Entropy numbers of some ergodic averages (Q2711150)

If \(X\) is a seminormed linear space and \(U\) a bounded linear operator on \(X\), we may consider the moving averages \(A_n= n^{-1} \sum^{n-1}_{j=0} U^j\), \(n= 1,2,\dots\). Given \(x\in X\), does a subsequence \(S\) of the sequence \(\{A_n(x)\}^\infty_{n=1}\) converge or cluster in some sense? The main thrust of this paper, building upon a result of Talagrand, is to estimate the minimum number of balls of radius \(\varepsilon\leq 1\) which have centers in \(S\) and which cover \(S\). The operator \(U\) is either a contraction on \(X= H=\) a Hilbert space, or the right shift on \(X=\) the Wiener space of correlated sequences. One of their results applies to the moving average \(B_n= n^{-1} \sum^{n^2+ n-1}_{j=n^2} U^j\) in place of \(A_n\), with an indication of extensions to more general moving averages. As an application, an estimate is given of the modulus of uniform continuity of the averages \(A_T(f)= T^{-1} \int^T_0 U_t(f) dt\), \(T\geq 1\), where \(H= L^2\) of a probability space, where \(f\in H\), and where \((U_t)_{t\geq 0}\) is a continuous flow of contractions on \(H\).