Equilibrium states for the random \(\beta\)-transformation through \(g\)-measures (Q2116372)

This paper considers the random \(\beta\)-transformation \(K_\beta\) introduced by \textit{K. Dajani} and \textit{C. Kraaikamp} [Ergodic Theory Dyn. Syst. 23, No. 2, 461--479 (2003; Zbl 1035.37006)]. \(g\)-measures were studied by \textit{M. Keane} [Invent. Math. 16, 309--324 (1972; Zbl 0241.28014)]. In the present paper the authors find an uncountable family of \(K_\beta\)-invariant exact \(g\)-measures for a particular set of algebraic \(\beta\)'s. The family of \(g\)-measures is explicitly given and the corresponding potentials are not locally constant.