Ergodic averages with deterministic weights

Fabien Durand (1) , Dominique Schneider (1)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

The purpose of this paper is to study ergodic averages with deterministic weights. More precisely we study the convergence of the ergodic averages of the type $\frac{1}{N} \sum_{k=0}^{N-1} \theta (k) f \circ T^{u_k}$ where $\theta = (\theta (k) ; k\in \NN)$ is a bounded sequence and $u = (u_k ; k\in \NN)$ a strictly increasing sequence of integers such that for some $\delta<1$ $$ S_N (\theta , u) := \sup_{\alpha \in \pRR} \left| \sum_{k=0}^{N-1} \theta (k) \exp (2i\pi \alpha u_k ) \right| = O (N^{\delta}) \ , \leqno{({\cal H}_1)} $$ i.e., there exists a constant $C$ such that $S_N (\theta , u) \leq C N^{\delta} $. We define $\delta (\theta , u)$ to be the infimum of the $\delta $ satisfying $\H_1$ for $\theta $ and $u$.

Mots clés: ergodic average

Domaines: Systèmes dynamiques [math.DS]

Fabien Durand, Dominique Schneider. Ergodic averages with deterministic weights. Annales de l'Institut Fourier, 2002, 52, pp.561-583. ⟨hal-00308799⟩