Extremal dichotomy for uniformly hyperbolic systems

TL;DR

This paper studies the extreme value statistics of hyperbolic toral automorphisms, showing how the nature of the maximum point (periodic or non-periodic) affects the extremal index and return time distributions.

Contribution

It provides a detailed analysis of how the extremal index varies depending on whether the maximum is at a periodic or non-periodic point, including a formula for the extremal index.

Findings

Return times are Poisson distributed at non-periodic points.

Return times are compound Poisson at periodic points.

Extremal index depends on the period and metric used.

Abstract

We consider the extreme value theory of a hyperbolic toral automorphism $T: \mathbb{T}^2 \to \mathbb{T}^2$ showing that if a H\"older observation $\phi$ 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 $\{\phi\circ T^i\}$ exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $\phi$ and with extremal index one. If however $\phi$ 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 $\phi$ 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.