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Dynamical Borel-Cantelli lemma for recurrence theory (2009.03515v1)
Published 8 Sep 2020 in math.DS and math.NT
Abstract: We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system $(X, \mu, T)$ with a compatible metric $d$. We prove that, under some regularity conditions, the $\mu$-measure of the following set [ R(\psi)= {x\in X : d(Tn x, x) < \psi(n)\ \text{for infinitely many}\ n\in\N } ] obeys a zero-full law according to the convergence or divergence of a certain series, where $\psi:\N\to\R+$. Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets.