Ergodic averages with deterministic weights

Durand, Fabien; Schneider, Dominique

doi:10.5802/aif.1894

Ergodic averages with deterministic weights
[Moyennes ergodiques pondérées par des poids déterministes]

Durand, Fabien ; Schneider, Dominique

Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 561-583.

Résumé
Abstract

Nous étudions la convergence de moyennes ergodiques $\frac{1}{N} \sum_{k = 0}^{N - 1} θ (k) f \circ T^{u_{k}}$ où $(θ (k))_{k \in ℕ}$ est une suite bornée et $(u_{k})_{k \in ℕ}$ une suite d’entiers strictement croissante tels que ${Sup}_{α \in ℝ} | \sum_{k = 0}^{N - 1} θ (k) \exp (2 i π α u_{k}) | = O (N^{δ})$ 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 $\frac{1}{N} \sum_{k = 0}^{N - 1} θ (k) f \circ T^{u_{k}}$ where $(θ (k))_{k \in ℕ}$ is a bounded sequence and $(u_{k})_{k \in ℕ}$ a strictly increasing sequence of integers such that ${Sup}_{α \in ℝ} | \sum_{k = 0}^{N - 1} θ (k) \exp (2 i π α u_{k}) | = O (N^{δ})$ 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.

MR 1906483 | Zbl 1054.37003

DOI : https://doi.org/10.5802/aif.1894
Classification : 37A05, 28D05, 11K99
Mots clés : moyennes ergodiques pondérées, théorème ..., convergence presque sûre, suites multiplicatives

q

, suites substitutives, suite de Thue-Morse généralisées

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     title = {Ergodic averages with deterministic weights},
     journal = {Annales de l'Institut Fourier},
     pages = {561--583},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {52},
     number = {2},
     year = {2002},
     doi = {10.5802/aif.1894},
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     zbl = {1054.37003},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1894/}
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.1894/

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