Ergodic averages with deterministic weights

Nous étudions la convergence de moyennes ergodiques $\frac{1}{N}{\sum }_{k=0}^{N-1}\theta \left(k\right)f\circ T^{u_k}$ où ${\left(\theta \left(k\right)\right)}_{k\in \mathbb{N}}$ est une suite bornée et ${\left(u_k\right)}_{k\in \mathbb{N}}$ une suite d’entiers strictement croissante tels que ${\mathrm{Sup}}_{\alpha \in ℝ}\left|{\sum }_{k=0}^{N-1}\theta \left(k\right)\exp \left(2i\pi \alpha u_k\right)\right|=O\left(N^{\delta }\right)$ 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.

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 ${\left(\theta \left(k\right)\right)}_{k\in \mathbb{N}}$ is a bounded sequence and ${\left(u_k\right)}_{k\in \mathbb{N}}$ a strictly increasing sequence of integers such that ${\mathrm{Sup}}_{\alpha \in ℝ}\left|{\sum }_{k=0}^{N-1}\theta \left(k\right)\exp \left(2i\pi \alpha u_k\right)\right|=O\left(N^{\delta }\right)$ 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.

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