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 _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

126

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'[