论文标题
复发理论的动力学鲍尔 - – cantelli引理
Dynamical Borel-Cantelli lemma for recurrence theory
论文作者
论文摘要
我们研究了具有兼容度量$ d $的动力学系统$（x，μ，t）$的重复集的动力学borel-cantelli引理。我们证明，在某些规律性条件下，$μ$ - 以下集合\ [ r（ψ）= \ {x \ in x：d（t^n x，x）<ψ（n）\ \ text {对于无限的许多} \ n \ in \ n \} \ in \ n \} \]根据某个系列的逆变或差异，其中$ψ：\ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n。我们主要定理的某些应用包括持续的分数动态系统，beta动力学系统和均匀的自相似集合。
We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system $(X, μ, T)$ with a compatible metric $d$. We prove that, under some regularity conditions, the $μ$-measure of the following set \[ R(ψ)= \{x\in X : d(T^n x, x) < ψ(n)\ \text{for infinitely many}\ n\in\N \} \] obeys a zero-full law according to the convergence or divergence of a certain series, where $ψ:\N\to\R^+$. Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets.
