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A Borel–Cantelli lemma for nonuniformly expanding dynamical systems

2010; IOP Publishing; Volume: 23; Issue: 8 Linguagem: Inglês

10.1088/0951-7715/23/8/010

1361-6544

Chinmaya Gupta, Matthew Nicol, William Ott,

Chaos control and synchronization

Let be a sequence of sets in a probability space such that . The classical Borel–Cantelli (BC) lemma states that if the sets An are independent, then μ({x ∊ X : x ∊ An for infinitely many values of n}) = 1. We present analogous dynamical BC lemmas for certain sequences of sets (An) 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.