On the speed of convergence in the ergodic theorem for shift operators

TL;DR

This paper investigates the convergence speed of ergodic averages for shift operators on probability spaces, showing they typically converge at a rate close to N^{-1/2} with some logarithmic adjustments, and explores applications to toral endomorphisms.

Contribution

It establishes near-optimal convergence rates for ergodic means of shift operators and applies these results to specific cases like toral endomorphisms.

Findings

Convergence rate of approximately N^{-1/2} for ergodic means.

Pointwise almost everywhere convergence of ergodic averages.

Applications to shifts associated with toral endomorphisms.

Abstract

Given a probability space $(X,\mu)$, a square integrable function $f$ on such space and a (unilateral or bilateral) shift operator $T$, we prove under suitable assumptions that the ergodic means $N^{-1}\sum_{n=0}^{N-1} T^nf$ converge pointwise almost everywhere to zero with a speed of convergence which, up to a small logarithmic transgression, is essentially of the order of $N^{-1/2}$. We also provide a few applications of our results, especially in the case of shifts associated with toral endomorphisms.