Extremal dichotomy for uniformly hyperbolic systems

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○T i } 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.