Almost sure convergence of maxima for chaotic dynamical systems

TL;DR

This paper studies the almost sure growth rate of maximum observations in chaotic dynamical systems, establishing conditions for the existence of almost sure limits based on the system's properties and the observable's behavior.

Contribution

It provides new results on the almost sure growth rate of maxima in non-uniformly hyperbolic systems, linking shrinking target estimates with observable regularity.

Findings

Existence of almost sure growth rate under certain conditions.

Examples where the almost sure limit does not exist.

Applicability to a broad class of chaotic systems.

Abstract

Suppose $(f,\mathcal{X},\nu)$ is a measure preserving dynamical system and $\phi:\mathcal{X}\to\mathbb{R}$ is an observable with some degree of regularity. We investigate the maximum process $M_n:=\max\{X_1,\ldots,X_n\}$, where $X_i=\phi\circ f^i$ is a time series of observations on the system. When $M_n\to\infty$ almost surely, we establish results on the almost sure growth rate, namely the existence (or otherwise) of a sequence $u_n\to\infty$ such that $M_n/u_n\to 1$ almost surely. The observables we consider will be functions of the distance to a distinguished point $\tilde{x}\in \mathcal{X}$. Our results are based on the interplay between shrinking target problem estimates at $\tilde{x}$ and the form of the observable (in particular polynomial or logarithmic) near $\tilde{x}$. We determine where such an almost sure limit exists and give examples where it does not. Our results apply to a wide class of non-uniformly hyperbolic dynamical systems, under mild assumptions on the rate of mixing, and on regularity of the invariant measure.