A ratio ergodic theorem for Borel actions of ℤd×ℝk

Abstract

AbstractWe prove a ratio ergodic theorem for free Borel actions of ℤd×ℝk on a standard Borel σ-finite measure space. The proof employs a lemma by Hochman involving coarse dimension, as well as the Besicovitch covering lemma. Due to possible singularity of the measure, we cannot use functional analytic arguments and therefore use Rudolph’s diffusion of the measure onto the orbits of the action. This diffused measure is denoted μx, and our averages are of the form (1/(μx(Bn)))∫ Bnf∘T−v(x) dμx(v). A Følner condition on the orbits of the action is shown, which is the main tool used in the proof of the ergodic theorem.