Dynamical Borel-Cantelli lemmas for Gibbs measures

Let T: X --> X be a deterministic dynamical system preserving a probability measure mu. A dynamical Borel-Cantelli lemma asserts that for certain sequ ences of subsets A(n) subset of X and mu -almost every point x is an elemen t of X the inclusion T(n)x is an element of A(n) holds for infinitely many n. We discuss here systems which are either symbolic (topological) Markov c hain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficien t conditions on sequences of cylinders and rectangles, respectively, that e nsure the dynamical Borel-Cantelli lemma.