Equilibrium states for the random $$\beta$$- transformation through $$g$$-measures

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 $$a_i\in \{0,1 , \ldots,\lfloor\beta\rfloor\}$$ }. This transformation was introduced in [3–5], 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 the 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. The construction of these $$g$$ -measures is explicit and the corresponding potentials are not locally constant.