Weak Gibbs measures for the natural extension of $(1/β, β)$-shifts

In this paper we consider the weak Gibbs measures for $(α, β)$-shifts. In the case of $α=0$, Pfister and Sullivan have given a necessary and sufficient condition on $β$ such that any equilibrium measure for a function of bounded total oscillations is a weak Gibbs measure in the natural extension of a $β$-shift. So it is natural to ask what happens when $α>0$. However, their proof cannot be applied to general $(α, β)$-shifts in a similar way. In this paper we consider the case of $α=1/β$ and give a criterion for the weak Gibbs property of equilibrium measures for $(1/β, β)$-shifts.