  Let $C$ and $A$ be two unital separable amenable simple C*-algebras with
tracial rank no more than one. Suppose that $C$ satisfies the Universal
Coefficient Theorem and suppose that $\phi_1, \phi_2: C\to A$ are two unital
monomorphisms. We show that there is a continuous path of unitaries $\{u_t:
t\in [0, \infty)\}$ of $A$ such that $$
\lim_{t\to\infty}u_t^*\phi_1(c)u_t=\phi_2(c)\tforal c\in C $$ if and only if
$[\phi_1]=[\phi_2]$ in $KK(C,A),$ $\phi_1^{\ddag}=\phi_2^{\ddag},$
$(\phi_1)_T=(\phi_2)_T$ and a rotation related map $\bar{R}_{\phi_1,\phi_2}$
associated with $\phi_1$ and $\phi_2$ is zero.
  Applying this result together with a result of W. Winter, we give a
classification theorem for a class ${\cal A}$ of unital separable simple
amenable \CA s which is strictly larger than the class of separable \CA s whose
tracial rank are zero or one. The class contains all unital simple ASH-algebras
whose state spaces of $K_0$ are the same as the tracial state spaces as well as
the simple inductive limits of dimension drop circle algebras. Moreover it
contains some unital simple ASH-algebras whose $K_0$-groups are not Riesz. One
consequence of the main result is that all unital simple AH-algebras which are
${\cal Z}$-stable are isomorphic to ones with no dimension growth.
