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Extremal dichotomy for uniformly hyperbolic systems
Ist Teil von
Dynamical systems (London, England), 2015-10, Vol.30 (4), p.383-403
Ort / Verlag
Taylor & Francis
Erscheinungsjahr
2015
Link zum Volltext
Quelle
EBSCOhost Business Source Ultimate
Beschreibungen/Notizen
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.