Extreme value theory for singular measures

Abstract

In this paper, we perform an analytical and numerical study of the extreme values of specific observables of dynamical systems possessing an invariant singular measure. Such observables are expressed as functions of the distance of the orbit of initial conditions with respect to a given point of the attractor. Using the block maxima approach, we show that the extremes are distributed according to the generalised extreme value distribution, where the parameters can be written as functions of the information dimension of the attractor. The numerical analysis is performed on a few low dimensional maps. For the Cantor ternary set and the Sierpinskij triangle, which can be constructed as iterated function systems, the inferred parameters show a very good agreement with the theoretical values. For strange attractors like those corresponding to the Lozi and Hènon maps, a slower convergence to the generalised extreme value distribution is observed. Nevertheless, the results are in good statistical agreement with the theoretical estimates. It is apparent that the analysis of extremes allows for capturing fundamental information of the geometrical structure of the attractor of the underlying dynamical system, the basic reason being that the chosen observables act as magnifying glass in the neighborhood of the point from which the distance is computed.

Extremes and Recurrence in Dynamical Systems

“…Instead, as discussed in the previous Chapters, deriving corresponding EVLs following the GEV approach requires assuming Д 0 (u n ) and Д 0 (u n ) conditions for the observable g(dist(x, ζ)) [77]. If such conditions apply, the GPD and GEV points of view on the extremes are, indeed, equivalent.…”

“…An especially important role is played by observables whose extremes correspond to close returns of the orbit near a reference point [73,49,77,78]. Interestingly, perturbing systems with noise allows the establishment of EVLs for such observables also in the case of deterministic quasi-periodic motion and removes clusters of extreme events when strong correlations are otherwise present [79].…”

“…9.4.5-9.4.7 and computing the angular coefficient ξ of a linear fit of data; for Baker, Hénon, and Lozi, maps, and the IFS generating the Cantor set. The Baker map is not discussed in this book; see [77] for details.…”

“…It first appeared in the pioneering paper of Collet, [72], which has been an inspiration for plenty of the research on this issue. Then the subject has been further addressed and developed in many subsequent contributions including [134,73,135,74,136,137,138,139,49,140,81,46,76,77,78,141,142,79,44].…”

“…, s are rare according to time scales equal toN or shorter. In order to achieve this, we exploit the results given in [46,77].…”