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

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.

### MSC:

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

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