The egodic theorem along a \(q\)-multiplicative sequence. (Le théorème ergodique le long d’une suite \(q\)-multiplicative.) (French) Zbl 0818.28006

Lesigne, E.; Mauduit, C.; Mosse, B.

Compos. Math. 93, No. 1, 49-79 (1994).

The authors study in this paper the \(q\)-multiplicative sequences as weight sequences for the ergodic theorem. We recall that a \(q\)- multiplicative sequence has the following property: \(\forall t\geq 1\), \(\forall a\geq 0\), \(\forall b< q^ t\), \(u(aq^ t+ b)= u(aq^ t) u(b)\).
The authors first show that the fibers of a \(q\)-multiplicative sequence with modulus 1 and taking only finitely many values are universally good for the ergodic theorem, i.e.: if \(\lambda\) is such that the set \(u^{- 1}(\lambda)= \{n_ 1< n_ 2< n_ 3\cdots\}\) is not empty, then, for every probability space \((\Omega, {\mathcal A}, \mu)\), for every measurable and measure-preserving map \(T\) of \(\Omega\) and for every \(f\in L^ 1(\mu)\) the sequence \(({1\over N} \sum^ N_{k= 1} f\circ T^ n k)\) converges in \(L^ 1(\mu)\) and almost everywhere.
Their second main result reads as follows: Let \(u\) be a \(q\)- multiplicative sequence taking its values in a closed subgroup \(G\) of \(\{| z|= 1\}\) such that for every non-trivial character \(\chi\) of \(G\) the sequence \(\chi(u)\) has an empty spectrum. Then, for every probability space \((\Omega, {\mathcal A}, \mu)\), for every measurable measure-preserving map \(T\), for every \(f\in L^ 1(\mu)\) and for every continuous function \(w\) on \(G\), the sequence \({1\over N} \sum^{N- 1}_{k= 0} w(u(h)) f\circ T^ h\) converges in \(L^ 1(\mu)\) and almost everywhere. Furthermore, if \(\int_ G w(g) dg= 0\), then this sequence tends to 0.
Note that a recent preprint of the first two authors gives more precise results.

Reviewer: J.-P.Allouche (Marseille)

MSC:

28D05	Measure-preserving transformations
11K70	Harmonic analysis and almost periodicity in probabilistic number theory
11B83	Special sequences and polynomials

Keywords:

\(q\)-multiplicative sequences; weight sequences; ergodic theorem; measure- preserving map

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