For more information about this meeting, contact Manfred Denker.

Title: | Dynamical Borel-Cantelli lemmas for some non-uniformly hyperbolic systems |

Seminar: | Seminar on Probability and its Application |

Speaker: | Matthew Nicol, University of Houston |

Abstract Link: | http://www.math.psu.edu/denker/nicol.pdf |

Abstract: |

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. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 09 / 30 / 2011 |

Time: | 02:20pm - 03:20pm |