Extremal behaviour of chaotic dynamics

Abstract

We present a review of recent results regarding the existence of extreme value laws for stochastic processes arising from dynamical systems. We gather all the conditions on the dependence structure of stationary stochastic processes in order to obtain both the distributional limit for partial maxima and the convergence of point processes of rare events. We also discuss the existence of clustering which can be detected by an extremal index less than 1 and relate it with the occurrence of rare events around periodic points. We also present the connection between the existence of extreme value laws for certain dynamically defined stationary stochastic processes and the existence of hitting times statistics (or return times statistics). Finally, we make a complete description of the extremal behaviour of expanding and piecewise expanding systems by giving a dichotomy regarding the types of extreme value laws that apply. Namely, we show that around periodic points we have an extremal index less than 1 and at very other point we have an extremal index equal to 1.

Keywords:

• extreme value laws
• hitting/return time statistics
• extremal index
• point processes
• expanding and piecewise expanding dynamical systems
• dichotomy

2000 Mathematics Subject Classification:

• 37A25
• 37A50
• 37C40
• 37B20
• 60G70
• 60G10

Acknowledgements

The author was partially supported by FCT (Portugal) grant SFRH/BPD/66040/2009, by FCT projects PTDC/MAT/099493/2008 and PTDC/MAT/120346/2010, which are financed by national and European Community structural funds through the programmes FEDER and COMPETE. The author also acknowledges the support of CMUP, which is financed by FCT (Portugal) through the programmes POCTI and POSI, with national and European Community structural funds. The author wishes to thank Mike Todd for his comments and suggestions.

Notes

1. i.e. the smallest such that f  ⁿ (ζ) = ζ. Clearly f ^ip (ζ) = ζ for any .