Extreme value laws in dynamical systems under physical observables

Extreme value theory for chaotic dynamical systems is a rapidly expanding area of research. Given a system and a real function (observable) defined on its phase space, extreme value theory studies the limit probabilistic laws obeyed by large values attained by the observable along orbits of the system. Based on this theory, the so-called block maximum method is often used in applications for statistical prediction of large value occurrences. In this method, one performs inference for the parameters of the Generalised Extreme Value (GEV) distribution, using maxima over blocks of regularly sampled observations along an orbit of the system. The observables studied so far in the theory are expressed as functions of the distance with respect to a point, which is assumed to be a density point of the system’s invariant measure. However, this is not the structure of the observables typically encountered in physical applications, such as windspeed or vorticity in atmospheric models. In this paper we consider extreme value limit laws for observables which are not functions of the distance from a density point of the dynamical system. In such cases, the limit laws are no longer determined by the functional form of the observable and the dimension of the invariant measure: they also depend on the specific geometry of the underlying attractor and of the observable’s level sets. We present a collection of analytical and numerical results, starting with a toral hyperbolic automorphism as a simple template to illustrate the main ideas. We then formulate our main results for a uniformly hyperbolic system, the solenoid map. We also discuss non-uniformly hyperbolic examples of maps (H'enon and Lozi maps) and of flows (the Lorenz63 and Lorenz84 models). Our purpose is to outline the main ideas and to highlight several serious problems found in the numerical estimation of the limit laws.

💡 Research Summary

This paper extends extreme‑value theory (EVT) for chaotic deterministic dynamical systems to the class of “physical” observables that are not simple functions of the distance to a reference point. Classical EVT for dynamical systems assumes an observable of the form φ(p)=g(dist(p,p_M)), with p_M a density point of the invariant (often SRB) measure. In many applications – wind speed, vorticity, total energy, etc. – the observable is linear or polynomial in the state variables and its level sets are hyperplanes, cusps or fractal‑like shapes rather than balls. The authors show that in such cases the tail index ξ of the limiting Generalised Extreme Value (GEV) distribution is no longer determined solely by the functional form of g and the dimension of the invariant measure; it also depends critically on the geometry of the attractor Λ and on the geometry of the observable’s level sets.

The paper proceeds through a series of increasingly complex examples:

1. Thom’s map (hyperbolic toral automorphism).
   Using two observables – a distance‑based φ_α and a non‑radial φ_{a,b} – the authors illustrate three regimes: (i) p_M inside the invariant square, where level sets are balls and the classical ξ=−1/α result holds; (ii) p_M outside the square, where the level set intersects the support of the invariant measure in a non‑ball shape; (iii) the non‑radial case where level sets are anisotropic. Analytical calculations reveal that ξ is given by a ratio of the effective dimension of the level set to the local dimension of the invariant measure. Numerical block‑maximum experiments confirm that insufficient block length leads to biased ξ estimates.

2. Solenoid map (uniformly hyperbolic attractor in ℝ³).
   The observable φ(x,y,z)=a x+b y+c is linear, so its level sets are planes. The solenoid’s attractor has a product structure: a one‑dimensional unstable manifold (d_u=1) and a fractal stable direction of Hausdorff dimension dim_H(Λ)−1. Assuming a quadratic tangency between the unstable manifold and the plane, the authors derive the universal formula
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