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Rates of Recurrence for Free Semigroup Actions

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Abstract

We consider finitely generated free semigroup actions on (X, d) and generalize Boshernitzan’s quantitative recurrence theorem to general free semigroup actions. Let G be a finitely generated free semigroup endowed with a Bernoulli probability measure \(\mathbb P_{\underline {a}}\) and \(\mathbb S\) be the corresponding continuous semigroup continuous semigroup action. Assume that, for some α > 0, the Hausdorff measure ν = Hα(X) as invariant by every generator in G. ν in X invariant by every generator in G. Then, for \(\mathbb P_{a}\)-almost every ω and ν-almost xX, one has the following:

$$\liminf\limits_{n\to\infty} n^{\frac{1}{\alpha}}d(x, f^{n}_{\omega}(x)) \leq 1 .$$

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Acknowledgments

The authors thank the editor and the anonymous referee for helpful comments and constructive suggestions that significantly improved the paper.

Funding

The second author was supported by National Natural Science Foundation of China (11901419).

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Correspondence to Cao Zhao.

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Liang, Y., Zhao, C. Rates of Recurrence for Free Semigroup Actions. J Dyn Control Syst 27, 417–425 (2021). https://doi.org/10.1007/s10883-020-09486-2

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  • DOI: https://doi.org/10.1007/s10883-020-09486-2

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