Entropy numbers of some ergodic averages

In this paper we extend a recent remarkable covering numbers estimate for averages of contractions in a Hilbert space $H$ due to Talagrand to some moving averages of contractions. By introducing a second regularization in Talagrand's spectral regularization, we find mild conditions on the spectral measure associated to any $x\in H$, allowing estimation of the number of Hilbertian balls of radius $0 < \varepsilon\le 1$, enough to cover the subset of $H$ defined by $\{B_n(x)=n^{-1}\sum_{j=n^2}^{n^2+n-1}U^jx,n\in\mathcal{N}\}$, where $U$ is a contraction of $H$ and $\mathcal{N}$ a geometric sequence. Moreover, we show that these conditions on the spectral measure ensure the existence of the modulus of continuity of $\{T^{-1}\int_0^Tf\circ U_t dt,T\ge1\}$, where $f$ is a contraction of $L^2(\mu)$ and $\{U_t,t\in\mathbb{R}\}$ is a flow which preserves the measure $\mu$. Finally, we give a covering numbers estimate in a non-Hilbertian case.