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We prove that the distributional limit of the normalised number of returns tosmall neighbourhoods of periodic points of non-uniformly hyperbolic dynamicalsystems is compound Poisson. The returns to small balls around a fixed point inthe phase space correspond to the occurrence of rare events, or exceedances ofhigh thresholds, so that there is a connection between the laws of Return TimesStatistics and Extreme Value Laws. The fact that the fixed point in the phasespace is a repelling periodic point implies that there is a tendency for theexceedances to appear in clusters whose average sizes is given by the ExtremalIndex, which depends on the expansion of the system at the periodic point.  We recall that for generic points, the exceedances, in the limit, aresingular and occur at Poisson times. However, around periodic points, thepicture is different: the respective point processes of exceedances converge toa compound Poisson process, so instead of single exceedances, we have entireclusters of exceedances occurring at Poisson times with a geometricdistribution ruling its multiplicity.  The systems to which our results apply include: general piecewise expandingmaps of the interval (Rychlik maps), maps with indifferent fixed points(Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.

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