Equilibrium States for the Random $\beta$-Transformation through $g$-Measures

TL;DR

This paper studies the random $eta$-transformation, identifying a large family of invariant measures for certain algebraic $eta$ values, expanding understanding of its ergodic properties.

Contribution

It introduces an uncountable family of invariant $g$-measures for the random $eta$-transformation on algebraic $eta$ values, extending prior measure classifications.

Findings

Identifies a large family of invariant $g$-measures.

Extends the understanding of ergodic measures for the random $eta$-transformation.

Focuses on algebraic $eta$'s, providing new measure constructions.

Abstract

We consider the random $\beta$-transformation $K_{\beta}$, defined on $\{0,1\}^{\mathbb N}\times[0, \frac{\lfloor\beta\rfloor}{\beta-1}]$, that generates all possible expansions of the form $x=\sum_{i=0}^{\infty}\frac{a_i}{\beta^i}$, where $a_i\in \{0,1,\cdots,\lfloor\beta\rfloor\}$. This transformation was first introduced by Dajani and Kraaikamp, and later studied by Dajani and de Vries, where two natural invariant ergodic measures were found. The first is the unique measure of maximal entropy, and the second is a measure of the form $m_p\times \mu_{\beta}$, with $m_p$ the Bernoulli $(p,1-p)$ product measure and $\mu_{\beta}$ is a measure equivalent to Lebesgue measure. In this paper, we give an uncountable family of $K_{\beta}$-invariant exact $g$-measures for a certain collection of algebraic $\beta$'s.