POISSON LIMIT THEOREMS IN DYNAMICAL SYSTEMS AND ERGODIC SUMS OF NON-INTEGRABLE OBSERVABLES

Abstract

This thesis investigates statistical properties of rare events in dynamical systems, with a partic-ular focus on recurrence statistics and limit laws for ergodic sums of heavy-tailed observables. A central theme is the Poisson Limit Theorem (PLT), which describes the limiting distribu- tion of return times to small sets, and its role in establishing asymptotics for extreme events. Concretely, this thesis addresses the following topics: First, we construct examples of non-mixing dynamical systems that still satisfy the PLT, demon- strating that strong statistical properties can persist even in the absence of mixing. This is significant because, until now, all known systems satisfying the PLT have been mixing, with existing proofs relying on mixing assumptions. Second, we establish strong limit laws for ergodic sums of non-integrable observables. In such cases, large fluctuations due to close visits to the singularity typically obstruct strong limit laws. To address this, we employ trimming, a well-known probabilistic technique, by systematically removing the closest visits to the singularity. We prove trimmed strong laws for irrational rotations, emphasizing once more that our results seem to be the first of their kind that do not rely on mixing. We also present in the conclusion some natural questions and directions a future research, including hitting time statistics, similar to the PLT setting, and distributional limit theorems for ergodic sums of non-integrable observables.