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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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