Remark 34.2 1) The diophantine condition (34.1) is shared by a large set of vectors in JRn. As an example, when n = 1, the set of real numbers p, G [0,1] such that \p, — p/q\ > K/qT, T > 2 for some K is dense in [0,1] and has measure going to 1 as K goes to 0. 2) The most common versions of KAM theorems concern Hamiltonian systems with a Legendre condition. In Chapter 7 we show the intimate relationship of such Hamiltonian systems with symplectic twist maps. It therefore comes as no surprise that KAM theorems have equivalents in both categories of systems. Note that there are isoenergetic versions of the KAM theorem for Hamiltonian systems, where the existence of many invariant tori is proven in a prescribed energy level (see Broer (1996), Delshams & Gutierrez (1996a)). 3) One important contribution in Moser (1962) was his treatment of the finitely differentiable case: he was able to show a version for n = 1 (twist maps) where /o and its perturbation
34. KAM Theory
125
are Cl, I > 333 instead of analytic. This was later improved to I > 3 by Herman (1983) and in higher dimension n, to I > 2n + 1 (at least if the original / 0 is analytic). 4) There is a version of the KAM for non symplectic perturbations of completely integrable maps of the annulus, called the Theorem of translated curves, due to Russmann (1970). It states that, around an invariant circle for /o whose rotation number u> satisfies the diophantine condition (34.1) (only one j in this case), there exists a circle invariant by ta ° / for a perturbation / of f0 and ta(x,y) = (x,y + a), which has same rotation number as the original. 5) One may wonder if, among all invariant tori of a symplectic twist map close to integrable, the KAM tori are typical. KAM theory says that in measure, they are. However Herman (1992a) (see also Yoccoz (1992)) shows that, for a generic symplectic twist map close to integrable, there is a residual set of invariant tori on which the (unique) invariant measure has a support of Hausdorff dimension 0 (and hence cannot be a KAM torus). Things get even worse when the differential Du in Theorem 34.1 is not positive definite: there may be many invariant tori that project onto, but are not graphs over the 0-section, and this for maps arbitrarily close to integrable (see Herman (1992 b)). 6) KAM theory implies the stability of orbits on the KAM tori, hence stability with high probability. But in "real situations" it is impossible to tell, for lack of infinite precision on the knowledge of initial conditions, whether motion actually takes place on a KAM torus. Nekhoroshev (1977) provides an estimate of how far a trajectory can drift in the momentum direction over long periods of time: If H(q,p) = h(p) + fe{q,p) is a real analytic Hamiltonian function on T*Tn with fe < e (a small parameter) and h(p) satisfies a certain condition (steepness) implied by convexity, then there exist constants eo,Ro, To and a such that, if e < eo, one has: 1*1 < r 0 exp[( £o /e) Q ] => \P(t) - p ( 0 ) | <
R0(e/s0)a.
With a (quasi) convexity condition instead of the steepness condition, Lochak (1992) and Poshel (1993) showed that the optimal a is ~. Delshams & Gutierrez (1996a) present unified proofs of the KAM theorem and Nekhoroshev estimates for analytic Hamiltonians. Whereas we cannot give a proof of the KAM theorem in this book, the following theorem (Arnold (1983)) offers a simple model in a related situation in which the KAM method can
126
6: INVARIANT MANIFOLDS
be applied in a less technical way.This will allow us to sketch very roughly the central ideas of the method. Theorem 34.3 There exists e > 0 depending only on K, p and a such that, if a is a 2TT -periodic analytic function
on a strip of width p, real on the real axis with
a(z) < e on the strip and such that the circle map defined by
/ : m i + 2-7r/i + a(x) is a diffeomorphism
with rotation number p satisfying the diophantine lM-p/?l>-2+^T,
condition:
Vp/?6Q
then f is analytically conjugate to a rotation R of angle 2irp Sketch of proof: We seek a change of coordinates H : S 1 —» S 1 such that: (34.2)
HoR
=
foH
write H(z) = z + h(z), with h(z + 2ir) = h(z). Then (34.2) is equivalent to (34.3)
h{z + 2irp) - h{z) = a(z + h(z)).
Since a(z) < s, h must be of order e as well and thus, in first approximation, (34.3) is equivalent to: (34.4)
h(z + 2-n\£) - h(z) = a{z)
Decomposing a(z) = Yl a,kei2vkz, h(z) = J2 bkelkz
in their Fourier series and equating
coefficients on both sides of (34.4) we obtain: *
pi2'nkjj, __ ^
where we see the problem of small divisors arise: the coefficients bk of h may become very big if p is not sufficiently rational. It turns out that, assuming the diophantine condition and using an infinite sequence of approximate conjugacies given by solutions of (34.4), one obtains sequences hn,an corresponding Hn,fn
= H~
l
and
o / o Hn which converge to H, R for some H. The domain
35. Properties of Invariant Tori
127
of hn and /„ is a strip that shrinks with n but in a controllable way. This iterative process of "linear" approximations to the conjugacy can be interpreted as a type of Newton's method for the implicit equation T{f, H) = H~l o / o H = R (given / , find H) and inherits the quadratic convergence of the classical Newton's method: R — T(fn, Hn) — 0(e2n) (see Hasselblat & Katok (1995) Section 2.7.b).
•
35. Properties of Invariant Tori The previous section showed the existence of many invariant tori for symplectic twist maps close to integrable. These tori are Lagrangian graphs with dynamics conjugated to quasiperiodic translations. In dimension 2, the Aubry-Mather theorem gives an answer to the question of what happens to these tori when they break down, eg. in large perturbations of integrable maps. In higher dimension, Mather's theory of minimal measure also provides an answer to that question (see Chapter 9). In this section, we look for properties that invariant tori may have whether they arise from KAM or not. We will see that certain attributes of KAM tori (eg. graphs with recurrent dynamics) imply their other attributes (eg. Lagrangian), as well as other properties not usually stated by the KAM theorems (minimality of orbits). A. Recurrent Invariant Toric Graphs Are Lagrangian
Theorem 35.1 (Herman (1990)) Let T be an invariant torus for a symplectic twist map f of T*Tn and suppose / _ is conjugated by a diffeomorphism h to a an irrational translation R on T n . Then T is Lagrangian. Proof. Since the restriction of the symplectic 2-form w| T is invariant under / L and since R = h~l o f\T o h, the 2-form h*w\T is invariant under R. Since R is recurrent, h*u>\T = J2i,j akjdxk A dxj must have constant coefficients akj. Integrating h*w\T over the Xk, Xj subtorus yields on one hand a^, on the other hand 0 by Stokes' theorem since h*ui\T = dh*\\T is exact. Hence h*uj\T = 0 = ui\T and the torus T is Lagrangian. •
128
6: INVARIANT MANIFOLDS
B. Orbits on Lagrangian Invariant Tori Are Minimizers The following theorem is attributed to Herman by MacKay & al. (1989), whose proof we reproduce here. Theorem 35.2 Let T be a Lagrangian torus, C1 graph over the zero section
ofT*Tn
which is invariant for a symplectic twist map f whose generating function S satisfies the following superlinearity
(35.1)
condition:
lim
§
^
- +oo
IIQ-
minimizing.
Note that Condition (35.1) is implied by the convexity condition (d\2S(q, Q).v, v) < —a \\v\\ as can easily be seen by the proof of Lemma 27.2. Proof.
Since T is Lagrangian, it is the graph of the differential of some function plus a
constant 1-form: T = dg(Tn) + (3 (see 57.4). Change coordinates so that T becomes the zero section: (q,p') = (q,p — dg(q) — f3). If F0(q,p)
= (Q, P), then, in the coordinates
(q,p'), we have Q' = Q,P' = P — dg(Q) — 0. Thus a possible generating function is: R(q, Q) = S(q, Q) + g(q) - g{Q) + /3{q - Q). Indeed -p' = 5 ^ ( 9 , Q) = d!S(q, Q) + dg{q) + (3 P' =d2R(q, Q) = d2S(q, Q) - dg(Q) - (3 We now show that R is constant on T, where it attains its minimum. We first note that: dxR{q, Q)=0^p
= dg{q) +p^Q
d2R(q, Q)=0&P
= dg(Q) -P^Q
where l(q) - nof(q,dg(q)
= l(q) = l(q)
+ /3) and n is the canonical projection. Hence R(q,l(q))
= Ro
is constant, and DR(q, Q) is non zero if Q ^ l(q). Since g is periodic, the superlinearity of S implies that R is coercive, i.e. \imnQ_qi^00R(q,Q)
—> oo. Since R has all its
critical points on T, it must attain its minimum Rmin there. It is now easy to see that the
35. Properties of Invariant Tori
q coordinates qn,...,qkof
129
any orbit segment on T must minimize the action. Indeed, let
,rk be another sequence of points of T n with qn = r „ , qk = rk. Then:
r„,...
fc-i
W(ri,.
..,rk)
= Y^ R{rj,rj+i)
+ g{qk) - g(qn) + P{qk -
qn)
j=n
>(k-
n)Rmin
+ g(qk) - g(qn) + (3(qk - qn) = W{qx,.
..,qk) D
Remark 35.3 Arnaud (1989) (see also Herman (1990)) has interesting examples which show that the condition that the graph be Lagrangian is essential in Theorem 35.2. Consider the Hamiltonians on T*T2 is given by: Hs(qi,q2,Pi,P2)
= - ( p i -Ecos(27r<j 2 )) 2 + -p%.
The torus {(^i, q2, e cos(27rg2), 0)} is made of fixed points for the corresponding Hamiltonian system, but it is not Lagrangian (exercise). A further perturbation Ge^{q,p) Hs(q,p)
=
+ S sin(27r<72), 0 < S < e of these Hamiltonians also provide counterexamples to
the strict generalization of the Aubry-Mather theorem to higher dimensions: such systems have no minimizers of rotation vector 0. All the fixed points for the time 1 map have non trivial elliptic part.
C. Birkhoff's Graph Theorem We now present a theorem of Birkhoff for two dimensional twist map which shows that invariant circles must be graphs. The topological proof we give, due to Katznelson & Ornstein (1997) is interesting in that it also offers a method of proof for the Aubry-Mather theorem, which we present in the next subsection. In subsection E, we sketch the generalization to higher dimensions of the graph theorem by Bialy and Polterovitch.
Theorem 35.4 (Birkhoff) Let f be a twist map of the cylinder A.
Then:
(1) (Graph Theorem) Any invariant circle which is homotopic to the circle Co = {y = 0} is a (Lipschitz) graph over Co-
130
6: INVARIANT MANIFOLDS
(2) If two invariant circles C_ and C+ homotopic to Co bound a region without other invariant circles, for any e, there are (uncountahly many) orbits going from e-close to C± to e-close to C7T.
This theorem was proved as two independent theorems by Birkhoff (1920).
Proof (after Katznelson
& Ornstein (1997)). For both (1) and (2), we can assume the
existence of an invariant circle, call it C + . Take any circle C which is a graph over Co and which lies under C+ . The image / ( C ) of this circle by / may not be a graph anymore, but one can make a pseudograph Uf(C) by trimmingh, a process that we denote by U. We now describe pseudo graphs and the trimming map. Take all the points of / ( C ) that can be "seen" vertically from above. This set forms the graph of a function which is continuous except for at most countably many jump discontinuities. Because of the positive twist condition, these jumps must always be downward as x increases: if C is given the rightward orientation, a vector tangent to C must avoid the cone <9+, by the ratchet phenomenon (see Lemma 12.1 in Chapter 2). Make a circle out of this graph by adjoining vertical segments at the jumps. This is Uf(C).
We call such a curve a right pseudograph: a curve made of the graph of
a function y = h(x) which is continuous except for downward jump discontinuities (the limit to the right h(x+) and the left h{x~) exist at each point and h{x~) > h(x+)), and by adjoining to this graph vertical segments to close the jumps. We can apply / to a pseudograph C and trim it as we did for a graph. Because of the positive twist condition, the horizontal part of Uf(C)
is made of images under / of
horizontal parts of C. Given a (right pseudo) graph C, we obtain a sequence of curves Cn =
(Uf)nC.
Lemma 35.5 C^ = l i m n ^ + 0 0 sup Cn is an f -invariant graph, where lim sup is taken in the sense of functions y = h(x) with the obvious allowance for vertical
Proof.
segments.
After one iteration of Uf on a (right pseudo) graph C, we get a pseudograph with
a downward modulus of continuity, the ratchet phenomenon and the vertical cuts implies that, for any pair of points z and z' in the lift of Uf(C),
z' - z is in a cone V of vectors
35. Properties of Invariant Tori 131
(x, y) with y > Sx if x < 0 and y < Sx if x > 0 (see Figure 35.1). This implies that Ct also has this modulus of continuity, and hence is a pseudograph.
Fig. 35.1. The cone defining the modulus of continuity at a point z of Uf(C). There is a partial order on circles homotopic to Co = {y = 0}: we say that C •< C if C is in the closure of the upper component of A\C, which we denote by A+ (C). Clearly / and U preserve this order, and C < U(C) for any circle C homotopic to {y = 0}. This implies that / n ( C ) X Ufn{C) < Coo for all n, and hence /(Coo) X UfiC^) X Coo- By area preservation /(Coo) = Uf(Coo) = CooIf Coo were not a graph, its vertical segments would be mapped by / inside A- (Coo) = A- (£//(Coo)), and A- (Coo) would contain A- (/(Coo)) as a proper subset. This contradicts the fact that / has zero flux. Hence C M is an /-invariant graph. D We now finish the proof of Birkhoff's theorems. Suppose that / admits an invariant circle C* homotopic to the zero section Co- We show that it is a (Lipschitz) graph. The region below C* is invariant. Let C m a l be the supremum of the invariant graphs in this region (under the partial order -<). By continuity, Cmax is an invariant circle which is a graph. But Proposition 12.3 implies that all invariant circles that are graphs are in fact Lipschitz graphs (again, the ratchet phenomenon). If Cmax / C„, then there exist a (not invariant) graph C with Cmax -< C < C,. Applying the trimming iteration process to C, we get an invariant (Lipschitz) graph C ^ with Cmax -< Coo -< C*. This contradicts the maximality of Cmax. Hence C* = Cmax is a Lipschitz graph. If / does not admit any invariant circle homotopic to C, other than the boundaries, the iteration process performed on any (right) pseudograph must converge to the upper boundary: we have C -< Uf(C). Since Coo C cZosure(U/ n (C)), on any graph e close to
132
6: INVARIANT MANIFOLDS
the lower boundary, there is a point whose w-limit set is in the upper boundary. We could have defined a trimming L of curves homotopic to the boundaries by taking their lower envelope (the points seen from below) instead of U. Then L(C) is a left pseudograph and L preserves the order of circles and L(C) -< C for any curve C homotopic to the boundaries. Using lim inf instead of lim sup in the argument above, we get an iteration process L o f which converges to an invariant graph, which must be the lower boundary this time. And on any graph e close to the upper boundary, there is a point whose w-limit set is in the lower boundary. • Remark 35.6 Performing both the Uf and Lf trimming processes on the same curve G yields points that come arbitrarily close to both boundaries in forward time. This fact was proven by Mather (1993) variationally and Hall (1989) topologically. See also LeCalvez (1990). The results of Mather and Hall are actually sharper. Mather alsofindsorbits whose a-limit set is in one boundary, the w- limit set in the same or the other boundary. Hall uses the existence of such solutions asymptotic to the boundaries to replace the area preserving condition. Both authorsfindorbits "shadowing" any prescribed sequence of Aubry-Mather sets in a region of instability (technically, Hall shadows periodic orbits). It would be interesting tofinda new proof of these results based on the trimming technique used above.
D*. Aubry-Mather Theorem Via Trimming
The above proof of Birkhoff's theorems appears as an aside in Katznelson & Ornstein (1997). They also recover the Aubry-Mather theorem with their trimming method. For this they define, abstractly, a different type of trimming operator, which they call proper trimming. Under proper trimming, the area below a curve is preserved. The main difficulty is to show the existence of such an operator. Once the existence is established, one takes limits of iterations under the map and the trimming operator. The limit is a pseudograph whose horizontal parts are forward invariant under / . The Aubry-Mather sets are the intersection of all the forward images (by the map) of these horizontal parts. Finally, they show the existence of Aubry-Mather sets of all rotation numbers by applying this trimming procedure simultaneously to all the horizontal circles in the annulus. Fathi (1997) offers some analog to this in higher dimension, where he considers a certain semiflow on graphs of differentials
35. Properties of Invariant Tori
133
on cotangent bundles (which are necessarily Lagrangian). In the limit, he recovers the generalized Aubry-Mather sets of Mather (which are described in Section 49).
E*. Generalizations of Birkhoff's Graph Theorem to Higher Dimensions This section surveys the work of Bialy, Polterovitch and, indirectly, Herman on invariant Lagrangian tori. It will require from the reader knowledge of material dispersed throughout the book, and more. Bialy & Polterovitch (1992a) prove the following generalization to Birkhoff's Graph Theorem. We explain the terminology in the sequel. Theorem 35.7 Let F be the time one map of an optical Hamiltonian
system of
T*Tn, and let L be a smooth invariant Lagrangian torus for F which satisfies the following
conditions:
1) L is homologous to the zero section 2) F\
ofT*Tn.
is either chain recurrent or preserves a measure which is positive on open
sets. Then L is a smooth graph (i.e. a section) over the 0-section. Optical (see Chapter 7) means that the Hamiltonian H is time periodic and convex in the fiber: Hpp is positive definite. Homologous to the zero section means that, together, the 0-section and the invariant torus bound a chain of degree n + 1, presumably some smooth manifold of dimension n+1 in our case. As for Condition 2), it suffices here to say that either chain recurrence or existence of an invariant Borel measures are satisfied when the invariant torus is of the type exhibited by the KAM theorem, where the map F | is conjugated to an irrational translation. In their paper, the authors use a condition that implies 2), as we show at the end of this section: 2') the suspension of F\.
admits no transversal codimension 1 cocycle homologous
to zero. This theorem is a culmination of efforts by these authors, as well as by Herman (1990) who gives a perturbative version of this result as well as some important Lipschitz estimates for invariant Lagrangian tori. We now give a very rough idea of the proof of Theorem 35.7. First reduce the theorem to the case of an autonomous Hamiltonian on TTn+1 by viewing time as an extra S 1 dimension, with the energy as its conjugate momentum (extended phase space).
134
6: INVARIANT MANIFOLDS
Assume by contradiction that the invariant torus L is not a graph. Consider the set S(L) of critical points of the projection TT\L. Generically, S(L) consists of an n — 1 dimension submanifold of L whose boundary is of dimension no more than n — 3. Assume we are in the generic case. Then S(L) can be cooriented by the flow: the Hamiltonian vector field is transverse to it on the invariant torus. This makes S(L) a cocycle, i.e. a representative of a cohomology class of the torus. It turns out that this cohomology class is dual to the Maslov class of the torus L. [The Maslov class of L is the pull-back of the generator of H1 (A(n)) by the Gauss map, where A(n) is the (Grassmanian) space of all Lagrangian planes in M 2n . Prosaically, this means the following: the oriented intersection of S(L) with any closed curve on L counts how many "turns" the Lagrangian tangent space of L makes along the curve. We explain that a little. The number of turns can be made quite precise because A(n) has one "hole" around which Lagrangian spaces can turn (Hi{A(n)) = Z)]. S(L) is the set of points on L where the Lagrangian tangent space becomes vertical in some direction. The tangent space, seen as a graph over the vertical fiber, is given by a bilinear form which is degenerate at points of S(L) and, thanks to the optical condition, decreases index (i.e. the dimension of the positive definite subspace increases) when following the flow at those points. Bialy and Polterovitch refer to Viterbo (1989) who proves that tori homologous to the zero section have Maslov class zero. Condition 2') now concludes: since it is homologous to zero, the cocycle S(L) must be empty, i.e. there is no singularity in the projection IT | and the torus is a graph. The non generic case follows by making a limit argument using uniform Lipschitz estimates for invariant tori proven by Herman (1990). Finally, let us show how the fact that F\ is measure preserving implies Condition 2'). Assume F is the time 1 map of an autonomous Hamiltonian system on T*Tn, L is an invariant torus and ft is the volume form on L preserved by the Hamiltonian vectorfieldXJJ. The Homotopy Formula (see 59.7) LxH^ = dixHQ + ixHdO implies that dixH& = 0. Assume XH is transversal to S, a codimension 1 cocycle homologous to zero and let C be an n-dimensional chain that S bounds. Transversality implies Js %xH J? ^ 0. On the other hand, Stokes' Theorem yields Js ix„ fi = j c dixH 0 = 0. This contradiction implies that 5 = 0. D
Remark 35.8 As noted by Bialy and Polterovitch, it is not clear that Theorem 35.7 is optimal: Condition 2) may be unnecessary, as is the case in dimension 2. One could imagine a new
36. (Un)Stable Manifolds and Heteroclinic orbits
135
proof of this theorem using higher dimensional trimming on Lagrangian pseudographs, which would not need this hypothesis...
36. (Un)Stable Manifolds and Heteroclinic orbits A. (Un)Stable Manifolds Consider two hyperbolic fixed point z* = (q*,p*),z**
= (g**,p**) for a symplectic twist
n
map F of T*T . We remind the reader that the stable and unstable manifolds at any fixed point z* are defined as:
Ws(z*) = {zeT*Tn\
lim Fn(z) = z*\, n—*+oo
Wu{z*)
= {z£
T*Tn | lim F~n{z)
= z*}
n—>+oo
Moreover the tangent space to Ws at z* is given by the vector subspace Es(z*)
of eigen-
vectors of eigenvalue of modulus less than 1, with a similar fact for W " and Eu. In our case, the differential DF at the points z* and z** has as many eigenvalues of modulus less than 1 as it has of modulus greater than 1. Hence the stable and unstable manifolds at these points have both dimension n. The following appears in Tabacman (1993):
Proposition 36.1 The (un)stable manifolds of a hyperbolic fixed point for a symplectic twist map
are Lagrangian.
Close to the hyperbolic fixed point, they are graphs of
the differentials of functions.
Proof.
Consider a point z on the stable manifold of the hyperbolic fixed point z*, and
two vectors v, w tangent to that manifold at z. Then: uz(v, w) = LJFHZ)(DFk(v),
DFk(w))
-> w*(0,0) = 0, as k -> oo.
which, since it has dimension n in T*T n , proves that the stable manifold is Lagrangian. The same argument, using F~k, applies to show that the unstable manifold is Lagrangian. We leave the proof of the second statement to the reader (Exercise 36.2).
D
In Exercise 36.3, the reader will show a generalization of this fact that makes it applicable to exact symplectic maps (not necessarily twist) of general cotangent bundles. The exercise
136
6: INVARIANT MANIFOLDS
will show that the (un)stable manifolds are in fact exact Lagrangian, i.e. the restriction of the canonical 1-form A to the (un)stable manifolds is exact.
Exercise 36.2
b) Prove that the local (un)stable manifold of a hyperbolic fixed point z* for a symplectic twist map F is a graph over the zero section {Hint, use the formula for the differential of F given in 25.5, and the twist condition det (d\2S) ^ 0 to show that the (un)stable subspace of DFZ* cannot have a vertical vector. To do this, expend LJZ- (DFW, W) assuming w = (0,u>) and show that necessarily w = 0.) c) Deduce from this that the (un)stable manifolds are graphs of differentials of functions ^ " , ^ s defined on a neighborhood of n(z*) in the zero section. Exercise 36.3 Let F is an exact symplectic map (not necessarily twist) of the cotangent bundle T*M of some manifold: F* \ — A = dS for some function S : M —> E. (A is the canonical 1 form on T*M). In Appendix 1, it is shown that any Hamiltonian map is exact symplectic, and any composition of exact symplectic map is exact symplectic. a) Show that the (un)stable manifolds W s '" of a fixed point z* are exact Lagrangian (immersed) submanifolds, i.e. A u = dLs'u for some functions L"'u : W3'u —> IR. (Hint. Show that the integral of A over any loop on Ws,u is 0). b) Show that if and W is an exact Lagrangian manifold invariant under the exact symplectic map .F, then: S(z) + constant = L(F(z)) - L(z), c) Conclude that L " ( 0 = J^ [S(Fk(z"))
- S(z*)} ,
Vp e W
Ls(zs) = - J2 [S(Fk(zs))
-
S(z*)].
For more on this approach, see Delshams & Ramirez-Ros (1997).
B. Variational Approach to Heteroclinic Orbits As a consequence of Proposition 36.1, we obtain a variational approach to heteroclinic orbits. Let z* = (q*, p*) be a hyperbolic fixed point. Let <&u, $" defined on a neighborhood U* of q* be the functions whose differentials have for graphs the (un)stable manifolds of z*. We can add appropriate constants to these functions and get s(q*) =
was
constant on the Lagrangian manifold Graph(dg + (3). Applying this
U
to g = $ or
36. (Un)Stable Manifolds and Heteroclinic orbits where F(q,$s(q))
= (Q,&S(Q))
137
(this makes sense in a subset of U*). Applying the
equation to (q*,q*) shows that the constant is S(q*,q*).
Hence
S(q,Q)-S(q*,q*)=
qk+1)
of the point (q, Q) = (q0, qx) to get:
£ [S(qk, qk+1) ~ S(q*, q*)} = £ [**(«fc+i) " *'(«*)] = **(«*) " *'(«o) fc=0
k=0
As JV —> oo, ^ ( q j y ) —* ^(g*) = 0 and thus the sum converges to —
(36.1)
5>(9fc,«Ifc+i)-S(g*,
Applying the same manipulations to the unstable manifold, using the fact that the generating function for F _ 1 is —S(Q, q), this leads to: Proposition 36.4 Let z* = (q*,p*),z**
= (q**,p**) be two hyperbolic fixed points
for the symplectic twist map F. Let U* and U** be neighborhoods of q* and q** on which the differentials of the functions
W(qQ,...,qN)=$u(q0)+J2
S{qk, qk+1) ~ $S(qN),
q0 £ U\qN € U"
fc=0
are segments of heteroclinic Proof.
orbits.
Left to the reader.
•
With this set-up, Tabacman (1995) shows that, in the 2 dimensional case, any two local minima (i. e. fixed points) £ and i\ of <j>{x) = S(x, x) such that cj>{£) = <j>{n) <
138
6: INVARIANT MANIFOLDS
(2) The matrix M = BA
1
is symmetric and Eu is the graph of the differential of the
quadratic form q i-> qtMq. This function is an approximation to $u (see 55.6.) (3) Perform similar steps to approximate # s at z**. (4) Pick TV (large enough) and use your favorite numerical method to search for critical points of the function W defined above, with points q0, qN suitably close to z* and z** respectively. (5) For more precision, make q0 and qN closer to z* and z** (resp.) and increase TV. Note that this algorithm can be substantially improved by starting, using normal forms, with an approximation of higher degree than linear for the local (un)stable manifolds (see Simo (1990)).
C. Splitting of Separatrices and Poincare-Melnikov Function In Hamiltonian systems, the Poincare-Melnikov function (actually an integral), measures how much the intersecting stable and unstable manifolds of two hyperbolic fixed points split. This kind of function has a long and rich history: Poincare (1899) introduced it as a way to prove non-integrability in Hamiltonian systems. It has then been used to prove the existence of chaos (transverse intersections of stable and unstable manifolds often lead to "horseshoe" subsystems), and to estimate the rate of diffusion of orbits in the momentum direction. The discrete, two dimensional case was considered by Easton (1984), Gambaudo (1985), Glasser & al. (1989), Delshams & Ramirez-Ros (1996). Here, following Lomeli (1997), we give a formula for a Poincare-Melnikov function for a higher dimensions symplectic twist map in terms of its generating function. A more general treatment, valid in general cotangent bundles, and which does not assume that the separatrix is a graph over the zero section, is given in Delshams & Ramirez-Ros (1997). Theorem 36.5 Let FQ be a symplectic twist map ofT*Tn
with hyperbolic fixed points
z* = (q*,p*),z**
= (q**,p**) such that a subset of Wu(z*)
containing z*,z**
is the graph p = i>{q) of a function
Let So be the generating function
= Ws(z**)
= W
ip over some open set.
of FQ. Consider a perturbation
Fe of FQ with
generating function Se = S0 + sSi such that Si (q*, q*, 0) = Si (q**, q**, 0) = 0 and ij\q=q.Si{q,q,e)
= 0 = j-q\q=q..Si{q,q,e).
Then the function L : W -
M:
36. (Un)Stable Manifolds and Heteroclinic orbits (36.2)
L(q) = J2Si(qk,Qk+i,Q) itez
where qk = n °
139
Fk(q,i>(q))
is well defined and differentiable. If L is not constant then, for e small enough, the (un)stable manifolds of the perturbed fixed points of Fe split. Their intersection
is
transverse at nondegenerate critical points of L.
Note that, whereas the condition Si(q*,q*,0)
= Si(q**,q**, 0) is essential, the fact
that their value is 0 is just normalization. Also, the condition on the nullity of derivatives can be discarded (see Delshams & Ramfrez-Ros (1997)).
Proof.
Work in the covering space M2n of T*T n . Let <£ : U C HT -> IR and V = d$
be such that Graph(ip) = W. As in the proof of Theorem 35.2, change coordinates so that W lies in the zero section: {q,p') = (q,p-ip{q)).
If F0(q,p)
!
coordinates (q,p ), we have q = q, p' = p - ip(q), Q' = Q,P'
= (Q, P), then, in the — P - ip(Q). Thus the
generating function becomes: Snew(q,
Q) = Sold(q, Q) + 0(q) - * ( Q ) .
Note that the first order term S i remains the same under this change of coordinates, since we only added terms which are independent of e. For s small enough, the (un)stable manifolds W " , W^ of the perturbed fixed points z*, z*e* (respectively) are graphs of the differentials V>",s = d$™'s for some functions $ " ' s of the base variable q. Clearly, the manifolds W" , s split for e small enough whenever the following Poincare-Melnikov function: M(q) =
(#(«)-#(«))
| £=0
is not constantly zero, and their intersection is transverse if the differential DM is invertible at the zeros. We will now show that:
where L(q) is the function defined in (36.2) , expressed in our new coordinates. Formula (36.1) gives us expressions for <£",s:
140
6: INVARIANT MANIFOLDS
^M
=£
[Ss(qt(eUt+i{e)) - Se(q",q")] ..
fc<0
^(9) =
fc>0
-Tt[S'^^),q%+l(e))-Se{q;q*)]
where q%(e) (resp g^fe)) is the q coordinate of Fj?(q, V""(g)) (resp. of F^(q, V>|(g))). We change the order of differentiation:
M(q) = J ^ ( ^ ) - ^ ) ) = | |
(C(g)-^(g)), e=0
and compute one of these terms, using Formula (36.1) and abbreviating g£ = 9fc(0):
de
C(
de
fc<0
!=0
L
£[S £ (g£(e),g£ + 1 ( £ ))-S £ (g**,q**)] it
5l5o(9j|,95| +1 )^g^(0) + d2S0{qt, guk+1)~qUk+1(0) 'de 'de
+ Siffi,
quk+l)
= £Sl(
37. KAM Theory
141
al. (1989), Gelfriech & al. (1994). In Gelfreich (1999), a precise formula for the estimate for the exponential splitting of separatrices in the standard map is proven, a culmination of years of work initiated by Lazutkin.
37.* Instability, Transport and Diffusion A*. Some Questions About Stability If one thinks of twist maps as local models for symplectic maps around elliptic fixed points, the problem of stability of these fixed points is directly related to the obstruction orbits of twist maps may encounter to drifting in the vertical (momentum) direction. In dimension 2, invariant circles obviously offer such obstructions. Three natural questions arise: Question 1 Are the invariant
circles the only obstruction for orbits of twist maps
to drift vertically? What if there are no invariant circles at all, can orbits drift to infinity on the cylinder? Question 2 Do the invariant tori of higher dimensional obstruction to the drift of orbits in the momentum
(eg. KAM)
tori offer any
direction, at least close to inte-
grate ? Question 3 How can we detect when a system does not have invariant
tori?
B*. Answer to Question 1: Shadowing of Aubry-Mather Sets Are the invariant circles the only obstruction for orbits of twist maps to drift vertically? What if there are no invariant circles at all, can orbits drift to infinity on the cylinder? The answer is: Yes and Yes. The answer to the first part of the question is already given by Part (2) of Birkhoff's Theorem 35.4 , which says that, in a region bounded by two invariant circles, which contains no other invariant circle, there exist orbits going from one circle to the other (in whichever order). This fact gave rise to the the name (Birkhoff) region of instability for such regions.
142
6: INVARIANT MANIFOLDS
The answer to the second part of the question (again Yes) follows from Mather (1993) and Hall (1989), who show that given any (infinite) sequence of Aubry-Mather sets, one can find an orbit that shadows it, i. e. stays at a prescribed distance from each Aubry-Mather set for a prescribed amount of time (the transition time is not controlled). In particular, for twist maps of the cylinder without any invariant circles, there exist orbits that are unbounded on the cylinder (take a sequence of Aubry-Mather sets going to infinity: such a sequence must exist if the twist is bounded from below). Note that Slijepcevic (1999a) has recently given a proof of these results using the gradient flow of the action methods of Chapter 3. Another approach to instability uses partial barriers: invariant sets made of stable and unstable manifolds of hyperbolic periodic orbits or Cantori. The theory of transport seeks to study the rate at which points cross these barriers. This theory was initiated by MacKay, Meiss & Percival (1984). The survey Meiss (1992) is beautifully written and encompasses the theory of twist maps of the annulus and transport theory. For other developments, see Rom-Kedar & Wiggins (1990) and Wiggins (1990). MacKay suggested that (the projection in the annulus of) ghost circles could be used as partial barriers. C*. Partial Answer to Question 2: Unbounded Orbits Do the invariant tori of higher dimensional to the drift of orbits in the momentum
(eg. KAM) tori offer any
obstruction
direction, at least close to integrable ?
The answer is: an ambiguous "No". Topologically, it is clear that, for n > 1, an n dimensional torus in T*Tn does not separate the space into two disjoint components. However, this does not offer a guarantee that orbits will drift in the momentum direction, especially in the presence of a set of high measure of invariant tori. But right after proving his version of the KAM theorem, Arnold (1964) gave a stunning example of a possible limitation to the stability offered by KAM tori. This was an example of a family of Hamiltonian systems, in which, even close to integrable where the KAM theorem implied the existence of many invariant tori, some orbits drifted in the momentum direction. One of the fundamen al questions remained: could the mechanism of diffusion detected in the highly nongeneric example of Arnold be found in generic systems? This paper has since generated an immense amount of work from mathematicians and physicists. In particular, much of the recent study of splitting of separatrices can be traced down to this paper, where chains of stable and unstable manifolds of invariant tori of lower dimensions ("whiskered tori") are used to
37. KAM Theory
143
construct drifting orbits. Also, as already mentioned, Nekhoroshev's theory measures the "exponential stickiness" of KAM tori, which implies that orbits must spend a very long time around these tori if they are caught in their neighborhood. To my knowledge, thefirstdecisive result in the direction of generic diffusion was given by Mather, who announced a striking result for Hamiltonian systems on T*T2: For a Cr (r > 2) generic Riemannian metric g on T 2 and Cr generic potential V periodic in time, the classical Hamiltonian system H{q,p, t) - \ \\p\\2g + V(q, t) possesses unbounded orbits. Mather's proof departs from the traditional methods used in this problem. It powerfully brings together the constrained variational methods developed in Mather (1993), the theory of minimal measures of Mather (1991b) (see also Chapter 9) as well as hyperbolic techniques. Delshams, de la Llave & Seara (2000) have given recently an alternate proof to this result, using hyperbolic techniques and methods of geometric perturbation. See also the related results of Bolotin & Treschev (1999), which use some mixture of variational and hyperbolic methods. Finally, de la Llave announced (Fall 1999) a generalization of this theorem to cotangents bundles of arbitrary compact manifolds. His method uses a generalizations of Fenichel's theory of perturbation of normally hyperbolic sets. Interestingly, the orbits found start at high energy levels, where the system is close to integrable. See also Xia (1998) , for some recent developments closer to the spirit of the original problem of diffusion and Bessi (1996), (1997) and (1998), a series of article where the full force of Mather's variational techniques are used to study diffusion, including the classical case of Arnold (1964). As of this writing, these problems are the subject of some of the most active research in Hamiltonian dynamics. D*. Partial Answer to Question 3: Converse KAM Theory
How can we detect when a system does not have invariant tori? Since invariant tori have been associated to long term stability, many people have studied the mechanisms that leads to the breaking of such tori. There is an extensive body of work in dimension 2. A popular tool is Green's criterium, which associates the breaking of an invariant torus of a certain rotation number with the instability (hyperbolicity) of periodic orbits of nearby rotation numbers. Thanks to this criterium, MacKay & Percival (1985) showed that the standard map fk (see Section 6) does not have any invariant circle for k > 63/64). Boyland & Hall (1987) shows that if there is a non cyclically ordered periodic
144
6: INVARIANT MANIFOLDS
orbit of rotation number w, then there are no invariant circles of rotation number in an interval given by the Farey neighbors of u>. Mather (1986) relates the non existence of invariant circles to the variation of the action function on Aubry-Mather sets (see also Gole (1992 a) for a related result which uses the total variation of the action on ghost circles). MacKay (1993), studies the question in great length and gives a renormalization method that explains the universality of scaling of the gaps of Aubry-Mather sets near the breaking point of an invariant circle. Finally, MacKay & al. (1989) starts the study of converse KAM theory in symplectic twist maps. Haro (1999) provides some more, interesting results.
7 HAMILTONIAN SYSTEMS VS. TWIST MAPS In this chapter, we explore the relationship between Hamiltonian systems and symplectic twist maps. We will assume that the reader is familiar with the material reviewed in Appendix 1, which introduces Hamiltonian systems in cotangent bundles and some of their fundamental properties. In the first part of this chapter, we show how to write Hamiltonian systems as compositions of symplectic twist maps. This is instrumental in setting up a simple variational approach to these systems, which is finite dimensional when one searches for periodic orbits. This method generalizes the classical method of broken geodesies of Riemannian geometry. Our main contribution is to make such a method available for Hamiltonian systems that do not satisfy the Legendre condition. We start in Section 38 with the geodesic flow, which serves as a reference model for Hamiltonian systems: it plays a role similar to that of the integrable map in the twist map theory. Almost no knowledge of Riemannian geometry is assumed here. In Section 39, we expend our approach to general Hamiltonian orLagrangian systems satisfying the Legendre condition (which we see as an analog to the twist condition). In Section 39. D we show that, whether or not the Legendre condition is satisfied, the time 1 map of a Hamiltonian system may be decomposed into finitely many symplectic twist maps. In Section 40, we see how symplectic twist maps also arise from Hamiltonian systems as Poincare section maps around elliptic periodic orbits. From an opposite perspective, we show in Section 41 that in many cases, a symplectic twist map may be written as the time 1 of a (time dependent) Hamiltonian system. Most of this last section is courtesy of M. Bialy and L. Polterovitch, who graciously let us publish their proof of suspension of symplectic twist maps for the first time in this book.
146
7: HAMILTONIANS VS. TWIST MAPS
38. Case Study: The Geodesic Flow A. A Few Facts About Riemannian Geometry
Hamiltonian Approach to the Geodesic Flow. Let (M, g) be a compact Riemannian manifold. This means that the tangent fibers TqM are endowed with symmetric, positive definite bilinear forms: (w, w') i-> 9(q) (v, v') for v, v' G TqM varying smoothly with the base point q. We will denote the norm induced by this metric by IMI : = \/9(q) (v,v). A curve q(t) in M is a geodesic if and only if it is an extremal of the action or energy functional:
A\\{q)= rhlqfdt. between any two of its points qr(ti) and qfe) among all absolutely continuous curves (3 : [ti, *2] —* M with same endpoints. Geodesies are usually thought of as length extremals, that is critical points of the functional / \ \q\ dt. But, thanks to the Cauchy-Schwartz inequality, action extremals are length extremals and vice versa (with the difference that action extremals come with a specified parameterization, see Milnor (1969)). One usually chooses to compute with the action, since it yields simpler calculations. For more detail on this, as well as a the more abstract definition of geodesic given in terms of a connection see e.g. Milnor (1969). The variational problem of finding critical points of A has the Lagrangian 1 1 2 Lo{q,v) = ^g(q)(v,v) = - | | g | | . Following the procedure of Section 59 of Appendix 1, we use the Legendre transform to compute the corresponding Hamiltonian function. In local coordinates q in M, we can write 9(q)(v,v) = {A(q]v,v), where (,) denotes the dot product in H n , and A7\ is a symmetric, positive definite matrix varying smoothly with the base point q. With this notation, we have dv
[q,v) - A{q)v,
^2
-A{q)
38. The Geodesic Flow
147
In particular, ^ ^ is nondegenerate. Hence the Legendre condition (see Appendix 1) is satisfied and the Legendre transformation is, in coordinates: C:(q,v)-*(q,p)
=
(q,A^v)
which transforms Lo into a Hamiltonian Ho'Ho(q,p)=pv-L0(q,v)
= (p,A(q)p)
- ^(A^q)A{q)p,
A{q)p)
=
~{A{q)p,p).
This Hamiltonian is a metric on the cotangent bundle: Ho{q,p) = ^{Aiq)p,p)
d
=
-gfq)(p,p).
We will also denote the norm associated to this metric by ||p|| = Jgf^{p,p)-
Note that
the Legendre transformation is in this case an isometry between the metrics g and g#: in particular, if (q,p) = £(q,v),ihen
\\p\\ = ||v||. Hence the Hamiltonian is half of the speed
and we retrieve, from conservation of energy in Hamiltonian systems, the fact well known by geometers that geodesies are parameterized at constant speed. The geodesic flow is the Hamiltonian flow h^ generated by HQ on T* M. It is not hard to see that the trajectories of the geodesic flow restricted to an energy level project to the same curves on M as the trajectories in any other energy level: the velocities are just multiplied by a scalar (See Exercise 38.1). For this reason, one often restricts the geodesic flow to the unit cotangent bundle T{M
= {(q,p)
€ T*M
| ||p|| = 1}. Traditionally, geometers
use the term geodesic flow to denote the conjugate £ - 1 / i g £ on TM of this Hamiltonian flow, as restricted to the unit tangent bundle. Remember that projections of trajectories of a Hamiltonian flow associated to a Lagrangian satisfying the Legendre condition are extremals of the action of the Lagrangian, and vice versa. (See Section 59 in Appendix 1). In the present case, if (q(t), p(t)) is a trajectory of the geodesic flow, then q(t) is a geodesic. Conversely, if q(t) is a geodesic, it is the projection on M of the solution (q(t),p(t))
of the
1
geodesic flow with initial condition (q0>Po) = (^(0), ^ q ( 0 ) ) . Exponential Map. We now want to establish a fundamental result of Riemannian geometry, which we will rephrase in the next subsection by saying that the time t of the geodesic flow is a symplectic twist map. The exponential map is defined by: expqo(tv)
= q(t),
148
7: HAMILTONIANS VS. TWIST MAPS
where q(t) is the geodesic such that q(0) = q0 and q(0) = v. Note that any geodesic can be written in this exponential notation. In terms of the geodesic flow, expqo (tv) = •K OKQO C{q0, v), where ir : T*M — i > M is the canonical projection. Theorem 38.1 Let M be a compact Riemannian M
manifold.
The map Exp : TM —*
xM
(38.1)
Exp : (q, v) ^ (q, Q) = (q,
defines a diffeomorphism
expq(v))
between a neighborhood of the 0-section in TM and some
neighborhood of the diagonal in M x M. Moreover, for (q, v) in that neighborhood: (38.2)
DiBte,ea:p„(t;)) = H | .
We remind the reader that the distance Dis(g, Q) between two points q and Q in a compact Riemannian manifold is given by the length of the shortest path between q and Q. One way to paraphrase this theorem is by saying that, any two close by points are joined by a unique, short enough, geodesic segment. Proof.
By definition, expq{0) = q and j^expq(sv)
= v at s = 0. Thus:
whose determinant is 1. Hence, Exp is a local diffeomorphism around each point of a compact neighborhood of the 0-section. By the compactness of M , there is an e such that Exp is a diffeomorphism between an e ball in TM around (q, 0) and a neighborhood in M x M around (q, q), where e is independent of q, We now show that Exp is an embedding when restricted to Ve = {(q,v)
€ TM \
\\v\\ < e}, where e is as above. Since we proved that Exp is a local diffeomorphism on Ve, it is enough to check the injectivity. Let two elements in V€ have the same image under Exp. Since the first factor of Exp gives the base point, this can only occur if they are in the same fiber of Ve. But, by our choice of Ve this implies these elements are the same. Finally, weshowthatDis(q, ea;pg(w)) = ||w|| whenever ||w|| < e. As a length minimizer, the shortest path giving the distance between two points is also an action minimizer, and
38. The Geodesic Flow
149
hence a geodesic. Since Exp is an embedding of Ve in M x M, exp is one to one on Ve n TqM and the unique geodesic that joins q and expq(v) in exp(Ve n T g M) is the curve t i-» q(t) = ea;pq(iw). The length of this curve is J 0 ||g|| dt = JQ \\v\\ dt = \\v\\ (see Exercise 38.2c)). The only way Formula (38.2) may fail is if there were a shorter geodesic joining q and expq(v) not in exp(Ve l~l TqM)). But this is impossible since this geodesic would be of the form expq(tw),
t G [0,1] with length ||u>|| > e. D
Exercise 38.2 a) Check that, in local coordinates, Hamilton's equations for the geodesic flow write: q = A{q)p (38.3)
.
/dAw
\
b) Verify that ft.o'(?,p) = ho(q, sp). (Hint, if (q(t),p(t)) is a trajectory of the geodesic flow, then (q(st),sp(st)) is also a trajectory). c) Show that if q(t) = expqo(tv), \\q(t)\\ = \\v\\ for all t. d) Show that Dis(q(0),g(t)) = \t\ ||p(0)|| Exercise 38.3 Show that the completely integrable twist map (x, y) i-> (x + y, y) is the time 1 map of the geodesic flow on the "flat" circle, i.e. the circle given the euclidean metric 9(*)(«.f) = v2-
B. The Geodesic Flow As A Twist Map Theorem 38.3 is the key to the following: Proposition
38.4
Ho{q,P) = 5 ||p||
The time 1 map h\ of the geodesic flow with
Hamiltonian
is a symplectic twist map on Ue = {(q,p)
| ||p|| < e},
£ T*M
for e small enough. More generally, given any R > 0, there is a to > 0 (or given any to there is an R) such that, for any t € [—to, to], h0 is a symplectic twist map on the set UR = {(q,p) | | ||p|| < R}. The generating function 2
of h0 is given by
S(q,Q)
= |Dis
(q,Q).
Proof.
Since /ij is a Hamiltonian map, it is exact symplectic (see Theorem 59.7 in Ap-
pendix 1). Define Exp*
= Exp o £ _ 1 . By Theorem 38.3, Exp*
is a diffeomorphism
150
7: HAMILTONIANS VS. TWIST MAPS
between Ue = {{q,p) \ \\p\\ = e} and a neighborhood of the diagonal in M x M. But Exp#(q,p) = (q,Q(q,p)), where Q = ir o h\{q,p). Hence HQ is a symplectic twist map on Ue, and W = Exp*. The more general statement derives from the fact that Exp#(q,tp)
= (q,g(t)), where A£(g,p) =
{q(t),p(t)).
We now show that |Dis 2 (q, Q) is the generating function of hi when it is a symplectic twist map on a domain U (the proof for /ig is identical). Since hi is a Hamiltonian map, (38.4)
(hl)*pdq-pdq
= dS,
with S(q,p) =
pdq - H0dt
where 7 is the curve h0(q,p), t G [0,1] (see Theorem 59.7 in Appendix 1). We now need to show that S, expressed as a function of q, Q is the one advertised. In this particular case, since q = A^p (see Exercise 38.2) and H0 = ^{A^p,p) = \ \\p\\2, the integral simplifies:
(38.5)
Jpdq-H0dt
= J \\\P(t)\\2 dt.
= ^ \(A(g)p,p)-±\\p(t)fdt
But the integrand is Ho, which is constant along 7. Hence, using Theorem 38.3, and the fact that £ is an isometry, we get:
S(q,p) = \ ||p||2 = \ INI* = ^Dis2(q, Q(q,p)), where (q,v) = C~l{q,p). This makes S the advertised differentiable function of q and Q in the region where (q,p) i-> (q, Q) is a diffeomorphism.
•
Remark 38.5 1) Note that the proof of Proposition 38.4 equates the action of a geodesic segment between two points to the generating function evaluated at this pair of points. 2) As a simple example of what makes hi cease to be a twist map when the domain U is extended too far, take M to be the unit circle with the arclength metric. In a chart 9 G (—e, 27T — e), we have: rv /n o\
i
®
D1S(O,0) = j 2 7 r _ 0
when
when
0^
w
e > n
2
As a result, the left derivative of |Dis (0,6) at 9 = n is 7r, whereas the right derivative is —7r: the function Dis2 is not differentiable at this point. The following will be instrumental in the proof of Theorem 43.1.
38. The Geodesic Flow Corollary 38.6 Let hs0(q,p) = (QS,PS)
Proof.
be the time s of the geodesic flow, then:
diDis(q, Qs) =-sign(s).r—- P and <92Dis(q, Qs) = sign(s) P\\
(38.6)
151
IIP.,
From Proposition 38.4, we get: - p = di^DtffaQJ
= D i s ^ . Q ^ a i D i s ^ . Q ! ) = ||p||
which proves <9iDis(q, Qx) = ~i&\- Using Qs=ito
diDisfaQJ
h\(q, sp), one may replacepby sp
in the previous computation to prove the first equality. For the second equality, the fact that Dis(q, Qs) = Dis(Q s , q), that 9 = 1 0 hl(Qs,-sPs)
(see Exercise 38.2) and the first
equality, enables us to write: <9 2 Dis(q,QJ = diDis{Qs,q)
ps = sign(s).-r5^T
C. The Method of Broken Geodesies We now draw the correspondence between the variational methods provided by symplectic twist maps and the classical method of broken geodesies (see Milnor (1969)). As before, let h\ be the time 1 map^14^ of the geodesic flow with Hamiltonian HQ. Fix some neighborhood U of the zero section in T*M. Proposition 38.4 implies that if we decompose h\ = (fyf ) N , 1
then for N big enough each h^ is a symplectic twist map in U. As a result, periodic orbits of period 1 for the geodesic flow, i. e. fixed points of h\ are given by the critical points of: N
W(q) = Y^ S^k,
Qk+i),
with
qN+l
= qu
k=i
where q belong to the set XN(U) of sequences in M such that (qfc, qk+1)
6 ip{U), where
we write tp = i> i • We now show that W is the action of a broken geodesic. ''o 1
Since h^ is a symplectic twist map, the twist condition implies that, given (qfc, qk+1) in ip(U), there is a unique (pk,Pk) such that/i0N {qk,pk) = (q f c + 1 ,Pfc), i.e., there is exactly one trajectory ck: [jj, ^ } —> T*M of the geodesic flow that joins (qk,pk)
to
(qk+1,Pk)
The following discussion remains valid if we replace the time 1 map by any time T.
152
7: HAMILTONIANS VS. TWIST MAPS
The projection ir(ck) on M is a geodesic, parameterized at constant speed equal to the norm of pk. As noted in Remark 38.5, S(qk, qk+i) S(qk, qk+i)
= \0^2(qk^
qk+1) is also the action of ck:
= Jc pdq — Hdt. Hence W is the sum of the actions of the c^'s, i.e. the
action of the curve C obtained by the concatenation of the Cfc's. The curve C can be described as a broken geodesic, in general it has a "corner" at the pointq fc whenever Pfc-i 7^ p fc : via the Legendre transformation, P f c _i and pk correspond to the left derivative and right derivative of the curve C at qk. If q is a critical point of W, Pk = Pfc+i ( s e e Remark 23.3 and Exercise 26.4), and thus the left and right derivatives coincide: in this case C is a closed, smooth geodesic. In conclusion, the function W(q) can be interpreted as the restriction of the action functional A(c) to the finite dimensional subspace of broken geodesies, which is parameterized by elements of XN(U),
in the (infinite dimensional) loop space of T*M. One can further
justify this method by showing that the finite dimensional space X^(U)
is a deformation
15
retract' ^ of a subset of the loop space and that it contains all the critical loops of that subset. This was Morse's way to study the topology of the loop space (see Section 16 in Milnor (1969)). Conversely, and this is the point of view in this book (and more generally that of symplectic topology), knowing the topology of certain subsets of the loop space, one can gain information about the dynamics of the geodesic flow or, as we will see, of many Hamiltonian systems.
D. The Standard Map on Cotangent Bundles of Hyperbolic Manifolds In this subsection, we use our understanding of the relation between geodesic flow and symplectic twist maps to define the Standard Map on the cotangent bundle of any compact hyperbolic manifold. Recall that a hyperbolic manifold M of dimension n is a manifold that can be covered by the hyperbolic half space H n = {(x\,...,xn) 2
given the Riemannian metric ds Geodesies on H
n
e M" | xn > 0}
= ^t £ ) " dx\, which has constant negative curvature.
are open semi circles or straight lines perpendicular to the boundary
{xn = 0}. Here, the relevant property of the geometry of H™, and hence of any hyperbolic manifold, is that the exponential map at each point is a global diffeomorphism between the fiber and H n , a corollary of the Hopf-Rinow Theorem (Gallot, Hulin and Lafontaine (1987), 'This retraction can be obtained by a piecewise curve shortening method.
39. The Geodesic Flow 153 p. 99). The generalization of the standard map that we present now is in fact valid on any Riemannian manifold with the property that T*M = M x M. Proposition 38.7 Let S : Mn x IH n -> IR be given by: S(q,Q)
= \vis2(q,Q)
where V : H™ —* IR is some C2 function, hyperbolic metric.
+ V(q), and Dis is the distance given by the
Then S is the generating function for a symplectic twist map
that we call the generalized standard map on H n . Furthermore,
if V is equivariant
under a group of isornetries E of H n representing the fundamental
group of the
hyperbolic manifold M = H " / E , then S is the generating function for a lift of a symplectic twist map on T*M. Proof.
We show that S complies with the hypothesis of Proposition 26.2 where we take n
M = m ,U
= T*Mn ^ M n x IR n . Let hl{q,p)
= (Q, P) be the time 1 map of the
n
geodesic flow on T * H . The assumption that the exponential is a global diffeomorphism for this metric means that p —> Q(q0,p)
is a global diffeomorphism {q0} x E
n
—> IH n for
each fixed q0 and thus /ij is a (global) symplectic twist map . Likewise P —> q(Q0, P) is a diffeomorphism because h^1, the inverse of a symplectic twist map is a symplectic twist map itself. Since, according to Proposition 38.4, ^Dis 2 is the generating function for /ij, we have established that the maps Q >—> e*i^Dis 2 (q 0 , Q) and q i-» d2^T)is2(q, Q0) are both diffeomorphisms for each fixed q0,Q0.
Coming back to our full generating function,
we have proven that:
q^d2S(q,Q0)
=
d2^is2(q,Q0)
is a diffeomorphism. Q -> d1S(qQ,Q)
=
d1^DiS2(q0,Q)+dV(q0)
must also be a diffeomorphism H n —> TQoMn since we added a translation by the constant dV(q0) to a diffeomorphism. Proposition 26.2 concludes the proof that 5 is the generating function for a twist map of T*Mn. The last statement of the proposition is an easy consequence of Exercise 26.5.
•
154
7: HAMILTONIANS VS. TWIST MAPS
39. Decomposition of Hamiltonian Maps into Twist Maps In Subsection A, we generalize Theorem 38.4 by proving that Hamiltonian maps satisfying the Legendre condition are symplectic twist maps, provided appropriate restrictions on the domain of the map. We then reformulate this result in the Lagrangian setting (Subsection B), giving a generalization of the fundamental Theorem 38.1. In Subsection C, we focus on T*T n , where, given further conditions on the Hamiltonian, we extend the domain of these symplectic twist maps to the whole space. Finally, in Subsection D we prove a theorem of decomposition of Hamiltonian maps into symplectic twist maps , whether or not they satisfy the Legendre condition.
A. Legendre Condition Vs. Twist Condition Heuristics. Remember that Hamiltonian maps, which are time t maps of Hamiltonian systems, are exact symplectic (Theorem 59.7) and, through the flow, isotopic to Id. Therefore, to show that a certain Hamiltonian map is a symplectic twist map, we need only check the twist condition. Clearly, not all Hamiltonian maps satisfy it. Take F(q,p)
= (q + m, p) on
the cotangent bundle of the torus, for example: it is the time one map of H(q, p) =
m.p,
and it is definitely not twist. Here is a heuristic argument, which appeared in Moser (1986a) in the context of twist maps, to guide us in our search of the twist condition for Hamiltonian maps. The Taylor series with respect to e of the time e map of a Hamiltonian system with Hamiltonian H is: q(e) = q(0) + e.Hp + o(e 2 ) p(e)=p(0)-e.Hq 2
Thus, up to order e , dq(e)/dp(0)
+
o(J)'
= e.-ffpp. This shows that whenever HW is nondegen-
erate, the time e map is a symplectic twist map in some neighborhood of
q(0),p(0).
The problem is to extend this argument to given regions of the cotangent bundle: the term o(e 2 ) might get large as the initial condition varies. Rigorous Argument. We now present a rigorous version of this argument, valid on compact subsets of the cotangent bundle of an arbitrary compact manifold. We say that a Hamiltonian H : T*M x IR —> 1R satisfies the global Legendre condition if the map: (39.1)
p^
Hp{q,p,t)
39. Decomposition of Hamiltonian Maps
155
is a diffeomorphism from T*M i-> TqM for each q and t. We will say that H satisfies the Legendre embedding condition if the m a p p >—> Hp is an embedding (i.e. a 1-1, local diffeomorphism). We let the reader check that, although we have written it in a chart of conjugate coordinates in T*M, this condition is coordinate independent. We give examples of systems satisfying these conditions after the proof of the theorem.
Theorem 39.1 Let M be a compact, smooth manifold and H : T* M x H be a smooth Hamiltonian function which satisfies either the global Legendre condition (39.1) the Legendre embedding condition. Then, given any compact neighborhood U
or
inT*M
and starting time a, there exists e0 > 0 (depending on U) such that, for all e < e0 the time e map of the Hamiltonian flow of H is a symplectic twist map on U. Proof.
Choose a Riemannian metric g on M. Define the compact ball bundles: U(K) =
{(q,p)eT*M\\\p\\
The nested union of these sets covers T*M. Hence any compact set U is contained in a U(K) for some K large enough, and we may restrict the proof of the theorem to the case U = U(K). Since the Hamiltonian vector field of H is uniformly Lipschitz on compact sets, there is a time T such that the Hamiltonian flow h%+t (z) of H is defined on the interval t € [0, T] whenever z € U{K). In the rest of this section, we fix a and abbreviate h^+t by hl. By continuity of the flow, h\°'T\U(K))
is a compact set. We now show that we can
work in appropriately chosen charts of T*M. Since M is compact, we can find a real r > 0 such that T*M is trivial above each ball of radius 1r in M. (Indeed, there exist such a ball around each point. If one had a sequence of points whose corresponding maximum such r converged to zero, a limit point of this sequence would not have a trivializing neighborhood, a contradiction). Take a finite covering {Bi} of M by balls of radius r, and let B[ be the ball of radius 2r with same center as Bi. Choose e3 < T such that n o /jl°'£3) (7T_1 (Bi) n U(K)) C B[. Such an e 3 exists since there are finitely many Bi's and the flow is continuous. From now on, we may work in any of the charts n~1(Bi) 1
~ Bi x H " ,
and know that for the time interval [0, e 3 ], we will remain in the charts TV^ (B^) ~ B[ x M n . We let (q,p) denote conjugate coordinates in these charts.
156
7: HAMILTONIANS VS. TWIST MAPS
Let e < £3 and write hc(q,p)
= (q(e),p(e)).
Consider the map Vv
:
(q,P) •-»
(q, q(e)). We need to show that iph.' is an embedding of U(K) in M x M . By compactness, it suffices to show that iph' is a local diffeomorphism which is 1-1 on U(K): the inverse is then automatically continuous. Write the second order Taylor formula for q(e) with respect to e (this is a smooth function since the flow is smooth): q(e) = q + eHp(q,p,
e2R{q,p,e).
a) +
The smoothness of the Hamiltonian flow guarantees that R is smooth in all its variables. Indeed, its precise expression is (see eg. Lang (1983), p. 116):
R(q,p,e) =
j\l-t)^^ldt
and the integrand is smooth since the flow is. The differential ofiph* with respect to (q,p) is of the form: Diph<(q,p) = (
^J,
+
A = eHpp(q,p,a)
+
e2Rp(q,p,e).
Since det Hpp / 0 by the Legendre condition and since Rp is continuous and hence bounded on the compact set U(K) x [0, £3], there exists £2 in (0, £3] such that det Dtph> = det 4 ^ 0 on U(K) x (0, £2] (we have used the fact that there are finitely many of our charts Bi covering U(K)). Hence iph* is a local diffeomorphism for all e e (0, £2]. We now show that, by maybe shrinking further the interval of e, ip^ is one to one on U(K). Suppose not and iph'(Q,P) = i>h<(Q',P') for some (q,p),(q',p')
6 U(K). The definition of iph'
immediately implies that q = q'. Also, since i/j^e is a local diffeomorphism on U(K), we can assume that |[p — p'\\ > 5 for some 5 > 0. Using Taylor's formula, we have: 9(e) - q'(e) = e(Hp(q,p,a) Define the compact set P(K)
- Hp(q,p',a))
:= {(q,p,q,P')
+ e2(R(q,p,
e) - R(q,p',
€ U{K) x U(K)
\ \\p-pi\\
e)). > 6}.
Since p 1—> Hp is a diffeomorphism, the continuous function ||-Hp(g, p, a) — Hp(q, p', a) || is bounded below by some K\ e
e
ll-ft(g>Pi ) ~ R(Q>P'i )\\
is
> 0 on P{K).
The continuous function (q,p,e)
bounded, say by K2, on P(K)
x [0, e2] and hence
||g(£)-9'(£)||>(eK1-£2^2)>0
i->
39. Decomposition of Hamiltonian Maps
157
whenever e e (0, ei] and ei is small enough. Now choosing e0 = min{ei, e 2 } finishes the proof of the theorem.
•
Examples 39.2 We give two classes of examples. In the first class, the Hamiltonian is not assumed to be convex. We characterize the Hamiltonians in local charts of a cotangent bundles. Again, the following conditions are coordinate independent. • Let H(q,p,t)
= \{A{^t)p,p)
+ V(q,t)
This is simply because p — i > Hp = A^^p
and det A^t)
/ 0, then H satisfies (39.1).
is linear and nonsingular. Note that no convexity
is assumed here, only nondegeneracy of i / p p (and its independence of p). Hence this class contains, but is substantially larger than, the classical mechanical systems. • If Hpp(q, p , t) is definite positive, and its smallest eigenvalue is uniformly bounded below by a strictly positive constant, then H satisfies the global Legendre condition. This is a direct consequence of Lemma 25.4. If we remove the lower bound on the smallest eigenvalue, one can show (see Exercise 39.3) that the map p t-» Hp is not necessarily a diffeomorphism any more, but remains an embedding and thus H satisfies the Legendre embedding condition. Such an embedding condition, and a version of Theorem 39.1, are also satisfied if H p p is positive on a compact set U invariant under the flow (see Exercise 39.4).
Exercise 39.3 Show t h a t a C 1 map / : H n >-> R n which satisfies (Dfx • v, v) > 0 for all v and x in R n is an embedding, i.e. it is injective with continuous and differentiable inverse. Deduce t h a t a Hamiltonian such t h a t Hpp is positive definite satisfies the Legendre embedding condition. Give an example where this embedding is not onto. Exercise 39.4 Let U be a compact region which is invariant under the flow of a Hamiltonian H. Assume also t h a t Hpp is positive definite on U. Show t h a t the time t map is a symplectic twist map for all t > 0 sufficiently small. (Hint. First prove, as in the previous exercise, t h a t p i-» Hp is an embedding of T*M f~l U for each q. Then adapt the proof of Theorem 39.1).
B. Lagrangian Formulation Of Theorem 39.1 The following proposition, which is a reformulation of Theorem 39.4 in Lagrangian terms, is a generalization of the fundamental Theorem 38.1. It guarantees the existence and uniqueness of Euler-Lagrange solutions between any two close by points. A time that the solution is traversed has to be specified within a compact interval. In Chapter 9, we will encounter
158
7: HAMILTONIANS VS. TWIST MAPS
Tonelli's theorem which implies, for fiber convex Lagrangian systems, that these solutions can also be assumed to be action minimizers. Proposition 39.5 Let M be a compact manifold and L : TM x M —> IR be a Lagrangian function diffeomorphism.
satisfying the global Legendre condition: v i—• Lv(q,v,t)
is a
Then, for all starting time a and bound K on the velocity, there ex-
ists an interval of time [a, a+eo] such that, for all e < eo, there exists a neighborhood O of the diagonal in M x M such that whenever (q, Q) C O, there exists a unique solution q(t) of the Euler-Lagrange equations such that q = q(a), Q = q(a + e) and \\q(a)\\
The Legendre condition enables us to define the Legendre transform C : (q, v) —>
(q,p = Lv) and the Hamiltonian function H(q,p,t) 1
derstood that q = q o C' (q,p) Legendre condition and C~1(q,p)
= pq — L(q,q,t),
where it is un-
(see Section 59 of Appendix 1). H satisfies the global = (q,Hp)
(see Remark 59.1). In particular Theorem
39.1 applies to the Hamiltonian H. Let V(K) =
{(q,p)\\\Hp(q,p,a)\\
This set is compact since it corresponds, under the Legendre transformation, to
/:- 1 (i/(^)) = {(g I g)lllg(«)ll<^} in the tangent bundle. Theorem 39.1 tells us that, for all e e (0, e0] with e0 small enough, the map hc is a symplectic twist map on V(K). 0 =
Define
tl>h.(V(K)).
39. Decomposition of Hamiltonian Maps
159
We now show, maybe by decreasing eo, that O is a neighborhood of the diagonal in M x M. = ir^1 (q)C\V (K) and write hl(q,p)
LetVq(K)
= (q(t),p{t))
where, as before, h* denotes
+t
h° . The curve q(t) is a solution of the Euler-Lagrange equation satisfying q = q(a) and if (q,p) € Vq(K), then ||q(a)|| = \\HP\\ < K. As in the proof of Theorem 39.1, we write the Taylor approximation of the solution: n°h'(q,p)
= g(e) = q + eHp +
e2R{q,p,e).
At first order in e, the image of Vq(K) under 7r o he is {q + eHp(q,p)
| (q,p) €
Vq(K)},
which is a solid ball centered at q. When adding the second order term e2R, q still is in n o he(Vq(K)),
provided that e is small enough. By compactness e can be chosen to work
for all q. Thus {q, q) € hc{V{K))
= O for all q e M , as claimed.
The rest of the proof is a pure translation of the statements of Theorem 39.1: by construction, if {q,Q) e O, then (q, Q) = {q, q(e)) where q(t) = •noh\q,p)
and (q,p) G V(AT).
Hence q(t) is a solution to the Euler-Lagrange equation starting at q at time a, landing on Q at time a + e. Moreover, since (q,p) € ^ ( i f ) , ||g(a)|| = \\Hp(q,p,a)\\
< K. Finally,
this solution is unique. Otherwise, by the uniqueness of solutions of O.D.E.'s, there would be p ^ p' such that Trohe(q,p)
= TTO he(q, p ' ) , a contradiction to the twist condition.
•
C. Global Twist: The Case Of The Torus When the configuration manifold is T™, there is hope to show that the time t map of a Hamiltonian system is a symplectic twist map on the whole cotangent bundle. We present here some conditions under which this is true. No doubt one could find other, maybe weaker conditions which would also work. Assumption 1 (Uniform opticity) H(q,p,t)
= Ht(z)
is a twice differentiable function on T*Tn x JR and satisfies the
following: (1)
sup\\V2Ht\\
(2) C||v||
< {Hpp(z, t)v, v) < C~l \\v\\ for some positive C independent of (z, t) and
Sometimes Hamiltonian systems such that i 7 p p is definite positive are called optical. This is why we refer to Assumption 1 as one of uniform opticity.
160
7: HAMILTONIANS VS. TWIST MAPS
Assumption 2 (Asymptotic quadraticity) H(q, p, t) is a C 2 function on T*Tn satisfying the following: (1) det
Hpp^O.
(2) For ||p|| > Ku
H{q,p,t)
= (Ap,p)
+ c.p, A1 = A, det A ^ 0.
Here A denotes a constant matrix, c a constant in IR" and K\ a positive real. We stress that, in Assumption 2, A (and hence i7 p p ) is not necessarily positive definite. Theorem 39.7 Let he be the time € of a Hamiltonian flow for a Hamiltonian tion satisfying any of the Assumptions symplectic twist map ofT*Tn
func-
1 or 2. Then, for small enough e, he is a
(or on U,
respectively).
Remark 39.8 Proposition 39.7 holds for h°+e whenever it does for tf: /i" + £ is the time e of the Hamiltonian G(z, s) = H(z, t + s), which satisfies all the assumptions H does. Proof.
We prove the proposition with Assumption 1, and indicate how to adapt the proof
to Assumption 2. The strategy is to estimate - § ~
and use Proposition 25.3 to turn the
local Assumptions 1 and 2 into global twist condition. We can work in the covering space IR 2n of T*T n , to which the flow lifts. The differentia] of hl at a point z = (q, p) is solution of the linear variational equation '16^ (39.2)
U(t) = -JV2H(ht(z))U(t),
U(0) = Id,
J =
/ 0 [id
-Id\ 0
)
We first prove that U(e) is not too far from Id: Lemma 39.9 Consider the linear equation: U(t) = A(t)U{t),
U(t0) = U0
where U and A are n x n matrices and ||j4(t)|| < K, Vt. Then : \\U(t)-U0\\
In general, if 4>* i s solution of the O.D.E. z = Xt(z) then Dcp1 is solution of U(t) = r>X ( (^'a)L''(t),f/(0) = Id. Heuristically, this can be seen by differentiating ^t4>l(,z) = Xtitfiz)) with respect to z (see e.g. Hirsh & Smale (1974)).
39. Decomposition of Hamiltonian Maps Proof.
161
Let V(t) = U(t) - U0, so that V(t0) = 0. We have: V{t) = A{t) (U(t) - U0) + A{t)U0 = A(t)V(t)
+ A(t)U0
and hence: ||V(t)|| = ||V(t) - V(to)\\ < \t - to\K \\U0\\ + f For all \t — to| < £? w e
can
K \\V{s)\\ ds
apply Gronwall's inequality (see Hirsh & Smale (1974)) to get: WVm^eKWUoWe^-^
and we conclude by setting e = \t — to\.
•
We now proceed with the proof of Theorem 39.7 . By Lemma 39.9 we can write: U(s) = Id + Oi(s) where ||Oi(s)|| < 2Ks, for s small enough (i.e. such that eKs < 2). Integrating Equation (39.2) on both sides then yields: (39.3) Let (q(t),p(t))
[/(e) = Id+ = hl(q,p)
[ JV2H(hs{z)).(Id Jo
+ Oi(s))ds
= hl{z). The matrix b€(z) = dq(e)/dp,
is the upper right
n x n matrix of U(e). From Equation (39.3), and Assumption 1 (1) we know it is of the form: (39.4)
be{z)=
f Hpp(hs(z))ds+ Jo
f Jo
02(s)ds
where || JJ 0 2 ( s ) d s | | < K2e2. From this, and the fact that (39.5)
C\\v\\2<(Hpp(z)v,v)
we deduce that: (39.6)
(eC - K2e2) \\v\\2 < (be(z)v, v) < ( e C " 1 + K2e2) \\v\\2
so that in particular be(z) is nondegenerate for small enough e. Since be(q,p) in q, the set of nonsingular matrices {be(z)}zeU2n (39.7)
sup
zeiR 2 "
is periodic
is included in a compact set and thus:
\\b;\z)\\
162
7: HAMILTONIANS VS. TWIST MAPS
for some positive K'. We can now apply Proposition 25.3 to show that he is a symplectic twist map with a generating function S defined on all of M 2n . Remark 39.10 The above proof shows that h€ satisfies a certain convexity condition :
(39.8)
<6«_1».»> = ( ( ^ ( £ ) ) ' w . i A ^ a N I 2 ,
V« e 1RB.
where a is a positive constant. To see that it is the case, note that, denoting by inf ||67 1 (z)|| IH=i, zeR2" and M the corresponding sup, (39.6) implies: m=
m(eC - K2e2)\\vf
< {b-\z)v,v)
< M(eC" 1 + K2e2) \\vf .
We now adapt the above proof to Assumption 2. Note that under this assumption, we can still derive (39.4): the boundary condition (2) implies that V2H is bounded. Since H is C 2 , and Hpp = A outside a compact set, Hpp(hsz) is uniformly close to ff pp (z) for small s, and thus the first matrix integral in (39.4) is non singular for z and small s. Thus b€(z) is also nonsingular for small e. Since be(z) = eA outside of the compact set ||p|| < K\, the set of matrices {be(z) \ z G H n } is compact and hence Inequality (39.7) holds and, again, we can apply Proposition 25.3 . • D. Decomposition Of Hamiitonian Maps Into Twist Maps
When the time e maps of a Hamiitonian system are symplectic twist maps for e < e*, one can readily decompose the time 1 map into such twist maps. Take a time independent Hamiitonian, for example. Its time 1 map h1 can be written:
and, for N > 1/e*, each h^ is a symplectic twist map. It is only slightly more complicated when H is time dependent. In this case we can write: (39.9)
h1=h\,-1 —FT-
o(ft^t)o.../i^1o...h* —FT
77
39. Decomposition of Hamiltonian Maps 163 fc+i
and each /iT 7 " is a symplectic twist map by assumption on our Hamiltonian. as the next N
Proposition shows. What may be more surprising, and gives a greater scope to the use of symplectic twist maps, is that there is a large class of Hamiltonian systems which, even though their time e is not twist, can be decomposed into a product of symplectic twist maps. This is a generalization of an idea that LeCalvez (1991) applied in his variational proof of the Poincare-Birkhoff Theorem (see Chapter 1). This will work with either of the following broad assumptions: Assumption 3. H is a C 2 function o n T ' M x [0,1], and the domain U is a compact neighborhood in T*M.
Assumption 4. H(z, i) = Ht(z) is a function on T * T " x HI satisfying sup ||V 2 H t \\ < K. The domain U = T*Tn. Proposition 39.11 (Decomposition) LetH(z,t) Assumptions
be a Hamiltonian function
satisfying
3 or 4, or the hypothesis of either Theorem 39.1 or Theorem 39.7. Then
l
h , the time 1 map of its corresponding Hamiltonian system, can be decomposed into a finite product of symplectic twist maps (defined on the domain U corresponding to the various
assumptions): h1 = F2N o .. .0 Fi.
Proof.
We have given the trivial proof above for Hamiltonians that satisfies the hypothesis
of Theorems 39. land 39.7. We now prove the proposition when H satisfies Assumption 3. Pick a ball bundle U(K) = {(q,p)
| ||p|| < K) with K large enough so that U C U{K).
Let G be the time s of the geodesic flow, where s is chosen so that G is a symplectic twist map on U(K). That such an s exists is proven in Proposition 38.4. We can write: (39.10) hl =Go
(G~1 O /I 1 JV _ 1 \ o G o . . . o (G~1 O hTj
= F2N o . ..o Fi.
0...0G0
(G-1
ohf)
164
7: HAMILTONIANS VS. TWIST MAPS
One can check that, at each successive step of the composition, the points remain in U(K). The map G is a symplectic twist map, by assumption, and G o h£ is also a symplectic N
twist map by openness of the set of twist maps on a compact neighborhood (see Exercise 26.6). Suppose now that H satisfies Assumption 4. Let G(q,p) = (q + p,p), our favorite symplectic twist map on T*Tn. Decompose h1 as in Equation (39.10). We now show that G _ 1 o h£ is also a symplectic twist map. Lemma 39.6 implies that h\+€ satisfies ||-D/4+£ - Id\\ < tKeKc. Hence
DG-1.Dh"f
-DG~
for some positive constant C. Thus G^1 o / i ^ is twist for N large enough, since the N
sufficient conditions det dQ/dp ^ 0 and || (dQ/dp)-11| to the C 1 norm.
< oo are both open with respect D
40. Return Maps in Hamiltonian Systems We show that return maps around a periodic orbit of a Hamiltonian system is exact symplectic. If the periodic orbit is elliptic, the return map has an elliptic fixed point, and thus, generically, it is a symplectic twist map around this point (see Section 91). Consider a time independent Hamiltonian on IR 2n+2 , with the standard symplectic structure i?o = Yl^o^Qk A dpk- Assume that we have a periodic trajectory 7 for the Hamiltonian flow. It must then lie in the energy level {H = H0) where H0 = #(7(0)). Take any 2n + 1 dimensional open disk E which is transverse to 7 at 7(0), and such that E intersects 7 only at 7(0). Such a disk clearly always exists, if 7 is not a fixed point. In fact, one can assume that, in a local Darboux chart, E is the hyperplane with equation qo = 0: this is because in the construction of Darboux coordinates, one can start by choosing an arbitrary nonsingular differentiable function as one of the coordinate function (see Arnold (1978), section 43, or Weinstein (1979), Extension Theorem, Lecture 5). Define E = E fl {H = H0}. It is a standard fact (true for periodic orbits of any C1 flow ) that the Hamiltonian flow hl admits a Poincare return map TZ, defined on E around z§, by 1l(z) = ht(z\z), where t(z) is the first return time of z to E under the flow (see Hirsh
41. Suspension of Symplectic Twist Maps
165
& Smale (1974), Chapter 13). We claim that V, is symplectic, with the symplectic structure induced by fin on E. Since E is transverse to 7, we may assume that: g0 = — ^ 0 opo on E. Hence, by the Implicit Function Theorem, the equation H{0,qi...,qn,pa,...,pn)
= H0
implies that p0 is a function of (qi,..., qn,Pi,- • • ,Pn)- This makes the latter variables a system of local coordinates for E. We will now work in a simply connected neighborhood O of z0 in E, parameterized by (QI , . . . , qn, p\,..., pn). Since dq0 = 0 in O, the restriction of fin is in fact n
w = J? 0 | o = ^dqk
Adpk-
To prove that 1Z is exact symplectic, use Formula 59.9 of Appendix lwhich states that, for any closed curve in O, or more generally for any closed 1-chain c in O, I pdq — Hdt = I pdq — Hdt JTZc
JC
since c and TZc are on the same trajectory tube. We now show that J1Zc Hdt = Jc Hdt — 0. This is due to the fact that d(Hdt) = dH A dt = 0 A dt = 0 since H = H0 on O. Since O is simply connected, Poincare's Lemma shows that Hdt is an exact form and hence its integrals along the closed curves c and Tic are null. Now we have Jnc pdq = Jc pdq for any closed curve o in O and Exercise 58.6 implies that 11 is exact symplectic.
41. Suspension of Symplectic Twist Maps by Hamiltonian Flows Moser (1986a) showed how to suspend a twist map of the annulus into a time 1 map of a (time dependent) Hamiltonian system satisfying the fiber convexity Hpp > 0. In subsection A we present a suspension theorem for higher dimensional symplectic twist maps announced in Bialy & Polterovitch (1992b), which implies Moser's theorem in two dimensions. These authors kindly agreed to let their complete proof appear for the first time in this book. In subsection B, we give the proof, due to the author, of a suspension theorem where we let go of a symmetry condition assumed by Bialy and Polterovitch. The price we pay is the loss of thefiberconvexity of the suspending Hamiltonian.
166
7: HAMILTONIANS VS. TWIST MAPS
A. Suspension With Fiber Convexity
Theorem 41.1 (Bialy and Polterovitch) Let F be a symplectic twist map with generating function S satisfying: (41.1)
di2S(q,Q)
is symmetric
and negative
Then there exists a smooth Hamiltonian
function
nondegenerate. H(q,p,t)
on T*Tn x [0,1]
convex in the fiber (i.e. Hpp is positive definite) such that F is the time 1 map of the Hamiltonian flow generated by H. The Hamiltonian function H can also be made periodic in the time t. Proof.
Following Moser, we will construct a Lagrangian function L(q, v, t) on IR 2n x
[0,1] with the following properties: (41.2) (a) The corresponding solutions of the Euler-Lagrange equations connecting the points q and Q in the covering space K n in the time interval [0,1] are straight lines q + t(Q - q)\ (41.2) (b) S(q,Q)
= f L{q + Jo
t(Q-q),Q-q,t)dt;
(41.2) (c) L is strictly convex with respect to v : %Ji is positive definite. (41.2) (d) L(q + m, v, t) = L(q, v, t) for all m in Z n . If such a function L is constructed, its Legendre transform H satisfies the conclusion of Theorem 41.1: (41.2) (a) and (b) imply that F is the time 1 map of the Hamiltonian H, (41.2) (c) implies that Hpp is convex (see Exercise 59.2) and (41.2) (d) that the Euler-Lagrange flow of L takes place on TTn and hence the Hamiltonian flow of H is defined on T*Tn. Note that if (41.2) (c) is satisfied then (41.2) (a) is equivalent to the following equation: d2L
/ „ , ^ / /N 4L2)(a)
d2L
V+
<
7MTq
dL
Lemma 41.2 Set Rij(q, v, t) = holds: (41.3) (a) Ri:j = Rji; (4L3) (b)
^
= -dv~>
n
lMrt-8q-=0a
da2s^
dqidQj
(q -tv,q
+ (l — t)v). Then the following
41. Suspension of Symplectic Twist Maps
(41,)(d)fi+E^
167
=o
for all i,j, k. o2 c-
The proof is straightforward and uses the fact that the matrix g ^ g is symmetric.
Lemma 41.3 Set L(q,v,t)
= / (1 - A) ^TRij(q,
Xv,t)viVjd\.
Then the following
holds: (41.4) (a) — = /
'Y^Rii(q,TV,t)vjdT 3
rt2T
(41.4) (c) L satisfies Equation (41.2) (a') Proof.
Rewrite L as follows:
(41-5) L(q,v,t)
= / / ds^^Rijiq^VityviVjdX Jo Jx .• „-
J o
ds Jo
S2_\Ri3{cliSTVit)viVjdT=
ds 10
=
dX ^ ^ Rjjjq, Xv, t)vjVj Jo
/
y^jViai(q,
sv,t)d.
.j
where a,(q, v, y) = / y^^Rij(q,TV,t)vjdT. Jo i as a path integral: / y2viai(q,sv,t)ds Jo ,•
We can rewrite the last integral of (41.5)
= / ^ A v
ajdvj
where 7(s) = (q, sv,t). Fixing q and i, Equation (41.3) (b) implies that the form J2i oiidvi is closed, and, because v e IR™, exact, say J2i aidvi = dA for some function A(w) on IR™. Then the Fundamental Theorem of Calculus yields: L(q,v,t) Since J2i &idvi = dA = ^dv,
= A(v) - A{0).
Equation (41.4) (a) follows. The proof of (41.4) (b) is
similar. We now prove (41.4) (c). In view of (41.4) (a), the left hand side / of (41.2) (a') can
168
7: HAMILTONIANS VS. TWIST MAPS
be written as follows: I=
^2VlJ
Y^~d^^TV'tS>vJdT + J
Jo
Ui
Y^~dT^TV't^dT
dqi
=ai +a2 - o 3 , where a^ is the kth integral in the above expression. Rewrite 03 using (41.3) (c) as follows: f \~^ dRij Jo j - 1 dQl
f ^-^ dRi 4 J0 ^ dQi
Thefirstterm is equal to a\. Therefore: 1=
J Yl vA ~d^^q'™'*) + J2-g^~(q'TV'^TVl \ dT-
Equation (41.3) (d) implies that the bracket, and hence I, vanish. Given any function L(q, v, t), set L{q,Q)= I L(q + Jo
•
t(Q-q),Q-q, t)dt.
Lemma 41.4 Assume that L satisfies (41.2) (a'). Then the following holds: (41.6) (a)
f g = _g ( g ,Q_ 9 ! 0 ) ;
(41.6) (6)
H= g ( Q , Q _ g , 1 ) ;
41 6 (c
< -> >
{q Q q 0)
^kr-£k ' - ' -
Proof. Equation (41.6) (c) is a consequence of (41.6) (a), which we now prove. The same argument also proves (41.6) (b). If L satisfies (41.2) (a') or equivalently (a) then:
{{^t{Q-q),Q-q,t)) Therefore,
= ^q +
t{Q-q),Q-q,t).
41. Suspension of Symplectic Twist Maps
169
r{-^ +i(Q - 9) ' Q - 9 ' i)+(i - i) i(& +t(Q - 9) ' g - 9 ' i) )}^ =/i{( i -o + t ( g - , ) ' g - 9 ' t ) } | f t = -^ ( , ' o - 9 ' o ) D Given any two differentiable functions L{q, v, t), f(q, t), set:
Lf(q,v,t) = L(q,v,t) + -J-(q,t)v + ^(«,*)Lemma 41.5 (41.7) (a) Lf(q,Q)
= L(q,Q)
+
f(Q,l)-f(q,0);
(41.7) (b) If L satisfies (41.2) (a') then Lf satisfies it as well, for all f. The proof of this lemma is straightforward. We are now in position to finish the proof of Theorem 41.1. Let L be the function defined in Lemma 41.3. From (41.6) (c) and (41.4) (b), we get: &l t ™ dqidQj '
d2L
,
dvidvj
r>
,rt
'
d2S
i
/^^
dqidQ
and therefore L(q,Q)
= S(q,Q)
+ a(q) + b(Q)
for some differentiable functions a and b. Set
f(q,t) =
(l-t)a(q)-tb(q).
We claim that the function Lf satisfies (41.2) (a)-(d). We prove these properties one by one. 1. We proved in (41.4) (c) that L satisfies (41.2) (a'), and hence (41.2) (a). Statement (41.7) (b) proves that Lf does as well. 2. From (41.7) (a), we get: Lf(q, which proves (41.2) (b).
Q) = L(q, Q) - b(Q) - a(q) = S(q, Q),
170
7: HAMILTONIANS VS. TWIST MAPS
„ d2Ls d2L ,„ , d2S , NN 3. = ——j = (itjj) = — (g — i v , q + (1 — £)«). Since this last matrix is 2 positive definite by Hypothesis (41.1), so is the first one. 4. Since S(q + m,Q
+ m) = S(q, Q), the function L is periodic in q. We need to check
that | £ and §£ are also periodic in q. Using the definitions and (41.6) (a) and (b), one can easily check that L(q,q)
= -^(q,q)
=
^(q,q)=0.
From the definitions of the functions a and b we obtain that a(q)
+
b(q) = -S(q,q),
-
= - —
{q,q),
-
{ q )
= _ _ ( , ,
q).
Because of the periodicity of S, all these functions are periodic in q. Since df da Tt^^-^dq-%^
3b
df dq=-a-b>
both |j£ and | ^ are periodic. This finishes the proof of our claim, and hence that of Theorem 41.1.
•
B. Suspension Without Convexity If we let go of the symmetry of # ^ (but keep some form of definiteness) in Theorem 41.1 , we can still suspend the twist map F by a Hamiltonian flow. The cost is relatively high however: we can no longer insure that the Hamiltonian is convex in the fiber. The proof, quite different from that of Theorem 41.1, first appeared in Gole (1994c). I am indebted to F. Tangermann for a useful discussion about this theorem.
Theorem 41.6 Let F(q,p) differential b(z) = (41.8)
= (Q,P)
be a symplectic twist map of T*Tn whose
Q£' satisfies:
inf n{b-\z)v,v)
>a\\v\\,
a > 0, Vw / 0 € H " .
Then F is the time 1 map of a (time dependent) Hamiltonian
H.
Remark 41.7 Condition (41.8) tells us that F does not twist infinitely much. Note that (41.8) holds when H satisfies Assumptions 1 and 2 (see Remark 39.10).
41. Suspension of Symplectic Twist Maps 171
Proof. Let S(q, Q) be the generating function of F. Since p = -diS(q, Q), we have that b = dQ/dp = - (di2S(q, Q))'1. Hence equation (41.8) translates to: (41.9)
inf
(<j,Q)6lR 2 "
{-d12S{q,Q)v,v)>a\\v\\,
a > 0,Vv ^ 0 e HT.
The following lemma show that (41.9) implies the hypothesis of Proposition 25.2, which in turn shows that whenever we have a function on R 2 n which is suitably periodic and satisfies (41.9), it is the generating function for some symplectic twist map. Lemma 41.7 Let
{AX}XEA
be a family ofnxn
inf \(Axv,v)\>a\\vf>
real matrices satisfying: Vt> / 0 € HT.
xEA
Then : det Ax
T^ 0
and sup H-A"1 II < a~l. x£A
We postpone the proof of this lemma. We now construct a differentiable family St, t € [0,1] of generating functions, with Si = S, and then show how to make a Hamiltonian vector field out of it, whose time 1 map is F. Let S(aO)
^
= hafmQ-q^ \ W ( i ) | | Q - q | | 2 + (l-/(*))£(
forO
where / is a smooth positive functions, / ( l ) = /'(1/2) = 0, /(1/2) = 1 and lim t ^ 0 + f(t) = +oo. We will ask also that l / / ( t ) , which can be extended continuously to l//(0) = 0, be differentiable at 0. The choice of / has been made so that St is differentiable with respect to t, for t G (0,1]. Furthermore, it is easy to verify that: inf
(g.Q)elR 2 "
(-d12St{q,Q)v,v)
> a\\vf ,
a > 0, Vv ^ 0 G JRn, t € (0,1].
Hence St generates a smooth family Ft, t € (0,1] of symplectic twist maps. In fact Ft(q,p) = (q- (a/(£)) - 1 P,p),
t < 1/2, so that lim t ^ 0 + Ft = Id. Define:
st(q,p) =
St°i>Ft(q,p),
where %j)Ft is the change of coordinates given by the fact that Ft is twist. Since ipFt (q,p) = (q,q-{af(t))-lp), t < 1/2,
172
7: HAMILTONIANS VS. TWIST MAPS
st(q,p)
l(af(t))~2\\p\\2
=
In particular, by our assumption on 1/f(t), st can be differentiably continued for all t G [0,1], with so = 0. Hence, in the q,p coordinates, we can write: Ffpdq - pdq = dst,
te
[0,1].
By Theorem 59.7, Ft is a Hamiltonian isotopy. Proof of Lemma
•
Jj.1.1.. That det Ax ^ 0 is obvious from the assumption: A has no
kernel. For all non zero v G IR n , we have: Vt; G Mn - {0},
inf K ^ ^ l
>
a
Also: inf \\Axv\\>
inf \{Axv,v)\=
inf
K i 4 * t , ' p >l
so that m£xeA inf||»||=i \\Axv\\ > a. But:
inf
H^ii =
IH|=i"
inf
i^M =
„eiR"-{o}
||«||
inf
\W
«€iR»-{o} H ^ ^ l l
so that, finally: suplU-^^finf
inf
J ^ L )
' <<*-!.
8 PERIODIC ORBITS FOR HAMILTON IAN SYSTEMS
We present in this chapter some results of existence and multiplicity of periodic orbits in Hamiltonian systems on cotangent bundles. Our main goal is to show the power, and relative simplicity of the method of decomposition by symplectic twist map as presented in Chapter 7, which results into finite dimensional variational problems. Some of the results in this chapter are not optimal. They could probably be improved using methods similar to the ones presented here. In some case, these results have recently been improved by other authors, using usually substantially more complicated methods. In Section 42, we present two theorems of existence of periodic orbits for Hamiltonian systems in the cotangent bundle of the torus. They are relatively direct applications of Theorem 27.1 of Chapter 5. In Section 43, we prove a theorem of existence of periodic orbits for systems in cotangent bundles of arbitrary compact manifolds. The boundary condition that we impose (that the Hamiltonian flow be the geodesic flow outside a compact set) is inspired by a similar theorem of Conley & Zehnder (1983) for systems on the cotangent bundle of the torus. That theorem was itself inspired by a conjecture of Arnold (1965), where he proposes an entirely topological generalization (the linking of certain spheres) of the boundary twist condition of the theorem of Poincare-Birkhoff. In Section 44, we explore this linking of sphere condition and prove Arnold's conjecture in the simple case when the map is a symplectic twist map .
174
8: PERIODIC ORBITS FOR HAMILTONIAN SYSTEMS.
42. Periodic Orbits in the Cotangent of the n-Torus We present here two results of existence and multiplicity of periodic orbits for Hamiltonian systems in T*Tn. The first one concerns a certain class of optical systems, the second one Hamiltonians that are quadratic nondegenerate outside of a bounded set. A. Optical Hamiltonians
Assumption 42.1 (Uniform Opticity)
H(q,p,t)
= Ht{z) is a twice differentiable function on T*Tn x Et which satisfies the
following: (l)sup||V2tft||<X (2) The matrices Hpp(z, t) are positive definite and their smallest eigenvalues are uniformly bounded below by C > 0. Theorem 42.2 Let H(q,p,t) be a Hamiltonian function on T*Tn x H satisfying Assumption J^HA. Then the time 1 map h1 of the associated Hamiltonian flow has at least n + 1 periodic orbits of type m, d, for each prime m, d, and 2n when they are all nondegenerate. Proof. We can decompose the time 1 map: fc+i
i
h1 = h\,-x o ... o h7r o ... o hF.
and each of the maps h~^~ is the time jj of the (extended) flow, starting at time j - . Proposition 39.11 shows that, for ./V big enough, such maps are symplectic twist maps. Moreover, we noted in Remark 39.10 that these maps also satisfy a convexity condition which, together with Lemma 27.2 (see (27.5) in its proof) allows us to show that the generating function S is coercive. The result follows from Theorem 27.1.
•
42. Periodic Orbits in T*T"
175
B. Asymptotically Quadratic Hamiltonians We now turn to systems that are not necessarily optical, but satisfy a certain quadratic "boundary condition" which makes them completely integrable outside a compact set: Theorem 42.3 Let H :T*Tn (42.1)
H(q,p,t)
x IR —> Ht satisfy the following boundary
= -(Ap,p)
+ c-p,
condition:
A1 = A, det A ^ 0 when ||p|| > K,
where A is an n x n matrix, c £ JRn and K is a positive real. Then, for all m, d in Z n x 7L, the time-1 h1 map of the Hamiltonian flow has at least n + 1 distinct m,d-orbits,
and at least 2 n when they are all nondegenerate (i.e. generically).
Fur-
thermore, such an orbit lays entirely in the set ||p|| < K if and only if the rotation vector m/d
belongs to the ellipsoid: £ = {x£TRn\
Proof.
WA-^x-c^KK}.
The boundary condition (42.1) is Assumption 2 preceding Theorem 39.7, in
which it is proven that the time e of such Hamiltonians are twist maps. Hence, as remarked in Proposition 39.11, the time 1 map can be decomposed into symplectic twist maps. We now want to apply Theorem 27.1. To insure that these twist maps satisfy the conditions of that theorem, we note that, in the proof of Proposition 39.11, instead of G{q,p) = (q+p,p),
we
can take G{q, p) = (q + Ap + c, p), the time 1 map of H0(q, p) = \ (Ap, p) + c.p. This map is clearly a symplectic twist map . With this minor change, outside the set ||p|| < K, the maps i^fcji^fc-i of the decomposition are respectively the time 1 and the time (jj — 1) of the Hamiltonian flow associated to Ho, that is: F2k (q, p) = (q + Ap + c, p) Fik-i(q,p)
= (q + (l/N - l){Ap +
c),p).
These maps satisfy the conditions of Theorem (27.2) , which proves the existence of the advertised number of m, d orbits. To localize these orbits, note that an orbit starting in ||p|| > K must stay there, and the map h1 on such an orbit is just G. The rotation number of such an orbit is thus (Q - q) = Ap + c from which we conclude that, in this case, m/d is in the complement of £.
D
176
8: PERIODIC ORBITS FOR HAMILTONIAN SYSTEMS.
C. Remarks about the Above Results Periodic Orbits for the Map vs. Periodic Orbits for the Flow. There is a distinction between periodic orbits of h1 and periodic orbits of the Hamiltonian equations: for a general time dependent Hamiltonian flow, (ft 1 )" ^ hn, and hence an m, d periodic orbit for h1 is not necessarily one for the O.D.E. (which should satisfy ht+d(z)
= ^(z)
+ (km, 0) for all
t e [kd, (k + l)d), k G Z). However, if H is periodic in time, of period 1, the equality (hl)n
= hn does hold, and in this case the two notions coincide. In particular, this holds
trivially for time independent Hamiltonians. Unfortunately, these cases are degenerate in our setting, since Dhd(z)
preserves the vector field XH, which is thus an eigenvector with
eigenvalue one. So in these cases, we can only claim the cuplength estimates for the number of periodic orbits for the Hamiltonian flow in either Theorems 42.2 or 42.3. We think that some further argument should yield, even in the time periodic case, the sum of the betti number estimate for the number of flow periodic orbits, when the periodic orbits are nondegenerate as orbits of the flow, i.e., when the only eigenvector of eigenvalue one for Dhd(z)
is in the direction of the vector field XH-
Possible Improvements. Note that the full strength of Theorem 27.1 was not brought to bear in the proof of Theorem 42.3: the symplectic twist maps that we obtained in the decomposition of h1 are linear outside a bounded set, whereas Theorem 27.1 can deal with asymptotic linearity. It is very conceivable that one could cover a larger class of systems using this method, including classical mechanical systems on the torus. Related Results in the Literature. Conley & Zehnder (1983) contains a theorem of existence of multiple homotopically trivial periodic orbits, with a boundary condition similar to that of Theorem 42.3. In an impressive and technically difficult piece of work, Josellis (1994), (1994b) gives an improved version of Theorem 42.3 in that the Hamiltonian flow is only asymptotically quadratic. See also Felmer (1992) for related results using a mountain pass lemma. In Benci, V. (1986), it is shown that fiberwise convex, time periodic Lagrangian systems on arbitrary compact manifolds have at least one periodic orbit of any given free homotopy class. This result, which assumes also certain assumptions on the first and second derivative of the Lagrangian implies, via the Legendre transformation, the existence of at least one m , 1 orbit for the optical systems we consider in Theorem 42.2. Conversely, via the Legendre transformation, Theorem 42.2 applies to Lagrangian systems
43. Periodic Orbits In General Cotangent Spaces
177
whose Lagrangian function satisfies the same conditions as H in our theorem (it is not hard to see that these conditions translate under the Legendre transformation). Hence Theorem 42.2 extends some existing theorems for such systems (see, e.g., Mawhin & Willem(1989), Theorem 9.3).
43. Periodic Orbits in General Cotangent Spaces
We now turn to the study of Hamiltonian systems in cotangent spaces of arbitrary compact manifolds. Our main result, which first appeared in Gole (1994) is:
Theorem 43.1 Let (M, g) be a compact Riemannian ||-||. Let F : T*M —» T*M on B*M,
manifold, with associated norm
be the time 1 map of a time dependent Hamiltonian
2
where H is a C function satisfying the boundary
H{q,p,t)
= ±\\pf
for
H
condition:
\\p\\>C.
where C is strictly smaller than the injectivity radius. Then F has at least cl(M) distinct fixed points and at least sb(M)
if they are all nondegenerate.
Moreover,
these fixed points lie inside the set {||p|| < C} and can all be chosen to correspond to homotopically trivial closed orbits of the Hamiltonian
flow.
The injectivity radius on a Riemannian manifold is defined as
inf
sup
•! r
exp\ „,
.
is infective
> .
The rest of this section is devoted to the proof of this theorem. Note that Cielieback (1992) provides a similar theorem, with asymptotically quadratic conditions. His proof uses a version of Floer cohomology.
178
8: PERIODIC ORBITS FOR HAMILTONIAN SYSTEMS.
A. The Discrete Variational Setting Define B'M
= {(q,p)
€ T*M
| g(q)(p,p)
= \\pf
< C2}.
Let 7r denote the canonical projection w : B*M —> M. Let F be as in Theorem 43.1. From Proposition 39.11 we can decompose F into a product of symplectic twist maps : F = F2N ° • • • ° F1, where F2k restricted to the boundary dB*M of B*M is the time 1 map hg of the geodesic 1 N
2
flow with Hamiltonian H0(q,p)
~
= \ \\p\\ • Likewise, F2k-\
N
is h0
on dB'M.
Let
5/t be the generating function for the twist map Fk and t/jk = tpFk the diffeomorphism {QIP) ~* (9; Q) induced by the twist condition on Fk- We can assume that ipk is defined on a neighborhood U of B*M in T*M where [/ = { ( q , p ) e r * M \\\p\\
+ S}.
Our variational study will take place in the set:
0={q= (43.1)
e M2N
{qlt...,q2N)
\(qk,qk+1)
S i/)k(U) and
Proposition 43.2 T/ie sei O can 6e described as: 0 = {qeM2N\
Dis(qk,
qk+1)
< \ak\(C + 6), Dis(q2N,qi)
< (C + 6)}
where , .„ „.
f
1
if k is even
/n particular, O contains the set of constant sequences in
Proof.
M2N.
This is an easy application of the twist condition, using the fact that the map Fk
equal the time ak of the geodesic flow on the boundary of U.
•
43. Periodic Orbits In General Cotangent Spaces
179
We note that U and O are independent of the map F, as long as F satisfies the boundary condition of Theorem 43.1. Next, define : 2N
(43.3)
W(q) =
Y/Sk(qk,qk+1), fc=0
where we have set q2Ar+i 2N
q € M , {qk,pk)
=
Qi- Choosing to work in some local coordinates around
we let pk = -9iS f c (q f c ,q A : + 1 ) and Pk
€ T*kM is such that M^Pk)
such that Fk(qk,pk)
= d2Sk(qk,qk+1).
= («fc,9fc+i)
and
In other words,
(
is
= (qk+i, Pk)- We let the reader check that the following proofs can
be written in coordinate free notation (see Remark 26.3). By Exercise 26.4, a sequence q of O is a critical point of W if and only if the sequence {(
= 5Zfc = 1 (Pk-i — Pk)dqk
•Pfc(
=
which is null exactly when Pk-\
= pk, i.e. when
(9fciPfc)- Now remember that we assumed that q 2 iv+i
=
Qi-
Hence, to prove Theorem 43.1, we need to find enough critical points for W. As before, we will study the gradient flow of W (where the gradient will be given in terms of the metric g) and use the boundary condition to find an isolating block. The main difference with the previous situations on T*Tn is that we cannot put W in the general framework of generating phases quadratic at infinity. Nonetheless, thanks to the boundary condition we imposed on the Hamiltonian, we are able to construct an isolating block and use Floer's theorem of continuation (Theorem 63.7in Appendix 2) to get a grasp on the topology of the invariant set, and hence on the number of critical points. B. The Isolating Block In this subsection we prove that the set B defined as follows: (43.4)
B = {q € O | Dis(qfc, qk+1)
< \ak\C}
is an isolating block for the gradient flow of W, where O is defined in (43.1), C is as in the hypotheses of Theorem 43.1, and ak is defined in (43.2). Proposition 43.3 B is an isolating block for the gradient flow ofW.
180
8: PERIODIC ORBITS FOR HAMILTONIAN SYSTEMS.
Proof.
Suppose that the point q of U is on the boundary of B. This means that
Dis(qk, qk+i) to
l|Pfcll
=
= \ak\C for at least one k. Using the boundary condition, this is equivalent
C> where as usual pk = —diSk(qk,qk+1).
We want to show that
Dis(qk,qk+1)
increases either in positive or negative time along the gradient flow of W. The kth component of the gradient vector field is given by: (43.5)
qk = Ak(Pk^
- pk) =
VWk(q)
where Ak = A(qk) is the inverse of the matrix of coefficients of the metric g at the point qk. We used that, on a Riemannian manifold, the gradient of a function / is given by
=
{qk+1,Pk):
Dis(9fc,g fc+ i) = <9iDis( 9fc , 9fc+1 ) • VWk{q) dt t=o + d2Dis(qk,qk+1) • VWk+l{q) = sign(ak)-—| + sign{ak)j~
• Ak(Pk-X
- pk)
• Ak+1(Pk
-
pk+1)
We now need a simple linear algebra lemma to treat this equation. Lemma 43.4 Let (, ) denote a positive definite bilinear form in JRn, and ||.|| its corresponding norm. Suppose that p and p' are in IR" ,that \\p\\ = C and that ||p'|| < C. Then : (p,
P'-p)<0.
Moreover, equality occurs if and only if p' = p. Proof.
From the positive definiteness of the metric, we get: {pf-p,p'-p)>0,
with equality occurring if and only if p' = p. From this, we get:
43. Periodic Orbits In General Cotangent Spaces
181
2 ( p , p ' ) < ( p ' , p ' ) +
Finally,
<(P'-P),P> = ( P ' . P } - < P . P > < ^ ( ( P ' . P ' > - < P . P » < O with equality occurring if and only if p' = p.
•
Applying Lemma 43.4 to each of the right hand side terms in (43.6), we can deduce that ^Dis(q f c , qk+l)
is positive when k is even, negative when k is odd. Indeed, because of
the boundary condition in the hypothesis of the theorem, we have ||-Pfc|| = ||p fe || whenever ||p fc || = C: the boundary dB*M is invariant under F and all the Fks. On the other hand q £ B => \\pt\\ < C and ||P(|| < C, for alU, by invariance of B*M. Finally, ak is positive when k is even, negative when k is odd. The problem is that we have not shown yet that ~Dis(qk,qk+1)
cannot be 0. This
problem is confined to the "edges" of dB, i.e. the sets of points q such that more than one pk has norm C. The problem at these edges occurs when k is in an interval {/,..., m} such that, for all j in this interval, ||pj|| = C = \\Pj\\ and VWj(q) = 0. It is now crucial to note that {I,..., m} can not cover all of { 0 , . . . , 2iV}: this would mean that q is a critical point corresponding to a fixed point of /ij in dB*M. But such a fixed point is forbidden by our choice of C: orbits of our Hamiltonian on the set ||p|| = C are geodesies, but geodesies in that energy level cannot be rest points since C > 0, and they cannot close up in time one either since C is less than the injectivity radius. We now let k — m in (43.6) and see that exactly one of the 2 terms in the right hand side of Equation (43.6) is nonzero. Hence the flow must definitely escape the set B at q in either positive or negative time, from the mth face of B.
•
Remark 43.5 If the Hamiltonian considered is optical and we decompose its time 1 map into a product of N twist maps as in 39.11, all the i*Vs coincide with h^ on the boundary of B*M. In that case, all the a^'s in the above proof are positive, and B is a repeller block.
182
8: PERIODIC ORBITS FOR HAMILTONIAN SYSTEMS.
C. End of Proof of Theorem 43.1 To finish the proof of Theorem 43.1 we use Floer's theorem 63.7 of continuation of normally hyperbolic invariant sets. We consider the family F\ of time 1 maps of the Hamiltonians: Hx = (1 - \)H0 + \H, where H is as in Theorem 43.1 and H0(q,p)
= \ ||p|| 2 . Corresponding to this is a family
of gradient flows C\, solution of
where W\ is the discrete action corresponding to the decomposition in symplectic twist maps of the map F\. We take care that this decomposition has the same number of steps, say 27V, for each A. As before, the manifold on which we consider these (local) flows is O, an open neighborhood of B in M2N.
Each F\ satisfies the hypothesis of Theorem 43.1,
and thus Proposition 43.3 applies to Qlx for all A in [0,1]: B is an isolating block for each of these flows. Hence the maximum invariant sets G\ for the flows C\ m B are related by continuation. The part of Floer's Theorem that we need to check is that Go is a normally hyperbolic invariant manifold for CoLemma 43.5 Let G$ = {q 6 B \ qk = q1,Vk}.
Then GQ is a normally
hyperbolic
invariant set for Co- Go is a retract of O and it is the maximal invariant set in B. Proof.
The only critical points for Wo in B are the points of Go which correspond to
restpoints of the geodesic flow, i.e. the zero section. Indeed, critical points of Wo in B corresponds to periodic points of period 1 for the geodesic flow in B*M. Our definition of that sets precludes nontrivial periodic geodesies in B*M. We now show that the maximum invariant set for Co in B is included in Go. Since Co is a gradient flow, such an invariant set is formed by critical points and connecting orbits between them. The only critical points of Wo in B are the points of Go- If there were a connecting orbit entirely in B, it would have to connect two points in Go, which is absurd since W0 = 0 on Go, whereas W0 should increase along non constant orbits. Go is a retract of M2N
under the composition of the
maps: 9 = ( 9 i , - - - , 9 2 ^ ) ~>Qi - " (9i><7i,-••>
43. Periodic Orbits In General Cotangent Spaces
183
which is obviously continuous and fixes the points of Go- It remains to show that Go is normally hyperbolic. Since Go = M is an n-dimensional manifold made of critical points, saying that it is normally hyperbolic is equivalent to saying that for q in Go, kerV2Wo(q) has dimension n: indeed, if it is the case, the only possible vectors in this kernel must be tangent to Go, and thus he differential of theflowis nondegenerate on the normal space to TGo- In the present situation, the second variation formula of Lemma 29.4 says that the 1-eigenspace of Dh\ is isomorphic to the kernel of V 2 Wo. Hence it is enough to check that at a point (q1,0) £ B*M corresponding to q, 1 is an eigenvalue of multiplicity exactly n for Dh\(q^, 0). Let us compute Dh})(q1,0) in local coordinates. It is the solution at time 1 of the linearized (or variation) equation: U= along the constant solution (q(t),p(t)) tic matrix I 2
-JV2H0(qi,0)U = (9i,0), where J denotes the usual symplec-
1. An operator solution for the above equation is given by U(t) =
exp (-tJV Ho{q1,0))
• On the other hand:
V^o(9l,0)=(S
A^)
which we computed from Ho(q,p) = \A(q)p.p, the zero terms appearing atp = Obecause they are either quadratic or linear in p. From this, Dhl(qi,0)
= exp (JW2Ho(qi,0))
=
0
I
is easily derived. This matrix has exactly n independent eigenvectors of eigenvalue 1 (it has in fact no other eigenvector). Hence, from Lemma 29.4, V 2 W(q) has exactly n vectors with eigenvalue 0, as was to be shown. D We now conclude the proof of Theorem 43.1. We have proved that the gradient flow £', has an invariant set G\ with H*(M) ^-> H*(Gi). From this we get in particular: cZ(Gi) > cl(M) and s6(Gi) > sb{M). Theorem 61.2 and the remark following it tell us that C' must have at leastd(Gi) rest points in the set Gj, and sfo(Gi) if all rest points are nondegenerate. ButLemma 29.4 (which, as the reader can readily check, is valid in general cotangent bundles) tells us that nondegeneracy
184
8: PERIODIC ORBITS FOR HAMILTONIAN SYSTEMS.
for V2W at a critical point is the same thing as nondegeneracy of a fixed point for F (no eigenvector of eigenvalue 1). This proves the existence of the advertised number of fixed points of the map F. In the following section, we will see how to insure that all the fixed points of the time 1 map that we find correspond to homotopically trivial periodic orbits. This concludes the proof of Theorem 43.1.
•
D. Periodic Orbits of Different Homotopy Classes To determine the topological type of a periodic orbit for a map on T*M, we consider the free homotopy class of a curve built from a given sequence of points on the manifold, in the fashion of broken geodesies. Free Homotopy Classes. The free homotopy class of a curve is an equivalence class of all curves that are homotopic without a fixed base point. As a result, free homotopy classes can be seen as conjugacy classes in -K\ (M) (the conjugacy is by concatenation with a curve that goes from the starting point of the given curve to a given base point, and back). Thus this set of free homotopy classes can not be endowed with a natural algebraic structure. Two elements of a free class give the same element in Hi(M). form a set smaller than -KI(M),
bigger than Hi(M).
Hence free homotopy classes
All these sets coincide if TTI(M) is
abelian. Construction of the Broken Solutions. For each Fk in the proof of Theorem 43.1 , define ipk to be the inverse map of the diffeomorphism Q —> —diSk(q,Q). the map that make correspond p to Q according to Fk(q,p)
That is, fixing q,
= (Q, P). Since each Fk
is a symplectic twist map equal to hgk on dB*M for some positive or negative ak, the set ipk(B*M)
is a ball of radius \ak\ centered at q (in the sense of distance induced by
the Riemannian metric). In particular q € ipk{B*M).
Since B*M
—>
diffeomorphism, we can define a path ck(q, Q) between q and a point Q of ipk(B*M) 1
by
1
taking the image by (pk of the oriented line segment between
. If we look for periodic orbits of period d and in
a given free homotopy class, we decompose Fd into 2Nd twist maps, by decomposing F into 2N. Analogously to Equation (43.1), we then define :
43. Periodic Orbits In General Cotangent Spaces Od = {q = (qx,..
.,q2Nd)
G M2Nd \(qk,qk+1)
G MU)
185
and
(U is as before a neighborhood of B*M). To each element q in Od, we can associate a closed curve c(q), made by joining up each pair (qk, qk+l)
with the curve ck(qk,
qk+1)
uniquely defined as above. This "broken solution" c(q) is a piecewise differentiable loop, and it depends continuously on q, and so do its derivatives (left and right). In the case of the decomposition of h\ , taking Fk=h\, this is exactly the construction of the broken geodesies (see Section 38). Now any closed curve in M belongs to some free homotopy class that we denote by m . To any d periodic point for F, we can associate a sequence q(x) G Od of q coordinates of the orbit of this point under the successive Fk's in the decomposition of Fd. Definition 43.6 Let z be a periodic point of period d for F. Let q be the sequence in Od corresponding to x. We say that x is an m, d-point if c(~q) is in the free homotopy class m. This definition has the advantage to make sense for any map F of T*M which can be decomposed into the product of symplectic twist maps . If F is also the time 1 map of a Hamiltonian, it agrees with the obvious definition: Proposition 43.7 If z is an m,d periodic point, then the projection ir(z(t)),t
G [0, d]
of the orbit of z under the Hamiltonian flow is a closed curve in the free homotopy class m. Proof.
Left as an exercise {Hint. Use the geodesic flow to construct the homotopy between
c(q(z)) and n(z{t))).
D
Let (43.7)
0m4
= {q G Od | c(q) G m }
Since c(q) depends continuously on q G Od, Om^ is a connected component of Od. The reader who wants to make sure that, in the proof of Theorem 43.1, the orbits found are homotopically trivial, can check that the proof we gave in last section works identically when one replace the space O, by its connected component O e ,i, where e is the homotopy class of the trivial curve. Another place where one uses this decomposition of O in different homotopy components is the following:
186
8: PERIODIC ORBITS FOR HAMILTON IAN SYSTEMS.
Theorem 43.8 Let (M,g) be a Riemannian
manifold of negative curvature and H be
as in Theorem 1. If'-ym denotes the (unique) closed geodesic of free homotopy class m, F has at least 2 m, d-orbits in B*M when length(-fm)
< dC .
The proof of Theorem 43.8 (see Gole (1994), Theorem 2) has the same broad outline as that of Theorem 43.1. We work in O m ,d instead of O. The normally hyperbolic invariant set that we continue to in this setting is given by the set Go of critical sequences corresponding to the orbits under the h^k 's of the points on 7 m . The normal hyperbolicity of Go derives this time from the hyperbolicity of the geodesic flow in negative curvature.
44. Linking of Spheres: Toward a Generalization of the Theorem of Poincare And Birkhoff
This section goes back to the original motivation of Theorem 43.1, namely the following conjecture of Arnold (1965) (which appears as Remark D there) that generalizes the Theorem of Poincare-Birkhoff. We will define and explore the notion of linking of spheres in the sequel.
Conjecture 44.1 (Arnold) Let F be a globally canonical diffeomorphism Bn x T n such that, for all q e Tn, the spheres dBn x q and F(dBn in dBn
x HT (EC1 being the universal cover ofTn).
points in B*Tn
(compted with
of B*Tn =
x q) are linked
Then F has at least 2™ fixed
multiplicity).
Arnold defines globally canonical as exact symplectic and homotopic to the Identity. In Banyaga & Gole (1993) we proposed the following generalization of this conjecture:
Conjecture 44.2 Let M be a compact manifold, and F be a Hamiltonian ball bundle B*M in T*M. Suppose that each sphere dB*M F in dB*M. they are
map of a
links with its image by
Then F has at least cl(M) distinct fixed points, and at least sb(M) if
nondegenerate.
44. Linking of Spheres
187
In Banyaga & Gole (1993) (see also Gole (1994)), we proved the following simple case. We will give the proof in the case of M = Tn.
Theorem 44.3 Let F be a symplectic twist map of B*M which links spheres on the boundary dB*M.
Then F satisfies the generalized Conjecture
44-2-
Linking of Spheres and the Boundary Twist Condition. If you remove a circle C\ from 3-space, the hole it leaves out creates some topology. In particular, another circle C2 can "go around that hole" or not. If it does, the circles C\ and Ci link. In mathematical terms, the complement of C\ has a new generator in first homology that the 3-space did not have. The first homology class of C2 in the complement of C\ measures the linking of the 2 circles. Likewise, removing an n — 1 sphere in IR2™-1 creates a new generator in H n _ j , and the homology class of another sphere in that group measures the linking of the two spheres. We will adapt this notion to the setting where one sphere is the boundary dAq of a fiber Aq of the ball bundle B*M, and the other sphere is F(dAq).
We will go into more detail
later on these concepts, when we prove that, at least in the case of symplectic twist maps of T*T n , the linking of these two n — 1-spheres in the 2n — 1 dimensional boundary of B*M is equivalent to:
Definition 44.4 (Fiber Intersection Property) We say that a map F : B*M —> B*M satisfies the Fiber Intersection Property if each fiber Aq = 7r _1 (q) intersects its image F(Aq)
with a nonzero algebraic intersection number (i.e. the number of intersections
counted with orientation).
Note that, in the case of twist maps of the annulus, this property is clearly equivalent to the boundary twist condition of the Poincare-Birkhoff Theorem 7.1: If points on the two boundary components of the annulus go in opposite directions under F then the vertical fiber {x = xo} and its image by F should have a nonzero algebraic intersection number. Before going through the rigorous definition of sphere linking and its equivalence with the Fiber Intersection Property, we give the proof of Theorem 44.3.
188
8: PERIODIC ORBITS FOR HAMILTONIAN SYSTEMS.
Proof of Theorem 44.3. We assume for now the equivalence of the linking of spheres condition and the Fiber Intersection Property. If F is a symplectic twist map, a fiber Aq and its image under F may intersect at most once. Hence the Fiber Intersection Property means in this case that each fiber intersects its image exactly once. Fixed points of F correspond to critical points of q —» S(q, q). This function is well defined since, by the above, the diagonal in M x M is in the image of B*M by the embedding 1pp. Hence F has as many fixed points as the function q —> S(q, q) has critical points on M. Morse and LyusternickSchnirelman's theories (See Theorem 61.2 and the remark following it) give the advertised estimates.
•
Equivalence of Sphere Linking and Fiber Intersection. We now show that, in the case considered by Arnold, the Fiber Intersection Property is indeed equivalent to linking of boundary spheres. For the case of more general manifolds than T ' T " , we refer the reader to Bany aga & Gole (1993) or Gole (1994). The reader may already be aware of a connection between linking and intersection: going back to the example of 2 circles in JR.3, their linking can be measured by the algebraic intersection number of one circle with any disk bounded by the other one. This correspondence breaks down in S 1 x K 2 = dB*T the circles dAq and F{dAq)
however: there,
do not bound any disks. We can still define their linking
number homologically, and relate it to the Fiber Intersection property, which takes place in - 2
the full space B* T . The important point is that the linking of spheres is a purely topological condition which can be read entirely in the boundary. We first remind the reader of the definition of linking of spheres from algebraic topology. Let Aq be a fiber of B*t"
^ S n _ 1 x IR n . Then dAq ^ S n _ 1 . It make sense to talk about
its linking with its image F(dAq)
in dB*T
: the latter set has dimension 2n — 1 and the
dimensions of the spheres add up to 2n - 2. The linking number F{dAq) by the class [F(dAq)\
n
€ Hn^{dB*f \dAq;
with dAq is given
2 ) (from now to the end of this chapter, we
only consider homology with integer coefficients). More precisely, we have: Hn.^dB^XdA,,) (
44
-i)
=* ffn_! ( S " - 1 x (Et B - {0})) Kunneth
^
. Jr7„_1(r-
1
f
•
)ffiJtfn_1(IR"-{0})
Thus, removing dAq from dB* f ™ creates a new generator in the (n - l ) s t homology (with integer coefficients) of that set, i.e. a generator, call it 6, of Hn-i(JRn
- { 0 } ) = 2 . As any
44. Linking of Spheres sphere of dimension n — 1 in dB*T
\dAq,
F(dAq)
189
represents an n — 1 cohomology class
in that set that we can write: [F(dAq)] = aa®/3b in the final decomposition in (44.1) . We call the integer /3 the linking number of the spheres F(dAq) and F{dAq)
and dAq. If the linking number is nonzero, we say that the spheres dAq
link. Finally, if dAq and F(dAq)
link for all qr £ M, we say that F satisfies
the Linking of Spheres Condition. Lemma 44.5 / / F is the lift of a diffeomorphism Intersection
of B*Tn = Tn x Bn,
Property is equivalent to the Linking of Spheres Condition. More pre-
cisely, the algebraic intersection number #{Aq n F(Aq)) the spheres dAq and F(dAq) Proof.
the Fiber
and the linking number of
are equal.
We complete (44.1) into the following commutative diagram: Hn^(dB*Tn\dAq)
~
ifn_1(]Rn-{0})©frn_1(Sn)
I ** Hn_!(B*f " \ 4 , )
I J^
tf„-i
n
({M - {0}} x Bn))
where i, j are inclusion maps and Bn is the n-ball. It is clear that jtb generates ffn_i ((W1 - {0}) x £ " ) ^ # „ _ ! ({Mn - {0}} x I T ) . If S is any n - 1 sphere in {lR n - {0}} x HT, the class [5] G Hn^
({M n - {0}} x H n )
(i.e. , an integer) measures the (usual) linking number of a sphere with the fiber Aq in B*T
= IR 2n . But it is well known that such a number is the intersection number of any
ball bounded by the sphere with the fiber Aq, counted with orientation (see Rolfsen (1976) page 132).
•
CHAPTER 9* GENERALIZATIONS OF THE AUBRYMATHER THEOREM There are, strictly speaking, no full generalizations of the Aubry-Mather Theorem in higher dimensions: we will see in this chapter examples of fiber convex Lagrangian systems whose set of minimizers achieves only very few rotation directions. However some attempts of generalizations in higher dimensions are quite successful in what they try to achieve. In Section 45, we survey some results by de la Llave and his collaborators. Their setting is explicitly non dynamical but generalizes naturally the Frenkel-Kontorova model to functions on lattices of any dimension. They are entirely successful in proving an AubryMather type theorem in this setting, as well as in some PDE cases, as well as in the context of minimal surface laminations. In Section 46, we review the work of Angenent (1990) which generalizes twist maps to a certain type of maps of S 1 x IR n and proves, among other results, an Aubry-Mather type theorem for these maps. In Section 47, we look at the work of MacKay & Meiss (1992) who construct higher dimensional analogs of Aubry-Mather sets in symplectic twist maps that are close to the anti-integrable limit: one where the potential term in the generating function of a standard type map dominates. In Section 48, we survey the work of Mather on minimal measures in convex Lagrangian systems. This is the closest to a generalization of the Aubry-Mather theory as one can get in the setting of general convex Lagrangian systems (as well as symplectic twist maps). We start this section by introducing the notion of minimizers and reviewing some ergodic theory. We then survey Mather's fundamental graph theorem and finish the section by pointing at the limitations of the theory. Section 49 surveys the work of Boyland and the author which shows that some of these limitations can be alleviated if one considers systems on cotangent bundles of hyperbolic manifolds.
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9: GENERALIZATIONS OF AUBRY-MATHER
45/ Functions on Lattices, PDE's and Minimal Surfaces A*. Functions on Lattices Remember from Chapter 1 that the Frenkel-Kontorova model describes configurations of interacting particles in a periodic potential. For simplicity, these configurations are assumed to be one dimensional, and the interactions only involve nearest neighbors. The resulting action function is the familiar:
W(x) = i j > f c - xk+lf - J2 V(xk) fcez fcez
where the potential function V has period 1. W coincides with the action function for the standard map with generating function S(x, X) = | (X — x)2 — V(x). As noted in Section 14, the variational equation S7W = 0 for this action function is
(-Ax)k - V'(xk) = 0 where A(x)k = —2xk + xk-\ + xk+i is the discretized Laplacian. Note that the configuration x can be seen as a function 7L —> IR which to the integer k makes correspond the real xk. One obtains (see Blank (1989), Koch & al. (1994), Candel & de la Llave (1997), de la Llave (1999)) a natural generalization of this model, relevant to Statistical Mechanics, by asking that x : Z d —> IR be a function on a lattice of dimension d. We assume nearest neighbor interaction here. The energy becomes:
w{x) = \
Y, (xk-xtf-Ynxk). d
{(kj)ez \ |fc-j|=i} kez.d Again V is of period 1 and the corresponding variational equation is still of the form: (45.1)
(-AB) fc - V'(xk) = 0
where (Ax)k = ^i f c _,i = 1 Xj — 2dxk is the d-dimensional discrete Laplacian. In fact, the theory in Candel & de la Llave (1997) applies to substantially more general settings, where A; can belong to a set A on which a certain type of groups acts in a mildly prescribed way, and where the interactions involves not just nearest neighbors, but all possible pairs of particles (with some decay condition at infinity). Remember that the solutions x : 7L —> IR found by Aubry and Mather for the FrenkelKontorova model are such that \xk -kw\<<x>. One way to express this is by saying that the
45. Functions on Lattices and PDE's
193
graph of x : 7L —> IR is at bounded distance from a line of slope u> in IR x IR. Likewise, the following generalization of the Aubry-Mather Theorem finds configurations whose graphs are at bounded distances from hyperplanes of "slopes" u> e Md. This version is taken from Candel & de la Llave (1997): Theorem 45.1 For every w e H d , there exists a solution of (45.1)
such that
sup \xk — u • k\ < oo. fcezd
The method of proof is very similar to the proof of the Aubry-Mather Theorem presented in Chapter 3. One considers the analog of CO sequences, called Birkhoff configurations by these authors. In complete analogy to the CO sequences, they satisfy: Xk+j + I > Xk, Vfc e 7Ld or
xk+j+l<xk,VkeV.d
The analog to the set of CO sequences of rotation number u>, which we denoted by C0U in Chapter 3 is: Bu = {x | x is Birkhoff and sup \xk — k • OJ\ < oo} k€Zd
In a way analogous to the proof of Theorem 15.1, one shows that the gradient flow of W (that these authors, justifiably, call the heat flow) preserves order among configurations and is suitably periodic, so that the set Bu is invariant under the flow. The same argument as in the proof of Theorem 15.1 is then used to show that W must have a critical point inside Bu. So, as in the classical Aubry-Mather Theorem, one not only finds solutions that have asymptotic slope u>, but these solutions have strong order properties, expressed here in terms of non intersection: they are Birkhoff. B*. PDE's As Equation (45.1) suggests, the above theory smells of discretized PDE's. It is therefore not too surprising that the same kind of methods can be applied to certain PDE problems. The main ingredients necessary are some translation invariance and a heat flow that satisfies a comparison principle u > v => 4>fu >
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solutions whose graphs are at bounded distance from planes with prescribed slopes, and have nonintersection properties: (45.2)
Au+V'(x,u)
=0
where V{x + e,u + l) = V{x, u) V i e K % e R , e e Z.d, < 6 l fc
(45.3)
J2L1 + V'(X'U) = ° t=i
where Lt are Z d periodic vector fields satisfying Hormander's hypoelhpticity conditions and V is as in the previous case.
(45.4)
(-A)1/2u +
V'(x,u)=0
with V as above, de la Llave (1999) also looks at the following PDE: (45.5)
Ou = uu- uxx = -V(u) + f(x, t) u(x + l,t)=u{x,t
+ T) = u(x,t)
where the function / also has the periodicity: (45.6)
f(x + l,t)=f(x,t
+
T)=f(x,t).
We say that the real number T is of constant type if its continued fraction expansion is bounded. For instance, noble numbers are of constant type. Theorem 45.2 (de la Llave) Let T be a number of constant type, let f 6 L2 satisfy (45-6) and let V : H —> 1R satisfy (i) 0 < a < V < P where a is any positive number and (3 only depends on T (in an explicit manner) (ii) \Vn{x)\
46. Functions on Lattices and PDE's
195
The method of proof is different from that of the above PDE's, but still involves a variational approach. C*. Laminations by Minimal Surfaces
Moser (1986b) proposed a generalization of the Aubry-Mather theory to certain minimizing hypersurfaces in IRn (according to some specific variational problem). There, Moser asks if such a result would also work when "minimizing" is understood in the classical sense of minimal area, i.e. minimal surfaces. Caffarelli & de la Llave (2000) answers the question positively in the following theorem: Theorem 45.3 (Caffarelli & de la Llave) Let g be a C2 strictly positive metric in H™ invariant under integer translations. Then we can find a number M depending only on the oscillation properties of the metric such that, for every n — 1 dimensional hyperplane II, we can find a minimal surface S such that Dis(X', 77) < M. Down on the torus T n = M n / Z n , each surface £ gives rise to a lamination, periodic if the plane II has rational slope, "quasiperiodic" otherwise. The full result in Caffarelli & de la Llave (2000) is actually much more general, in that it deals with a large class of periodic minimization problems, those involving integrals with elliptic integrands. Moreover, as in Theorem 45.1, this result holds for manifolds whose fundamental group satisfies mild conditions, in particular manifolds of negative curvature. Minimal Geodesies. An important special case of the above theorem is when n = 2. The minimal surfaces are then minimal geodesies (in the sense that they globally minimize length between any two of their points) for periodic metrics, eg. metrics on the torus T 2 . This case had been thoroughly studied by Hedlund (1932), who had obtained a similar result to the above in this case. Morse (1924) had previously treated the case of surfaces of negative curvature (see Section 49B for a generalization of this to Lagrangian systems). The analogy of Hedlund's work with the Aubry-Mather theorem is made clear in Bangert (1988), where he shows that both theories are part of a third one. We will see in a later section that Mather (1991b) also arrives at the same kind of conclusion, with his theory of minimal measures.
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46.* Monotone Recurrence Relations Angenent (1990) proposes a generalization of twist maps of the annulus to maps of S 1 x IR^ which are defined by solving a recurrence relation: (47.1)
A{xk_u...,xk+m)=Q
which generalizes &2S{xk-\, xk) + d\S(xk,xk+i)
= 0 in twist maps, where k = I = 1.
The function A is required to satisfy the conditions: a.) Monotonicity A(x_i,...
,x+m)
is anon decreasing function of all the xk except possibly
for k = 0. Moreover, it is strictly increasing in the variables x_; and xm. b) Periodicity A(xk-i,
•••, xk+m)
c) Coercion lim a ; i _ ± 0 0 A(x-t,...,
= A(xk-i
+ 1 , . . . , xk+m
+ 1).
xm) = l i m X m ^ ± 0 O A'x^i,...,
xm) = ±oo.
Under these conditions, Angenent calls (47.1) a monotone recurrence relation. Conditions a) and c) imply that one can solve for xk+m
in terms of a given (xk_i,...,
Hence this defines a map FA : (xk-i, •••, xk+m-i)
>-> (xk-i+i,-
xk+m-\).
• •, xk+m) 1
to itself. Condition b) implies that this maps descends to a map on S x ] R
from IR' + m
;+m_1
. Hence
the N above is N = I + m — 1. The notion of CO configurations, rotation number and partial order on sequences etc... of Chapter 2 and Chapter 3 are still entirely valid here, since the variables xk are 1 dimensional (Angenent also calls CO sequences Birkhoff). An interesting notion that Angenent (1990) introduces, inspired by PDE methods, is that of sub— or supersolution of the monotone recurrence relation (47.1): x is a subsolution if A(xk_i,...
,xk+m)
< 0, VA; € 2 and a
Theorem 47.1 (Angenent) Let x, x be sub- and supersolutions
respectively, which
supersolution if A(xk_h
... ,xk+m)
>0,Vk€
7L.
are ordered: x < x. Then there is at least one solution of (47.1), say x, for which x<x<x
holds.
Using this theorem (whose proof is simple), Angenent (1990) is able to generalize a theorem of Hall (1984), itself a generalization of the Aubry-Mather theorem: if a twist map of the annulus, which is not necessarily area preserving, has a (m, n)-periodic orbit, then it must have a CO (m, n)-periodic orbit. If the map is also area preserving, this implies,
47. Anti—Integrable Limit
197
taking limits, the existence of CO orbits of all rotation numbers. Analogously, Angenent proves that if there is an orbit of FA with rotation number u> € H, then FA must also have a CO orbit of rotation number ui. Suppose that two solutions x and w of (46.1) "exchange rotation numbers" in the sense that: lim Xk/k > u>i > lim uik/k n—>+oo
n—*— oo
and lim Wk/k < uo < lim Xk/k n~*+oo
n—>—oo
holds for some wo < ^i • Then Angenent proves that there must be CO orbits of any rotation number u S [wo> w{\. Moreover this exchange of rotation numbers condition implies chaos: the topological entropy htop {FA ) > 0, in that there is a compact invariant set semi conjugate to a Bernouilli shift. This also generalizes shadowing results of Hall (1989) and Mather (1991a). Angenent (1992) uses similar techniques to prove the following beautiful theorem: Theorem 46.2 Let f be a twist map of the compact annulus. Let po < Pi be the rotation number of f restricted to the boundaries. If the topological entropy htop(f) of f vanishes, then f must have an invariant circle of rotation number u) e [po,Pi]-
47/ Anti-lntegrable Limit MacKay & Meiss (1992) explore the existence of Aubry-Mather sets (as well as many other possible configurations) close to the anti-integrable limit, where the potential of a standard like map becomes all powerful. Consider a family Fe of symplectic twist maps of T*Tn given by the generating functions: Se(q,Q) =
eT(q,Q)+V(q)
where, for simplicity, we can assume
T(q,Q) =
\(Q-q)2,
although more general T"s can be considered. As usual, orbits of Fe correspond to solutions of
198
(47.1)
9: GENERALIZATIONS OF AUBRY-MATHER
d2Se{qk-n
= 0.
Even though _F0 is not defined, it is perfectly acceptable to set e = 0 in Formula (47.1). This is called the anti-integrable limit, a notion that seems to have appeared independently in Aubry & Abramovici (1990) and Tangerman & Veerman (1991). The force of this concept is that the solutions of (47.1) at e = 0 are perfectly understood: they are simply allocations of qk to one of the critical points of V: (47.1) is just dV(qk) = 0 when e = 0. If V is a Morse function, it has finitely many critical points modulo 2 n and they are all nondegenerate. This has the following consequence: Theorem 47.1 (MacKay-Meiss) Any solution q(0) of (47,1) for e = 0 continues to a solution q(e) when e is small. Proof. Rewrite the infinite system of equations (47.1) in the form G(e,q) = 0 where G : IR x X -> (1R")Z is given by G(e,q) = d2Se(qk_1,qk) + ftSe(9fc,gk+1) and X is the Banach space of sequences such that supfc \\qk — qf(.(0)|| < oo. The Implicit Function Theorem on Banach spaces (see Lang (1983)) applies here to find, for small e, a q(e) such that G(e, q(e)) = 0 as long as f£(0, q(0)) is invertible. But this is indeed the case since rIC ^(0,q(0))k = V"(qk(0)) so that f | ( 0 , q(0)) is an infinite block diagonal matrix with the n x n diagonal blocks V"(qk(Q>)) all invertible and uniformly bounded. Indeed these matrices are chosen among afiniteset, since qk(Q) is necessarily a critical point of V, of which there arefinitelymany mod Z n , by the assumption that V is Morse.
•
One can simultaneously continue compact sets of stationary solutions from the antiintegrable limit. Such sets can be quite complicated, since the set of all stationary configurations of the anti-integrable limit can be seen as a shift on as many symbols as there are critical points. In particular, one can find invariant Cantor sets for Fe. One can also get orbits with all rotation vectors u G IR™. To do so, consider the anti-integrable stationary solution q(0) which is such that q0(0) is at some arbitrarily chosen critical point of V and
48. Minimal Measures
199
qk(0) = [ku] + qo(0),
where [ku] is the integer part of the vector kw. Each 9^(0) is thus on the same critical point as qQ(0), but translated by the integer vector [u]. Since |gfc(0) — ku\ < y/n, q(0) has rotation vector w. Now use Theorem 47.1 to continue this to an orbit of Fe with rotation vector u>. One can also continue simultaneously all anti-integrable solutions as the above with rotation vectors in a compact set: they themselves form a compact set. Even though this seems almost too easy, the anti-integrable limit is a very useful concept in order to understand the spectrum of all possible dynamics of symplectic twist maps. It is fair to say that, to this date, the least understood cases are those that are neither close to integrable nor to anti-integrable.
4 8 / Mather's Theory of Minimal Measures We now come to Mather's theory of existence and regularity of minimizers. This theory is quite general: it covers a wide class of convex Lagrangian systems on tangent bundles of arbitrary compact manifolds. Note that similar, but less developed theories were created by Bangert (1989) in the setting of minimal geodesies on compact manifolds and Katok (1992) in the setting of perturbations of integrable symplectic twist maps. There is no doubt that Mather's theory could be worked out for general symplectic twist maps. Even now, the correspondences between Lagrangian systems and symplectic twist maps given in Chapter 7 (see in particular the Bialy-Polterovitch suspension theorem 41.1) should allow an ample transfer of Mather's results to the symplectic twist maps case. The lesson we get from Mather's work is that, yes, minimizers in general manifolds behave very much like those on the circle (the realm of the classical Aubry-Mather theory), in that they satisfy a graph property. The bad news is that minimizers may be much scarcer than in the circle case: Hedlund (1932) had already constructed a Riemannian metric on T 3 (a setting encompassed by Mather's) which is very small along 3 non intersecting geodesies which generate Hi(T3). All other minimizers of a certain length are then bound to spend a good portion of their time close to these geodesies. In particular, these three geodesies are the only possible recurrent minimizers. This limits the possible rotation vectors of minimizers to these three directions only. Bangert (1989) (geodesic setting) and Mather (1991b) (Lagrangian setting) show that, in a precise sense, this is the worst case scenario:
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there should be at least as many rotation directions represented by minimizers as there are dimensions in Hi(M,IR). And, to end on an optimistic note, Levi (1997) constructs, in this worst case scenario of Hedlund's example, "shadowing" locally minimizing orbits that spend any prescribed proportion of time close to each of the minimizers. In particular, he constructs locally minimizing orbits of all rotation vectors. In order to introduce Mather's theory of minimal measures, we need to define the notion of Lagrangian minimizers, similar to that of Aubry minimizers for twist maps. In passing, we note the connection between the notion of action minimizing and hyperbolicity. A*. Lagrangian Minimizers
Throughout this section and the next, we consider time-periodic Lagrangian systems determined by a C2-Lagrangian functions L : TM x S 1 —> 1R, where M is a compact manifold given a Riemannian metric g. Remember (see Appendix 1 and Chapter 7) that extremals of the action
A{i)= f L(y,j,t)dt satisfies the Euler-Lagrange equations ^ f § — ff = 0. Using local coordinates these equations yield a first order time-periodic differential equation on TM, and thus in the standard way, a vector field on TM xS 1 . This can be viewed as the Hamiltonian vector field corresponding to the Lagrangian system, pulled back to TM by the Legendre transformation. Since TM x S 1 is not compact it is possible that trajectories of this vector field are not defined for all time in IR and thus do notfittogether to give a global flow (i.e. an IR-action). When the flow does exist, it is called the Euler-Lagrange (or E-L) flow. The following quite general hypotheses are the setting of Mather (1991b). Mather's Hypotheses.
L is a C 2 function L : TM x S 1 -> IR that satisfies: (a) Convexity: %^ is positive definite. (b) Completeness: The Euler-Lagrange (global) flow determined by L exists. (c) Superlinear: L V?,' t ' —> oo when \\v\\ —> +oo.
48. Minimal Measures
201
Mather's Hypotheses are satisfied by mechanical Lagrangians, i.e. those of the form L(x,v,t) =
±\\v\\2-V(x,t),
where the norm is taken with respect to any Riemannian metric on the manifold. (In fact, one may allow the norm to vary with time, under some conditions, see Mane (1991), page 44). Minimizers. We know that, for twist maps, orbits on Aubry-Mather sets are minimizers in the sense of Aubry. We have also seen in Chapter 6 that orbits on KAM tori are minimizers for symplectic twist maps . These are natural reasons to look for minimizers in convex Lagrangian systems. Lagrangian minimizers are defined in a way analogous to the discrete case. If M is a covering space of M (see Appendix 2), L lifts to a real valued function (also called L) defined on TM x S1. A curve segment 7 : [a, b] —* M is called a M-minimizing segment or an M-minimizer if it minimizes the action among all absolutely continuous curves (3 : [a, b] —• M which have the same endpoints as 7. A curve 7 : IR —> M is also called a minimizer if 7| is a minimizer for all [a, b] C IR. When the domain of definition of a curve is not explicitly given it is assumed to be IR. In practice, the two main covering spaces that we will consider are the universal cover (in next section) and the universal abelian cover (in this section, see Appendix 2 for the definitions of these covering spaces). A fundamental theorem of Tonelli (see Mather (1991b) or Mane (1991)) implies that if L satisfies Mather's Hypotheses, then given a < b and two distinct points xa,Xb G M there always is a minimizer 7 with 7(0) = xa and 7(6) = Xb- Moreover such a 7 is automatically C 2 and satisfies the Euler-Lagrange equations (this uses the completeness of the E-L flow). Hence its differential dj(t) = (•y{t), j(t)) yields a solution (dj(t), t) of the E-L flow. B*. Ergodic Theory
Most of the material in this subsection can be found in Hasselblat & Katok (1995), where it is thoroughly developped. We start by motivating this theory by the following trivial remark: if F is a map of T*Tn and <j>(z) = ir(F(z)) - ir(z) (TT : T*T n -> Tn is the canonical projection) then, when it exists: hm - > d>(Fk(z)) = hm - i rnoon'-' fc=l
n—00
^—^ n
v
v
" =
pF(z), fry
j,
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9: GENERALIZATIONS OF AUBRY-MATHER
the rotation vector of z under F. The expression lirrin^oo £ Y%=i 4>(Fk(z)) f° r a general continuous function
exists for p — a.e. z. Moreover, if p(X) < oo, Jx cprdp = fx <j>dp. Remember that a Borel measure on a topological space is one whose sigma-algebra of measurable sets is generated by the open sets. The above theorem actually does not require the measure to be Borel, but we will assume it for the rest of this section. That F is measure preserving means fi{F~1 (A)) = n{A) for any Borel subset of X. An immediate corollary of Birkhoff's theorem is (one needs to compactify T*Tn with a point at oo): Corollary 48.2 LetF be a volume preserving map (eg. symplectic) ofTn. The rotation vector PF is defined on a subset of full Lebesgue measure ofT*Tn. It turns out that the Lebesgue measure is only one of the many measures that a symplectic map F preserves. Take z e T * T " to be a iV-periodic point of F, for instance, and let: 1
N
where the Dirac measure 5W is the (Borel) probability measure concentrated at the point w ( 5W(A) is 1 if w e A and it is 0 if not). Since SFk^(F~1(A)) = 5Fk+i^(A), 77 is invariant under F. One of the many differences between 77 and the Lebesgue measure is their supports. In general, the support of a Borel measure p, is defined as: Supp p — {z e X I p(U) > 0 whenever z eU, U open } Clearly, the support of the measure n constructed above is the orbit of the periodic point z, whereas the support of the Lebesgue measure is all of T*Tn. Hence, the support of
48. Minimal Measures
203
invariant measures is another way to conceptualize invariant sets. Let F : X —> X be continuous. Then the support of any F-invariant Borel measure p is closed, F-invariant and its complement has zero /x-measure. If p(X) < oo, Poincare's Recurrence Theorem implies that Supp p is contained in the set of F—recurrent points. In fact, if a point z is in Supp p then its w-limit set UJ{Z) is included in Supp p. Hence, to find recurrent orbits in a dynamical system, as we have been doing in this book, one can look for (supports of) invariant measures. Coming back to rotation vectors, and the measure r? supported on a periodic orbit, the rotation vector PF{Z) not only exist r] — a.e., but it is constant on Supp 77. In fact, it can easily be checked that the time average 4>T is constant on Supp ?? for any function 4> G L 1 (T*T n , 77): the measure 77 is ergodic. Definition 48.3 An F-invariant probability measure /iona space X is ergodic if it satisfies one of the following equivalent properties: 1) Every F-invariant set has p measure 0 or 1. 2) If <j> 6 L1 (X, p) is F-invariant then
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as supports of ergodic measures, pull-back of measures on S 1 invariant under circle diffeomorphisms. Indeed, take the set n(M) in Theorem 12.8: it is the omega limit set i?(T) for a circle diffeomorphism T. Now, pick x € J2(T) and take the weak* limit fi of the probability measures fi^ = 2N-I 2 _ j v ^Tk(xy the limiting measure p, defines an ergodic measure for T, and its pull back by 7r is ergodic for F with support the Aubry-Mather set M [the weak* limit is defined by pn A p, iff / x <j>p,n —> J"x 0 ^ for all continuous 0]. Hence our main objects of study in this book, periodic orbits and Aubry-Mather sets, are all supports of ergodic probability measures, part of the larger set MF of all F-invariant Borel probability measures. If X is a compact metric space, it turns out that the set M of all Borel probability measures is convex and compact under the weak* topology. Moreover MF itself is a compact and convex subset of M for this topology. A theorem of convex analysis (Krei-Millman) says that MF is then in the convex hull of its extreme points : those p, e MF which cannot be written as tp\ + (1 — t)p2 for two distinct p\,pi
€ MF- Finally, the extreme points are all
ergodic measures. We will see in the next subsection that there is a strong correspondence between the (strict) convexity of a certain projection of MF and the Aubry-Mather theorem. As we will see in next section, Mather (1991b), (1993) considers measures that are invariant under the Euler-Lagrange (E-L) flow instead of a symplectic twist map . In the light of the suspension theorem of Bialy-Polterovitch (Chapter Chapter 7), his setting encompasses a large class of symplectic twist maps. All the statements that we made above are valid for E-L flows on TTn provided one compactifies TT™ (as Mather does) in order to use the compactness of the space of E-L-invariant probability measures.
C*. Minimal Measures For a more detailed exposition the reader is urged to consult Mather (1991b), Mane (1991). There is also a very nice survey of this theory in Mather (1993). Invariant Measures, their Action and Rotation Vector. Given a E-L invariant probability measure with compact support p on TM x S 1 , one can define its rotation vector l
as follows: let Pi, fo,..., Pn be a basis of H (M) forms with [Ai] = Pi in DeRham cohomology. 17
(17)
p(fi)
and let A i , . . . , A„ be closed one-
We refer the reader uncomfortable with
When homology and cohomology coefficients are unspecified they are assumed to be IR, so the notation Hi(M) means Hi(M;TR), etc.
48. Minimal Measures
205
(co)homology to Appendix 2 and urge her/him to read through this section thinking of the case M = T n , taking [A*] = [dxi], as a basis for H 1(Tn) ~ IRn, where ( n , . . . , xn) are angular coordinates on T*Tn. Define the ith component of the rotation vector p(p) as i(M) = /
-MA*-
Note that this integral makes sense when one looks at A, as inducing a function from TM x S1 to IR byfirstprojecting TM x S1 onto TM, and then treating the form as a function on TM that is linear on fibers. The rotation vector does depend on the choice of basis /%, but because these 1-forms are closed, Pi(p) does not depend on the choice of representative Aj with [Aj] = /?;. Since the rotation vector is dual to forms (with pairing < A,/x > = J Ad//), it can be viewed as an element of Hi(M). In the case M = T n , if 7 (0) is a generic point of an ergodic measure p, the natural definition of rotation vector of a curve 7 coincides with p(p): Pi(7) =
lim - 5 — b-a—>oo
-—'- =
0—a
lim
/
b-a—too 0 — a Jd-y\,a
dxi = b]
dxidp = pi{p) J
where 7 is a lift of 7 to IR" and the third equality uses the Ergodic Theorem (again, dxt is seen as a function TM x S 1 - * IR). This prompts the following formula for the (ith coordinate of the) rotation vector of a curve 7 : IR —> M for a general manifold M: Pi (7) =
lim
7—- /
b-a—too O — a
Aj,
d Jj. Tl[a,6]
if the limit exists. As before, if 7(0) is a generic point for an ergodic measure p, p(j) exists and coincides with p(p) (and this is not necessarily the case for non ergodic measures). Next we define the average action of a E-L invariant probability on TM x S 1 :
m = JLdp, i.e. the space average of L which, when p is ergodic, equals the time average along p,-a.e. orbit 7: 1 fb A(p) = lim / LH, j)dt, p - a.e. orbit 7. 6-a-»oo 0 — a
Ja
The Set of Minimal Measures and the Function /?. The set of E-L invariant probability measures, denoted byJWi, is a convex set in the vector space of all measures, as we have
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seen in the previous subsection (it is also compact for the weak* topology if, as Mather does, one compactifies TM) and the extreme points of ML are the ergodic measures (see Mane (1987)). Consider the map ML f
-> Hi (M) x IR given by: H
(P(MM(A*))-
This map is trivially linear and hence maps ML to a convex set UL whose extreme points are images of extreme points of ML, »-e. images of ergodic measures. Mather shows, by taking limits of measures supported on long minimizers representing rational homology classes, that for each u>, there exists an invariant (but not necessarily ergodic) measure p such that p(p) = u and A(p) < ooS18^ Since L is bounded below, the action coordinate is bounded below on UL. Hence we can define a map (3 : Hi (M) —> H by (3{UJ) = ini{A(p)
| p e ML,p(p)
= w},
which is bounded below and convex: the graph of f3 is the boundary of UL- We say that a probability measure p € ML is a minimal measure if the point (p(p),A(p))
is on
the graph of /?. In other words, a measure p with p(p) = to is a minimal measure iff it minimizes the average action subject to the constraint that it have rotation vector w. By construction, an extreme point {LJ,(3{U>)) of graph(P) corresponds to at least one minimal ergodic measure of rotation vector u. It turns out that if p is minimal, /U-a.e. orbit lifts to a E-L minimizer in the universal abelian cover M of M (whose deck transformation group is H\(M; Z)/torsion, see Appendix 2). Conversely, if p is an ergodic probability measure whose support consists of M-minimizers, then p is a minimal measure. Hence, each time we prove the existence of an extreme point (w, P(OJ)), we find at least one recurrent orbit of rotation vector u> which is a
M-minimizer.
Another important property of/3 is that it is superlinear, i.e. ^ r —> oowhen||a;|| —> co. We motivate this in the simple case where L = \ ||x|| 2 - V(x) and ||-|| comes from f le Euclidean metric on the torus. If p is any invariant probability measure, then 18
The impatient reader may be tempted to proclaim, from this fact, the existence of orbits of all rotation vectors. Alas, as we noted in Remark 48.4, we can guarantee that the rotation vector of orbits in the support of a measure p are equal to p(p) only when p is ergodic
48. Minimal Measures
A(j*)= ILdn> ^
> -^l
f
207
l^--Vmax\dn
i dfi\ - Vmax
= ^\p(lJ-)\2
-
Vmax,
where we used the Cauchy-Schwartz inequality for the second inequality. We see that in this particular but important case, /3 grows at least quadratically with the rotation vector. The superlinearity of ft implies the existence of many extreme points for graph(P) (although in most cases still too few, as we will see in the next subsection), as we explain now. Countably Many Minimal Measures: Is This Enough? As the boundary of the convex set UL, the graph of /3 is made of flat faces, or linear domains. Each of these linear domains, which we denote by Sc, is a convex set, intersection of UL with a supporting hyperplane of UL of "slope" c. [Since c acts linearly on homology classes w to give the equation c-ui = a of the supporting hyperplane, it can be seen as an element of first cohomology.] Let Xc be the projection on Hi{M) of Sc. The sets Xc are convex domains which tile the space .Hi(M). Now the growth condition on /? implies that its graph cannot have a linear domain going to infinity: the sets Xc must be compact. Extreme points of Xc are projections of extreme points of Sc, themselves extreme points of UL- Hence there are infinitely many such extreme points, and infinitely many outside any compact set. Their convex hull is Hi (M), and in particular, they must span H\ (M) as a vector space. Since these extreme points are the rotation vectors of minimal ergodic measures, we have found: Theorem 48.5 There exist at least countably many minimal ergodic measures and at least n = dim H\ (M) of them with distinct rotation directions. In particular there are at least n rotation directions represented by minimal measures for a E-L flow on T*Tn. We will see in Hedlund's example that this lower bound is attained by some systems. Finally, the generalized Mather sets are defined as Mc = Support(Mc),
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where M.c is the set of all minimal measures whose rotation vectors lies in Xc. Let 7r : TM x S 1 —> M x S 1 denote the projection. The main result in Mather (1991b) is the following theorem: Theorem 48.6 (Mather's Lipschitz Graph Theorem) For all c G Hl{M), compact, non-empty subset ofTM
1
Mc is a
x S . The restriction O/TT to Mc is infective.
The
inverse mapping 7r _1 : 7r(Mc) —> Mc is Lipschitz. In the case M = T n , Mather proves that, when they exist, KAM tori coincide with the sets Mc (see also Katok (1992) for some related results in the symplectic twist maps context). The proof of the Graph Theorem (see Mather (1991b) or Mane (1991)), which is quite involved, uses a curve shortening argument: if two minimizing curves in n(Mc) were too close to crossing transversally, one could "cut corners" and, because of recurrence, construct a closed curve in Mc with lesser action than the minimal action of measures in Mc, a contradiction. This argument by surgery is reminiscent of the proof of Aubry's Fundamental Lemma in Chapter 2. The Autonomous Case. An important special case is that of autonomous systems (i.e. with time independent L). In this case, one can discard the time component and view Mc as a compact subset of TM. Mather's theorem implies that Mc is a Lipschitz graph for the projection ir : TM —> M. To see this, suppose that two curves x(t) and y(t) in 7r(Mc) have x
(0) — y(s) for some s. Mather's theorem rules out immediately the possibility that s is an integer, unless x = y is a periodic orbit. For a general s, consider
the curve z(t) = y(t + s). Then, z(t) = y(t + s) and, by time-invariance of the Lagrangian, (z(t), z(t)) is a solution of the E-L flow. It has same average action and rotation vector as (y, y) and hence it is also in Mc. But then z(0) = x(0) is impossible, by Mather's theorem, unless i(0) = y(s) = x(0) and thus, by uniqueness of solutions of ODEs, x(t) — y(t + s). The Symplectic Twist Map Case. By using Theorem 41.1, one can translate the results of Mather to the realm of symplectic twist maps (see Exercise 48.7) and deduce the existence of many invariant sets that are graphs over the base and are made of minimizers. As noted before, one could also redo all of Mather's theory in the setting of symplectic twist maps (see Katok (1992), who considered the near integrable case).
48. Minimal Measures
209
D*. Examples and Counterexamples
Recovering Past Results. When Mather's function /? (see previous section) is strictly convex, each point on graph(j3) is an extreme point and there are ergodic minimal measures (and hence minimal orbits) of all rotation vectors. One can prove that this is true when M = S1, and Mather (1991b) shows how his Lipschitz Graph Theorem implies the classical Aubry-Mather Theorem, by taking a E-L flow that suspends the twist map. The fact that Mc is a graph nicely translates into the fact that orbits in an Aubry-Mather set are cyclically ordered: as pointed out by Hall (1984), the CO property corresponds to trivial braiding of the suspended orbit, itself guaranteed by the graph property. The graph of (3 is also strictly convex when L is a Riemannian metric on T 2 , and hence there are minimal geodesies of all rotation vectors for any metric on the torus. This was known by Hedlund (1932), who had basically worked out the same results as Aubry and Mather in that setting, albeit in a different language. [See Bangert (1988) for a unified approach of the two theories]. Hence one could hope, as a generalization of the AubryMather theorem, that /3 is strictly convex for any Lagrangian systems satisfying Mather's hypotheses. This statement is false as we will see in the following examples. Examples of Gaps in the Rotation Vector Spectrum of Minimizers for Lagrangian on
T 2 . Take L : TT2 -> 1 , given by L(x, x) = \\x - xf where X is a vector field on T 2 . The integral curves x of X are automatically E-L minimizers since L s O o n these curves. Mane (1991) chooses the vector field X to be a (constant) vector field of irrational slope multiplied by a carefully chosen function on the torus which is zero at exactly one point q. The integral flow of X has the rest point q(t) = q, and all the other solutions are dense on the torus. The flow of X (and its lift to TT 2 by the differential) has exactly two ergodic measures: one is the Dirac measure supported on (q, 0), with zero rotation vector, the other is equivalent to the Lebesgue measure on T 2 and has nonzero rotation vector, say to. Mane checks that /3_1(0) (trivially always an Xc) is the interval {A w | A e [0,1]}, and that no ergodic measure has a rotation vectors strictly inside this interval. Thus the Mather set M0 is the union of the supports of the two above measures. Boyland & Gole (1996a) give an example of an autonomous mechanical Lagrangian on T which displays a similar phenomenon, although we also show in that paper that all autonomous Lagrangian systems satisfying Mather's Hypothesis do have minimizers of all
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rotation directions. We also give in this paper a very precise description of the /? function for such systems and show that the support of minimal ergodic measures have to be either a point, a closed curve, a suspension of a Denjoy Cantor set or the whole torus. Hedlund-Bangert's Counterexamples. Consider in IR3 the three nonintersecting lines given by the z-axis, the y-axis translated by (0,0,1/2) and the z-axis translated by (1/2,1/2,0). Construct a Z 3 - lattice of nonintersecting lines by translating each one of these three lines by integer vectors. Take a metric in IR3 which is the Euclidean metric everywhere except in small, nonintersecting tubes around each of the axes in the lattice. In these tubes, multiply the Euclidean metric by a positive function A which is 1 on the boundary and attains its (arbitrarily small) minimum along the points in the center of the tubes, i.e. at the axes of the lattice. Because the construction is W? periodic, this metric induces a Riemannian metric on T 3 . One can show (Bangert (1989)), if A is taken sufficiently small, that a minimal geodesic (which is a E-L minimizer in our context) can make at most three jumps between tubes. In particular, a recurrent E-L minimizer has to be one of the three disjoint periodic orbits which are the projection of the axes of the lattice. Thus there are only three rotation directions that minimizers can take in this example, or six if one counts positive and negative orientations. In terms of Mather's theory, the level sets of the function /? in I t 3 = Hi (T3) are octahedrons with vertices (±a, 0,0), (0, ±a, 0), (0,0, ±a) (we assume here that the function A is the same around each of the tubes). Since we are in the case of a metric, one can check that (3 is quadratic when restricted to a line through the origin (a minimizer of rotation vector aw is a reparameterization of a minimizer of rotation w). Hence a set Sc is either a face, an edge or a vertex of some level set {/? = b}, and the corresponding Mc is, respectively, the union of three, two (parameterized at same speed) or one of the minimal periodic orbits one gets by projecting the disjoint axes. Note that, instead of the function /? of Mather, Bangert uses the stable norm. Mather's function (3 is a generalization of that norm. Levi's Counter-Counterexample. It is important to note that the nonexistence of minimizers of a certain rotation vector w does not mean that there are no orbits of the E-L flow that have rotation vector u>. For example, Levi (1997) has shown the existence of orbits of all rotation vectors in the Hedlund example. He constructs, using some broken geodesic methods, local minimizers shadowing any curve made of segments (of sufficient length) of
49. Minimal Measures
211
the minimizing axes and jumps between the axes. This makes for extremely rich, chaotic dynamics.
Further Developments of Mather's Theory of Minimal Measures. As of this writing, a number of mathematicians are very active in research on minimizers in convex Lagrangian. Foremost is the group of young and talented researchers which formed in South America and Mexico around former students of the late Mane. This group has solved many problems posed by Mane soon before his death (see Mane (1996a) and Mane (1996b)), mainly on autonomous Lagrangian systems. Look for the names of Carneiro, Contreras, Delgado, Iturriaga, Paternain, Sanchez-Morgado and more. Recently this group, together with K.F. Siburg, has made strides in bridging this theory with the more geometric point of view of symplectic topologists: the minimal action function is related to symplectic capacities andHofer's energy (see Siburg (1998), Iturriaga & Sanchez-Morgado (2000)). Fathi (1997) recovers some of Mather's theory and creates ties with the KAM theory using super and sub solutions of Hamilton-Jacobi's equations. Contreras (2000) and Contreras et al. (2000), continue this work. As noted in Chapter 6, Mather has recently used the theory of minimal measures, together with some hyperbolic methods to prove the existence of unbounded orbits in Lagrangian systems, a great leap in the general problem of the so-called Arnold diffusion (see Delshams, de la Llave & Seara (2000)). Hence the theory has gone far beyond the task of generalizing the Aubry-Mather theory: it has given mathematicians new tools to study the global dynamical properties of Lagrangian systems.
Exercise. 48.7 Find hypotheses on the generating function of an symplectic twist map F which translate to Mather's hypotheses for the Lagrangian that suspends F (Hint. You may want to include Bialy and Polterovitch's conditions of Theorem 41.1 for F to have a convex suspension. Note that completeness of theflowis for free: F is defined everywhere.)
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49.* The Case Of Hyperbolic Manifolds
We start this section with another counterexample to the strict convexity of Mather's (3 function, and hence to the existence of orbits of all rotation vectors. The setting is that of a metric on the two-holed torus, the simplest example of a compact hyperbolic manifold. However, we finish the section on a positive note, by quoting a result of Boyland & Gole (1996b), in which we introduce another definition of rotation vector suited to hyperbolic manifolds and show the existence of minimal orbits of all rotation directions for a large class of Lagrangian systems on hyperbolic manifolds.
A*. Hyperbolic Counterexample
Take the metric of constant negative curvature on the surface of genus 2 (the two-holed torus) which has a long neck between the two holes (see Figure 49.1). A minimizer here is a minimizing geodesic for the hyperbolic metric. With a and b as shown, the minimal measure for the homology class [a] + [b] is a linear combination of the ergodic measures supported on Fa and i"),, where ra and T& are the closed geodesies in the homotopy classes of a and b, respectively. Indeed, ra and i~i, must "go around" the same holes as a and b, and any closed curve that crosses the neck will be longer than the sum of the lengths of ra and Tj. Hence ([a] + [b], /?([a] + [6])) cannot be an extreme point of graph((3).
Fig. 49.1. The surface of genus two and the loops a and b. No minimal measure with rotation vector [a] + [b] can have support passing through the long neck. In particular, a curve in the homotopy class of c cannot yield a minimizer in the abelian cover.
49. Hyperbolic Manifolds
213
B*. All Rotation Directions in Hyperbolic Manifolds As the previous example shows, the notion of minimizers in the abelian cover is maybe too restrictive, as it rules out many geodesies. Instead of working on the universal abelian cover, we work in the universal cover and define minimizers and rotation vectors with respect to that cover. All manifolds of dimension n which admit a hyperbolic metric of constant negative curvature have the Poincare n-disk as universal covering space Hi". Hence a hyperbolic manifold M is the quotient Mn/?ti(M)
where 7Ti(M) acts on H " as the group of deck
transformations. To visualize H n , assume n = 2, which covers any orientable surface of genus greater or equal to two. One model for H 2 is the usual Euclidean unit disk which is given the hyperbolic metric 1 ^ 3 .t+ y a\ • The ratio between the corresponding hyperbolic distance and the euclidean one tends exponentially to oo as points approach the boundary of the disk. Geodesies for the hyperbolic metric are arcs of (Euclidean) circles perpendicular to the boundary dJH2 of the disk. The minimizers we consider in this section lift to curves in the universal cover which minimize the action between any two of their points. We also assume that the Lagrangian L satisfies Mather's Hypotheses (time periodic C 2 function with (a) fiber convexity, (b) completeness of the E-L flow) except that we replace his condition (c) of superlinearity by one of superquadraticity: (c') superquadraticity. There exists a C > 0 such that L(x, v,t) >C \\v\\2. This, again, is satisfied by mechanical systems. Note that, without loss of generality, one can assume the Lagrangian L to be positive: being convex, it is bounded below, and adding a constant to L does not change the E-L solutions. We now state the two theorems that appear in Boyland & Gole (1996b). The first one finds minimizing solutions near any given geodesic: Theorem 49.1 (Boyland-Gole) Let (M, g) be a closed hyperbolic manifold.
Given a
Lagrangian L which satisfies Hypotheses (a), (b), (c'), there are sequences
ki,Ki,Ti
in 1R
depending only on L, with ki increasing to infinity, such that, for any hyper-
bolic geodesic To C H n = M (for the lifted metric), there are minimizers M with dist(~fi,F0) whenever d —
ji : IR —•
< m, 7i(±oo) = r 0 ( ± o o ) , and ki < 3 3 S disi(7 i (d),7 i (c)) < fci+1
c>Ti.
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The following theorem can be viewed as a (weak) generalization of the Aubry-Mather theorem for convex Lagrangian system on compact hyperbolic manifolds. Remember that, since the geodesic flow is the solution of the autonomous Lagrangian system with corresponding Hamiltonian \ ||p|| 2 , the energy levels ||v|| = c are all invariant sphere bundles that foliate the tangent bundle. In the case of M = S 1 with the Euclidean metric, these energy levels are pairs of flat invariant circles of the completely integrable map. Similarly to the Aubry-Mather theorem which states that traces of these invariant circles remain in the guises of invariant circles or Cantor sets whose dynamics is (semi)conjugate to circle homeomorphisms, the following theorem proves the existence of E-L invariant sets that are semiconjugate to the geodesic flow on these sphere bundles. The reason that we call this a "weak" generalization of the Aubry-Mather theorem is that we can only guarantee the existence countably many of these E-L invariant sets. Hence the geodesic flow is "weakly" topologically stable. Theorem 49.2 (Boyland-Gole) Let (M,g) be a closed hyperbolic manifold with geodesic flow gt- Given a Lagrangian L which satisfies Hypotheses (a), (b), (c') with E-L flow (f>t, there exists sequences ki and Tt with ki increasing to infinity, and a family of compact, 4>t-invariant sets Xi C TM so that for all i, {Xi,4>t) is semiconjugate to {T\M,gt) and ki < j;dist(
49. Hyperbolic Manifolds
215
geodesic r0 that goes to the first as t —> — oo, to the other one as t —• +00. We can declare a curve 7 to be asymptotically parallel to F0 iffy and To have same endpoints. This will insure that points of 7 are always at a bounded hyperbolic distance from r0. We also declare that the rotation vector exists ijf 7 has the same endpoints at ±00 as a geodesic i~o, and if the average \p{y)\ of ||7|| over t G IR exists, and we define the rotation vector to be the pair p(y) = (Jo, |p(7)|) (average direction and average speed). In that language, Theorem 49.1 states that, given any geodesic f, there are infinitely many E-L minimizers with r as a rotation direction. The naive definition of rotation vector that we just outlined has some major flaws: 1. p(y) (if it exists) does not belong to a linear space. 2. Two lifts of the same curve will have different rotation vectors. 3. Rotation direction is not constant p — a.e. for many ergodic measures for the geodesic flow. To remedy that, let -K\ (M), seen as deck transformation group, act on geodesies in IH2 and declare that two geodesies are parallel iff they belong to the closure of the same ni (M)orbit (of geodesies). Consider the set of tangent vectors at all points of all the geodesies in the closure of a TTI (M) orbit. This forms a closed subset of the unit tangent bundle of IH2. The projection by the differential of the covering map of this set on the unit tangent bundle of M is the support of a measure p which is invariant under the geodesicflow.Because of this, Boyland (1996) defines the rotation direction of a curve to be a measure invariant under the geodesic flow, weak* limit of measures supported by geodesies joining two points of the curve. This rotation vector being defined through ergodic theory, it is constant p — a.e. for any E-L ergodic p. Theorem 49.2 implies the existence of minimizer of all rotation directions, in this new, "homotopy", sense of the word. Note that there are many more such "homotopy" directions than there are "homology" directions. For instance the "long neck" metric of Figure 49.1 has no homology minimizer with rotation direction c, as argued in the previous subsection, but it will have infinitely many homotopy minimizers with that direction. On the negative side, the counterexamples of Mane (1991) and Boyland & Gole (1996a) on T 2 as reviewed in the previous section, where gaps in the rotation spectrum are found, probably have counterparts on hyperbolic manifolds, even with our new definition of rotation vector. Thus, we think there is little chance to prove the existence of global minimizers of
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all rotation vectors, even on these manifolds. However there should be good chances to find local minimizers of all rotation vectors, as we discuss in the next section. What we hoped to show in this section is that the universal cover is a more natural setting than the abelian universal cover when studying Lagrangian minimizers on hyperbolic manifolds.
50/ Concluding Remarks What, in the end, are the chances of finding orbits of all rotation vectors for symplectic twist maps or Lagrangian system, in say, T*T n ? Previous attempts at this problem yielded incomplete results. Bernstein & Katok (1987) "almost" found, for minimizing periodic orbits of symplectic twist maps close to integrable, some uniform modulus of continuity, which they hoped would unable them to take limits and get orbits with the limiting rotation vectors. In my thesis, I hoped that proving some regularity of the ghost tori (invariant set for the gradient flow of the periodic action) might enable one to do the same. This is how ghost circles came about. One thing is clear: one cannot hope for global minimizers to achieve all possible rotation vectors. However, the shadowing methods to construct local minimizers of all rotation vectors of Levi (1997) on the Hedlund counterexamples indicate a possible approach to the general case. The recent work of Mather on existence of unbounded orbits (seeDelshams, de la Llave & Seara (2000), Section 48 and the end of Chapter 6), also shows that, for general systems, hyperbolic and variational techniques can combine powerfully to construct orbits shadowing successive minimizers. One possibility to attack the problem of existence of orbits of all rotation vector would be to try to construct, in a manner analogous to Levi (1997), orbits shadowing the different supports of the ergodic measures which are extreme points of one generalized Mather set Mc- Doing so, one may manage to "fill in" the corresponding set of rotation vectors Xc with rotation vectors of actual orbits, may they be local minimizers.
10 GENERATING PHASES AND SYMPLECTIC TOPOLOGY
In Appendix 1, Section 5 8, we remark that the differential of a function W : M -+T*M gives rise to the Lagrangian submanifold dW(M) of T*M. As a generalization of this fact, one can construct Lagrangian submanifolds of T*M as symplectic reductions of graphs of differentials of generating phases, which are functions on vector bundles over M. Generating phases are the common geometric framework to the different discrete variational methods in Hamiltonian systems, including the method developed in this book. Applications of generating phases range from the search for periodic orbits to the Maslov index, symplectic capacities and singularities theory. Generating phases are a viable alternative to the use of heavy functional analytic variational methods in symplectic topology. This chapter intends to be a basic introduction to generating phases. We first present Chaperon's method, which he used to give an alternate proof of the theorem of Conley & Zehnder (1983). This theorem, which solved a conjecture by Arnold on the minimum number of periodic points of Hamiltonian maps of T 2 n , is considered by many as the starting point of symplectic topology, in the sense that it implies that the C° closure of the set of symplectic diffeomorphisms is distinct from the set of volume preserving diffeomorphisms. We then survey the abstract structure of generating phase, highlighting the common geometric frame for the symplectic twist maps method and that of Chaperon (as well as others).
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51. Chaperon's Method And The Theorem Of Conley-Zehnder Chaperon (1984) introduced a method "du type geodesiques brisees" for finding periodic orbits of Hamiltonians which did not make use of a decomposition by symplectic twist maps. This method has been the basis of later work by eg. Laudenbach & Sikorav (1985), Sikorav (1986), and Viterbo (1992). A. A New Action Function Until now, we have studied exact symplectic maps that come equipped with a generating function due to the twist condition. The concept of generating function is more general than this, however: we now show how an exact symplectic map of IR 2n which is uniformly C1 close to Id may have another kind of generating function. The small time t map of a large class of Hamiltonians satisfy this condition. Hence, the time one map of these Hamiltonians can be decomposed into maps that possess this kind of generating function, leading to a different variational setting for periodic orbits than the one we have used so far. Let F : M 2 n -> M2n
(q,p)-*(Q,P) be an exact symplectic diffeomorphism: (51.1)
PdQ - pdq = F*pdq - pdq = dS,
for some S : IR 2n —* H (remember that all symplectic diffeomorphism of IR 2n are in fact exact symplectic. We stress exact symplectic here in view of our later generalization to T*M). The following simple lemma is crucial here. Lemma 51.1 Let F : K 2 n —> M 2 n be an exact symplectic diffeomorphism. \\F — Id\\cl
Then, if
is small enough, the map --(q,p)-*(Q,p)
is a diffeomorphism Proof.
Q(q,p)
o/H2n .
is (uniformly) C 1 close to q and thus 0 is (uniformly) C 1 close to Id,
hence a diffeomorphism.
•
51. Chaperon's Method
219
Treating cj> as a change of coordinates, we can view S as a function of the variables Q,p. We now show how, in a way that is slightly different from the twist map case, F can be recovered from S. We define S(Q,p) = pq + S(Q,p),
where q = q{Q,p);
then (51.2)
dS = pdq + qdp + PdQ - pdq = PdQ + qdp
and thus S generates F, in the sense that:
P = g(QlP) (51.3)
°1
,-f(Q,P). Remark 51.2 Note that Id is not a symplectic twist map and thus it cannot be given a generating function in the twist map sense. One of the advantages of the present approach is that Id does have a generating function, which is S(Q,p)=PQ.
As an illustration,fixedpoints of F are given by the equations: OS „ p =~~ dQ= : t dS _ Q= dp = q which are equivalent to the following equation: d(S - pQ) = {P- p)dQ + {q- Q)dp = 0. Hence we have reduced the problem offindingfixedpoints of an exact symplectic diffeomorphism C 1 close to Id on K 2 n to the one of finding critical points for a real valued function. We now apply this method to give Hamiltonian maps of T 2 n afinitedimensional variational context. It can also be used for time one maps of Hamiltonians with compact
220
10: GENERATING PHASES
support in IR. n , or Hamiltonian maps that are C° close to Id in a compact symplectic manifold. Let H : H2™ x M be a C 2 function with variables (q, p, t). Assume H is 2 2 n periodic in the variables (q,p) (i.e., if is a function on T 2 n x M). As in Appendix 1, we denote by h\Q(q,p) = (q(t),p{t)) the solution of Hamilton's equations with initial conditions q(to) = q, p(to) = P- By assumption, /i' 0 can be seen as a Hamiltonian map on T2n. We know that h\0 is exact symplectic (see Theorem 59.7). Furthermore, by compactness of T2™, when \t — t0\ is small, h\0 is C1 close to Id (the Hamiltonian vector field of a C 2 function is C 1 , hence so is its flow). For \t — to\ small enough, we can apply Lemma 51.1 to get a generating function for h\Q. To make this argument global, we decompose h1 in smaller time maps: W-l
2
1
h1 = fiN-i o hJT2 o ... o hi o h£
(51.4)
and thus, for a large enough N, hl can be decomposed into N maps that satisfy Lemma 51.1. [The farther h1 is from Id, the bigger TV must be]. We can then apply the following proposition to h1: Proposition 51.3 Let F = F^ 0...0F1 where each Fk is exact symplectic in T*lRn, C1 close to Id, and has generating function Sk(Q,p)- The fixed points of F are in one to one correspondence with the critical points of : N
W {QltPi,
...,QN,pN)
= Y, Sk(Qk,Pk)
~
PkQk-i
fc=i
where we set Q0 = Q^. Proof. We will use the usual notation (Pk,Qk)
=
Fk(qk,pk)
where we know from (51.3) that Pk and qk are functions of Qk,pk. Then, using Equation (51.2),
51. Chaperon's Method
221
N
dW(Q,p)
= ^TPkdQk
+ qkdpk - pkdQk_x
fc=i JV-l
(51.5)
-
Qk^dpk
N
= ^2(pk~Pk+i)dQk
+ 5Z(gfc ~ Qk-i)dpk
fc=l fc=2
+ (PN - Pi)dQN
+ (qx -
QN)dp1.
This formula proves that {Q,p) is critical exactly when: Fk{qk,Pk) FN^NIPN)
= (
e { 1 , . . . ,N - 1},
(li,Pi),
that is, exactly when (q x , px) is a fixed point for F.
•
B. Interpretation Of W As Action Of Broken Geodesic When F is the time 1 map of some Hamiltonian and we decompose F as in (51.4), W has the interpretation of the action of a "broken" solution of the Hamiltonian equation. This is similar to the situation in Chapter 7. This time however, the jumps are both vertical and horizontal:
/', •'/
-
Pi • ' / /
PA-'/A
S
PfQu
tk
" ^
Y ^
tN s.
'k-l
Fig. 51.1. Interpretation of W as the action of a "broken" solution r, concatenation of the solution segments 7A, and "corners" in the t = tk planes. Each curve j {qk,pk,tk)
k
in Figure 51.1 is the unique solution of Hamilton's equations starting at
-l where tk = ^:NA and flowing for time 1/iV.
t
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10: GENERATING PHASES
Since Sk(Qk,pk) = Sk{qk,pk) +pkqk and Sk(qk,pk) 59.7), W measures the action of the broken solution F:
= f^ pdq - Hdt (see Theorem N
N
W(Ql,p1,...,QN,pN) (51.6)
= ^2pk{qk~Qk-i)
fc=i
C
+ J2 /
Pd<J-Hdt
/
fc=i- T* = / pdq — Hdt,
where we have used the fact that, on the "corner" segments, dt = 0, and on the vertical part of these corners, dq = 0. This is the definition given by Chaperon (1984) and (1989). C. The Conley-Zehnder Theorem
The following theorem solved a famous conjecture by Arnold (1978) in the case of the torus. It was hailed as the start of symplectic topology, as it shows that symplectic diffeomorphisms have dynamics necessarily different from that of general diffeomorphisms, or even volume preserving diffeomorphisms. The original proof of Conley & Zehnder (1983) also reduces the analysis to finite dimensions, but by truncating Fourier series of periodic orbits. Chaperon's proof avoids the functional analysis altogether. Theorem 51.4 (Conley-Zehnder) Let h1 be a Hamiltonian map o/T 2 n . Then h1 has at least 2n +1 distinct fixed points and at least 2 n of them if they all are nondegenerate. Proof. Let W be defined as in Proposition 51.3 for the decomposition of h1 into symplectic maps close to Id given by (51.4). We will show that W is equivalent to a g.p.q.i. on T 2 n , and hence, by Proposition 64.1, it has the prescribed number of critical points, corresponding to fixed points of h1. We refer the reader to Section 64 for the definition and properties of generating phases that are relevant here. We first note that W induces a function on (IR 2 ")^/^ 2 " where Z 2 n acts on (IR2™)^ by: (mq,mp).(Q1,p1,...
,QN,pN)
= (Qx +mq,p
+ mp,...
,QN+ mq,pN+
mp)
The fact that W is invariant under this action is most easily seen from (51.6). Indeed, since the Hamiltonian flow is a lift from one on T 2 n , the curve j k + (mq, mp, 0) is the solution between (qk + mq,pk +mp) and (Qk + mq,Pk + mp) starting at time &=LL of that flow. But
51. Chaperon's Method 223 /
pdq + Hdt =
J-fk+(mq,mp,0)
(p + mp)dq-Hdt
= mp(Qk-qk)
Jyk
+
pdq-Hdt J-fk
Hence the action of j k changes by mp(Qk - qk) under this transformation. On the other hand, under the same transformation, the sum J2k=i Pkilk ~ Qk-i) °f Formula (51.6) changes by J2k=i rnp(
an
(QN,PN)
d let x : E —• E be the bundle
diffeomorphism given by: X(QI,PI,--->QN'PN)
=
(al,bl,--->aN-l,bN-l,QN>PN)
where
ofc = Qk - Qk-i b
k
(Qo =
QN)
=pk~pN. 2n
In these new coordinates, the Z action only affects (QN, pN), so that \Vox~1 induces a function W on (lR 2n ) N ~ 1 x T 2 n . We now need to show that W is in fact a g.p.q.i. Define W0 (resp. V^o) to be the functions W (resp. W) obtained when setting the Hamiltonian to zero. Since Sk(Qk,pk) = pkQk in this case (see Remark 51.2), W0(Q,p) =J2k=iPk(Qk~ Qk-i) and hence a simple computation yields JV-l
W0(a,b,QN,pN)
= ) a
k
b
k
fc=i
which, as easily checked, is quadratic nondegenerate in thefiber.Finally, we show that J^ (W — Wo) is bounded, where v = (a, b). It is sufficient for this to check that d( W — Wo) is bounded. Using (51.5), we obtain: N
N
d(W - Wo) = Y,(PK - Pk+i)dQk + $ > * fc=i fc=i N
N
- ]CK?fc - Qk-i)dpk ~ 5](Pfe fc=i fc=i N
_
Pk+i)dQk
N
= ^2(lk ~ Qk)dPk + E ( fc=i fc=i
Qk-i)dpk
P
* - Pk)dQk
224
10: GENERATING PHASES
where we have set throughout Q0 =
QN>PN+I
= Pi- Since by definition (Qk, Pk) =
k
Fk (qk, Pk) where Fk = h^-i lifts a diffeomorphism of T , the coefficients of the above differential must be bounded. We can conclude by applying Proposition 64.1. • Remark 51.5 Since the lift of the orbits we find are closed, the orbits in T 2 n are contractible. In general, one cannot hope to find periodic orbits of different homotopy classes, as the example Ho = 0 shows. It would be interesting, however, to study the special properties of the set of rotation vectors that orbits of h1 may have, i.e., to find out if being Hamiltonian implies more properties on this set than those known for general diffeomorphisms of T 2 n .
52. Generating Phases And Symplectic Geometry We urge the reader to read Section 64, where we define generating phases as functions W : E —> H, where E is a vector bundle over the manifold M. We then give conditions under which lower estimates on the number of critical points of W can be obtained from the topology of M. In this section, we show how such functions give rise to Lagrangian submanifolds of T*M, hence the adjective "generating". In particular, we show that the action function obtained either in the symplectic twist map setting or in the Chaperon approach of last section generate a Lagrangian manifold canonically symplectomorphic to the graph of of the map F under consideration. More generally, this construction unifies the different finite, and even infinite, variational approaches in Hamiltonian dynamics. A. Generating Phases and Lagrangian Manifolds
Let Wbea differentiable function M —> H. In Section 58.C, we show that: dW(M) = {(q,dW(q))
\qeM}
is a Lagrangian submanifold of T*M. Note that this manifold is a graph over the zero section 0*M of T*M. Heuristically, we would like to make it possible to similarly "generate" Lagrangian submanifolds that are not graphs with some kind of function. One way to do this is to add auxiliary variables and see our Lagrangian manifold as an appropriate projection in T* M of a manifold in some bundle over M. This is what is behind the following construction.
52. Generating phases. 225 Let n : E —> M be a vector bundle over the manifold M. Let W(q, v) be a real valued function on an open subset of E. The derivative ^
: E -> £ * of W along the fiber of E
is well defined, in the sense that if U is a chart on M and V>i> ^2 : U x V —> 7r_1(C/) are two local trivializations of £ , and W\=W
°ip\, W2 = W o tp2, then
**-^-(q,u)du =
—(&{q,v))dv
where # = -02° V'f1 *s t n e c n a n g e °f trivialization. We assume that the map: (q, v) H-> ^ ^ (q, v) is transverse to 0. This means that the second derivative (in any coordinates) is o f
(fd^> 1$)
maximum rank at points (q,v) where ^ ( q , v) = 0. With this as-
sumption, the following set of fiber critical points is a manifold of same dimension as M: (
(52.1)
I dW
1
Ew = Uq,v)eE\—(q,v)=0J.
[For a proof of this general fact about transversality, see eg. the theorem p.28 in Guillemin & Pollack (1974) ] Define the map: iw • Zw -> T*M ( \ (q,v)->
(
dW
< \ lq,—(q,v)
Exercise 52.1 shows that this is an immersion. We now show directly that this immersion is Lagrangian: dW. i*wpdq = -^-{q,v)dq dq
= dW\
(q,v)
and hence: i*w(dqAdp)
=
d2W\Ew=0.
We will say that W is a generating phase for a Lagrangian immersion j : L —> T*M if j(L) =
iw(Ew).
Exercise 52,1 Show t h a t iw • Ew —» T*M is an immersion, i.e. t h a t Diw\ rank (Hint. Use t h e transversality condition t o show t h a t KerDiw
has full
H TSw = {0}).
226
10: GENERATING PHASES
B. Symplectic Properties of Generating Phases We start with the trivial, but important: Proposition 52.2 Suppose the Lagrangian submanifold L C T*M is generated by a function W : E —> IR. The points in the intersection
of L with the zero section 0*M
ofT*M
are in a one to one correspondence with the critical points
Proof.
i\v(q, v) is in L if and only if ^-(q,v)
ofW.
= O.ItisinO^ if andonlyif ^-(q,
0.
v) = •
In Section 64, we find that critical points persist under elementary operations on generating phases: if W\ : Ex -* K , and W2 : E2 —> IR are two generating phases such that W2 o <£ = Wx + C, or W2 : Ex x E2 -> IR and
W2(q,vx,v2)
= Wx(q,vx) + f(q,v2)
where ^ is a fiber preserving diffeomorphism, / is nondegenerate quadratic in v2 and C a constant, then Wx and W2 have the same number of critical points. The first operation is called equivalence, the second stabilization. This persistence is now geometrically explained by Proposition 52.2 and the following: Lemma 52.3 Two equivalent generating phases generate the same Lagrangian immersion. This is also true under Proof.
stabilization.
Let W2 o # = Wx + C where <£ is a fiber preserving diffeomorphism between
Ex -» M and E2 -» M. Writing $(q, v) = (q, <j>(q, v)) = (q, v'), where v -> <j>(q, v) is a diffeomorphism for each fixed q, we have: W2 (q,
dWx _ (dW* dv \ dv
oS\
d& J ' dv
This imphes that Ew2 = ^ ( - S V J . and we conclude the proof of the first assertion by noticing that:
52. Generating phases.
dWlf -w(q,v)
.
=
227
dW2/.f „ —(
Now let W2 (q, vi, v2) = Wi (q, v{)+f(q, v2) where / is quadratic and nondegenerate in v2. We have: dW2/dv = 0 «• v2 = 0 and dWi/d«i = 0 sothat£W2 = ZVi xO£ 2 ,whereO£ 2 isthezerosectionofE 2 .Moreoverd//9g| {i) . 2=0 j = 0 so that, at points (q, vi, 0) of S2,
q,—(q,v1,0)j =
[q,—(q,v1)j. D
C. The Action Function Generates the Graph of F We examine here the twist map case, and let the reader perform the analysis for the Chaperon case in Exercise 52.4. Let M be an n-dimensional manifold and F be a symplectic twist map on U C T*M, where U is of the form {(q, p) 6 T*M \ \\p\\ < K}. Let S(q, Q) be a generating function for F. S can be seen as a function on some open set V of M x M, diffeomorphic to U. (19) Since PdQ — pdq = dS(q, Q), we can describe the graph of F as: Graph(F) = { ( 9 , - g ( q , Q ) , Q, f g ( g , Q ) ) | (q,Q) g y j
C
(T*M) 2 ,
which is canonically symplectomorphic to (see the map j below):
{ (9, Q, | | ( 9 , Q), | ^ ( 9 , Q)) | ( 9 ) Q ) e y J c r ( M x M ) . One can easily check that this manifold has 5 as a generating phase. In other words the generating function of a symplectic twist map F is a generating phase for the graph of F. Consider now the more general case where F = FN ° • • • ° -Fi is a product of symplectic twist maps of U c T*M. This time, the candidate for a generating phase is: N
W{q) =
YJSk{qk,qk+l), fc=l
9
in the case where M = T " , and the m a p is defined on all of T * T " , we have V ^ U 9* R 2
228
10: GENERATING PHASES
where we do not identify q^+i
and q1 in any way. Then, writing
« = (92.---.9JV).
9 = (9I.9JV+I).
we show that W(q, v) is a generating phase for Graph(F)
u = {(qi,...,qN+1)eMN+1
|
c (T*M)2.
Let
(qk,qk+1)eMU)}
where ^ is the "Legendre transformation" attached to the twist map Fk. Let (3 : MN+1 M x M be the map defined by: (q1,...,
q^+i)
—>
—> (9IJ 9JV+I)- The bundle that we will
consider here is: U -> 0(U) C M x M. The Critical Action Principle (Proposition 23.2, and Exercise 26.4) states that ^ ^ (9, v) = 0 exactly when g = (g,v) is the 9 component of the orbit of ( 9 i , P i (91,92)) under the successive Fk's. This means that the set of orbits under the successive Fk 's is in bijection with the set Ew = {^j-(g, v) = 0} as defined in (52.1) . Since this set is parameterized by the starting point of an orbit, it is diffeomorphic to U, hence a manifold. For g € Ew, we have: F(q1,P1(q1,q2))
= (QN+U -FV+ifajv, 9jv+i))
but: dW Pi(9i,92) = - ^ ^ ( g j . g a ) = - ^ — ( 9 I , 9 J V + I . « )
aw -Pjv+i(9iv9jv+i) = d2SN(qN><JN+i) = -51 (gi-gAr+i,^) a gjv+i In other words, the graph of F in T*M x T*M can be expressed as:
K
dW
dW
9l
'~'dq'^q'V^9N+udq
1
\ ^,W7
(9.«)e^w|-
To finish our construction, we define the following symplectic map: j : (T*M x T*M, -QM
0 QM) -> ( r * ( M x M ) , ^ X M )
(9,P,Q,P)->(g,Q,-P,P). where Qx denotes the canonical symplectic structure on T*X. Clearly: j(Graph(F))=iw(Zw),
52. Generating phases. that is, W generates the Lagrangian manifold Graph(F). correspond to Graph(F) -diS\(quq2)
n A(T*M
= d2Siv(qN,qN+1),
229
Note that the fixed points of F
x T*M), i.e. to q e Sw
such that qx = qN+1
which are critical points of W L
and
, , a s we well
know. Exercise 52.4 Show that the generating function W of Chaperon (see Proposition 51.3) generates the graph of the Hamiltonian map F : T 2 " —> T 2 n . (Hint. If you are stuck, consult Laudenbach & Sikorav (1985)).
D. Symplectic Reduction We introduce yet another geometric point of view for the generating phase construction. We will see that if a Lagrangian manifold L C T*M is generated by the phase W : E —> 1R, than in fact L is the symplectic reduction of the Lagrangian manifold dW(E)
C T*E. We
introduce symplectic reduction in the linear case, and only sketch briefly the manifold case, referring the reader to Weinstein (1979) for more detail. Consider a symplectic vector space V, fio of dimension 2n. Let C be a coisotropic subspace of V. Let A(V) be the set of Lagrangian subspaces of V (a Grassmanian manifold). The process of symplectic reduction gives a natural map A(V) —> A (C/C-1)
that we now
describe. By Theorem 55.1, we know that we can find symplectic coordinates for V in which: C =
{(qi,...,q„,Pi,.-.,Pk)}
and we have CL = {(qk+i, • • •, qn)} C C. Then C/C1-
^{(qi,-..,qk,Pl,---,Pk)}
which is obviously symplectic. It is called the reduced symplectic space along C. We denote by Red the quotient map C —» C/C1-. The symplectic form Qc of C/C1
is natural in the
sense that it makes Red into a symplectic map: (52.2)
nc(Red(v),
Red{v'))
=
0(v,«').
Proposition 52.5 Let L C V be a Lagrangian subspace and C C V a coisotropic subspace.
Then
230
10: GENERATING PHASES
LC = Red(L n c) = L n C/L n c1is Lagrangian in C/C1-. We say that Lc is r/ie symplectic reduction of L along the coisotropic space C. Proof.
Formula (52.2) tells us that Lc is isotropic. We need to show that dimLc
\dimCJC^.
=
From linear algebra: dimLc
= dim(L D C ) - dim(L D Cx).
As would be the case with any nondegenerate bilinear form, the dimensions of a subspace and that of its orthogonal add up to the dimension of the ambient space. Also, the orthogonal of an intersection is the sum of the orthogonal. Hence: dimV = dim(L n C x ) + dim(L n C"1)1- = dim(L n Cx) + dim(L + C), since L x = L. Thus dimLc
= dim{L DC) — dimV + dim(L + C) = dimL + dimC — dimV = dimC
dirnV.
(52.3)
2 On the other hand: dim(C/CJ~)
= dimC — dimC~L = dimC — (dimV — dimC) = 2dimC - dimV
We conclude that dimLc
= ^dimiC/C-1)
(52.4)
by putting (52.3) and (52.4) together.
D
We now sketch the reduction construction in the manifold case. Let C be a coisotropic submanifold of a symplectic manifold (M, Q). Then TCX is a subbundle of TC (that is, the fibers are of same dimension and vary smoothly) so we can form the quotient bundle TC/TC1,
with base C and fiber the quotient TqCjTqCL
at each point q of C. It turns out
that this quotient bundle can actually be seen as the tangent bundle of a certain manifold C/G1-, whose points are leaves of the integrable foliation TC-1. Moreover one can show that the naturally induced form Qc is indeed symplectic on C/CL. red : C —> C/C
L
Finally, we define
as the projection. Its derivative is basically the map Red defined
52. Generating phases.
231
above. One can show that, if C intersect a Lagrangian submanifold L transversally, then Lc = red(L) is an immersed symplectic manifold of C/C±, which is the reduction of L along C. We now apply this new point of view to the generating function construction. Let E = M x IR^. We show that if L = iw(^w) C T*M is generated by the generating phase W : E —> M, then L is in fact the reduction of dW(E) c T*E along the coisotropic manifold C = {pv = 0}, where we have given T*E the coordinate (q, v,pq,pv). This is just a matter of checking through the construction. We know that dW{E) is Lagrangian in T*E. Its intersection with C is the set: f I dW(E) nC = |fo,w,p,,p„) G T*E I
Pq
dW dW 1 = -Qj-(q,v), Pv = - ^ (
=dW(Ew). where Ew is the set offibercritical points in E. Since by the transversality condition in our definition of generating phase Ew is a manifold, so is dW(E) fl C: for any W, the map dW : E —> JH'.E is an embedding. The bundle r C x is the one generated by the vector fields •^ andthusC/C-1-canbeidentifiedwithT*M = {(g,pg)}.Theimageofdiy(£;)nCunder the projection red : C —> C/C1- is exactly iw(ZV) = {( v ) = 0} = L. Note that because E = M x E w , the above argument is independent of the coordinate chosen (eg. C is well defined). With a little care, the argument extends to the case where E is a nontrivial bundle over M. Exercise 52.6 Show that, in the Darboux coordinate used above, t h e q-plane and the p-plane of V both reduce t o the q and p-plane (resp.) of C/C±.
E. Further Applications Of Generating Phases
The symplectic theory of generating phases does not only provide a unifying packaging for the different variational approaches to Hamiltonian systems. It can also serve as the basis of symplectic topology, where invariants called symplectic capacities play a crucial role. Roughly speaking, capacities are to symplectic geometry what volume is to Riemannian geometry: they provide obstructions for sets to be symplectomorphic, or for sets to be symplectically squeezed inside other sets. Viterbo (1992) uses generating phases to define such capacities, in contrast to prior approaches by Gromov (1985) who uses the theory of
232
10: GENERATING PHASES
pseudo-holomorphic curves. The basis of the definition of capacity in Viterbo (1992) is a converse statement to Lemma 52.3: Proposition 52.7 If Wi and Wi both generate ft*(0^), where hl is a Hamiltonian isotopy, then after stabilization W\ and Wi are equivalent. In view of this, Viterbo is able to define a capacity for a Lagrangian manifold L Hamiltonian isotopic to 0*M by choosing minimax values of a given (and hence any) generating phase for L. In another work, Viterbo (1987) shows that a certain integer function called Maslov Index on the set of paths in the Lagrangian Grassmannian is invariant under symplectic reduction. It can be shown that the Lagrangian Grassmanian A(V) has first fundamental group TTI(A(V)) = 7L. As mentioned in Chapter 6, we can roughly interpret this by saying that A(V) has a "hole" and the Maslov index measures the number of turns a curve makes around that hole. Now let Wt be the generating phases for a Hamiltonian isotopy h*. The set dWt{E) is Lagrangian in T*E and its reduction is the graph of hl (where Wt is the action function for a decomposition of h1). The Maslov Index in A(T*E) detects the change in Morse Index of the second derivative of Wt, whereas on the graph of /i*, it detects a non transverse intersection with the plane {(q,p) = (Q, P)}- This can be used to give a neat generalization to Lemma 29.4 and to explain the classical relationship discovered by Morse between the index of the second variation of the action function and the number of conjugate points (see Milnor (1969) for the classical, Riemannian geometry case, and Duistermaat (1976) for the more general convex Lagrangian case).
Appendix 1 OVERVIEW OF SYMPLECTIC GEOMETRY
Symplectic geometry is the language underlying the theory of Hamiltonian systems. This appendix is a short review of the main concepts, especially as they apply to Hamiltonian systems and symplectic maps in cotangent bundles. These spaces are natural when considering mechanical systems, where the base, or configuration space describes the position and the momentum belongs to the fiber of the cotangent bundle of the configuration space. In our optic of symplectic twist maps, one important concept studied in this chapter is that of exact symplectic map. Theorem 59.7 proves that Hamiltonian systems give rise to exact symplectic maps. We assume here some familiarity with the notions of manifold, vector bundle and differential form. The reader who is uncomfortable with these concepts should consult Guillemin & Pollack (1974) , Spivak (1970) or Arnold (1978). For more on symplectic geometry and Hamiltonian systems, see Arnold (1978), Weinstein (1979), Abraham & Marsden (1985) or McDuff & Salamon (1996).
53. Symplectic Vector Spaces In this section, we review some essentials of the linear theory of symplectic vector spaces and transformations. They will be our tools in understanding the infinitesimal behavior of symplectic maps and Hamiltonian systems in cotangent bundles. A symplectic form Q on a real vector space V is a bilinear form Q which is skew symmetric and nondegenerate:
234
Appendix 1: SYMPLECTIC GEOMETRY n(av + bv',w) f2(v,w)
= afi(v,w)
+ bf2(v',w),
(v,v',w
e V, a,b € IR).
= —£2(w,v)
v ^ 0 => 3u> such that f2(v, w) ^ 0 A symplectic vector space is a vector space V together with a symplectic form. Example 53.1 The determinant in IR2 is a symplectic form. More generally, the canonical symplectic form on Ht 2n , is given by: (53.1)
fio(v,w)
= (Jv, w),
J =
f0 \Id
-Id\ 0J
where the brackets ( , ) denote the usual dot product. We will see in the next theorem that all symplectic vector spaces "look" like this. In particular, their dimension is always even. Usually, one writes: n
f2o = dq Adp= YJ dqk A dpk fc=i
where it is understood that dqk,dpk are elements of the dual basis for the coordinates (<7ii • • •! Qn,Pi: • • • ,Pn) of IR2™. One can view dqk A dpk(v, w) as the determinant of the projections of v and w on the plane of coordinates (qk,Pk) (see Exercise 53.5). The symplectic space (M 2 n , Ho) can also be interpreted as TRn ® (IR™)*, equipped with the canonical symplectic form: fio(a © b, c 8 d) = d(a) - 6(c).
It is often convenient to view a bilinear form as a matrix. To do this, fix a basis ( e i , . . . , e„) of V, and set:
Ag = rt(ei)Cj-) Equivalently, if ( , ) is the dot product associated with the basis ( e i , . . . , e n ), then An is the matrix satisfying:
n(v,w) = (Anv,w). With any bilinear form Q on a vector space comes a notion of orthogonal subspace Wx to a given subspace (or vector) W :
wL = {v € v | n(v,w) = o,\/w e w}
53: Symplectic Vector Spaces
235
In the case of symplectic forms, the analogy with the usual notion of orthogonality can be quite misleading, as a subspace and its orthogonal will often intersect. We now show that all symplectic vector spaces are isomorphic to the canonical (lR 2n , f20). Theorem 53.2 (Linear Darboux) If (V, Q) is a symplectic space, one can find a basis for V in which the matrix An
of fi is given by An = J =
. ,
_
J.
Hence, the isomorphism that sends each vector in V to its coordinate vector in the basis given by the theorem will be an isomorphism between (V, fi) and (IR n , i?o). In classical notation, the coordinates in the Darboux coordinates are denoted by(20> (q,p) =
(qi,---,qn,pu---,pn)-
Note in particular that a symplectic space always has even
dimension.
Proof of Theorem 53.2. Since fi is nondegenerate, given any v / 0 € V, we can find a vector w G W such that fi(v, w) = —1. In particular, the plane P spanned by v and w is a symplectic plane and the bilinear form induced by fi on P with this basis has matrix: (53.2)
'0 V l
-1' Q
Since fi is nondegenerate on P, we must have P1- n P = {0}. Furthermore V = P + Px, since if u € V,
u — H(u,v)w + f2(u,w)v € P±. Q must be nondegenerate on the subspace PL of dimension dimV — 2, so we can proceed by induction, and decompose P x into J?-orthogonal planes on which the matrix of fi is as in (53.2). A permutation of the vectors of the basis we have found gives An = J.
•
Exercise 53.3 Show that the linear transformation whose matrix is J in the canonical basis is orthogonal (i.e belongs to 0(2n)), that it satisfies J2 = —Id (J is then called a complex structure) as well as u
In the literature, one also sees frequently (p, q), with —J = ( matrix.
" ,
) as canonical
236
Appendix 1: SYMPLECTIC GEOMETRY Qo(Jv, Jw) = Qo(v, w)
(that is, J is symplectic, see section 55). Exercise 53.4 Show that a one dimensional vector subspace in a symplectic vector space is included in its own orthogonal subspace. Exercise 53.5 Show that in a Darboux basis for a symplectic plane, Q{v, w) = det(v, w). Hence the form ^™ dqk A dpk applied to 2 vectors can be seen as the sum of minor determinants of these two vectors (see Arnold (1978)). Exercise 53.6 Prove that a general skew symmetric (not necessarily nondegenerate) form Q can be put, by an appropriate change of variable, into the "normal form": An =
/ 0fc Idk
-Idk 0fc
V
o,
where the integers k, I do not depend on the basis chosen.
54. Subspaces of a Symplectic Vector Space Let V be a symplectic vector space of dimension 2n, W C V a subspace, and flw the symplectic form restricted to W. The previous exercise shows that we can find a basis for W in which : Anw
=
/ 0fc Idk
\
-Idk 0fc
\
°«/
with
dimW = 2k + l
Furthermore, even though there may be many bases in which Qw has a matrix of this form, k and I are independent of the choice of such a basis. In other words, (W, flw) is determined up to isomorphism by its dimension and by k. We will say that W is: • null or isotropic if k = 0 (and / =
dimW),
• coisotropic if k + I = n. • Lagrangian if k = 0 and I = n (i.e. W is isotropic and coisotropic). • symplectic if I = 0 and k / 0. The rank of W is the integer 2k. The next theorem tells us that the qualitatively different subspaces of a symplectic space can be represented by coordinate subspaces in some Darboux coordinates.
54: Subspaces of Symplectic Spaces
237
Theorem 54.1 A subspace W of rank 2k and dimension 2k + I in a symplectic space can be represented, in appropriate Darboux coordinates, by the subspace of coordinates: (q1,...,qk+i,Pi,.-.,Pk)-
In particular, in some well chosen bases, an isotropic space is made entirely of q's and a coisotropic one must have at least n g's (the role of p's and g's can be reversed, of course) and a symplectic space has the same number of q's and p's.
Proof.
From the definition of the rank of W, there is a subspace U of W of dimen-
sion 2k which is symplectic, on which we can put Darboux coordinates. UL D W, the null space of ilw, is in the subspace U-1, which is symplectic (see Exercise 54.4). The next lemma shows that we can complete any basis of U1- n W with coordinates that we denote by (qic+i, • • • ,qk+i) into a symplectic basis of U1-, with coordinates (gfc+i,..., qk+i,Pk+i, • • • ,Pk+i)- The union of this basis and the one for U (of coordinates say (qi,...
,qk,Pi, • • -Pk)) is a symplectic basis, in which W can be expressed as
advertised.
•
Lemma 54.2 Let U be a null (i.e. isotropic) subspace of a symplectic space V Then one can complete any basis of U into a symplectic basis of V. Proof.
Without loss of generality, V is M 2 n with its standard dot product and canonical
symplectic form. Choose an prthonormal basis (ui,...,ui)
for U (in the sense of the dot
product). Using (53.1) and the results of Exercise 53.3, the reader can easily check that JU is orthogonal to U and that (u^,..., x
From Exercise 54.4, E
ui, Jui,...,
Jm) is a symplectic basis for E = U © JU.
1
®E = V and E - is symplectic. We can complete the symplectic
basis of E by any symplectic basis of EL and get a symplectic basis for V.
D
As a simple consequence of Theorem 54.1, we get: Corollary 54.3 / / U is an isotropic subspace of a symplectic space V, one can find a coisotropic W such that V = U @W. One can also find a Lagrangian subspace in which U is included.
238
Appendix 1: SYMPLECTIC GEOMETRY
This applies in particular to Lagrangian subspaces: given any Lagrangian subspace L, we can find another one L' such that V = L ® L'. In the normal coordinates of the theorem, L could be the q coordinate subspace and L' the p coordinate subspace.
Exercise 54.4 Let W be a subspace of a symplectic space V, Show that: W is symplectic •<=>• W ffi W1- = V <^=> Wx is symplectic (Hint, see the proof of the Linear Darboux theorem). Exercise 54.5 Show that: W isotropic -£=^ W C W^. W coisotropic •<=*• W1- C W. W is Lagrangian <=>• W is a maximal isotropic subspace, or minimal coisotropic subspace (for the inclusion). Note that the above equivalence are the definition most often seen in the literature. Exercise 54.6 This exercise shows how symmetric matrices can be used to locally parameterize the space of Lagrangian planes. Suppose you are given a basis vi,...vn for a Lagrangian subspace L of IR2". In the canonical coordinates (g,p), write vk = (xk,Vk)Let X and Y be the n x n matrices whose columns are the Xk's and yk's respectively. Suppose that L is a graph over the qr-plane. (a) Show that X is invertible and that the column vectors of the In x n matrix form a basis for L. (b) Show that the matrix YX~l is symmetric. (c) Deduce from this that the (Grassmanian) space of Lagrangian subspaces of ]R2" has dimension n(n + l)/2.
55. Symplectic Linear Maps The Symplectic Group. The Linear Darboux Theorem tells us that, up to changes of coordinates, all symplectic vector spaces are identical to (]R 2n , OQ). Therefore, as we define and study the transformations that preserve the symplectic form on a vector space, we need only consider the case (IR 2n , H0). Definition 55.1 A symplectic linear map
(55.1)
where $*QQ{v,w)
=
f20($v,$w).
55: Symplectic Linear Maps
239
The group formed by symplectic linear maps is called the symplectic group and is denoted by Sp(2n;JR), or in short Sp(2n). Because of (55.1)and (53.1), this group is naturally identified with the group of 2n x 2n real matrices $ that satisfy: (55.2)
& J$ = J
Examples 55.2 (a) The group Sp(2) is the group of 2 x 2 matrices of determinant 1. (b) The transformation F ( q , p) = (q+p,p), and so is any with matrix (
j is symplectic in IR In
withmatrix I
, I, where A* = A. These maps are called completely
integrable as they preserve the n dimensional foliation of (affine) Lagrangian planes {p = constant). (c) The examples of symplectic linear map this book is most concerned with are the differentials of symplectic twist maps and of the time 1 map of Hamiltonian flows.
Spectral Properties of Symplectic Maps. Theorem 55.3 Symplectic linear maps have determinant
1. If X is an eigenvalue of
l
a symplectic linear map, so is X~ , and they appear with the same multiplicity. -1
is a complex eigenvalue, then so are A , A, A Proof.
, all with the same
If X
multiplicity.
Let $ be a symplectic map. It is not hard to see that: (_l)["/2] dqi A . . . A dqn A dpi A . . . A dpn =
;
f}0 A . . . A £2Q
n\
where [n/2] is the integer part of n/2. Since # preserves the right hand side of this equation, it must preserves the left hand side, i.e., the volume. Hence det <£ = 1. The rest of the theorem is a consequence of the fact that the characteristic polynomial C(A) of a symplectic transformation ^ has real coefficients and that, by (55.2), $f is similar to <£_1:
•
240
Appendix 1: SYMPLECTIC GEOMETRY
Dynamical Type of the Origin as Fixed Point. The origin is a fixed point for any linear map. Its dynamical properties (i. e. how orbits under the iteration of the map of points close to the origin behave) are entirely given by its spectrum. In particular the stability (i. e. whether points near 0 stay in a neighborhood of 0) of the fixed point is function of the spectrum: if any eigenvalue is greater than 1 in modulus, the fixed point is unstable. Conversely, if all the eigenvalues are within the unit disk, the fixed point is stable. In symplectic maps, by Theorem 55.3, stability can only occur when all the eigenvalues are on the unit circle. In this case the fixed point is called elliptic if all its eigenvalues are distinct from ± 1 or parabolic if all its eigenvalues are ± 1 . On the other extreme, the origin is a hyperbolic fixed point for a symplectic linear map when no eigenvalue is on the unit circle. In this case, we can't have stability since, by Theorem 55.3 necessarily half of the eigenvalues have modulus greater than 1. Hence, in this case, the stable and unstable manifold (the n-dimensional union of eigensubspaces with eigenvalues larger (resp. smaller) than 1 in absolute value) are each n dimensional. These manifolds are also Lagrangian (see Proposition 36.1 and Exercise 36.3). Note that hyperbolicity can come in different flavors: Inversion hyperbolic (when all eigenvalues are negative) or spiral hyperbolic, when some of the eigenvalues are not real. Finally, a general fixed point exhibits a combination of elliptic, parabolic and hyperbolic behaviors, each in different even dimensional eigenspaces.
Exercise 55.4 (a) Show that if a 2n x 2n matrix $ is given by its n x n block representation:
•
-
(
-
then # is symplectic if and only if ab* = 6a', (b) Show t h a t
)
• l
cd = dcl,
ad* — be* =
Idn.
*-*-(-', ?)• In particular, if # is symplectic, so are
# e 0 ( 2 n ) & # ' # = Id
Show t h a t if <£ is in any two of t h e groups Sp(2n), 0 ( 2 n ) , Gl(n, C), it is in t h e third. Show that, in this case, we can write:
56: Symplectic Manifolds
* = (\b> ~ab)J
with {\aat\ab
241
=b
+ b'ltb = M Id
that is, the complex matrix a+ ib is in the unitary group U(n). Exercise 55.6 (a) Show that, when ±1 is an eigenvalue of
56. Symplectic Manifolds Let N be a differentiable manifold. A symplectic structure on N is a family of symplectic forms on the tangent spaces of N which depends smoothly on the base point and has a certain nondegeneracy condition. More precisely, a symplectic structure is given by a closed nondegenerate differential 2-form Q: dC2 = 0 and, for all z € M, v / 0 e TZM, 3io e T Z M such that n(v, w) ^ 0. Q is called a symplectic form and (M, J?) a symplectic manifold. A symplectic or symplectomorphism
map
between two symplectic manifolds (iVi,J?i) and (./V2, ^2) is a
differentiable map F : iVi —• JV2 such that: F*i? 2 = fii. In other words, the tangent space at each point of a symplectic manifold is a symplectic vector space, and the differential of a symplectic map at a point is a symplectic linear map between symplectic vector spaces.
242
Appendix 1: SYMPLECTIC GEOMETRY
Example 56.1 (a) Once again , the canonical example of symplectic manifold is given by (IR2n, Q0), where IR2n is thought of as a manifold. The tangent space at a point is identified with K 2 n itself, and the form fi0 is a constant differential form on this manifold. (b) Any surface with its area form is a symplectic manifold. Symplectic maps in dimension 2 are just area preserving maps. (c) Kahler manifolds (see McDuff & Salamon (1996)) are symplectic. (d) Cotangent bundles are non compact symplectic manifolds (see Section 57) and time 1 maps of Hamiltonian vectorfieldson them are symplectic maps. The fundamental theorem by Darboux (of which we have proven the linear version) says that locally, all symplectic manifolds are isomorphic to (M2™, fio). See Arnold (1978), Weinstein (1979) or McDuff & Salamon (1996) for a proof of this. Theorem 56.2 (Darboux) Let (TV, Q) be a symplectic manifold. Around each point of N, one can find a coordinate chart (q, p) such that : n
H = \_. dqk A dpk := dq A dp. l
Hence all 2n-dimensional symplectic manifolds are locally symplectomorphic. This is in sharp contrast with Riemannian geometry where, for example, the curvature is an obstruction for two manifolds to be locally isometric. Submanifolds Of Symplectic Manifolds. Submanifolds of a symplectic manifold can inherit the qualitative features of their tangent spaces: A submanifold Z C (TV, Q) is (co)isotropic if each of its tangent spaces is (co)isotropic in the symplectic tangent space of N. Hence a Lagrangian submanifoldis anisotropic submanifold of dimension n = ^dimN. Any curve on a surface is a Lagrangian submanifold. The 0-section and the fiber of the cotangent bundle of a manifold are Lagrangian submanifolds, and so is the graph of any closed differential form (see next Section 57.C). Exercise 56.3 Show the following: (a) Any symplectic manifold has even dimension.
57: Cotangent Bundles
243
(b) If (N, Q) is a In dimensional symplectic manifold, then Qn is a volume form. (c) A symplectomorphism is a volume preserving diffeomorphism. Exercise 56.4 Let (N, O) be a symplectic manifold and F : N —> N a symplectomorphism. Show that the set graph F is a Lagrangian submanifold of (iV x iV, i?ffi(—!?))
57. Cotangent Bundles A. Some Definitions Let M be a differentiable manifold of dimension n. Its cotangent bundleT*M
-^ M i s the
fiber bundle whose fiber T*M at a point q of M is the dual to the fiber TqM of the tangent bundle. The elements of T*M are cotangent vectors or linear 1-forms, based at q. Given local coordinates ( 9 1 , . . . , qn) in a chart of M , one usually denotes a tangent vector v by:
where -£- denotes the tangent vector to the k th coordinate line at the point q considered. A cotangent vector p at the point q takes the form:
(57.1)
p = ^2pkdqk
Where dqk denotes the 1-form dual to -^-: ,
f d \
.
def f
Once the system of coordinates q = (qi,...
Oifj^k,
,qn) is chosen, the coordinates p
=
(Pi> • • • ,Pn) for T*M as defined by (57.1) are uniquely determined, and we call them the conjugate coordinates. We will also refer to the pair (q, p) as a chart of conjugate coordinates. The cotangent bundle T*M as a smooth union of the fibers T*M is a differentiable manifold of dimension 2n, with local coordinates (q, p) as presented above. More precisely, if Q —> q is a coordinate change between two charts [/ and V of M , above which T*M is trivial, then :
#*(9,P) = (Q.-P) = ( S ^ M . ^ C P ) )
244
Appendix 1: SYMPLECTIC GEOMETRY
is a change of coordinates between the corresponding charts V x H™ and U x H™ of T*M. This law of change of coordinates is what distinguishes tangent vectors from cotangent vectors. More generally, given a (local) diffeomorphism F : M —> N between two manifolds M and TV, there is (locally) an induced pull-back map: F* : T*N —> T*M which can be written F* (q,p) = (F~1 (q), DF* (p)) in conjugate coordinates. Example 57.1 (a)H2™ = IR™ © (IR™)* can be seen as the cotangent bundle of the manifold IR n : this bundle is trivial, as any bundle over a contractible manifold. (b)The cotangent bundle of T n is T n x IR™. That T* T " is trivial is a consequence of the fact that T " = ET/Z™, where Z™ acts as a group of translations on IR™, whose differentials are the Id. See the following exercise. Exercise 57.2 More generally, if M = JRn/r where T is a group of diffeomorphisms of IR" acting properly discontinuously (i.e. around each point q of M there is a neighborhood U(q) such that U n (r\Id)(U) = 0), then TmM^
IR 2 n /r*
where r* is the set of pullback diffeomorphisms of ]R2n of the form 7*, where 7 e F.
B. Cotangent Bundles as Symplectic Manifold We now show that there is a natural symplectic structure on T*M. We first construct a canonical differential 1-form called the Liouville form which, as we will prove, has the following expression in any set of conjugate coordinates: n
A = ^Pkdqk
=pdq.
1
We then obtain a symplectic form by differentiating A: Q = — d\,
Q = dq A dp,
the latter holding in any conjugate coordinate system. We first present a coordinate free construction of A. To define a 1-form on T*M, it suffices to determine how it acts on any given tangent vector v in a fiber Ta(T*M)
of the tangent space of T*M. Since the base
point a is in T*M, it is a linear 1-form on T*M for q = ir(a). Let n : T*M -> M be the
57: Cotangent Bundles
245
canonical projection. The derivative 7r» : T{T*M) -> TM takes a vector v to the vector •K„V
in T g M. We can evaluate the 1-form a on that vector, and define: X(v) = a(n*v)
We now compute A in a local, conjugate coordinate chart (q,p) of T*M. Write: v, d d a = 2_^«*:%: and v = w q — + «P-~-Then nt(v) = v 9 ^ and a(-K,v) = ^ akVqk which exactly says that A = pdq.
D
Exact Symplectic Maps on Cotangent Bundles. That the symplectic form f2 is exact (i.e. the differential of 1-form, here A) on a cotangent bundle enables us to single out the important class of exact symplectic maps. Indeed, if a map F : T* M —> T* M is symplectic in T*M then the form F*X - A is closed: d{F"X - A) = F'dX - dX = -{F*Q - Q) = 0. This justifies the following Definition 57.3 A map F : T*M —> T*M is exact symplectic if the 1 form F* X - X is exact: F*X-X = dS for some real valued function S on T*M. We will see in Section 58 that time t maps of Hamiltonian flows are exact symplectic. So are most of the maps in this book. Note that in M 2n , since any closed form is exact, symplectic and exact symplectic are two equivalent properties. On the other hand, the map (a;, y) —> (x, y + a), a ^ 0, of the cylinder is an example of a map which is symplectic but not exact symplectic. Remark 57.4 1) The term exact diffeomorphism, or even exact symplectic diffeomorphism is sometimes used to denote the time 1 map of a (time dependent) Hamiltonian system. But it can be shown that the map (q,p) >-> (q + Ap,p), A = I
J is exact symplectic
according to our definition, but not isotopic to Id (true more generally whenever A is not homotopic (cannot be deformed) to I on T 2 , see Exercise 23.4). Hence these maps cannot be time-1 maps of Hamiltonians.
246
Appendix 1: SYMPLECTIC GEOMETRY
2) Cotangent bundles are just one example, albeit the most important one, of exact symplectic manifolds: symplectic manifolds whose symplectic form is exact. Many facts that are true for cotangent bundles also hold for exact symplectic manifolds. Exercise 57.5 Show t h a t t h e set of exact symplectic maps forms a group under composition. In particular, show t h a t if G*\ — A = SG and F*X — A = Sp then (F o G)*A - A = d[(SF oG) + SG}. Exercise 57.6 Show t h a t a m a p F of T * M is exact symplectic if and only if :
j pdq=
1pdq
iFy
for all differentiate closed curve 7.
C. Notable Lagrangian Submanifolds of Cotangent Bundles
It is not hard to see that the fibers of T*M are Lagrangian submanifolds: in coordinates they are given by {q = q0} and hence their tangent space is of the form {q = 0}. Likewise, the zero section 0 ^ of T*M is Lagrangian. Another class of example is of importance to us in Chapter 10. Consider a function W : M —* M. Its differential dW can be seen as a section of T*M, i.e. a map M —• T*M whose image dW(M) can be written as {(q,dW(q)) I q G M}. Hence, a basis for the tangent space of dW(M) at a point (q, dW(q)) is given by: 8
= —— -—-— - u, dqkdqi dqidqk so that dW(M) is a Lagrangian submanifold ofT*M. We can generalize this argument somewhat. Any 1-form a can be seen as a map from M to T*M, so we can ask the question: for what a is a(M) a Lagrangian manifold ? To answer this question, one can check (Exercise 57.7) the following formula: (57.2)
a*\ — a.
58: Hamiltonian Systems
247
where A is the Liouville form (the reader has to get used to the fact that we see a either as a form or a map, at our convenience. When seen as a map, a is actually an embedding of M into T*M). The manifold a(M) is Lagrangian exactly when: 0 = a*n = a*(-d\)
= -d(a*\)
= -da,
that is, exactly when a is a closed form. In particular, if the form a is exact with a = dW, this gives another proof that dW(M) is Lagrangian. W is the simplest instance of generating function for the Lagrangian manifold a(M) = dW(M) (generating phase or generating phase function is also used). We expend on this important notion of symplectic topology in Chapter 10. Exercise 57.7 Verify Formula (57.2) using local coordinates.
58. Hamiltonian Systems A. Lagrangian Systems Versus Hamiltonian Systems Euler-Lagrange Equations. A lot of mechanical systems can be put in terms of a variational problem. In these systems, under the principle of least action, trajectories are critical points of an action functional of the form: A ( 7 ) = I*
L{q,q,t)dt,
with boundary condition j(t0) = 9o>7(£i) = 9i (
(-g-(q,q,t)5q+
—(q,q,t)5q\
dt.
Performing an integration by parts on the second term of this integral, we get:
248
Appendix 1: SYMPLECTIC GEOMETRY
J t0
, dq
dt dq,
For 7 to be a critical point, the above integral must be 0. Since this should be true for any variation 6j, we must have: /,-„ -.\
(58 1}
-
dL
d dL
8q--It8q- = °>
which is a second order differential equation in q called the Euler-Lagrange equations. (The plural to "equations" just refers to the fact that the dimension is usually greater than 1). As an example, a large number of mechanical systems have a Lagrangian function of the form: L{q,q,t) = \\\q\\2~Vt{q). ("Kinetic - potential". The time dependence of V usually refers to some forcing) where V : IRn —> H. The Euler-Lagrange equations for such a system are: (58.2)
q + Wt(q)=0,
which, in mechanical cases, can be seen as the expression of Newton's Law: acceleration is proportional to the the force. From Lagrange to Hamilton. To solve the O.D.E. (58.2), one usually proceeds by introducing p — q to get a system of first order ODE's: q =p p = - W(g). As we will see presently, we have just put the Lagrangian problem into a Hamiltonian form. In general, if the following Legendre Condition: (58.3)
d2L det —-Tr ^ 0, dq
is satisfied, we can introduce 8L
V=
Tq
to transform the Euler-Lagrange equations (58.1) into a system of first order O.D.E.'s: because of the nondegeneracy condition (58.3), the implicit function theorem implies that, locally, we can make a change of variables :
58: Hamiltonian Systems
(58.4)
249
dL £:(q,q)-+(q,p=—)
This is, when q is seen as a point on a manifold M, a local diffeomorphism between TqM and T*M (see Exercise 58.2). This change of variables is called the Legendre transformation S21^ Define the Hamiltonian function by: H(q,p,t) =
pq-L(q,q,t),
Where it is understood that q = qo C~1(q,p). We can compute: dH _ dq _ dL_ _ dl^dq __ _ a L dq dq dq dq dq dq' dH _ . dq _dlLdq_ _ . dp dp dq dp But the Euler-Lagrange equations imply that: dt dq
dq
dq
Combining this with the previous formula yields Hamiltons equations: q = Hp (58.5) P = ~Hq. Remark 58.1 (a) The Legendre transformation is involutive: it is its own inverse, in the following sense. The map (q, q) —> (q, ^ = p) has inverse: (
\
i
d H
-\
(q,p)^(q,-gp= q) and L is the Legendre transformed of H in the sense that: L(q,Q,t)
=Pq-H{q,p,t)
where p — p(q, q, t) is given implicitly by ^ = q (to see that this is all legal, see Exercise 58.3 which shows that if L satisfies the Legendre condition, so does H). In the classical literature the term Legendre transformation refers t o t h e complete process of changing the Lagrangian L into the Hamiltonian H as shown in this section, and H is then called t h e Legendre transformation of L. It is grammatically less awkward to call H the Legendre transformed of L, which we do in this book.
250
Appendix 1: SYMPLECTIC GEOMETRY
(b) In the new coordinates, the action functional becomes: A(j)=
/ pdq - Hdt
where 7 is seen as a curve (q(t), p(t), t) in the extended phase space M 2 n x IR, or T*M x IR.
Symplectic formulation of Hamilton's Equations. Hamilton's equations have a natural expression in the symplectic setting. We assume for now that q is in IR". Using the notation Ht(Q,p) = H(q,p, t), we can rewrite (58.5) as
z=-JVHt(z)fJxH(z,t). where VHt = (^q) is the gradient of Ht with respect to the scalar product on IR 2n : (VHt,v)=dHt(v) Likewise, XJI , which we call the Hamiltonian vector field can be seen as the symplectic gradient of Ht: n0(XH,v)
= (~J2VHt,v)
= {VHt,v}
=
dHt(v).
This can also be written using the contraction operator on differential forms: ixH fl = dHt
Exercise 58.2 Show that, if <j>: U —> V is a change of coordinate charts in a manifold M, then ^ changes according to <j>* : V —+U. Hence ^ is a covector. Exercise 58.3 (a) Compute the Legendre transformed of L(q,q,t) = \{Aq,q) — V(q). (b) Show that, in general, if H is the Legendre transformed of L, then -^99
pp —
58: Hamiltonian Systems
251
B. Hamiltonian Systems on Symplectic Manifolds Motivated by the last expression that we found for the Hamiltonian vector field in H 2 r \ we extend the definition to symplectic manifolds: Definition 58.4 Let (N, Q) be a symplectic manifold and H(z,t) = Ht(z) be a Ck real valued function on N x H. The Hamiltonian vector field associated with H is the (time dependent) vector field XH defined by: (2{XH,v) = dHt(v),
Vv G TM.
Equivalently: iXuO = dHt The (possibly time dependent) O.D.E.: (58.6) is called Hamilton's
z =
XH(z,t)
equations.
In local Darboux coordinate charts (eg. in conjugate coordinates chart of a cotangent bundles), these equations take the form of (58.5) (see Exercise 58.5). If H is time independent, then (58.6) generates a (local)flowon N. If H is time dependent, then XH generates a (local)flowin the space NxTR, called the extended phase space in mechanics. Specifically, one solves the following time independent system on iV x H: z =
XH(z,s)
s=l which generates aflow0* in N x IR satisfying:
252
Appendix 1: SYMPLECTIC GEOMETRY
describing hl(z) is by saying that it is the unique solution z(t) of Hamilton's equation with initial condition z(s) = z. In practice, one often fixes s = 0 and denotes /ig by hl. The following exercise shows the one to one correspondence between time dependent vector fields and isotopies. It also shows that, even though the time 0 of a solution flow to a time dependent vector field is the Identity, the maps /i* do not in general form a 1-parameter group.
Exercise 58.5 Show that the equation (58.6) takes the form of (58.5) in local coordinates. Exercise 58.6 Let Xt be a vector field (not necessarily Hamiltonian) on a manifold N. Let hfs be the solution flow to the O.D.E. z = Xt(z), s — 1. Prove that: (i) h% = Id, Vs 6 M, (ii)hi = h\ o h*s, so that in particular hi = hl o (/i s ) _ 1 . Compute (hl)^1 and deduce that, in general, (foo)-1 m a y n o t be equal to (/ij) for any r: the maps hl do not form a group. (iii) Conversely, given any (sufficiently smooth) isotopy g* in N, with g° — Id, show that the time dependent vector field: 9
^
=
has solution hi = gl o (gB)
Tu
9t+no{9r\*) = (^{{9r\z))
1
.
C. Invariants of the Hamiltonian Flow Conservation of Energy. If G is a function on a differentiable manifold N, and X is a vector field, we recall that the Lie derivative of G along X is: L G
* ^
d G{4>\z)) = dt 4=0
dG{X{z))
where <*' is the flow solution for X . Theorem 58.7 Let H be a time independent Hamiltonian function on (M, Q) . Then H is constant under the Hamiltonian flow it generates:
LXHH
Proof.
LXHH
= dH(XH)
= O(XH,XH)
= 0.
= 0
•
58: Hamiltonian Systems 253
Remark 58.8 LxHG = Q{XG,XH) = -LxaH is also denoted by {G, H} and it is called the Poisson bracket of H and G. Hence, the poisson bracket measures how far the function G (resp. H) is from being constant along the flow of Xn (resp.Xc). When {H, G} = 0, one says that G (resp. H) is a first integral of the Hamiltonian flow of H (resp. G), or that the functions H and G are in involution. One can show (see eg. Arnold (1978), Abraham & Marsden (1985)) that the set of Hamiltonian vector fields form a Lie sub-algebra of the Lie algebra of vector fields on a manifold, in the sense that: X
{H,G} =
[XH,XG]-
In particular, the poisson bracket of two functions measures how far from commuting their Hamiltonian flows are.
Hamiltonian Flows are Exact Symplectic. We extend the notion of Lie derivative to differential forms. If X is any (possibly time dependent) vector field, and a a differential form, we define the Lie derivative of a in the direction of X by: Lxa=
Jt
t=Q
l
where g is the flow generated by X. At time t ^ 0, (58.7)
jt{gtya
= {gtTLxa.
The isotopy g* preserves the form a whenever the Lie derivative is zero: (ff*)*a = a, Vi <^> Lxa = 0. We have the important homotopy formula (see eg. Weinstein (1979) or McDuff & Salamon (1996)): (58.8)
Lxa = ixda + d(ixa)
and, at time t jt. o, jt(gtya = (gty(ixda + d(ixa)) A symplectic isotopy gl on (M, Q) is an isotopy such that g* is a symplectic map for all t. By the homotopy formula (and the fact that a symplectic form is closed), this can be reworded:
254
Appendix 1: SYMPLECTIC GEOMETRY gf is a symplectic isotopy •£=>• Lx fi = 0 <=>• d(ixf?)
(58.9)
where X is the (time dependent) vector field ^ - ((gt)~1(z))
= 0
(see Exercise 58.6). The
following theorem characterizes Hamiltonian isotopies, at least in cotangent bundles (or in any exact symplectic manifold, i.e. one whose symplectic form is exact) Theorem 58.9 (a) On any symplectic manifold, Hamiltonian
isotopies are symplec-
tic. (b) On a cotangent bundle T*M, is also exact
a Hamiltonian
isotopy with Hamiltonian
H(z, t)
symplectic: /i**A — A = hl*pdq — pdq — dSt,
where 7 is the curve (hT(z),r),
with St =
pdq — Hdr
r € [0,t] solution of Hamilton's
equations in the
extended phase space T*M x H , and z is the point at which the form is evaluated. (c) Conversely, if an isotopy g* is exact symplectic then it is Hamiltonian,
with the
Hamiltonian function given by:
iXtpdq-((gtr1r~(st).
Ht = where Xt(z) Proof.
^((gtr1(z)).
=
The first assertion (a) is an immediate consequence of (58.9): if hl is a Hamiltonian
isotopy then i(ht)£2 = dHt is exact, and therefore closed. In cotangent bundles, it is also a consequence of the second assertion. Now, look for ^(St) (58.10) jth*X
= h*t (iXHdX + d{ixHX))
in (b):
= h*d{-Ht + iXaX)
= dh\{-Ht
+
IXHX)
From this we get: (58.11)
h*tX-X
=d
h*(-HT + ixHX)dr
= f dSt
Jo that is, hl is exact symplectic. We leave it to the reader to rewrite the integral as the one advertised in the theorem. This finishes the proof of (b). To prove the converse (c), let g* be an exact symplectic isotopy: (gt)*pdq-pdq
= dSt
58: Hamiltonian Systems
255
for some St differentiable in all of (q, p, t). We claim that the (time dependent) vector field Xt(z) whoseflowis
jf(dSt) = J^Tpdq
= tfrLXtpdq = (g1)' (iXtd(pdq) + d{iXtpdq)) ,
from which we get
iXtdq Adp = d [iXtpdq - ( ( s T 1 ) * ^ ) ) = dHt, which exactly means that Xt is Hamiltonian with Ht as Hamiltonian function.
D
Integral Invariant of Poincare-Cartan. A less formal proof of (b) in the above theorem yields extra information. We follow Chapter 9 in Arnold (1978). We first prove that the vector field (XH, 1) in T ' M x E generates the kernel of the form d(pdq — Hdt) = dpAdq — Hqdq A dt — Hpdp A dt. The matrix of this form in the (Darboux) coordinate (q,p,t)is: ( 0 -Id Hq\ A = I Id 0 Hp \ , \-Hq -Hp 0 / where Hq, Hp are column vectors on the right and row vectors on the bottom of the matrix. Since the upper left In x 2n matrix is the nonsingular matrix J, A is of rank (at least) In. It is easy to see that the Hamiltonian vector field (Hp, —Hq, 1) = (XH, 1) generates its kernel. Now, take a closed curve 7 in T*M x H. The image under the Hamiltonian flow of 7 forms an embedded tube in T*M x JR. Since the tangent space to this tube at any of its point z contains the vector (XH(Z), 1), the form d(pdq — Hdt) restricted to this tube is null. As a result, because of Stokes' theorem, if 71 and 72 in T'M x IR encircle the same tube of orbits of the extendedflow,we must have:
(58.12)
J pdq-Hdt=
J
pdq-Hdt
since 71 — 72 is the boundary of a region of the tube. The form pdq — Hdt is called the integral invariant of Poincare-Cartan. As a particular case, if 71 is of the form (7, ti) and 72 = (Mi7i ^2), the form Hdt is null on these curves and hence Equation (58.12) reads: (58.13)
f Pdq=
f
pdq
256
Appendix 1: SYMPLECTIC GEOMETRY
This last equation implies the statement (b) in Theorem 58.9: it proves that the function (58.14)
St=
f
h^pdq-pdq
is well defined, i.e. the integral does not depend on the path chosen between zo and z. This proves in turn that h* is exact symplectic.
•
Conservation of Hamilton's Vector Field Under Symplectic Maps. The property that Hamilton's equations are preserved under symplectic maps characterizes these maps, which are for this reason called canonical transforma tions in some of the classical literature. Even though we do not need this theorem in this book, we include it here since it explains why symplectic geometry came to exist.
Theorem 58.10
Let F : (M,WM)
—• (N,UN)
be a diffeomorphism.
Then F is
symplectic if and only if for all function H : N —> H , (58.15)
F*XHoF
= XH.
In this case, F conjugates the Hamiltonian flows h* of H and gf of H o F: g* = F-1 o / i ' o F . This holds also when H is time Proof.
dependent.
Reminding the reader that by definition FtX(F(z))
= DFzX(z)
for any vector
l
field X, we also use the notation F*Y to mean {F~ )*Y. It is not hard to check that the following formula holds: (58.16)
F*ixa
=
iF-xF*a
for any vector field X and differential form a. Coming back to our statement, we have on one hand: F*ixHUN
= F*dH =
dHoF,
and on the other hand, F*ixHUN
= iF*X„F*U>N =
IF'XH^M
58: Hamiltonian Systems
257
because of (58.16) and the fact that F is symplectic. This proves (58.15). Conversely, if (58.15) holds for any H, the same kind of computation shows that,
and since any tangent vector at a point of M is of the form XHOF for some H, we must have F*OJN = LJM, i-e. F is symplectic. The conjugacy statement, a general fact about O.D.E.'s, is left to the reader, as well as checking that everything still works with time dependent systems.
•
Exercise 58.11 The Lie derivative of a function can be denned, in the obvious way, along any differentiable isotopy. What fails in Theorem 58.7 when H is time dependent? Exercise 58.12 Show that in Darboux coordinates: sH
G \ - — — - —
dq dp
—
dp dq
Exercise 58.13 Prove that the function St denned in (58.14) satisfies: St(z) =
pdq- Hdt
+ C(z0, t),
for some C, and 7 as in Theorem 58.12. (Hint. Apply Stokes on the appropriate surface.) Exercise 58.14 Prove that hi is exact symplectic (i.e. even for s ^ 0), where h\(z) is, as in subsection B, the solution of Hamilton's equation such that z(s) = z. Exercise 58.15 Let H be autonomous, or of period T. Show that XH(Z) is preserved by DhT(z), i.e. XH is an eigenvector of DhT with eigenvalue 1.
Appendix 2 SOME TOPOLOGICAL TOOLS In order to estimate the minimum number of periodic orbits for a symplectic twist map or a Hamiltonian system, we need an estimate on the minimum number of critical points for the energy function of the corresponding variational problem. Estimating the number of critical points of functions on compact manifolds is the jurisdiction of Morse Theory and Lyustemick-Schnirelman Theory. Given the gradient flow of a real valued function / on a compact manifold M, Morse Theory rebuilds M from the unstable manifolds of the critical points of / . The combinatorial data of this construction gives a relationship between the set of critical points and the topology of M, in the guise of its homology. Unfortunately, the space on which the energy function W is defined is not compact. However, it usually is a vector bundle over a compact manifold M, and reasonably natural boundary conditions on the map or Hamiltonian system translates into some conditions of "asymptotic hyperbolicity" for W. This is a situation where Conley's theory, which studies the relationship between the recurrent dynamics of generalflowsand the topology of (pieces of) their phase spaces was brought to bear with great success. For the reader who has no background in algebraic topology, we start in Section 59 by outlining, following some examples, an easy way to compute the homology of a manifold by decomposing it into cells. We then illustrate Morse theory in Section 60, by considering the cells given by the unstable manifolds of critical points of a real valued function on the manifold. We hope that this will give such a reader at least a flavor of the rest of this chapter. Starting in Section 61, we assume familiarity with algebraic topology. We give some of the basic definitions of Conley's theory and state results on estimates of number of critical points in isolated invariant sets for gradient flows. In Section 62, we prove most of these
260
Appendix 2: TOPOLOGICAL TOOLS
results. In Section 63, we apply these results to functions on vector bundles whose gradient flow are asymptotically hyperbolic.
59.* Hands On Introduction to Homology Theory To a manifold, or to certain subspaces of it, we want to associate some algebraic objects called homology groups that are invariant under homeomorphisms or other natural topological deformations. Usually, the best way to calculate these groups (but not the best way to show their invariance properties), is to decompose the spaces studied into well understood pieces, and then define the groups from the combinatorial data describing how these pieces fit together. In this introduction we decompose spaces into cells, which are discs of different dimensions, and show how to compute cellular homology. A*. Finite Cell Complexes
Given a topological space X (e.g. a differentiable manifold) we can construct a new one by attaching a cell of dimension n. This is done by choosing an attaching map f from the bounding sphere S n _ 1 of the cellDn (a disk of dimension n) to X. The new space, denoted by X U/ Dn is given by the union of X and Dn where each point of dDn is identified with its image by / i n X. The topology on XUfDn is that of the quotient XUDn/{x ~ f(x)}. Example 59.1 One can construct the sphere S2 by attaching the disc D2 to a point p. The space X — {p} is a manifold of dimension 0, and the attaching map / sends each point of the boundary circle of D2 to p. One can also construct a sphere by attaching a disk to another one (what is the attaching map?). These constructions have obvious generalization to higher dimensions. A cellular space is a space built by successively attaching a finite number of cells, starting from a finite number of points (cells of dimension 0). If in this process each cell is attached to cells of lower dimensions, the space obtained is called a finite cell complex or CW complex. The union of all cells of dimension less than A; in afinitecell complex is called the k-skeleton. Thus the k + 1-skeleton is built by attaching cells of dimension k + 1
59. Homology Theory
261
to thefc-skeleton.The dimension of the cell of maximum dimension in a cellular space X (and hence of a CW complex) is called the dimension of X. Examples 59.2 Figure 59.1 shows how the torus can be decomposed (this is not the only way!) into afinitecell complex: its 0-skeleton is the point z. To get the 1-skeleton we attach both extremities of the "equator" a and the "meridian" b to z. The attaching maps send the boundaries - 1 and 1 of the 1-cells a = [-1,1] = b to the point z. Finally, the 2-skeleton is obtained by attaching the disk D (stretched to a square) to the 1-skeleton as indicated by the "fiat" picture of the torus. Note that the 1-skeleton looks like a "bouquet of two circles".
Cells dim:
It
I)
->
•
D
^*u. -> dim:
*
0
Skeletons F i g . 5 9 . 1 . T h e torus T 2 as a finite C W complex.
One can generalize this construction to surfaces of any genus g (spheres with g handles) by gluing a 2 cell to a polygon with 4g sides and identifying all vertices to a single point, and edges two by two as indicated by their name and orientations on the following figure (g is 2 here):
262
Appendix 2: TOPOLOGICAL TOOLS
Fig. 59.2. The double torus (surface of genus 2) as a finite CW complex. Identify edges according to their names and orientations, and identify all vertices to one point. When cutting the octagon in half through the curve 7 we obtain two handles, which are tori with a disk (bounded by the curve 7) removed in each. Any compact surface can be described with this method. More generally, we will show in the next section that any compact manifold is homeomorphic to a finite cell complex. Exercise 59.3 Decompose $ " , T n , I R P n and the Klein bottle into finite cell complexes. Remember that IRIP" can be defined as ID"/ ~, where the relation ~ identifies any two antipodal points on the boundary of the n-disk ID". The Klein bottle is [—1, l ] 2 / ~ where (l,»)~(-lI-»)and(a:>l)~(*,-l).
B*. Cellular Homology Bouquets of Spheres. When we "crush" the (k-2)-skeleton Xk-2 of a finite cell complex X to a point inside the (k-l)-skeleton Xk-i, that point. Hence each (k-l)-cell of Xk-i
the boundary of each (k-l)-cell crushes to becomes a (k-l)-sphere in Xk-i/Xk-2-
All
these spheres meet at exactly one point, where the crushed Xk-2 collapsed: we say that Xk-i/Xk-2
is a bouquet of spheres. The attaching map / of a fc-cell to Xk-i k l
to a map / : S ~
—• Xk-i/Xk-2,
gives rise
by composition with the quotient map. Hence we have
a map / from a sphere of dimension k — 1 to a bouquet of spheres, all of dimension fc — 1.
59. Homology Theory
263
Digression on Degree and Homotopy. Any continuous map from a sphere Si to a sphere S2 of same dimension comes equipped with a degree which, informally, is an integer which measures the number of times Si "wraps around" S2 under this map. This integer can be negative, as we keep track of orientation. Since the proper topological definition of degree requires homology (which we are in the process of defining), we restrict ourselves to differentiable maps. The degree of a differentiable map / between two manifolds of same dimension is given by: (59.1)
deg{f)=
£
1 • {sign det Dfx)
where z is any regular value of / , i. e. no determinant in the above sum is zero (by Sard's theorem, almost all values of a smooth map are regular). It turns out that the above number is independent of the (regular) point z. The degree of a map is also invariant under homotopy of the map. [Two continuous maps /o and f\ between the manifold M and the manifold N are homotopic if there is a continuous map (called homotopy) F : [0,1] x M —> N such that F(0, z) = f0(z), F(l, z) = fx(z) for all z in M]. Back to Horticulture. The attaching map / : Sfc_1 —> Xk-i/Xk-2 has a multiple degree: on each sphere Si in the bouquet one can compute the oriented number of preimages under / of a regular point as in (59.1) (without loss of generality, we can assume that / is differentiable except at the common point of the spheres). Suppose that C j - 1 , . . . , cktf1_ denote the (k — l)-cells of the cell complex and cf,..., c^fe itsfc-cells.We now form an Nk-i^Nk integer matrix dk whose entry dfc (ij) is the degree of the attaching map from dcj to the ith sphere of the bouquet, i.e. cf _ 1 /9cf _ 1 . The matrices dk, for k 6 { 1 , . . . ,dimX} essentially give all the combinatorial data describing how the complex X is pieced together from our collection of cells. Chain Complexes. We now want to view the matrices dk as those of linear maps between finite dimensional vector spaces, or modules. To do this, one thinks of c\,..., c^ as the basis vectors of an abstract vector space (or free module) Ck whose elements are formal sums of the form 1
264
Appendix 2: TOPOLOGICAL TOOLS
where a,j is an element of some "coefficient" field (or ring) K (usually 2 2 , Z, Q or IR). Hence Ck is generated by the k-cells and dimCk = Nk. For convenience, we define 9 o s 0 on Co. Lemma 59.4
(59.2)
dfc-i o dk = 0.
The proof of this crucial lemma, which we will not give here (see, eg. Dubrovin & al. (1987)) usually uses the long exact sequence of a triple and a pair in simplicial homology. A chain of maps and vector spaces (or modules): n —' ^n-1
—*•• • <-^fc —* Ofc_i — + . . . — > O o
satisfying (59.2) is called a chain complex. Definition 59.5 Thefcthhomology group of the finite cell complex X with coefficients in a ring or field K is given by:
Hk(X;K)=Kerdk/Imdk+1. where, by convention, do = 0 = dn+i This definition makes sense since, by Lemma 59.4, Im dk+\ C Ker dk- Note that Hk(X) = 0 whenever k > dim X or k < 0, since for such k, Xk = 0Homology of the Circle. The circle S 1 is a CW complex: we start with a pointp and attach to it an interval I = [0,1]; the boundary points of I become identified to p under the attaching map. Using IR as coefficients in our chain complex, we get Co = M.p = IR, C\ = IR.7 = M. The map d\ = 0: p has the two preimages {0} and {1} under the attaching map, but they come with opposite orientations under the orientation induced by I. Hence the degree of the attaching map is 0 here and the homology of the circle is given by: Hk(S\K)
={
fc = 0,1, ** otherwise
59. Homology Theory
265
Homology of the Torus. Figure 59.1 above gives the generators for a chain complex for the torus: C0 = H • z, C\ = IR • a 0IR • b, C2 = IR • D. All the boundary maps are 0 in this case: d^a = 0 because it geometrically yields z twice but with opposite orientation. Likewise for dib. As for 82D = a + b-a-b
= 0, again due to orientation. Hence
Ker dk/Im dk+i = Ck for k = 1,2,3. We have shown: (
IR M2 0
Hk(T2,M)^l [
k = 0,2 k=l otherwise.
Clearly, this result remains valid if we replace IR by any coefficient ring K. Homology of the Klein Bottle. A less trivial example is given by the Klein bottle. This non orientable surface is a torus with a twist and it cannot be embedded in IR . We build it with the same cells z, a, b and D as the torus. The only change occurs in the definition of <92: instead of gluing D to two copies of b in opposite orientation, we give them the same orientation (according to that of the boundary of D):
i
«1
»
'z
F i g . 5 9 . 3 . A cell decomposition for the Klein bottle. T h e only difference with t h a t of the torus is the orientation of one of the segments 6.
Let us use the integers 7L as our coefficient ring. As a result of the orientation change, the matrix of di is now I
1 and Ker 82 = {0}. From this we immediately get that
H2(Klein,~Z.) = 0. As in the case of the torus, di = 0 so that Ker d\ = C\ = a - Z 0 b - Z . Since Im d2 = {0} • a © 2Z • 6, H^Klein, Z) ^ Z ® Z 2 . As in the case of the torus, Ho(Klein, Z) = Z (in fact, the rank of Ho always gives the number of connected components of a manifold). Now let's reexamine the above computation with coefficients K — Z2 instead: the map 82 = 0 in this case since 2=0 in this ring. Thus, in this case we are back to the same situation as with the torus: H0(Klein, Z 2 ) = Z 2 , H\{Klein, Z 2 ) = Z2©Z2,ZZ2(.K7eOT,Z2) ^ Z 2 . Finally, let us choose K = IR. Since Ker 82 = {0} in this
266
Appendix 2: TOPOLOGICAL TOOLS
case again, H2(Klein,M) = {0}. SinceR./2IR = M/M = {0},Hi(Klein,~!R) ^ M . A s before H0(Klein, M) ^ IR. Some General Properties and Definitions Related to Homology. Let X be a compact
manifold of dimension n. As we will see in next section, it can always be decomposed into afiniteCW complex. •dimHk(M,JR) = rank Hk(M, T€) = bk is the kth Betti number of M. •2fc=i( — ) fc ^fc = x(M) is the Euler characteristic of M .
•Neither bk nor x{M) depend on the chain decomposition chosen for M. •bo gives the number of connected component of M. •bn = 1 if M is orientable, bn = 0 if M is not orientable. •The importance of homology stems in great part from its invariance under topological equivalences. One topological equivalence is that of homeomorphism. A coarser equivalence (see Exercise 59.7) is that of homotopy type. Two topological spaces M and N have the same homotopy type if there are continuous maps <j> : M —> N,ip : N —* M such that 4> o ip and ip o (f> are homotopic to the Identity maps of M and N respectively. In other words M can be deformed into iV and vice-versa. Theorem 59.6 // the two manifolds M and N are homeomorphic, or have the same homotopy type, then they have same homology: H* (M) — H* (N) (the star * stands for any integer).
C. Cohomology
Roughly speaking, cohomology is dual to homology. For readers of this book, it might be easier to see it through differential forms, which are dual to chains of cells in the sense that the integral < c , w > = J w o f a form wona chain c is a linear, real valued function of the variable c (it is also linear in u>). The duality bracket given by integration also satisfies: < d C,LJ > = < c, doj >
where d is the exterior differentiation on forms. This formal equality is a general requirement for defining cohomology. In the case of forms it is simply given by Stokes' Theorem. Finally, we can define the cochain complex
60. Morse Theory
267
where C£ = Ak is the vector space offc-formsand dk is exterior differentiation. As with homology, we can define the DeRham cohomology group as: Hk{M,M) = Ker dk+l/Im
dk,
i.e. this cohomology is the quotient of closed forms over exact forms. One notable difference between homology and cohomology is the direction of the arrows in the complexes that defines them. Another notable difference, which makes the use of cohomology often preferable, is the existence of a natural product operation in cohomology, called the cup product. In DeRham cohomology, this cup product takes the form of wedge product of the forms: [o>i] U [1J2] = [wi A w2]
where the notation [w] denotes the class of the closed form w and U the cup product. There are many different ways to define cohomology, but it can be shown that (given some normalization requirements), they all give the same result on compact manifolds. Poincare, for instance, introduced cohomology (not under that name) by geometrically constructing a dual complex to a triangulation (a special CW chain decomposition). In the next section, where unstable manifolds of critical points of a Morse function provide us with a chain decomposition, the dual decomposition can be taken to be that of stable manifolds. Exercise 59.7 Show that the circle and the cylinder have same homotopy type but are not homeomorphic. Exercise 59.8 Using Exercise 59.3, compute the homology of Sn,Tn,]RJP".
60.* Morse Theory We now show how any compact manifold can be described as a cellular space, with cells given by the unstable manifolds of the critical points of a Morse function. This immediately yields a relationship between critical points and homology, in the guise of the Morse Inequalities. We first define some of these terms. For more details on the material in this section, the reader should consult Milnor (1969).
268
Appendix 2: TOPOLOGICAL TOOLS
Gradient Flow and Morse Functions. Let / : M —> JR be a differentiable function on a manifold M. A critical point for / is a point z at which the differential of / is zero: df{z) = 0. If / is twice differentiable, the critical point z is called nondegenerate if d
(60.1)
e
t
^ ^ °
where this second derivative is taken with respect to any local coordinates x around z on M. The function / is a Morse function if all its critical points are nondegenerate. One can show that, on any given manifold, Morse functions are generic in the set of twice differentiable functions (see eg. Guillemin & Pollack (1974) or Milnor (1969)). Note that Condition (60.1) is independent of the coordinate system. Indeed, at a critical point z, d2f(z) dy2
_
dxtd2f{z)dx dy dx2 dy'
This last formula also implies that the number of negative eigenvalues of the real, symmetric matrix Jji does not depend on the coordinate system chosen around the critical point z. This number is called the Morse index of z. Qualitatively, the level set portrait of a function around a nondegenerate critical point is entirely determined by the index of the critical point. Indeed: Lemma 60.1 (Morse Lemma) Let z be a nondegenerate critical point for a function f on a manifold of dimension n. There is a coordinate system x around z such that: f(x) = /(*) -x\-...-x\
+ x2k+1 + ... + x2n.
We refer the reader to Milnor (1969) for a proof of this lemma, which generalizes the diagonalization process (Gram-Schmidt) for bilinear forms. Since the Morse Lemma clearly implies that the critical points of a Morse functions are isolated, we have: Corollary 60.2 A Morse function on a compact manifold has a finite number of critical points. The gradient flow of a function / is the solution flow for the O.D.E.: (60.2)
z = -V/(z).
60. Morse Theory
269
The gradient V / is defined here by (V/,.) = df(.), where the brackets denotes some chosen Riemannian metric (on H " or T n , one usually uses the dot product). / decreases along the flow:
| / K ' ) ) = -{V/(z(i))} 2 <0 with equality occurring exactly at the critical points. The eigenvectors corresponding to the negative eigenvalues of JJi^ span a subspace of TZM which is tangent to the unstable manifold at z of the gradient flow. [We remind the reader that the unstable manifold of a restpoint for a flow ft is the manifold Wu (z*) of points whose backward orbit is asymptotic to the restpoint z„: Wu{z*) = {z | limt^.oo ft(z) = «»}]. In the case of the gradient flow of a Morse function, the unstable manifold of a critical point is tangent to the x\,..., Xk plane given by the Morse Lemma. Hence the Morse index of o nondegenerate critical point of a Morse function is the dimension of its unstable manifold. Remark 60.3 Note that if the metric chosen to define the gradient is the euclidean one in the Morse coordinate chart, the (x\,..., Xk) plane is itself the (local) unstable manifold of the critical point, at least in that chart. This can always be arranged, by a local perturbation of the metric, and we will assume from now on that this is the case.
Topology of Sublevel Sets. The gist of Morse theory consists in studying how the topology of the sublevel set. Ma = {xeM\
f(x) < a}
changes as a varies. Theorem 60.4 // there are no critical points in / _ 1 [a, b], then Ma and Mb are diffeomorphic. The inclusion of Ma in Mb is a deformation retraction. Proof. Deform Mb into Ma by flowing down the trajectories of the gradientflow,with appropriate speed (controlled by reparameterization of theflow)and during an appropriate time interval. This is possible as long as there are no critical value in [a, b]. D
270
Appendix 2: TOPOLOGICAL TOOLS
Pig. 60.1. Deformation of a sublevel set Mb into the sublevel set M° when there are no critical points in f~l [a, b]. The lines with arrows represent trajectories of the gradient flow.
Theorem 60.5 Suppose f~1[a,
b] is compact and has exactly one critical point in its
interior, which is degenerate and of index k. Then Mb has the homotopy type of Ma with a cell of dimension k attached, namely, a ball in the unstable manifold of the critical point. Proof,
(sketch) Let z be the critical point, c = f{z) and e > 0 be a small real number.
By the previous theorem, M c + e has the same homotopy type as Mb and likewise for M c _ £ and Ma. Hence, we just have to show that M c + £ has the homotopy type of Mc~£ with a cell attached.
Fig. 60.1. A neighborhood of a Morse critical point z. A suitable parameterization of the flow retracts Mc+€ onto M c _ £ U S, which itself can be deformed into M°~e U B.
60. Morse Theory
271
We have represented in Figure 60.1 the sets Mc±e within a Morse neighborhood. The drawing makes it intuitively clear that some reparameterization of the gradient flow (which we have represented by some arrows) will collapse Mc+€ into Mc~e U S, where the set S is given by: S = {/ < c + e,x 2 k+1 + ... + x\+n < 6}. S can obviously be deformed into: B = {f
= 0},
that is, a ball in the unstable manifold of z. In other words, Mc+e
_
Mc-e
u
B
a Any cellular space X is homotopically equivalent to a finite cell (CW) complex Y, where X and Y have the same number of cells in each dimension (one deforms each of the attaching maps defining X into one that attaches to cells of lower dimensions than its own, see Dubrovin & al. (1987), Section 4). This and the previous theorems yield: Theorem 60.6 Any sublevel set Ma of a Morse function
on a compact
manifold
M has the homotopy type of a finite CW complex, whose cells correspond to the unstable manifolds of the critical points.
Morse Inequalities. Since there always is a Morse function on any given manifold, Theorem 60.6 yields: Corollary 60.7 Any compact manifold has the homotopy type of a finite CW complex. Theorem 60.8 (Morse Inequalities)
Given any Morse function
f on a compact
manifold M, the homology of M is generated by a finite complex {Ck, dk}{\,... dimM} whose generators correspond to the critical points of index k of f. In particular, if Cfc = dimCk is the number of critical points of index k, (60.3)
cfc > 6 f c = d i m f f f c ( M , l R )
272
Appendix 2: TOPOLOGICAL TOOLS
and, better: (60.4)
ck - cfc_i + ... ± c0 > bk - 6fc_i + ... ± 60,
with equality holding for k = n. Proof. Thefirststatement in the theorem is somewhat of a tautology for us, since we have defined the homology of M as the cellular homology of any cellular complex representing M, and thus in particular, we can choose the complex generated by the unstable manifolds of the critical points. Formula (60.3) is then trivial, since
Hk(M) =
Kerdk/Imdk+1,
and Ker dk is a subspace of Ck. The inequalities (60.4) are a consequence of (60.3) and their proof, left to the reader, only involves linear algebra. • Floer-Witten Complex. One can give a nice geometric interpretation to the maps dk in the context of Morse theory. Assume that the gradient flow <£>* of our chosen Morse function is Morse-Smale, i.e. that for any given pair of critical points x, z, their respective stable and unstable manifold meet transversally. This is again a generic situation, which has the following implications: the set
M(x,z) =
Wu(x)nW{z),
which is the union of all orbits connecting x and z, is a manifold and
dimM(x, z) = index(x) — index(z). In particular, if index(x) - index(z) — 1, M(x, z) is a one dimensional manifold made of a finite number of arcs that one can count, with ± according to a certain rule of intersection. This intersection number m(x, z) gives the coefficient in the generator z of d(x), i.e.
dkx = \ J
m(x,z).z.
One can also define cohomology in this fashion: just take the same complex, but defined for the function - / . What was a stable manifold becomes an unstable manifold and Ck
61. Topology of Invariant Sets
273
becomes Cn_fc. This not only gives us a geometric way to see cohomology, but a trivial proof of Poincare's duality theorem for compact manifolds: Hn-k(M,lR)
^
Hk(M,JR).
For more details on this chain complex, which is sometimes called the Floer-Witten complex but dates back to J. Milnor's book on cobordism, see eg. Salamon (1990). For a proof of Poincare's duality using the Morse complex, see Dubrovin & al. (1987).
61. Controlling the Topology of Invariant Sets The relationship revealed by Morse between the critical point data of a function and the topology of the underlying manifold has a very wide generalization in the theory of Conley, which brings about a similar relationship for general continuous flows on locally compact topological spaces. The relationship is between local topological data of recurrent parts of the flow and the global topology of well chosen regions of the space. We will outline this theory in Section 63.C. For now, we make a small step toward this generalization. Here, and for the rest of this chapter, the cohomology used is the Cech cohomology with coefficients in IR (see Spanier (1966)). We do not need to define this cohomology here: it is enough to state that it is well defined on compact sets. Furthermore it has a continuity property: If X = DXn, then H* (X) = lim H* (Xn), where the latter expresses the inductive limit of the cohomology modules. Otherwise, Cech cohomology satisfies all the usual axioms and rules of cohomology and coincides with other cohomologies on compact manifolds. In this section, we present some of the fundamental concepts and results of Conley's theory, which are used in this book. The next section goes into more detail and provides proofs for the theorems presented in this section, as well as more results. Isolating Blocks. Consider a compact set / which is invariant under the gradient flow of a function W on some finite dimensional manifold. If W is a Morse function, then necessarily I is made of critical points and the intersections of all their stable and unstable manifolds (prove it as an exercise!). Exactly as we did for manifolds, consider the Floer-Witten chain complex, generated by the critical points and with boundary maps given by the stableunstable manifolds intersection data. It turns out (see Floer (1989a) or Salamon (1990)) that
274
Appendix 2: TOPOLOGICAL TOOLS
this complex gives the cohomology Conley Index of I, a topological/dynamical invariant of I that we define below. In certain cases, as in what follows, one can evaluate the cohomology Conley index and hence give lower estimates on the number of critical points. We use these results in Section 63 to estimate the number of critical points of functions on vector bundles. Definition 61.1 Let M be afinitedimensional manifold. A compact neighborhood B in M is called an isolating block for a (continuous) flow ' if points on the boundary dB of B immediately leave B under the flow, in positive or negative time: zedB=>
4>{0>e) C Bc
or <^-£'°) C Bc
for some e = e(z) > 0.
The exit set B~ of B is defined as the set of points in dB which immediatelyflowout of B in positive time. Given an isolating block B for theflow*, define 1(B) to be the maximal invariant set included in B ("maximal" is in the sense of inclusion here). Alternatively,
i(B) = n tgR 0*(s).
Cohomology Conley Index and Cuplength. There are two classical ways to measure the topological complexity of an invariant set 1(B). One is its cohomology Conley index: h(I) =
H*(B,B~).
The bigger the dimension of this vector space, the more complex the topology of I. Note that in the notation "h(I)n, we have deliberately omitted the mention of B: this is because the vector spaces H*(B, B~) are isomorphic for all isolating block B such that I = 1(B) (Conley & Zehnder (1984)). Hence h(I) is an invariant of the set I. In practice, the size of h(I) is often measured by the sum of the Serf/ numbers
sb(h(I)) = Y, dimHk(B, B~). k
This again is an invariant of I. A second, somewhat rougher way to measure the complexity of an invariant set / (or any topological space which admits continuous (semi)flows and a cohomology) is the cuplength which is defined as: cl(I) = l+sup{A;e]N|3wi,...a; f e , ujj £ Hn>(I), rij > 1, and ux U ... U u t ^ 0}
61. Topology of Invariant Sets
275
Generalized Morse and Lyusternick-Schnirelman Theories. The following results are consequences of the much more general theory of Conley for (semi)flows that we sketch in the next section. Theorem 61.2 Let I be a compact isolated invariant set for the gradient flow of a function W on some manifold. If the function points in I is greater or equal to sb(h(I)).
is Morse, the number of critical
In general, the number of critical points
in I is at least equal to cl(I).
Remark 61.3 If one applies Theorem 61.2 to the case I = M = B, B~ = 0, then h(I) = H*(M),
cl(I) = cl(M) and we retrieve both the critical point estimates of Morse
and Lyusternick-Schnirelman. Historically, the first time Theorem 61.2 was applied in a significant way was in the proof of the following proposition, which appeared in several parts in Conley & Zehnder (1983): Proposition 61.4 Let M be a compact manifold and W be a real valued function
on
M x I R " xJR m . Suppose that the gradient flow ofW admits an isolating block B of the form B ~ MxD+xD-
with exit set M x D+ x dD~, where D+ C M n , D~ C H m
are homeomorphic to the unit balls. If W is a Morse function,
it has at least sb(M)
critical points in B. In general, W has at least cl(M) critical points.
Fig. 61.0. The isolating neighborhood in Proposition 61.4.
276
Appendix 2: TOPOLOGICAL TOOLS
Conley and Zehnder applied this theorem in the case M = Tn, where sb(Tn) = 2™, and cl(Tn) = n +1. In the following section we will give a proof of Theorem 61.2 (we will only sketch the sum of betti number estimate, but give a complete proof of the cuplength estimate) as well as of Proposition 61.4.
62. Topological Proofs A. Proof of the Cuplength Estimate in Theorem 61.2
Conley & Zehnder (1983) prove a cuplength estimate (their Theorem 5) that is valid for a compact invariant set 7 of a general flow (f>1. We follow their proof. Define a Morse decomposition for 7 to be a finite collection {Mp}p€p of disjoint compact and invariant subsets of 7, which can be ordered in such a way that any x not in U p6 pM p is a-asymptotic to an Mj and w-asymptotic to an Mt, with i < j (x is a-asymptotic (resp. ur-asymptotic) to Mj if limj-.-oo (resp. +oo) *(£) £ AT,). One can show that a compact invariant set always has such a Morse decomposition. We now state Theorem 5 of Conley & Zehnder (1983): Theorem 62.1 (Conley-Zehnder) Let I be any compact invariant set for a continuous flow, and let {Mp}pep be a Morse decomposition for I. Then
(62.1)
cl(I) < J2 cl(Mp).
The relevant example for us is when ' is the gradient flow of a function with a finite number of (not necessarily nondegenerate) critical points on a compact invariant set 7: it easy to check that these critical points form a Morse decomposition. Since an isolated point has trivial cohomology, cl(Mp) = 1 for each p in this example, and we have proven the cuplength estimate in Theorem 61.2. The case when the critical points are not isolated is trivial in that theorem: cZ(7) < oo is always true... Proof of Theorem 62.1. Note that if ( M i , . . . , M*) is a Morse decomposition, then (Mi fc_i, Mfc) is also a Morse decomposition, where Mi,...^-! is formed by the union of Mi U... U Mfc_i and of all the connecting orbits between these sets. Hence, by induction, we only need to consider the case where k = 2, and (Mi, M2) is a Morse decomposition
62. Topological Proofs
277
for I. From the definition of a Morse decomposition, we can deduce the existence of two compact neighborhoods h of M\ and I2 of Mi in I with h UI2 = I and such that M\ — nt>o<^'(/i) and M2 = nt
= 1,2. Thus the proof of (62.1)reduces to that of the inequality
cl(Ii) + cl(I2) > cl(I) whenever 7i U I2 = I are three compact sets. The next lemma is devoid of dynamics: Lemma 62.2 Let I\ U I2 C I be three compact sets. If i\ : I\ —> I, i2 : I2 —> I and i : h U I2 —•I are the inclusion maps, then, for any a,f3€
H*(I),
i\a = 0 and i ^ = 0 ^ i * ( a U / 3 ) = 0 .
Proof. We chase the diagram: H*(I,h) i 31 H*(I) lit H*(h)
® H*(I,I2) i f2 ® H*(I) 1*5 ® H*(I2)
Z -^ ^
H*(I,hUl2) I f H*(I) J- i* H*{h\Jl2).
The vertical sequences are exact sequences of pairs, the horizontal lines are given by Kiinneth Formula. Starting on the second line of the diagram with a, (3 G H*(I), suppose i\a = 0 = i\& then there must be a e H*(I, h) with j\a = a, J3 € H*{I, I2) with fifi = /?. Now j*(al) J3) = aU (3 and hence i* (a U /?) = V o j * (a U /3) = 0, by exactness. D Tofinishthe proof of Theorem 62.1, let a.\,..., ai be in H* (I) and ct\ U... U ai ^= 0. Let this product be maximum, so that cl(I) = / + 1. Order the a's in such a way that ai U ... U ar is the longest product not in the kernel of i\. In particular cl(Ii) > r + 1 and it(ai U . . . U a r U ar+1) = 0. Lemma 62.2 forces i2(a r + i U ... U a;) ^ 0 (i* is one-to-one here, since h D I2 = I). Thus cl(I2) > I - (r + 1) + 1 = I — r, and d ( / 1 ) + c Z ( 7 2 ) > / + l = cZ(7). a
278
Appendix 2: TOPOLOGICAL TOOLS
B*. The Betti Number Estimate of Theorem 61.2 and Conley's Theory
We have proven in Theorem 62.1 that, for a general function W, the number of critical points in an invariant set / for the gradient flow of W is greater than cl(I). We now show that if W is a Morse function, the number of critical points in / is greater than sb(h(I)). To do so, one can either follow Floer (1989a) in his generalization of the Witten complex (of unstable manifolds of critical points for gradient flows, see the end of Section 60) to invariant sets. His proof relies in part on Conley's theory. Alternatively, one can use Conley's generalized Morse inequalities that we state in this subsection. Let / be a compact invariant set for a continuous flow 0* on some locally compact topological space. Let (Mi,... ,Mk) be a Morse decomposition for I. Analogously to the cuplength estimates of Theorem 62.1, Conley-Morse inequalities relate certain betti numbers of the Morse sets Mj to the corresponding betti numbers of /. To define the adequate betti numbers, we need to generalize the notion of isolating block to that of index pair for isolated invariant sets. A compact set / is an isolated invariant set if there is a neighborhood N of I such that / = I(N) is the maximal invariant subset in N. An index pair for an isolated invariant set is a generalization of an isolating block. Roughly, I is a pair of compact spaces (iVi, N2) such that Ni\N2 is a neighborhood of I and / = I(Ni\N2). Moreover iV2 plays the role of the exit set: to exit NiU N2, a. point of Ni must first go through N2, see Conley (1978), Conley & Zehnder (1984). The fundamental property of these sets is that the homotopy type [N1/N2, *] is independent of the choice of index pair for / and hence defines a topological invariant called the Conley index of the invariant set I. Giving less information, but easier to manipulate is the cohomology Conley index H*{Ni,N2) = h{I), again an invariant of I. If (Ni,N2) = (B,B~) for an isolating block B, this definition of h(I) is the same as we have given in Section 61. One way to encode the information given by h(I) is via the coefficients of the Poincare polynomial:
p{t,h(I)) :=
YJtidimH:i{Nl,N2).
In Conley & Zehnder (1984), it is proven that, given a Morse decomposition ( M i , . . . , Mk) for an invariant set / of a continuous flow *, there is a filtration N0 C iV"i c ... C Nk such that (Nj, iV,-_i) is an index pair for Mj. This is instrumental in proving the following (Conley & Zehnder (1984):
62. Topological Proofs
279
Theorem 62.3 (Conley-Morse Inequalities) (62.2)
£ > ( * , KMj)) = p(t, /»(/)) + (1 + t)Q(t),
where Q(t) is a polynomial with positive coefficients This theorem is an extraordinary generalization of the classical Morse inequalities: it is valid for any continuous flow on a locally compact space (not necessarily a manifold!). End of Proof of Theorem 61.2. To retrieve the betti number estimates of Theorem 61.2, we use the Morse decomposition of the invariant set I made of the (isolated) critical points zi,..., zN. Thanks to the Morse Lemma, we can construct an isolating block for each Zj, and show that the Conley index of z, has the homotopy type of a pointed sphere made by collapsing the boundary of the local unstable manifold of Zj to a point. The isolating block is denoted by S in Figure 62.1. XI, , Xk (Unstable)
Xk+h
i Xn
(Stable)
Fig. 62.1. The index pair (5, S~) retracts on (B, B ), a pair made of the local unstable manifold of z and its boundary (a disk of dimension k equal to the index of the critical point z and its bounding sphere). Thus h(z) = H*(S,S~) S* H*(B,B~) S H*(Sk,*) which has exactly one generator in dimension k.
280
Appendix 2: TOPOLOGICAL TOOLS — tu', where Uj is the Morse index of z3•,. Now the pair (J, 0) is an
Hencep(t, h(zj))
isolating pair for I (no points exit / ) , and thus p(t, h{I)) = J2 tkdimHk(I).
The positivity
of the coefficients of Q in (62.2) therefore insures that there are at least dimHk (I) critical points of index kin I.
•
C. Floer's Lemma The following lemma, proven in Floer (1987) gives a situation where one can get a handle on the topology of an invariant set i\ It is central to the proofs of several topological results we will use, including Proposition 61.4. Lemma 62.4 (Floer's Lemma) Let B be an isolating block for a flow ' on a finite dimensional manifold, and I be its maximal invariant set. Suppose that there is a retraction a : B —» P, where P is some compact subset of B. If there is a class u € H*(B,B~)
such that : « H t ) U a » : H*{P) - • H*(B,
is an isomorphism,
B~)
then af:H*{P)-+H*{I)
is injective, where a / denotes the restriction of a to I. (IfNcM
are two topological spaces and i : N —> M is the inclusion map, a retraction
is a map r : M —• N such that r oi = Idx, that is r restricts to Id on N). Corollary 62.5 Let B, I, P be as in Lemma
62.4,
an
d let the flow 4>l in that lemma
be the gradient of some function W. Then the number of critical points of W is at least cl(P). Proof.
Since H*(P)
-+ H*{I) is injective, cl(I)
immediate consequence of Theorem 61.2. Proof of Lemma negative time.
> cl{P) and the Corollary is an •
624. Define B°° = n t >o^*B, the set of points that stay in B for all
62. Topological Proofs
281
Lemma 62.6 1) H*(B, B°° U B~) = 0 2) I* : #*(B°°) -> H*(I(B))
is an isomorphism, where I : 1(B) -> B°° is t/ie
inclusion. Before proving this lemma, we use it tofinishthe proof of Lemma 62.4. Consider the diagram: H*(B,B~) I Id H*(B,B~)
® H*{B,B°°) | j* ® H*(B)
a*
^
H*(B,B°°\JB-) I k* -H H*(B,B-) r ^
H*(B°°)
=0
H*(I)
where all vertical maps are induced by inclusions, and the twofirsthorizontal maps are given by Kunneth Formula. Suppose a/*v = 0 for some v e H* (P). Since I* is an isomorphism and ai = aB°° °l ,0 = aj* = l*(a.B°°)*v =$ (aB°°)*v = 0. Since a s » = a o i, 0 = aB=o *v = i*a*v. The middle, vertical sequence is the exact sequence of a pair. Hence thereisau;€ff*(J5,B 00 )suchthatj*w = a*v.ButMUa*w = k*(ul)w) = fc*(0) = 0. The hypothesis of Lemma 62.4 forces v = 0. D Proof of Lemma 62.6. LetB* = *(£) andB°° = n t > 0 B ' as before. Note in particular that, in the Hausdorff topology, limj^+oo Bl = B°°, and limt_^0 B* = B- To the triple of spaces (B, Bl U B~,B~) corresponds the exact sequence: (62.3) ... ^H*{B, Bl U B~) -> H*{B, B~)^H*(Bl U B~, B'^H^^B, Bl U B " ) . . . , (see e#. Dubrovin & al. (1987)). We now show that i* is an isomorphism. Consider the diagram: (B'UB-.B") (B',B-nB')
-^>
(S,B-)
The excision theorem implies that i | is an isomorphism, and the continuity of the Cech cohomology implies that i\ is an isomorphism. Since the diagram commutes, i* must be
282
Appendix 2: TOPOLOGICAL TOOLS
an isomorphism. But this forces H*(B, B1 U B~) = 0 in (62.3). Taking the limit of this equality as t —* oo proves 2). Using the long exact sequence of the pair (B°°, I), the map I* induced by the inclusion 1:1—* B°° is an isomorphism whenever H*(B°°,I) = 0, which we proceed to show. Note that ^B00 c B°° and, by definition, / = D^o^iB00). Consider the maps:
(B00,^-*^00) C {(t>-tB00,4,'tB'x') +> (B00,^-^00), where j is the inclusion. The map j o 0 - t is clearly homotopic to Id, hence H"{BX, -*B°°, tfr'S00) = 0. Since this is true for all t, the continuity of the Cech cohomology concludes.
•
D. Proof of Proposition 61.4
To prove this proposition, we let the manifold M play the role of P in Lemma 62.4. The retraction a of that lemma is given here by the canonical projection a : B —> M. Clearly the projection of B onto M x D~ is a deformation retract, which deforms B~ onto M x a D - . Hence ff*(£, B~) ^H*(MxD~,Mx 3D-). Now, Kunneth Formula gives an isomorphism: u
H*(M)®H*(D-,dD~)^H*(MxD-,MxdD-) where, as suggested by the notation, one gets all of the classes in the right hand side vector space as cup products of classes in the two left hand side spaces (with the appropriate identifications given by the inclusion maps). But, letting n = dimD~, we have H*{D~,dD~) = H*(Sn, •), which has exactly one generator u in dimension n. Hence H*+n{B,B~) = H*{M) and sb(h(I)) = sb(M) where I is the maximal invariant set in B. This and Theorem 61.2 yield the Betti number estimate. The homeomorphism H*(M) = H*(B, B~) is of the type prescribed by Lemma 62.4. This implies that the induced map H*(M) -+ H*(I) is injective and hence cl(I) > cl(M). This fact and Theorem 61.2 give the cuplength estimate. D
63. Topological Proofs
283
E*. Floer's Theorem of Global Continuation of Hyperbolic Invariant Sets.
Floer's Lemma 62.4 is the cornerstone to the proof of the following theorem, where he makes good use of the powerful property of "invariance under continuation" of the Conley Index. This theorem illustrates the power of Conley's theory, and shows the historical root of Hoer Cohomology. Note that, in the theory of dynamical systems, the hyperbolicity of an invariant set for a dynamical system is intimately related to its persistence under small perturbations of the system: this relationship is the core of many theorems on structural stability. What is interesting about the following theorem (and Conley's theory in general) is that it provides situations when the (rough) persistence of an invariant set can be proven for arbitrarily big perturbations. The notion of continuation of invariant sets makes use of the simple following fact: an index pair for a flow <£* will remain an index pair for allflowsthat are C° close to '. Two isolated invariant sets for two different flows 0O> <j>\ are related by continuation if there is a continuous curve 4>\ of flows joining them which can be (finitely) covered by intervals (in A) of flows having the same index pair. The following theorem appeared as Theorem 2 in Floer (1987)to which we refer the reader for a proof. It can be seen as an instance of weak, but global, stability of normally hyperbolic invariant sets, which we now define. An invariant set G is normally hyperbolic for aflow0* on a manifold N if there is a decomposition:
TN\n=TG
®E+ ®E~
which is invariant under the covariant linearization of the vector field V corresponding to 4>* with respect to some metric (, ), so that for some constant m > 0: (62.4)
(f, DV£) < -m(t, f) (resp. > +m<£, 0 ) for f e ET (resp. E+)
Theorem 62.7 (Floer) Let
284
Appendix 2: TOPOLOGICAL TOOLS
In this precise sense, normally hyperbolic invariant sets continue globally: their topology can only get more complicated as the parameter varies away from 0.
63. Generating Phases Quadratic at Infinity A. Generating Phases On Product Spaces
The following proposition serves a key role in various proofs in this book, as well as in symplectic topology. Proposition 63.1 Let M be a compact manifold, and W a real-valued function on M x ]RK satisfying: 1
^
I'dW
\ (9 V) dSH =0
„4f-ooR(^ ' "
) '
where Q(v) is a nondegenerate quadratic form on 1R . Then W has at least cl(M) critical points. If W is a Morse function, then it has at least sb(M) critical points. The function W of Proposition 63.1 is a special case of a class of function called generating phases. We develop this notion in the next subsection. See also Chapter 10 for a symplectic viewpoint on generating phases. Proof. In an appropriate orthonormal basis ( e i , . . . , e#) of 1R , (Ql Q(v) = (Av,v)
with A-
'•.
V
\ J,
aKJ
and aj ^ 0 for all j . Let (vi,..., VK) be the coordinates of an element v 6 IR^ in this basis. We claim that: B(C) = {(q, v)eMxJRK
\ sup \vd\ < C}
is an isolating block for the gradient flow of z = -VW(z),
when C is large enough. To
prove this, note that B(C) is a compact neighborhood. Thus, in order to show that the flow exits in small positive or negative time at the boundary of B(C), it suffices to check that, on each "face" {VJ = C} of dB(C), the dot product of VW with the normal vector to this
63. GPQI
285
face is non zero. The same argument will apply to the face {VJ = — C}. The normal unit vector pointing out at a point z = (q, v) of {VJ = C} is ej. But: dW, . dQ - ^ ( * ) - e , = -^-ej
fdW dQ\ + ( ^ - — j .e, 1 fdW dQs
= C a +
{ i C\dv
dv
This last expression must be of the sign of aj, for large C as the last term inside the bracket tends to zero when C —> oo (we use (63.1) here: ||v|| is of the order of C). The same proof works for the face {vj = — C}, since the outward normal vector is — ej on this face. We have proved that, for all C larger than some Co, the set B(C) is an isolating block. The exit set (see Definition 61.1) B~ is the union of the faces {VJ = ±C} such that the corresponding eigenvalue aj is negative. Hence B = M x D+ © D~ where the disks D+, D~ are respectively the intersections of the positive and negative eigenspaces of Q with the set {sup,- \VJ\ < C}, and the exit set is B~ = M x D+ ® dD~. We are exactly in the situation of Proposition 61.4 which gives us the appropriate estimates for the number of critical points inside B. • B. Generating Phases on Vector Bundles
Definition. Proposition 63.1 is a cornerstone in the theory of generating phases. We now develop this theory a little and prove a generalization of Proposition 63.1 for functions on non trivial bundles which we will need. In Chapter 10, we indicate how this theory gives an approach to symplectic topology. Definition 63.2 A generating phase is a function
where E is the total space of a vector bundle E —• M and M is a manifold. If moreover W satisfies: (63.2)
lim -L(-^-(WQ)\ = 0, IMI-oo ||v|| \dvK J where, for each q, Q(q, v) is a nondegenerate quadratic form with respect to the fiber v, then we say W is a generating phase quadratic at infinity, abbreviated g.p.q.iS22^ 22
This definition is slightly more general than the one encountered in the literature, see Remark 63.8.
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Appendix 2: TOPOLOGICAL TOOLS
We show in Chapter 10 that the term "generating" refers to the fact that, provided they satisfy a generic condition in their derivative, generating phases generate Lagrangian manifolds of T*M. Generating phases are also called genera ting functions when associated to the Lagrangian manifold that they generate, or generating phase function. We show in Chapter 10 that twist maps generating functions are generating functions in this sense. Equivalence and Stabilization. We now define some elementary operations on generating phases. These will enable us to extend Proposition 63.1 to cover general g.p.q.i.'s. See Chapter lOfor the use of these operations in symplectic topology. In the following, assume that both Ei and E2 are bundles over the same manifold M. Definition 63.3 Let W\ : E\ —> IR, and W2 : E2 -* M be two generating phases. We say that W\ and W2 are equivalent if there is a fiber preserving diffeomorphism s? : Ei —* E2 such that:
W2o$ = Wi+ est. Definition 63.4 Let W1 : E1 -> IR be a g.p.q.i. The function W2 : Et © E2 -> IR defined by:
W2{q,v1,v2)
= Wi(q,vi) +
f(q,v2),
where / : E2 —»IR is a nondegenerate quadratic form in the fibers (i. e. in the variable V2), is called a stabilization
ofWj.
Proposition 63.5 If the generating phase W\ is equivalent to W2, or is a stabilization of W2 (or both) then the critical points of W\ are mapped bijectively into those of W2 and their sets of critical values are the same, up to a shift by a constant. Proof.
Let W2 o <£ = Wx + C as in Definition 63.3. Then, dWi = $*dW2
Hence the set of critical points of W\ is sent bijectively to that of W2 by
W2(q, vl,v2)
= Wi(q, «i) + f(q, v2)
63. GPQI
287
be as in Definition 63.4. Critical points of a generating phases W satisfy, in particular, dW/dv
= 0. But here,
dv
\ovi
0V2J
and since any point (q, 0) of E2 is critical for / , the critical points of W2 correspond exactly to those of W\. It is easy to see that the critical values of W\ and W2 are the same at the corresponding critical points.
D
Critical Points Estimates for GPQI's. Proposition 63.6 Let M be a compact manifold and W : E —• IR be a g.p.q.i. on a fiber bundle E —» M. Then W has at least cl{M) critical points. If W is a Morse function, Proof.
then it has at least sb(M) critical points. It is a corollary of Proposition 63.1 and of the following:
Lemma 63.7 Let W : E —> IR be a g.p.q.i. Then it is equivalent, after
stabilization,
to a g.p.q.i. g W : M x IR
of the base
—> K whose quadratic part Q is independent
point. Proof.
(We follow Theret (1999)) There exists a fiber bundle F such that E © F is trivial
(eg. take F to be the dual of E, see Klingenberg (1982)). Stabilize W by endowing F with a nondegenerate quadratic form Qi- Since E © F is trivial, there is a fiber bundle diffeomorphism $ : E © F —* M x TRK. A fiber bundle diffeomorphism being linear in each fiber, (W © Q 2 ) ° # _ 1 is a g.p.q.i. on M x IR^. We now show that any g.p.q.i. W(q, v) on a trivial bundle M x IR^ is equivalent to one with a quadratic part which is independent of the base point q. Let Q be the quadratic part of W and write Q(q, v) = (A(q)v, v), where ( ,) denotes the dot product on MK. Let JB+ © E~ = Eq be the decomposition of Eq into the positive and negative eigenspaces of A(q). If the fiber bundles E+ and E~ were trivial, the Gram-Schmidt diagonalization process would make Q equivalent to a constant quadratic form with matrix I
I on
E+ © E~, and the resulting fiber bundle diffeomorphism would make W equivalent to a
288
Appendix 2: TOPOLOGICAL TOOLS
g.p.q.i. such as we advertised. To arrive to this situation, stabilize Q\E+ (resp. Q\E-) to a positive definite Q+ (resp. negative definite Q~~) on a trivial bundle £+ (resp. £~). D Remark 63.8 Our definition of g.p.q.i. is more general than the one commonly found in the literature (i.e. Sikorav (1986), Laudenbach & Sikorav (1985), Chaperon (1989), Theret (1999), Viterbo (1992)). Usually one asks that W be equal to its quadratic part Q outside of a compact set. One can show (see Theret (1999)) that if W — Q is bounded outside of a compact set, then W is equivalent, after stabilization, to such a function quadratic outside a bounded set. It is not clear to us that the same would hold with our more general asymptotic condition. In that sense, Proposition 63.6 is slightly stronger than any we know of in the literature.
64.* Covering Spaces, Lifts and Fundamental Group Covering spaces. The simple notation T 2 = IR 2 /2 2 is rich in geometric and algebraic meaning. The quotient map proj : H 2 —> H 2 / Z 2 is an instance of a covering map. Such a map is a local homeomorphism such that each point in the torus has an evenly covered neighborhood U. This in turn means that proj-1!/ is made out of disjoint copies of U (as an example of evenly covered nneighborhood, take a disk of radius less than 1 around any point z). The covering map proj makes IR2 a covering space of T 2 . These notions generalize in a straightforward fashion to covering maps proj : M —> M, where M, M are topological spaces (often manifolds). Lifting of Curves. One can lift curves from a space M to a covering space M in a well prescribed way: if the curve 7 starts at ZQ in M, choose a point ZQ € proj~] (ZQ) C M to start the lift of 7, i.e. a curve 7 such that proj(j)
= 7. Above an evenly covered
neighborhood U of z0, there is only one way to define 7, since there is only one copy of U containing our choice ZQ. One then proceed by continuity, covering 7 with afinitenumber of overlapping evenly covered neighborhoods. A curve has as many distinct lifts as there are preimages of its starting point. Lifting of Maps and Deck Transformations. If / : N —* M is a continuous map on topological spaces, and M p^> M is a covering space, a lift f : N —> M of / is a
64. Covering Spaces
289
map that satisfies proj o f = f. If there is such a lift that maps a given point ZQ to some z\ 6 proj~1(f(zo)), it is unique. On the other hand, certain conditions must be satisfied for a lift to exist. But if N is simply connected (see below) the lift always exists. A very important example of lift is when N = M and / = proj. Then necessarily, a lift 4> of proj is a homeomorphism that permutes points in each fiber proj~l(z), for any z in M. The lifts of proj form a group under composition called the group of deck transformations, or also covering transformations. For instance, Z 2 is the group of deck transformations of the covering space H 2 —» T 2 . The most relevant example to this book is a notion of lift related to the above. Let N = M here and f : M —> M. Again, let M P™J M be a covering space. Then we call a lift of / a map / : M —* M such that proj o / = / o proj. In other words, / is a lift, in the old sense, of the map / o proj : M —* M. Again, given So G M and a choice of one element z\ in thefiberabove f(proj(zo)), there is a unique way to define a lift / such that f(zo) = z.\. Such lifts always exist if M is simply connected, i.e. when it is the universal cover of M (see below). Two lifts / i , /z of the same map / necessarily differ by a deck transformation <j>, in the sense that /a = <po / j . Classification of Covering Spaces for T 2 . We can construct other covering spaces of the torus than IR2, with groups of deck transformations other than Z. For instance, the cylinder JR. x S 1 = H 2 / Z is a covering space of the torus with deck transformation group Z. IR 2 /(2Z 0 3Z) -* IR 2 /Z 2 is also a covering space which is itself a torus, but "6 times as big" as the standard one it covers. It has Z2 © Z3 as group of deck transformations, reflecting the finite number of elements afiberproj~1 (z) has. In general, given any subgroup G of Z 2 , you get a covering space H 2 / G —• T 2 with deck transformations group Z 2 /G. In fact these are all the possible covering spaces of the torus! Universal Cover and Fundamental Group. The above classification of covering spaces generalizes to any manifolds, as we will see. Given a connected manifold M, the first step towards this classification is to find a covering space M which serves the role IR2 does for T 2 . It turns out that the defining feature of M2 in this context is that it is connected and simply connected: any loop in IR2 is homotopic to a point, or constant loop. This makes IR2 the universal cover of T 2 : it is the unique (up to homeomorphism) covering space of T 2 which is simply connected. Its uniqueness comes from a construction, which we now
290
Appendix 2: TOPOLOGICAL TOOLS
sketch on an arbitrary connected manifold M. Choose some point z0 on M. Declare that two curves starting at ZQ are equivalent if they have same endpoint and are homotopic. Define the covering space M as the set of all such equivalence classes. If [7] 6 M is one such equivalence class, define the covering map as proj([y]) = 7(1) (its endpoint). One can indeed show that, with the appropriate topology, this is a covering space (see Greenberg (1967) or Munkres (1975)). The deck transformations of this universal cover form a group called the fundamental group of M, denoted by 7ri(M, ZQ) or ir\{M) in short (changing the base point yields isomorphic groups). Since a deck transformation must permute points in a fiber, -K\ (M) is the group of all homotopy classes of loops based at a chosen point, with group law given by concatenation of two loops (i.e. follow one, then the next, which is possible since they have same endpoints). The inverse of a loop is the same loop traversed backwards. As an example, since Z 2 is the deck transformation for the universal cover 1R2 of T 2 , we must have TT^T 2 ) = Z 2 . More generally TT^T") = Z™. On the other hand ^ ( S " ) = {0}, since the sphere is itself simply connected (and thus it is its own universal cover).
Classification of Covering Spaces of any Manifold. As stated above in the case of the torus, we can use the universal cover M to classify all covering spaces of the (connected) manifold M: any other covering space TV of M is of the form TV = M/G where G is a subgroup of 7Ti(M). Furthermore, G = iri(N) and if G is a normal subgroup of 7r1(M), then the deck transformations of TV —> M form the group -rri (M)/G. [Remember that G is normal if aGa"1 — G for any a e TT\(M). As an example, any subgroup of TT\(T2) = Z 2 is normal, since Z 2 is abelian.] Fundamental Group vs. Homology. Note that ^ ( T 2 ) =* Z 2 9* Hi(T 2 , Z). This is not a coincidence: both groups were constructed as equivalence classes based on closed loops. In general, a theorem of Poincare (1895) says that H\ (M, Z) is the abelianization of -K\ (M): it is the fundamental group made commutative. To abelianize a group G one takes its quotient with the subgroup [G, G] of its commutators, which are of the form xyx~^y~x. Hence we can write Poincare's theorem as:
fli(M,Z) ^ ( M V M M ^ T T ^ M ) ] .
64. Covering Spaces
291
To see how the case M = T 2 fits here, note that Z 2 is already abelian. In general, ir\ (M) can be much more complicated than H\{M, Z ) . Finally, this leads us to an important case of covering space, called universal abelian cover of a manifold M . It is the covering M/[7Ti(M), 7ri(M)] -> M which, since the subgroup [TTI(M), ITI(M)] has (abelian) deck transformation group Hi(M, Z ) =
is always normal,
in(M)/[iri(M),n\(M)].
Exercise 64.1 Convince yourself, looking at Figure 59.2 that the fundamental group of the double torus is < a, 6,c, d;aba~1b~1cdc~1d~1 >, i.e. the group generated by the elements a, b, c, d together with the relation that aba~1b~1cdc~ld~1 = e, (e is the neutral element). What is the first homology group of the double torus? Repeat the question for surfaces of genus g.
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Index A A (annulus), 11 A (strip, covering of the annulus), 11 action, 19, 146, 247 action-angle variables, 79 action minimizer, 36 a-asymptotic, 276 a limit set, a ( x ) , 49 annulus, 11 anti-integrable, 197 Aubry diagram, 35 Aubry-Mather sets, 32, 44, 44 attaching a cell, 260 attaching map, 260 average action, 205
B base, 12 B (Mather's minimal action function), 206 Betti number, 266 billiard map, 25 Birkhoff orbit, 34 boundary twist condition, 13, 132 bouquet of spheres, 262 broken geodesic, 152
c A (cylinder) 11 canonical, - transformations, 256 - symplectic form, 234 Cantor set, 50 cantorus, cantori, 45 cell, 260 cellular space, 260 chain complex, 264 CO, cyclically ordered, 34 cyclically ordered orbits, 34 coercive function, coercion condition, 32 configuration space, 233 cohomology, 267 cohomology Conley index, 274, 278 coisotropic, - submanifold, 242 - subspace, 236 conjugate, - coordinates, 243 topologically -, 33 Conley index, 278 contractible, 289 convexity -of Lagrangians, 200 completely integrable, 1, 22, 239 completeness (of E-L flow), 200 cotangent, - bundle, 243 - vector, 243 covering, - map, 11, 288
- space, 288 - transformation, 289 critical point, 267 crossing of sequences. 35, 56 cuplength, 274 cup product, 267 CW complex, 260 cyclically ordered, CO, 34 cylinder, 11
D deck transformation, 289 degree, 11, 262 <5jk, Kroenecker symbol, 243 DeRham cohomology group, 267 dimension, -of a cell complex, 260 distance, 148
E elliptic - fixed point, 23 - periodic orbit, 239 embedding, 13, 98 equivalent (gpqi's), 286 ergodic, 203 Euler characteristic, 266 Euler-Lagrange (E-L), - equations, 248 - flow, 200 evenly covered, 288 exact Lagrangian, 135 exact symplectic, - form, 245 - manifold, 245 - map 15, 245 exit set, 274 exponential map, 147 extended phase space, 251 extreme points, 204
F factor, 33 fiber, 12 - critical points (manifold of), 225 - Intersection Property, 187 filtration, 278 finite cell complex, 260 flux, 14 first integral, 252 P-ordered, 41 free homotopy, 184 fundamental group, 289
G generalized Mather sets, 207 generating function - of a twist map, 17 - of a symplectic twist map , 99 - of a Lagrangian manifold, 247, 285
generating phase, 284, 285 - for Lagrangian manifolds, 225, 247 - quadratic at infinity, 285 generic, 112 geodesic, 152 - flow, 147 ghost torus, 117 gpqi, 285 gradient, 268 gradient flow, 268
- transformation, 248 Lie derivative, - of a function 252 - of a form, 253 lift, 288 linear domain, 207 - embedding condition, 154 global - condition, 154 - transformation, 248Liouville form, 244 linking number, 189 linking of spheres, 189
H
M
Hamiltonian, - equations, 249, 251 - function, 248 - map, 251 - vector field, 250, 251 height, 71 heteroclinic orbit, 49 homoclinic orbit, 49 homology group, 264 homotopic, 263 homotopy, 263 - formula, 253 - type, 266 hyperbolic fixed point and periodic orbit, 23, 239 hyperbolic manifold, 152
Maslov - class, 133 - index, 232 maximal - invariant set, 274 - skeleton, 65 minimally linked (skeleton), 68 minimal measure, 206 minimal set, 49 minimizer, -for twist maps, 36 -for a Lagrangian system, 201 minimizing orbits, segments, 36 (m, n), - minimizer, 36 -(periodic) orbits, 20 - sequences, 20 m , d-periodic point, 103 ML (E-L invariant measures), 205 monotone orbits, 34 - of the gradient flow, 63 monotone flow, 53 monotone recurrence relation, 196 Morse decomposition, 276 Morse function, 268 Morse index, 268 Morse-Smale function, 272
I index pair, 278 induced, 12 injectivity radius, 177 integral invariant of Poincare, 255 intersection index, 56 involutive, 249 isolating block, 274 isolated invariant set, 278 isotopy, 251 isotopic to Id, 16 isotropic, - subspace, 236 - submanifold, 242
N nondegenerate -periodic orbit, 111 -critical point, 267 normally hyperbolic (invariant set), 283 null (subspace), 236
Jacobi matrix, 64
K Kroenecker Symbol, 5jk, 243
L Lagrangian, - function, 247 - subspace, 236 - submanifold, 242 Legendre - condition, 154, 154, 248
oj-asymptotic, 276 Omega limit set u>(x), 49 optical Hamiltonian system, 159 orbit, 18 order interval (in sequence space), 55 orthogonal subspace, 234 parabolic, 239 partial order (on sequences), 34 periodic action, W m n , 20
P C O „ , periodic CO sequences 73 phonon gap, 120 7ri(M,*o), T i ( M ) , 289 Poincare-Melnikov function, Poincar6-Melnikov , 139 Poincare polynomial, 278 pointwise convergence, 34 Poisson bracket, 252 polar decomposition, 241 Positive cone, 55 principle of least action, 247 proj (projection from covering space), 11 pseudograph, 130 •0 (twist change of coordinates), 13 pull-back (of a map), 243 ratchet phenomenon, 41 recurrent, 39 reduced symplectic space, 229 region of instability, 141 related by continuation, 283 residual, 112 residue, 119 retraction, 280 Riemannian manifold, 146 rotation number, 21 backward and forward - , 2 1 - of a point, 21 - of a ghost circle, 60 rotation vector -of a curve on a manifold, 205 -of a curve on a hyperbolic manifold, 214 -of an invariant measure, 203, 204 -of a periodic orbit of a map, 103 -of a sequence, 103 S , circle, 11 semiconjugacy, 33 shift map, 54 shear map, 22 simply connected, 289 skeleton - of critical sequences, 65 - of a CW-complex, 260 space average, 203 stable, - fixed points, 239 - manifold, 239 - norm, 210 stabilization (of gpqi's), 286 standard map, 21 generalized - (on hyperbolic manifolds), 153 strip, 11 sublevel sets, 269 subsolution (for a monotone recurrence relation), 196 sum of the Betti numbers, 274 superlinearity, 200
superquadraticity, 213 support (of a measure), 202 symplectic - capacity, 231 - form, 15, 233, 241 - group, 238 - isotopy, 253 - map, 15, 236, 238 - structure, 241 - vector space, 234, 236 - twist map, 3, 88, 99 - reduction, 230 T, integer translation, 11 T n , torus, 88 tangency (of sequences), 56 tilt map, 20 time average, 201 Tonelli's theorem, 201 topological stability, 33 trace, 119 transverse intersection - of sequences, 56 - of manifolds, 225 trimming, 130 twist condition, - for annulus maps, 1, 12 - vs. gen. functions, 18 - for maps of T * T n - for maps of T*M, 99 twist map - of the annulus, 12 - positive or negative. 13 symplectic, 3, 88, 99
u U L , 206 unitary group, 240 unit cotangent bundle, 147 universal cover, 289 universal abelian cover, 290 unstable manifold, 269, 239 Variation equation, 160
w weak* limit, 203 well ordered (orbit), 34 W m n , 20 - minimizer, 36 A
X m , n , (m,n sequences), 20 X , (m, dN sequences), 107 X , quotient of X , 108
