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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, \(| z| =1\).

37A99 Ergodic theory