Extremal dichotomy for toral hyperbolic automorphisms

Abstract. We consider the extreme value theory of a hyperbolic toral automorphism T : T2 ! T2 showing that, if a Hölder observation is a function of a Euclidean-type distance to a non-periodic point ⇣ and is strictly maximized at ⇣, then the corresponding time series { T i} exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as and with extremal index one. If, however, 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 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 centered at nonperiodic points follow a Poisson law, whereas the law is compound Poisson if the balls are centered at periodic points.