Dynamical Borel-Cantelli lemmas and applications

Sammanfattning: The classical Borel–Cantelli lemma is a beautiful discovery with wide applications in the mathematical field. The Borel–Cantelli lemmas in dynamical systems are particularly fascinating. Here, D. Kleinbock and G. Margulis have given an important sufficient condition for the strongly Borel–Cantelli sequence, which is based on the work of W. M. Schmidt. This Master’s thesis deals with an improvement of Kleinbock’s and Margulis’ theorem and obtains a weaker sufficient condition for the strongly Borel–Cantelli sequences. Several versions of the dynamical Borel–Cantelli lemmas will be deduced by extending another useful theorem by W. M. Schmidt, W. J. LeVeque, and W. Philipp. Furthermore, some applications of our theorems will be discussed. Firstly, a characterization of the strongly Borel–Cantelli sequences in one-dimensional Gibbs–Markov systems will be established. This will improve the theorem of C. Gupta, M. Nicol, and W. Ott. Secondly, N. Haydn, M. Nicol, T. Persson, and S. Vaienti proved the strong Borel–Cantelli property in sequences of balls in terms of a polynomial decay of correlations for Lipschitz observables. Our theorems will be applied to relax their inequality assumption. Finally, as a result of Y. Guivarch’s and A. Raugi’s findings, we know that the weakly mixing property could be characterized by Borel–Cantelli sequences that only contain a finite number of distinct sets, each with positive measure. This is a Borel–Cantelli result, although not strong. So a weakly beta-mixing property will be introduced to imply the strong Borel–Cantelli property.