Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

TL;DR

This paper investigates the almost sure asymptotic behavior of Birkhoff sums in infinite measure-preserving dynamical systems, establishing conditions under which normalized sums converge to 1 for almost every point.

Contribution

It introduces new normalization sequences and conditions for convergence of Birkhoff sums in infinite ergodic systems with strong mixing properties.

Findings

Existence of normalization sequences for sums of integrable observables.

Almost sure convergence of normalized sums under strong mixing assumptions.

Conditions for convergence when observables are not integrable.

Abstract

We consider a conservative ergodic measure-preserving transformation $T$ of a $\sigma$-finite measure space $(X,\mathcal{B},\mu)$ with $\mu(X)=\infty$. Given an observable $f:X\to \mathbb{R}$ we study the almost sure asymptotic behaviour of the Birkhoff sums $S_Nf(x) := \sum_{j=1}^N\, (f\circ T^{j-1})(x)$. In infinite ergodic theory it is well known that the asymptotic behaviour of $S_Nf(x)$ strongly depends on the point $x\in X$, and if $f\in L^1(X,\mu)$, then there exists no real valued sequence $(b(N))$ such that $\lim_{N\to\infty} S_Nf(x)/b(N)=1$ almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence $(\alpha(N))$ and $m\colon X\times \mathbb{N}\to\mathbb{N}$ such that for $f\in L^1(X,\mu)$ we have $\lim_{N\to\infty} S_{N+m(x,N)}f(x)/\alpha(N)=1$ for $\mu$-a.e. $x\in X$. Moreover if $f\not\in L^1(X,\mu)$ we give conditions on the induced observable such that there exists a sequence $(G(N))$ depending on $f$, for which $\lim_{N\to\infty} S_{N}f(x)/G(N)=1$ holds for $\mu$-a.e. $x\in X$.