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 x ∈ X, one has the following:
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The authors thank the editor and the anonymous referee for helpful comments and constructive suggestions that significantly improved the paper.
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The second author was supported by National Natural Science Foundation of China (11901419).
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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