On the KMS states for the Bernoulli shift

Let $\Omega:=\{0,1\}^{\mathbb{Z}}$ be the Cantor space, and let $\tau:\Omega \to \Omega$ be the Bernoulli shift. For the flow on the crossed product $C(\Omega)\rtimes_\tau \mathbb{Z}$ determined by a potential that depends on only one coordinate, we show that for every $\beta \neq 0$, there is an extremal $\beta$-KMS state on $C(\Omega)\rtimes_\tau \mathbb{Z}$ of type $II_\infty$. 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 $III_\lambda$.