Non-singular ℤ^d-actions: an ergodic theorem over rectangles with application to the critical dimensions

2016/06/06 by Anthony H. Dooley, A. H. Dooley, Dooley, Anthony H. +2
Mathematics · #37A35 (Secondary) #37A40 (Primary) 37A30 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.DS #msc:37A30 #msc:37A35 #msc:37A40

paper · pdf · doi:10.48550/arxiv.1606.01620

26 pages, submitted to Ergodic Theory and Dynamical Systems

arxiv created 2016/06/06 · openalex publication_date 2016/06/06 · arxiv updated 2016/06/07 · openalex created_date 2022/09/14 · openalex updated_date 2026/07/28

Abstract

We adapt techniques of Hochman to prove a non-singular ergodic theorem for ℤ^d-actions where the sums are over rectangles with side lengths increasing at arbitrary rates, and in particular are not necessarily balls of a norm. This result is applied to show that the critical dimensions with respect to sequences of such rectangles are invariants of metric isomorphism. These invariants are calculated for a class of product actions.

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