2023/02/13 by Aidan Young, Young, Aidan
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2302.06139
openalex publication_date 2023/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we extend the notion of temporo-spatial differentiation problems to the setting of actions of more general topological groups. The problem can be expressed as follows: Given an action T of an amenable discrete group G on a probability space (X, μ) by automorphisms, let (Fk)k = 1^∞ be a Følner sequence for G, and let (Ck)k = 1^∞ be a sequence of measurable subsets of X with positive probability μ(Ck). What is the limiting behavior of the sequence ( (1)/(μ(Ck)) ∫Ck (1)/(|Fk|) ∑g ∈ Fk f(Tg x) d μ(x) )k = 1^∞ for f ∈ L^∞(X, μ)? We provide some positive convergence results for temporo-spatial differentiations with respect to ergodic averages over Følner sequences, as well as with respect to ergodic averages over subsequences of the integers (e.g. polynomials), multiple ergodic averages, and weighted ergodic averages.