An Investigation of the Statistical Properties of Certain Chaotic Dynamical Systems Through Extremes and Recurrence: A Theoretical and Applied Approach

Motivated by proofs in extreme value theory, we investigate the statistical properties of certain chaotic dynamical systems, including the well-known dispersing billiard model. In particular, we prove the existence of a maximal probability distribution and rare event point process in the setting of two-dimensional hyperbolic systems with singularities. We also obtain bounds on the growth rates of Birkhoff sums with non-integrable observables, where the Birkhoff ergodic theorem fails, by using the recurrence properties of the system to a point of maximization. We end with an analysis of extreme temperatures across Texas where we find compelling evidence that the probability of observing higher summer temperature extremes has increased.