A new mean ergodic theorem for tori and recurrences

Abstract

Let$X$be a finite-dimensional connected compact abelian group equipped with the normalized Haar measure$\unicode[STIX]{x1D707}$. We obtain the following mean ergodic theorem over ‘thin’ phase sets. Fix$k\geq 1$and, for every$n\geq 1$, let$A_{n}$be a subset of$\mathbb{Z}^{k}\cap [-n,n]^{k}$. Assume that$(A_{n})_{n\geq 1}$has$\unicode[STIX]{x1D714}(1/n)$density in the sense that$\lim _{n\rightarrow \infty }(|A_{n}|/n^{k-1})=\infty$. Let$T_{1},\ldots ,T_{k}$be ergodic automorphisms of$X$. We have$$\begin{eqnarray}\frac{1}{|A_{n}|}\mathop{\sum }_{(n_{1},\ldots ,n_{k})\in A_{n}}f_{1}(T_{1}^{n_{1}}(x))\cdots f_{k}(T_{k}^{n_{k}}(x))\stackrel{L_{\unicode[STIX]{x1D707}}^{2}}{\longrightarrow }\int f_{1}\,d\unicode[STIX]{x1D707}\cdots \int f_{k}\,d\unicode[STIX]{x1D707},\end{eqnarray}$$for any$f_{1},\ldots ,f_{k}\in L_{\unicode[STIX]{x1D707}}^{\infty }$. When the$T_{i}$are ergodic epimorphisms, the same conclusion holds under the further assumption that$A_{n}$is a subset of$[0,n]^{k}$for every$n$. The density assumption on the$A_{i}$is necessary. Immediate applications include certain Poincaré style recurrence results.