Dynamical Borel-Cantelli lemmas (Q1995567)

The classical Borel-Cantelli lemma says that if the sum of the probabilities of a collection of independent events is infinite, then the probability of the occurrence of infinitely many of these events must be one. There is a number of generalizations of this result for dynamical systems, when the assumption about independence cannot take place. In particular, \textit{D. Y. Kleinbock} and \textit{G. A. Margulis} [Invent. Math. 138, No. 3, 451--494 (1999; Zbl 0934.22016)] have given a useful sufficient condition for strongly Borel-Cantelli sequences, based on the work of \textit{W. Schmidt} [Can. J. Math. 12, 619--631 (1960; Zbl 0097.26205)]. The author obtains a weaker sufficient condition for the strongly Borel-Cantelli sequences, which allows to extend known results of Borel-Cantelli lemma type for dependent variables. Applications to one-dimensional Gibbs-Markov systems are established.