0-1-Sequences of Toeplitz Type

Z. Wahrscheinlichkeitstheorie verw. Geb. 13, 123-131 (1969)

0-1-Sequences of Toeplitz Type KONRAD JACOBS a n d M I C H A E L K E A N E

Summary. 0-1-sequences are constructed by successive insertion of a periodic sequence of symbols 0, 1 and "hole" into the "holes" of the sequence already constructed. Assuming that finally all "holes" are filled with symbols 0, 1, an almost periodic point in shift space results. Under certain conditions, it is even strictly ergodic. It is proved that the attached invariant measure has pure point spectrum, and a rather explicit expression for eigenvectors is obtained.

Introduction A device used by Toeplitz [7] for the explicit construction of almost periodic functions on the line is generalized and modified such as to yield a construction method for sequences of zero's and one's (w3). The sequences thus obtained are called Toeplitz sequences. They are all almost periodic (Theorem 4). A known example of Oxtoby [-6] fits into our theory and shows that Toeplitz sequences are not always strictly transitive; other examples could be provided easily. Theorem 5 gives a sufficient condition (called regularity) for the strict transitivity of a Toeplitz sequence. In order to elucidate the technique applied here, we develop in w167 1 and 2 a theory of uniform summability and quasi-uniform convergence in a bit more general fashion than would be needed for Theorem 5. In w4 we prove that the spectrum of the unique invariant measure in shift space which is attached to a regular Toeplitz sequence, is discrete and even rational. For other construction methods for strictly transitive 0-1-sequences see Kakutani [-4] and Keane [-5]. In spite of Hahn-Katznelson [2] and Jacobs [3] the search for a "machinal" construction method which yields strictly transitive 0-1-sequences with a continuous spectrum for the attached invariant measure seems to have failed as yet,

w 1. Uniform Summability and Quasi-Uniform Convergence Let F be the linear space of bounded complex-valued functions on the group of integers Z. For f 6 F we set []f II = sup If(z)l, z~77

1 k+n-1

Sk( f, n ) = - I'1

~

f(z)

(k6Z, n6N),

z= k

[l/][* = sup ISk(f, n)l

(nEN),

k~Z

IIf [[* = lim tlsup IIf I[*. We shall say that f s F is uniformly summable with sum f if f = lim Sk( f, n) n

124

K. J a c o b s a n d M. K e a n e :

exists unformly in ks7l. Denoting the collection of uniformly summable functions by L, we see that L is a linear subspace of F, the map f ~ f is a linear form on L, and f e F belongs to L if and only if there exists a constant a ( = f ) such that

Ijf-~ll*=0. Suppose that f, fl, f2, ... e F. Then fl, fa, ... converges quasi-uniformly to f if 1) limfj(z)=f(z)(zeZ), and J

2) lim I I f - f j l l * = 0 . J

The first condition is made to assure uniqueness of a quasi-uniform limit, and we shall not use this condition in the following proof.

Theorem 1. If the sequence fl, f2 .... of uniformly summable functions converges quasi-uniformly to the function f, then f is uniformly summable and f = lira fj. J

Proof. First we see that e -- lira fj exists, since for any two uniformly summable functions g and h we have J Ig,-hl = tlg-hll * Let s > 0 be arbitrary. Then there exists a j e N such that and IIf-f~H* <e. The latter inequality and uniform summability of fj imply that there exists an n o e N such that for all n>n o and all ksZ,

Ifs-S~(fj, n)l <5 and

Ilf-~ll*<~. Since

lif-fjll~*

= sup keZ

ISk(f, n)--Sk(f j, n)l,

the following estimate is valid for all n>n o and all k6Z: I ~ - Sk(f, n)l_--
f= ~ = lim j~.. J w 2. Quasi-Uniform Convergence in Compact Systems Let X be a compact metric space and ~ a homeomorphism of X. The pair (X, cp) is called a compact (dynamical) system. If X o is a non-empty compact ~p-invariant subset of X, then (X o, ~p) is called a subsystem of (X, ~p), where we make no distinction between cp and its restriction to X o. We denote the orbit closure of x e X by

0(x)= {r

0 - 1 - S e q u e n c e s of T o e p l i t z T y p e

125

Subsystems of (X, ~o) can be minimal in two respects, topologically and measure theoretically. A subset X o of X is called minimal (-topologically minimal) if (Xo, q~) is a subsystem of (X, q~) having no proper subsystem. It is easy to see that X o is minimal if and only if 0 (x) = X o for each x ~ X o . The basic criterion for minimality of an orbit closure was given by Gottschalk [1]: 0(x) is minimal if and only if x is almost periodic, i.e. for any neighborhood U of x there exists an L e N such that for any z e Z at least one of the points ~0~(x),~0z+l(x).... , (pz+L(x) belongs to U. In particular, x is almost periodic if for any neighborhood U there exists an L E N such that ~oZL(x)~U for all zeZ, that is, if x returns periodically to any neighborhood. A subset X o of X is called uniquely ergodic (-measure theoretically minimal) if (Xo, q)) is a subsystem and if there exists exactly one qo-invariant probability measure #o on X such that/~o is concentrated on X o (Ito (Xo) = 1). Of course,/*o is ergodic. For any function f on X and x e X , we define the function fxeF by

f~(z):= f(~pZ(x))

(z~Z).

A point x e X is called strictly transitive if fx is uniformly summable for each continuous function f on X. This gives us a useful condition for unique ergodicity of orbit closures: 0(x) is uniquely ergodic if and only if x is strictly transitive (see Oxtoby [6]). If 0 (x) is uniquely ergodic, let us denote the unique ~o-invariant probability measure concentrated on 0(x) by #x. Now q0-invariant measures and (p-invariant linear forms on the space of continuous functions stand in one-to-one correspondence with each other, and f ~ f ~ is such a linear form. Thus we obtain a method for calculating the integrals of continuous functions f by means of Cesaro sums:

5 f d#x=J~. Putting the two concepts of minimality together, a subset of X is called strictly ergodic if it is minimal and uniquely ergodic. Now let (X, qg) be a fixed compact system and denote by [., .] a bounded metric on X, For x, yEX and 6 > 0 define the functions dx, y and fix,y in F by

dx, y(z)= I~o~(x), q~Z(y)l

(z ~TZ),

6x, y(z)= l {ex,, _>_a}(z)

(zeZ).

We say that the sequence x,,x= .... e X converges quasi-uniformly to x e X if d .... , dx, ~2, "" converges quasi-uniformly to 0. Theorem 2. xl, x2, ... converges quasi-uniformly to x if and only if lim x j = x and lim 1{6x,x~I{*= 0 for each 6>0. a J

Proof. Since ~o is continuous, we have limd x' x,(z)=0 (zeT7), if l ijm x ; = x . j If M denotes a bound for the metric on X, the inequalities a. ll~x,x,II*< ]ldx,x, ll* < a + M [tax,x, ll* for each 6 > 0 and j e N imply the equivalence of lim IId~, x~ll*= 0 and lim ]1fix,~jl]* = 0 (6>0). J a

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K. Jacobs and M. Keane:

Theorem 3. I f xl, x 2 . . . . is a sequence of strictly transitive points which converges quasi-uniformly to x, then x is strictly transitive. Proof Because of Theorem 1, it suffices to show that fx~, fx2, "" converges to f~ quasi-uniformly for each continuous function f Now lim x j = x implies lim f~j(z)=fx(Z), since f and q~ are continuous. J J

We set ]l/H = s u p [f(y)[. For e > 0 choose a 3 > 0 such that y , y ' e X , [y,y'[<6 yeX

implies I f ( y ) - f ( y ' ) [ < f .

Choosing j and n so large that I[~,xj[[,

~

for

each n > n o, we see that for n > n o 1 k+n-1

[]fx-f~jll*=sup ~

]

Z

k~Z

{f(q;(x))-f(cp'(xj))}

t=k

1

__<sup-keTl n

k+.-i

Y,

If(<~'(~))-f(~o'%))l

t=k 6~, x i(t)= 0 l

k+n-1

+ sup--

Z

k~Z

t=k 5x, x j ( t ) = l

n

Ifko'(x))-f(~o'%))l

-<--s + 2 llfll 9 IlG, x~ll* <e. -2

It follows that l l f x - f , jN* < s for large j, and f~,, fx2, ... converges to fx quasiuniformly. It would be interesting to know whether a quasi-uniform limit of almost periodic points is almost periodic, either with or without the assumption of strict transitivity.

w3. Toeplitz Sequences We now want to construct examples of strictly ergodic systems using the results of w2. Set f2=I- [ {0, 1} = {(0=(..., (0_1 , (00, (01 . . . . )l(0kE{O, 1} ( k E ~ ) } . Z

Provided with the product topology, • is a compact totally-disconnected Hausdorff space, a metric inducing the topology being given by 1(0, ~/1= 2-1ne{I"ll'~"*n")

((0, ~/~Q).

(,)

An open and closed base for the topology is given by the finite-dimensional

cylinders k[b] = {(0~f~l(0k=bl ..... cok+,_ 1 =b,}, where k runs over I and b = bl...b, is any block (i. e. finite sequence of zeroes and ones). The shift T on f~ is defined by

(T(0)k=~k+ 1

(k~Z);

0-1-Sequences of Toeplitz Type

127

since T takes cylinders into cylinders, it is a homeomorphism of f2. Similarly we may construct ~ = [ - I {0, 1, ~ } = {~=(...,O5o, ~ . ...)l~ke{O, 1, ~ } (k~Z)} Z

with the shift T. Then (*) defines also a metric on ~. Obviously ~2 may be considered as a compact T-invariant subspace of t~, whereby T induces T on t2. Let ch and ~/be any two points of i}. We say that o5 is a completion of ~ if, for each keZT,/'/ke {0, 1} implies Chk=~lk. Pictorially, one may imagine ~ as a sequence of zeroes and ones in which there are some "holes" (i. e. oo's). Completions of ~/ are obtained by filling some of these "holes" with zeroes and ones. We shall be especially interested in the case where c5 and ~ are periodic. In this case we obtain completions of ~/ by inserting another periodic sequence into the "holes" of ~. For example, if ~/= (..., 0, oe, 0, oe, ...) with period 2, and if we insert the periodic sequence (.... 1, Go, 1, 0% ...) with period 2 into the holes of ~, we obtain a completion o5=( .... 0, 1, 0, oo, 0, 1, 0, oo, ...) with period 4. In the following we shall drop the notation . . . . . for points of s and for the shift, since no confusion will result. Now let tf ~), ~/(2), ... be a sequence of periodic points of ~ such that 1. t/(j+l) is a completion of~/(j) for e a c h j e N , and 2. for each ke7Z there exists a j e N such that t/~J)e{0, 1}. Then for each k e g , ~/(k j) is either constantly zero or one for all largej and we may define a limit sequence tlk = lim q(J) J

(k eZ)

with t/e f2. r/is obviously a completion of t/~j) for each j eN. Sequences t/obtained in this manner are called Toeplitz sequences, because of the similarity of their construction with a construction in Toeplitz [7]. Theorem 4. Every Toeplitz sequence is almost periodic.

Proof Let r/be a Toeplitz sequence and k[b] a cylinder with qek[b]. Because of 2., there exists a j with tlcJ)ek[b], and due to the periodicity of q(J) and the completion property, q returns to ,[b] periodically under T. Thus q is almost periodic. Now we shall separate out a class of Toeplitz sequences which are strictly transitive. Let us denote for periodic points e ) e ~ the relative frequency of 0o in co by roo(co), i.e. if co has period n, then r~((~) = 1~. (number of cc's among coz, ..., co,). n

A Toeplitz sequence t/ is called regular, if it can be obtained by approximating sequences t/(j) (jeN) such that lira % (17(j)) = 0. J

128

K. Jacobs and M. Keane:

Theorem 5. Every regular Toeplitz sequence is strictly transitive. ProoJl Let r/ be a regular Toeplitz sequence with approximating sequences q(J) (jeN) such that lira r~(~/~J))=0. It is trivial that the periodic sequences ry ) J are strictly transitive. Therefore it suffices to show that ~/(~),t/(2).... converges quasi-uniformly to t/, because of Theorem 3. For n e N and periodic cos f2 with period p, we set r2o(co) I{kll --< k___ p, cok+n. = oo for at least one n' with - n < n' < n}[ P r~ (co) is the relative frequency of occurrence of an oe within n steps of the given place k in co. We have

r~(co)<(2n+ l)r~(co)

(co~(]),

so that for any n6N, lira r~ (t/(J))= O. J Now let n be fixed. Then certainly because of completion,

d,, n(j>(k) = ]r k rl, T k t/(J)[< 2-" if oe does not occur within n steps of k in t y ). It follows, since t/(j) is periodic and thus all relative frequency calculations only depending on t/~j) are uniform, that [Id., .,j, II* < 2- n + rn (t/(j))

(the metric on f2 is bounded by 1). Hence lim IId,, ,,~,II* = 0 J

and t/(~), t/(e).... converges quasi-uniformly to r/. Corollary. I f tI ii a regular Toeplitz sequence, then O(tl) is strictly ergodic.

w4. The Spectrum of Regular Toeplitz Sequences In this paragraph we prove that if r/is a regular Toeplitz sequence, then the transformation induced by T on the Hilbert space L 2 of square-integrable functions with respect to p~ has discrete spectrum. We denote this transformation also by Z Let r/ be a regular Toeplitz sequence and r/(1), t/~2), ... an approximating sequence with lira r~ (r/(J))=0. Suppose that the smallest period of ~/r is Pi" We J modify t/(j) in the following manner: 1. For every k with 1 <_k<_pj and r/~j)= ~ , inspect the sequence (..., t/k, r/k+., ~lk+2pj,"')" If this sequence contains both a zero and a one, we do not change r/(3. If however this sequence consists only of zeroes, then we replace the ~ ' s in 9

IIk_pj,

Ilk

~ rlk4-lyj~

9

by zeroes. If it consists of one's only, we replace the oe's in it by one's.

0-1-Sequences of ToeplitzType

129

2. After having completed 1. for each 1 < k < pj, determine the smallest period p~ of the modified sequence. If p} < pj, then we repeat 1. using pj in the place of pj. If p} = pj, then the modification is finished. For each j ~ N we modify the sequence /7(J) in the above manner until no further modification is possible. The end state will be reached for each j after a finite number of steps. Let us denote by nj the smallest period of the modified sequence/7(J) ( j e ] N ) , Then/7(J) possesses the property that if 1 <_k <_nj and/7(J) ~--- GO, then there exist mo and m~ such that /7k+mo,j=0, /7k+m~,j=l. Also it is obvious that /7(~),/7(2), ... still converges quasi-uniformly to r/ after modification and li.m r~o(/7(J))= 0. Moreover n~~ oo iff 7? is nonperiodic, and/7(~)=/7(2) . . . . . /7 iff 17 J

is periodic. Theorem 6. T has discrete spectrum on L2 . Proof. 1. We shall begin by constructing eigenfunctions of T. Let j e N be fixed.

a) There exists an e > 0 such that if ITk/7, Tk'/TI <e, then k=-k ' rood nj. To see this, we choose an i e N such that for each I with 1 < l < n i and/71i)= oo there exist moz and mlt such that 1 < l + m o l n j < n i, 1 < l + m l l nj
This is possible because of the modification we made above. Now set e = 2 -'~"j and let k, k'e2g with ]Tk/7, Tk'/71 < e . Then /Tk,/Tk+l' ' " , / 7 k + n i n j - l = / 7 k ' ,

/7k'+l, " ' , / T k ' + n i n j - 1 ,

and thus if l<_l<_nj and '~k+l, "(j) V(J)tk'+le{O,1}, then 'lk+l "(j) =/7(kJ)+z" However, if 1 < l-- nj and -(J) ,k+~ = o0, then by the choice of i there exist mo and m 1 with 0=< l + m o nj <=n i nj-- 1, 0 -<- l + m 1 nj ~ n~ n~- 1, and ,(i) r l k + l + m o n j q4-,(0 ' - q k + l + m l nj k ~l+mon d,'lk+l+mlnj

t

It follows that /Tk + l +mo nj=l=/7k + l +ml nj

and thus /7k" +t +monj=~ /Tk' +l +m~ nj 9

Therefore/7[J)+l= oe. This implies Tk/7(J)= Tk'11 (j). Since nj is the smallest period of/7(J), we have k-= k' rood nj. b) We set for 0 < k < nj Ak : = { Tk +""J (/7)1m ~7Z}.

Then Age_O(~7) and it follows from a) that 1)

Ak~Ak,=~)

(k4:k').

nj -- 1

2) U

k=O

3) T A k = A k + a ( m o d nj) (O<=k
130

K. Jacobs and M. Keane:

Now set 2k:=exp (2~ i k ) (o<-<_k
fk = ~ J.~1a,

(O<=k
n=O

Then nj - l

rfk = ~' "~ 1a.-1 + 1a.j-1 n=l nj-1

='~k F, ;o~ 1a. =,~ L n=0

and

fk is an

eigenfunction with eigenvalue 2 k (0 ~ k < nj).

2. We now show that the sets of eigenfunctions constructed in 1. (for all j) span the space L 2 . Since the functions 1A~ (0_--__k < nj) can be written as linear combination of thefk's, and since the cylinders form a base for the topology on t2, it obviously suffices to show that cylinder sets can be approximated by the Ak'S. Let b=bl ..... b, be a 01-block and kE2g. Set B--k[b ]. For fixed j, we form n)-i

U A,, k=O Ak ~_ B

where the Ak'S are the sets constructed in 1. a). Then and closed, and thus

Bj~_B, Bj and

B are open

/~. (B- Bj) = ~ 1 B d/~.- ~ 1Bj d#. = lim 1 t-~m

t

= lira 1

t~ t

i [l~(TSt/) - 1Bj(TSt/)] s=l

(number o f s with

l<_s<_t and TSqeB-B).

Now TStleB-Bi only if one of the symbols .is '~(j), ,is+1, "tj) ..., ,s+,-l"~J) is an o% since if all of t/~j), " ' ' , ~ l-(J) s + n - - 1 is 0 or 1, then T s+,"j t/eB (v = 0, + 1.... ) and thus A s _ B. Therefore

~,,(B- B)_-
lira #~(B - B j) = O. J

Therefore the eigenfunctions constructed in 1. span L 2 and T has discrete spectrum.

0-1-Sequences of Toeplitz Type

131

References 1. Gottschalk, W.: Almost periodic points with respect to transformation semigroups. Ann. of Math., II. Ser. 47, 762-766 (1946). 2. Hahn, F., and Y. Katznelson: On the entropy of uniquely ergodic transformations. Trans. Amer. math. Soc. 126, 335-360 (1967). 3. Jacobs, K.: Syst~mes dynamiques Riemanniens (/t paraRre). 4. Kakutani, S.: Ergodic theory of shift transformations. Proc. V. Berkeley Sympos. math. Statist. Probab. II, 2, 405-414 (1967). 5. Keane, M.: Generalized Morse sequences. Z. Wahrscheinlichkeitstheorie verw. Geb. 10, 335-353 (1968). 6. Oxtoby, J. C.: Ergodic sets. Bull. Amer. math. Soc. 58, 116-136 (1952). 7. Toeplitz, O.: Beispiele zur Theorie der fastperiodischen Funktionen. Math. Ann. 98, 281-295 (1928). Professor M. Keane Yale University Department of Mathematics Box 2155, Yale Station New Haven, Conn., USA

Professor Dr. K. Jacobs Mathematisches Institut der Universit~it Erlangen-Niirnberg 8520 Erlangen Bismarckstrage 189

(Received October 5, 1968)

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