Weak Gibbs measures for the natural extension of $(1/β, β)$-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 $\alpha=1/\beta$ and give a criterion for the weak Gibbs property of equilibrium measures for $(1/\beta, \beta)$-shifts.