  Let $A$ be a unital simple C*-algebra with tracial rank zero and $X$ be a
compact metric space. Suppose that $h_1, h_2: C(X)\to A$ are two unital
monomorphisms. We show that $h_1$ and $h_2$ are approximately unitarily
equivalent if and only if $$ [h_1]=[h_2] {\rm in} KL(C(X),A) {\rm and}
\tau\circ h_1(f)=\tau\circ h_2(f) $$ for every $f\in C(X)$ and every trace
$\tau$ of $A.$ Adopting a theorem of Tomiyama, we introduce a notion of
approximate conjugacy for minimal dynamical systems. Let $X$ be a compact
metric space and $\alpha, \beta: X\to X$ be two minimal homeomorphisms. Using
the above mentioned result, we show that two dynamical systems are
approximately conjugate in that sense if and only if a $K$-theoretical
condition is satisfied. In the case that $X$ is the Cantor set, this notion
coincides with strong orbit equivalence of Giordano, Putnam and Skau and the
$K$-theoretical condition is equivalent to saying that the associate crossed
product C*-algebras are isomorphic.
  Another application of the above mentioned result is given for
$C^*$-dynamical systems related to a problem of Kishimoto. Let $A$ be a unital
simple AH-algebra with no dimension growth and with real rank zero, and let
$\alpha\in Aut(A).$ We prove that if $\alpha^r$ fixes a large subgroup of
$K_0(A)$ and has the tracial Rokhlin property then $A\rtimes_{\alpha}\Z$ is
again a unital simple AH-algebra with no dimension growth and with real rank
zero.
