Given a probability space (X,μ), 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-1n=0N-1 Tnf 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.
Chalmoukis, N., Colzani, L., Gariboldi, B., Monguzzi, A. (2024). On the speed of convergence in the ergodic theorem for shift operators. CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 1-19 [10.4153/s0008414x24000658].
On the speed of convergence in the ergodic theorem for shift operators
Chalmoukis, Nikolaos;Colzani, Leonardo;Gariboldi, Bianca;Monguzzi, Alessandro
2024
Abstract
Given a probability space (X,μ), 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-1n=0N-1 Tnf 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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