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Let $C$ and $A$ be two unital separable amenable simple C*-algebras withtracial rank no more than one. Suppose that $C$ satisfies the UniversalCoefficient Theorem and suppose that $\phi_1, \phi_2: C\to A$ are two unitalmonomorphisms. 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 aclassification theorem for a class ${\cal A}$ of unital separable simpleamenable \CA s which is strictly larger than the class of separable \CA s whosetracial rank are zero or one. The class contains all unital simple ASH-algebraswhose state spaces of $K_0$ are the same as the tracial state spaces as well asthe simple inductive limits of dimension drop circle algebras. Moreover itcontains some unital simple ASH-algebras whose $K_0$-groups are not Riesz. Oneconsequence of the main result is that all unital simple AH-algebras which are${\cal Z}$-stable are isomorphic to ones with no dimension growth.

Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras