Abstract
We consider the extreme value theory of a hyperbolic toral automorphism showing that, if a Hölder observation φ is a function of a Euclidean-type distance to a non-periodic point ζ and is strictly maximized at ζ, then the corresponding time series {φ○Ti} exhibits extreme value statistics corresponding to an independent identically distributed (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 to small balls centred at non-periodic points follow a Poisson law, whereas the law is compound Poisson if the balls are centred at periodic points.
Acknowledgements
Jorge Milhazes Freitas would like to thank Mike Todd for helpful comments and suggestions.
Disclosure statement
No potential conflict of interest was reported by the authors.