# Meeting Details

Title: Dynamical Borel-Cantelli lemmas for some non-uniformly hyperbolic systems Seminar on Probability and its Application Matthew Nicol, University of Houston http://www.math.psu.edu/denker/nicol.pdf Dynamical Borel-Cantelli lemmas for some non-uniformly hyperbolic systems (joint with Nicolai Haydn, Tomas Persson and Sandro Vaienti) Suppose $(T,X,\mu)$ is a dynamical system and $(B_i)$ is a sequence of sets in $X$. We consider whether $T^i x\in B_i$~i.~o.for $\mu$ a.e.\ $x\in X$ and if so, is there an asymptotic estimate on the rate of entry. If $T^i x\in B_i$ i.~o. for $\mu$ a.e.\ $x$ we call the sequence $(B_i)$ a Borel--Cantelli sequence. If the sets $B_i:= B(p,r_i)$ are nested balls of radius $r_i$ about a point $p$ then the question of whether $T^i x\in B_i$~i.~o. for $\mu$ a.e.\ $x$ is often called the shrinking target problem. We show, under certain assumptions on the measure $\mu$, that if $\frac{C_1}{i}\le \mu (B_i)\le \frac{C_2}{i}$ then exponential decay of correlations implies that the sequence of balls $(B_i)$ is Borel--Cantelli. We give conditions in terms of return time statistics which imply quantitative Borel-Cantelli results for sequences of balls such that $\mu (B_i)\ge \frac{C}{i}$. Corollaries of our results are that for planar dispersing billiards sequences of nested balls $B(p,1/\sqrt{i})$ are Borel--Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems.