Induced transformations of recurrent a.m.s. dynamical systems

This note proves that an induced transformation with respect to a finite measure set of a recurrent asymptotically mean stationary dynamical system with a sigma-finite measure is asymptotically mean stationary. Consequently, the Shannon–McMillan–Breiman theorem, as well as the Shannon–McMillan theorem, holds for all reduced processes of any finite-state recurrent asymptotically mean stationary random process. As a by-product, a ratio ergodic theorem for asymptotically mean stationary dynamical systems is presented.

A ratio ergodic theorem for Borel actions of ℤd×ℝk

We 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.