The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics

• Type: Article in International Scientific Journal
• Date: 2013

Authors

Ana Cristina M Moreira Freitas (Author) FEP; Jorge Milhazes Freitas (Author) FCUP; Mike Todd (Author) Other

Journal

Title: Communications in Mathematical Physics

Vol. 321 No. 2

Pages: 483-527

ISSN: 0010-3616

Publisher: Springer Nature

Indexing

ISI Web of Knowledge - 35 Citations

ISI Web of Science

Scopus - 39 Citations

Scientific classification

FOS: Natural sciences > Physical sciences

Other information

Authenticus ID: P-004-ZVQ

DOI: 10.1007/s00220-013-1695-0

Abstract (EN): We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of certain 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.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 45