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 $ϕ$ which is a function of a Euclidean-type distance to a non-periodic point $ζ$ is strictly maximized at $ζ$ then the corresponding time series $\{ϕ\circ T^i\}$ exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $ϕ$ and with extremal index one. If however $ϕ$ 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 $ϕ$ 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.