On the KMS states for the Bernoulli shift

Let $\mathrm{\Omega }:=\{0,1{\}}^{\mathbb{Z}}$ be the Cantor space, and let $\tau :\mathrm{\Omega }\to \mathrm{\Omega }$ be the Bernoulli shift. For the flow on the crossed product $C(\mathrm{\Omega }){⋊}_{\tau }\mathbb{Z}$ determined by a potential that depends on only one coordinate, we show that for every $\beta \mathrm{\ne }0$, there is an extremal $\beta$-KMS state on $C(\mathrm{\Omega }){⋊}_{\tau }\mathbb{Z}$ of type $I{I}_{\mathrm{\infty }}$. Also, when the potential takes values that are rationally dependent, we determine the values of $\lambda \in (0,1)$ for which there is a an extremal $\beta$-KMS state of type $II{I}_{\lambda }$.