Statistical properties of dynamical systems: large deviations, extremes, return time statistics and dynamical Borel-Cantelli lemmas.

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This project seeks to improve our understanding of certain key statistics of non-uniformly hyperbolic dynamical systems: return time statistics, extreme value statistics, dynamical Borel-Cantelli lemmas and large deviations. While there has been progress in describing these statistical properties for uniformly hyperbolic systems, uniform hyperbolicity seldom holds for physical systems and models based on non-uniform hyperbolicity are more realistic. Return time statistics quantify probabilities of a physical system transforming into or returning to a state in a certain period of time. Extreme value statistics give limit laws for successive maxima of a time series of observations on a system. Borel-Cantelli lemmas are a fundamental tool in establishing the almost-sure behavior of stochastic processes. The aim is to extend our understanding of these statistical properties to non-uniformly hyperbolic systems including those with polynomial correlation decay rate. A further goal is to obtain large deviation estimates for functions on dynamical systems with discontinuities (or singularities) or which give rise to multiplicative cocycles, such as matrix valued functions.Some physical systems are so complex that, although they may be modeled by mathematical equations, they are best described from the viewpoint of statistics. For example, knowing the probabilities of a hurricane located offshore affecting certain towns on the coast is useful information and more certain information usually cannot be given. However, it is a major problem that the classical statistical assumption of complete randomness (independence of successive observations) is seldom satisfied for realistic models of physical systems since successive observations are highly correlated. Extreme value theory estimates the likelihood of observing an event of a certain magnitude, while large deviations estimates the probability of an outlier or rare event. This project aims to improve predictions of complex physical systems by providing an understanding of extreme value theory, large deviations and other statistics for a wide class of physically realistic mathematical models. The PI will also continue to work towards improving science education at all levels and by informing the public through outreach activities.