# zbMATH — the first resource for mathematics

Temps de retour pour les systèmes dynamiques. (Return times for dynamical systems). (French. English summary) Zbl 0655.58022
Summary: Let ($$\Omega$$,$${\mathcal B},\mu,T)$$ be a dynamical system, $$\mu$$ a finite measure, and assume T ergodic. Let $$A\in {\mathcal B}$$ be a set of positive measure. It is shown that for almost all $$\omega\in \Omega$$, the return- time sequence $$\Lambda_{\omega}=\{n\in {\mathbb{Z}}_+|$$ T $$n\omega\in A\}$$ is a good sequence for the pointwise ergodic theorem. Similarly as in the author’s papers [C. R. Acad. Sci., Paris, Sér. I 305, 397-402 (1987; Zbl 0634.28008); Isr. J. Math. 60 (to appear); Israel Functional Analysis Seminar 87, Springer Lect. Notes Math. (to appear)], this result is obtained from certain inequalities relative to the shift model ($${\mathbb{Z}},S)$$ and proved by methods of Fourier analysis. In particular, the argument is of finite nature and gives precise information about the structure of the sequences $$\Lambda_{\omega}$$, namely the behaviour of the associated polynomials (1/n)$$\sum_{j\leq n}(T$$ $$j\chi_ A)(\omega)z$$ n, $$| z| =1$$.

##### MSC:
 37A99 Ergodic theory
##### Keywords:
dynamical system; finite measure; ergodic; Fourier analysis