Extremal dichotomy for uniformly hyperbolic systems

Abstract

We consider the extreme value theory of a hyperbolic toral automorphism $T:{\mathbb{T}}^2\to {\mathbb{T}}^2$ showing that if a Hölder observation $\varphi$ which is a function of a Euclidean-type distance to a non-periodic point $\zeta$ is strictly maximized at $\zeta$ then the corresponding time series $\{\varphi \circ T^i\}$ exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $\varphi$ and with extremal index one. If however $\varphi$ is strictly maximized at a periodic point $q$ then the corresponding time-series exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $\varphi$ but with extremal index not equal to one. We give a formula for the extremal index (which depends upon the metric used and the period of $q$). These results imply that return times are Poisson to small balls centered at non-periodic points and compound Poisson for small balls centered at periodic points.

21 pages, 4 figures

References in corpus (2)

• Rare Events for the Manneville-Pomeau map
• Borel Cantelli Lemmas and Extreme Value Theory for Geometric Lorenz Models

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