  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 $III_\lambda$.
