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Rates of Recurrence for Free Semigroup Actions

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

Keywords: free semigroup; semigroup; recurrence free; rates recurrence; semigroup actions

Journal Title: Journal of Dynamical and Control Systems
Year Published: 2020

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