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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 ononly one coordinate, we show that for every $\beta \neq 0$, there is anextremal $\beta$-KMS state on $C(\Omega)\rtimes_\tau \mathbb{Z}$ of type$II_\infty$. Also, when the potential takes values that are rationallydependent, we determine the values of $\lambda \in (0,1)$ for which there is aan extremal $\beta$-KMS state of type $III_\lambda$.

On the KMS states for the Bernoulli shift