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 = H_α(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:

我们考虑在（X，d）上有限生成的自由半群动作，并将Boshernitzan的定量递归定理推广到一般的自由半群动作。令G为具有伯努利概率测度\（\ mathbb P _ {\下划线{a}} \）的有限生成的自由半群，而\（\ mathbb S \）为对应的连续半群连续半群动作。假设，对于一些α > 0时，Hausdorff测度ν = ħ _α（X），通过在每一个发电机不变ģ。ν在X不变通过在每一个发电机摹。然后，对于\（\ mathbb P_ {A} \） -almost每ω和ν -almost X ∈ X，一个具有如下：