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Ergodic properties of \(q\)-multiplicative sequences. (Propriétés ergodiques des suites \(q\)-multiplicatives.) (French) Zbl 0853.11064

A sequence \((u (n))_{n \geq 0}\) with values in the multiplicative group of complex numbers of modulus 1 is said to be \(q\)-multiplicative if, for every positive integer \(t\) and for every non-negative integers \(a\) and \(b\) with \(b< q^t\), one has \(u(aq^t+ b)= u(aq^t) u(b)\). By studying the uniform convergence of exponential sums, the authors of this paper give a characterization of the strictly ergodic \(q\)-multiplicative sequences and of the \(q\)-multiplicative sequences \(u\) which are good sequences for the ergodic theorem, i.e., for every continuous function \(\rho\), \(\rho (u)\) is a good sequence of weight for the ergodic theorem. In particular, good sequences are proved to be either of empty spectrum (this case has been studied by the authors and B. Mossé in [Compos. Math. 93, 49-79 (1994; Zbl 0818.28006)]) or Besicovitch almost periodic sequences. Furthermore, the authors show the independence of these two notions (strict ergodicity and good sequences) by exhibiting some examples of sequences.

MSC:

11K31 Special sequences
28D05 Measure-preserving transformations

Citations:

Zbl 0818.28006
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References:

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