  Let $X$ be a compact metric space and let $\af$ be a homeomorphism on $X.$
Related to a theorem of Pimsner, we show that $C(X)\rtimes_{\af}\Z$ can be
embedded into a unital simple AF-algebra if and only if there is a strictly
positive $\af$-invariant Borel probability measure.
  Suppose that $\Lambda$ is a $\Z^d$ action on $X.$ If
$C(X)\rtimes_{\Lambda}\Z$ can be embedded into a unital simple AF-algebra, then
there must exist a strictly positive $\Lambda$-invariant Borel probability
measure. We show that, if in addition, there is a generator $\af_1$ of
$\Lambda$ such that $(X, \af_1)$ is minimal and unique ergodic, then
$C(X)\rtimes_{\Lambda}\Z^d$ can be embedded into a unital simple AF-algebra
with a unique tracial state.
  Let $A$ be a unital separable amenable simple \CA with tracial rank zero and
with a unique tracial state which satisfies the Universal Coefficient Theorem
and let $G$ be a finitely generated discrete abelian group. Suppose $\Lambda:
G\to Aut(A)$ is a \hm. Then $A\rtimes_{\Lambda} G$ can always be embedded into
a unital simple AF-algebra.
