Extremal dichotomy for uniformly hyperbolic systems

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.

Keywords:

• extreme value theory
• return time statistics
• stationary stochastic processes
• point processes
• extremal index

2000 Mathematics subject classification:

• 37A50
• 60G70
• 37B20
• 60G10
• 37C25

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.

Additional information

Funding

Ana Cristina Moreira Freitas was partially supported by FCT [grant SFRH/BPD/66174/2009]. Jorge Milhazes Freitas was partially supported by FCT [grant SFRH/BPD/66040/2009]. Both these grants are financially supported by the program POPH/FSE. Ana Cristina Moreira Freitas and Jorge Milhazes Freitas are supported by FCT (Portugal) project [PTDC/MAT/120346/2010], which is financed by national and European structural funds through the programs FEDER and COMPETE. Maria Carvalho, Ana Cristina Moreira Freitas and Jorge Milhazes Freitas are also partially supported by CMUP [UID/MAT/00144/2013], which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement [PT2020].