Ergodic averages with deterministic weights
[Moyennes ergodiques pondérées par des poids déterministes]
Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 561-583.

Nous étudions la convergence de moyennes ergodiques 1 N k=0 N-1 θ(k)fT u k (θ(k)) k est une suite bornée et (u k ) k une suite d’entiers strictement croissante tels que Sup α | k=0 N-1 θ(k) exp (2iπαu k )|=O(N δ ) avec δ<1. De plus nous donnons des exemples explicites de telles suites θ et u puis, nous nous intéressons aux cas où θ est une suite q- multiplicative.

We study the convergence of the ergodic averages 1 N k=0 N-1 θ(k)fT u k where (θ(k)) k is a bounded sequence and (u k ) k a strictly increasing sequence of integers such that Sup α | k=0 N-1 θ(k) exp (2iπαu k )|=O(N δ ) for some δ<1. Moreover we give explicit such sequences θ and u and we investigate in particular the case where θ is a q-multiplicative sequence.

DOI : 10.5802/aif.1894
Classification : 37A05, 28D05, 11K99
Keywords: weighted ergodic averages, central limit theorem, almost sure convergence, $q$-multiplicative sequences, substitutive sequences, generalized Thue-Morse sequences
Mot clés : moyennes ergodiques pondérées, théorème ..., convergence presque sûre, suites multiplicatives $q$, suites substitutives, suite de Thue-Morse généralisées
Durand, Fabien 1 ; Schneider, Dominique 2

1 Universidad de Chile, Centro de Modelamiento Matemático, Casilla 170-3, Correo 3, Santiago (chili) \& Université de Picardie Jules Verne, Faculté de Mathématiques \& d'Informatique, Pôle de Saint-Leu, 33 rue Saint-Leu, 80039 Amiens Cedex 1 (France)
2 Université de Picardie Jules Verne, Faculté de Mathématiques \& d'Informatique, Pôle de Saint-Leu, 33 rue Saint-Leu, 80039 Amiens Cedex 1 (France)
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Durand, Fabien; Schneider, Dominique. Ergodic averages with deterministic weights. Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 561-583. doi : 10.5802/aif.1894. http://archive.numdam.org/articles/10.5802/aif.1894/

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