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arXiv · 1501.05023

Extremal dichotomy for uniformly hyperbolic systems

Maria Carvalho·Ana Cristina Moreira Freitas·Jorge Milhazes Freitas·Mark Holland·Matthew Nicol
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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.

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Maria Carvalho, Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, Mark Holland, Matthew Nicol. 2015-01-27. Extremal dichotomy for uniformly hyperbolic systems. https://doi.org/10.1080/14689367.2015.1056722

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