Ergodic Properties of the Random Walk Adic Transformation over the Beta Transformation

TL;DR

This paper investigates the ergodic properties of a random walk adic transformation linked to a beta transformation, demonstrating its conservativity, ergodicity, and distributional stability in an infinite measure setting.

Contribution

It introduces a new class of random walk adic transformations over beta transformations and establishes their fundamental ergodic properties, extending prior work on subshifts.

Findings

The transformation is conservative and ergodic.

It is asymptotically distributionally stable.

It exhibits bounded rational ergodicity.

Abstract

We define a random walk adic transformation associated to an aperiodic random walk on $G=\mathbb{Z}^{k}\times\mathbb{R}^{D-k}$ driven by a $\beta$-transformation and study its ergodic properties. In particular, this transformation is conservative, ergodic, infinite measure preserving and we prove that it is asymptotically distributionally stable and bounded rationally ergodic. Related earlier work appears in [AS] and [ANSS] for random walk adic transformations associated to an aperiodic random walk driven by a subshift of finite type.