Equilibrium states for the random β - transformation through g -measures

We consider the random β-transformation Kβ, defined on {0,1}N×[0,⌊β⌋]β-1]], that generates all possible expansions of the form x=∑i=0∞aiβi, whereai∈ { 0 , 1 , … , ⌊ β⌋ } }. This transformation was introduced in [3–5], where two naturalinvariant ergodic measures were found. The first is the unique measure ofmaximal entropy, and the second is a measure of the form mp× μβ, with mpthe Bernoulli (p, 1 - p) product measure and μβ is a measure equivalent to theLebesgue measure. In this paper, we give an uncountable family of Kβ-invariantexact g-measures for a certain collection of algebraic β’s. The construction of theseg-measures is explicit and the corresponding potentials are not locally constant.