Mathematics – Dynamical Systems

Scientific paper

[
0.00
] – not rated yet
Voters
0
Comments 0

2012-04-10

Mathematics

Dynamical Systems

Scientific paper

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.

**Freitas Jorge Milhazes**

Mathematics – Dynamical Systems

Scientist

**Moreira Freitas Ana Cristina**

Mathematics – Dynamical Systems

Scientist

**Todd Mike**

Mathematics – Dynamical Systems

Scientist

No associations

LandOfFree

If you have personal experience with

The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamicsdoes not yet have a rating. At this time, there are no reviews or comments for this scientific paper.The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics will most certainly appreciate the feedback.

Profile ID: LFWR-SCP-O-716724

Use Google custom search:

All data on this website is collected from public sources.
Our data reflects the most accurate information available at the time of publication.