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



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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.



Dynamical systems, Chaotic Dynamics, Borel-Cantelli, Probability and Statistics, Ergodic, Hyperbolic, Climate Science, Extreme Value theory


Portions of this document appear in: Carney, M., and Nicol, M. (2017) Dynamical Borel-Cantelli lemmas and rates of growth of Birkhoff sums of non-integrable observables on chaotic dynamical systems. Nonlinearity, 30(7), 2854-2870. And in: Carney, M., Nicol, M., and Zhang, H.K. (2017) Compound Poisson Law for Hitting Times to Periodic Orbits in Two-Dimensional Hyperbolic Systems with Singularities. Journal of Statistical Physics, 169(4), 804-823.