N. Chernov, D. Kleinbock
Dynamical Borel-Cantelli lemmas for Gibbs measures
(63K, LATeX)
ABSTRACT. Let $T:\, X\mapsto X$ be a deterministic
dynamical system preserving a probability measure $\mu$.
A dynamical Borel-Cantelli lemma asserts that for certain
sequences of subsets $A_n\subset X$ and $\mu$-almost every
point $x\in X$ the inclusion $T^nx\in A_n$ holds for
infinitely many $n$. We discuss here systems which are
either symbolic (topological) Markov chain or Anosov
diffeomorphisms preserving Gibbs measures. We find
sufficient conditions on sequences of cylinders and
rectangles, respectively, that ensure the dynamical
Borel-Cantelli lemma.