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

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 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], two natural invariant ergodic measures were found. The first is unique measure maximal entropy, and second a $$m_p\times \mu_{\beta}$$ with $$m_p$$ Bernoulli $$(p,1-p)$$ product $$\mu_{\beta}$$ equivalent to Lebesgue measure. In this paper, we give an uncountable family -invariant exact $$g$$ -measures for certain collection algebraic ’s. construction these explicit corresponding potentials are not locally constant.