The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics

TL;DR

This paper demonstrates that in non-uniformly hyperbolic dynamical systems, the distribution of return times near periodic points follows a compound Poisson law, revealing clustering behavior of rare events linked to extreme value phenomena.

Contribution

It establishes the convergence of return time distributions to a compound Poisson process around periodic points in non-uniformly hyperbolic systems, extending understanding of extreme events in such dynamics.

Findings

Returns near periodic points form clusters with geometrically distributed sizes.

Exceedances follow a compound Poisson process, contrasting with Poisson behavior at generic points.

Results apply to various systems including Rychlik maps, Manneville-Pomeau maps, and quadratic maps.

Abstract

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.