  Let $\varphi:X\to X$ 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 $C^*$-algebra
$C(X)\rtimes_{\varphi}\mathbb{Z}$ defined such that
$\alpha^{F}_{t}(fU)=fUe^{-itF}$ 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 $C(X) \rtimes_{\phi} \mathbb Z$ is simple this set is either $\{0\}$
or the whole line $\mathbb R$.
