Shrinking random $\beta$-transformation

TL;DR

This paper introduces the shrinking random $eta$-transformation for specific $eta$ values, studies its invariant measures, and reveals differences in ergodic measures between the transformation and its induced system.

Contribution

It defines the shrinking random $eta$-transformation and analyzes the uniqueness and properties of its invariant measures and entropy.

Findings

Both transformations have a unique measure of maximal entropy.

The induced system's measure differs from the original system's ergodic measure.

The paper characterizes the invariant measures for the transformation and its induced version.

Abstract

For any $n\geq 3$, let $1<\beta<2$ be the largest positive real number satisfying the equation $$\beta^n=\beta^{n-2}+\beta^{n-3}+\cdots+\beta+1.$$ In this paper we define the shrinking random $\beta$-transformation $K$ and investigate natural invariant measures for $K$, and the induced tranformation of $K$ on a special subset of the domain. We prove that both transformations have a unique measure of maximal entropy. However, the measure induced from the intrinsically ergodic measure for $K$ is not the intrinsically ergodic measure for the induced system.