A Borel-Cantelli lemma for nonuniformly expanding dynamical systems

作者：Gupta Chinmaya^*; Nicol Matthew; Ott William

来源：Nonlinearity, 2010, 23(8): 1991-2008.

DOI：10.1088/0951-7715/23/8/010

Let (A(n))(n=1)(infinity) be a sequence of sets in a probability space (X, B, mu) such that Sigma(infinity)(n=1) mu(A(n)) = infinity. The classical Borel-Cantelli (BC) lemma states that if the sets A(n) are independent, then mu({x epsilon X : x epsilon A(n) for infinitely many values of n}) = 1. We present analogous dynamical BC lemmas for certain sequences of sets (A(n)) in X (including nested balls) for a class of deterministic dynamical systems T : X -> X with invariant probability measures. Our results apply to a class of Gibbs-Markov maps and one-dimensional nonuniformly expanding systems modelled by Young towers. We discuss some applications of our results to the extreme value theory of deterministic dynamical systems.

• 出版日期2010-8