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

Karma Dajani, Kieran Power

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_…