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arXiv · 2312.08922

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

Nikolaos Chalmoukis·Leonardo Colzani·Bianca Gariboldi·Alessandro Monguzzi
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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.

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Nikolaos Chalmoukis, Leonardo Colzani, Bianca Gariboldi, Alessandro Monguzzi. 2023-12-14. On the speed of convergence in the ergodic theorem for shift operators. https://doi.org/10.4153/s0008414x24000658

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