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A new mean ergodic theorem for tori and recurrences. (English) Zbl 07277635
Summary: Let $$X$$ be a finite-dimensional connected compact abelian group equipped with the normalized Haar measure $$\mu$$. 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 $$\omega (1/n)$$ density in the sense that $$\lim_{n\rightarrow\infty}(|A_n|/n^{k-1})=\infty$$. Let $$T_1,\dots,T_k$$ be ergodic automorphisms of $$X$$. We have $\frac{1}{|A_n|}\mathop{\sum}_{(n_1,\dots ,n_k)\in A_n}f_1(T_1^{n_1}(x))\cdots f_k(T_k^{n_k}(x))\stackrel{L_\mu^2}{\longrightarrow }\int f_1\,d\mu\cdots \int f_k\,d\mu,$ for any $$f_1,\dots ,f_k\in L_\mu^{\infty}$$. 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.
##### MSC:
 37A30 Ergodic theorems, spectral theory, Markov operators 37A05 Dynamical aspects of measure-preserving transformations 37A44 Relations between ergodic theory and number theory 37A46 Relations between ergodic theory and harmonic analysis 11D45 Counting solutions of Diophantine equations
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