Equilibrium States for the Random $β$-Transformation through $g$-Measures

Abstract

We consider the random $β$-transformation $K_β$, defined on $\{0,1\}^{\mathbb N}\times[0, \frac{\lfloorβ\rfloor}{β-1}]$, that generates all possible expansions of the form $x=\sum_{i=0}^{\infty}\frac{a_i}{β^i}$, where $a_i\in \{0,1,\cdots,\lfloorβ\rfloor\}$. 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 $m_p\times μ_β$, with $m_p$ the Bernoulli $(p,1-p)$ product measure and $μ_β$ is a measure equivalent to Lebesgue measure. In this paper, we give an uncountable family of $K_β$-invariant exact $g$-measures for a certain collection of algebraic $β$'s.