Recurrence and dynamical Borel-Cantelli results in dynamical systems

Abstract

In probability theory, the Borel-Cantelli lemma can be used to determine the probability that infinitely many events occur in some sequence of events. In a dynamical setting, we may establish analogous properties to sequences of functions or sets, known as dynamical Borel-Cantelli properties.

In this thesis, we study and establish dynamical Borel Cantelli properties in various settings. We do this, firstly, in the context of maps characterised by the presence of indifferent fixed points, known as intermittent maps, and then in the setting of unimodal maps with a degenerate critical point. In the final section, using minimal dynamical assumptions, we establish conditions for which dynamical Borel-Cantelli properties hold for sequences shrinking to various limit sets.