title: Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems.
abstract: In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.
authors.affiliation: Institute of Meteorology, Klimacampus, University of Hamburg, Grindelberg 5, 20144  Hamburg, Germany ; Department of Mathematics and Statistics, University of Reading, Reading, UK ; Walker Institute for Climate Change Research, University of Reading, Reading, UK.
authors.forename: Valerio
authors.initials: V
authors.lastname: Lucarini
authors.affiliation: Institute of Meteorology, Klimacampus, University of Hamburg, Grindelberg 5, 20144  Hamburg, Germany ; Service de Physique de l'Etat Condensé, DSM, CEA Saclay, CNRS URA 2464, Gif-sur-Yvette, France.
authors.forename: Davide
authors.initials: D
authors.lastname: Faranda
authors.affiliation: Institute of Meteorology, Klimacampus, University of Hamburg, Grindelberg 5, 20144  Hamburg, Germany.
authors.forename: Jeroen
authors.initials: J
authors.lastname: Wouters
authors.affiliation: Department of Mathematics and Statistics, University of Reading, Reading, UK.
authors.forename: Tobias
authors.initials: T
authors.lastname: Kuna
keywords: Axiom A; Chaotic systems; Extreme value theory; Henon map; Kaplan–Yorke dimension; Response theory