KMS states on the crossed product $C^{\ast }$-algebra of a homeomorphism

Let $\phi :X\to X$ be a homeomorphism of a compact metric space $X$. For any continuous function $F:X\to \mathbb{R}$ there is a one-parameter group ${\alpha }^F$ of automorphisms on the crossed product $C^{\ast }$-algebra $C(X){⋊}_{\phi }\mathbb{Z}$ defined such that ${\alpha }_t^F(fU)=fU{e}^{-itF}$ when $f\in C(X)$ and $U$ is the canonical unitary in the construction of the crossed product. In this paper we study the KMS states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS-states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when $C(X){⋊}_{\varphi }\mathbb{Z}$ is simple this set is either $\{0\}$ or the whole line $\mathbb{R}$.