  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.
