Universal Behaviour of Extreme Value Statistics for Selected Observables of Dynamical Systems

The main results of the extreme value theory developed for the investigation of the observables of dynamical systems rely, up to now, on the block maxima approach. In this framework, extremes are identified with the block maxima of the time series of the chosen observable, in the limit of infinitely long blocks. It has been proved that, assuming suitable mixing conditions for the underlying dynamical systems, the extremes of a specific class of observables are distributed according to the so called Generalised Extreme Value (GEV) distribution. Direct calculations show that in the case of quasi-periodic dynamics the block maxima are not distributed according to the GEV distribution. In this paper we show that considering the exceedances over a given threshold instead of the block-maxima approach it is possible to obtain a Generalised Pareto Distribution also for extremes computed in systems which do not satisfy mixing conditions. Requiring that the invariant measure locally scales with a well defined exponent—the local dimension—, we show that the limiting distribution for the exceedances of the observables previously studied with the block maxima approach is a Generalised Pareto distribution where the parameters depend only on the local dimensions and the values of the threshold but not on the number of observations considered. We also provide connections with the results obtained with the block maxima approach. In order to provide further support to our findings, we present the results of numerical experiments carried out considering the well-known Chirikov standard map.