Ergodic averages with deterministic weights
Annales de l'Institut Fourier, Volume 52 (2002) no. 2, p. 561-583

We study the convergence of the ergodic averages $\frac{1}{N}{\sum }_{k=0}^{N-1}\theta \left(k\right)f\circ {T}^{{u}_{k}}$ where ${\left(\theta \left(k\right)\right)}_{k\in ℕ}$ is a bounded sequence and ${\left({u}_{k}\right)}_{k\in ℕ}$ a strictly increasing sequence of integers such that ${\mathrm{Sup}}_{\alpha \in ℝ}|{\sum }_{k=0}^{N-1}\theta \left(k\right)\mathrm{exp}\left(2i\pi \alpha {u}_{k}\right)|=O\left({N}^{\delta }\right)$ for some $\delta <1$. Moreover we give explicit such sequences $\theta$ and $u$ and we investigate in particular the case where $\theta$ is a $q$-multiplicative sequence.

Nous étudions la convergence de moyennes ergodiques $\frac{1}{N}{\sum }_{k=0}^{N-1}\theta \left(k\right)f\circ {T}^{{u}_{k}}$${\left(\theta \left(k\right)\right)}_{k\in ℕ}$ est une suite bornée et ${\left({u}_{k}\right)}_{k\in ℕ}$ une suite d’entiers strictement croissante tels que ${\mathrm{Sup}}_{\alpha \in ℝ}|{\sum }_{k=0}^{N-1}\theta \left(k\right)\mathrm{exp}\left(2i\pi \alpha {u}_{k}\right)|=O\left({N}^{\delta }\right)$ avec $\delta <1$. De plus nous donnons des exemples explicites de telles suites $\theta$ et $u$ puis, nous nous intéressons aux cas où $\theta$ est une suite $q$- multiplicative.

DOI : https://doi.org/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
@article{AIF_2002__52_2_561_0,
author = {Durand, Fabien and Schneider, Dominique},
title = {Ergodic averages with deterministic weights},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {52},
number = {2},
year = {2002},
pages = {561-583},
doi = {10.5802/aif.1894},
zbl = {1054.37003},
mrnumber = {1906483},
language = {en},
url = {http://www.numdam.org/item/AIF_2002__52_2_561_0}
}

Durand, Fabien; Schneider, Dominique. Ergodic averages with deterministic weights. Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 561-583. doi : 10.5802/aif.1894. http://www.numdam.org/item/AIF_2002__52_2_561_0/

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