Weak Gibbs measures for the natural extension of $(\kappa /\beta ,\beta )$-shifts

Abstract

In this paper we consider the weak Gibbs measures for $(\alpha ,\beta )$-shifts. In the case of $\alpha =0$, Pfister and Sullivan have given a necessary and sufficient condition on $\beta$ such that any equilibrium measure for a function of bounded total oscillations is a weak Gibbs measure in the natural extension of a $\beta$-shift. So it is natural to ask what happens when $\alpha >0$. However, their proof cannot be applied to general $(\alpha ,\beta )$-shifts in a similar way. In this paper we consider the case of $\alpha =\kappa /\beta$ and give a criterion for the weak Gibbs property of equilibrium measures for $(\kappa /\beta ,\beta )$-shifts.

24 pages, 2 figure