Rates of Recurrence for Free Semigroup Actions

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