Extremal dichotomy for uniformly hyperbolic systems.
Carvalho, M; Moreira Freitas, Ana Cristina; Milhazes Freitas, J; et al.Holland, MP; Nicol, M
Date: 22 July 2015
Article
Journal
Dynamical Systems
Publisher
Taylor & Francis
Publisher DOI
Related links
Abstract
We consider the extreme value theory of a hyperbolic toral automorphism T : T2 → T2
showing that, if a Holder 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 ...
We consider the extreme value theory of a hyperbolic toral automorphism T : T2 → T2
showing that, if a Holder 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.
Mathematics and Statistics
Faculty of Environment, Science and Economy
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