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Let $A$ be a unital separable simple amenable $C^*$-algebra with finitetracial rank which satisfies the Universal Coefficient Theorem (UCT). Suppose$\af$ and $\bt$ are two automorphisms with the Rokhlin property that {inducethe same action on the $K$-theoretical data of $A$.} We show that $\af$ and$\bt$ are strongly cocycle conjugate and uniformly approximately conjugate,that is, there exists a sequence of unitaries $\{u_n\}\subset A$ and a sequenceof strongly asymptotically inner automorphisms $\sigma_n$ such that $$ \af={\rmAd}\, u_n\circ \sigma_n\circ \bt\circ \sigma_n^{-1}\andeqn\lim_{n\to\infty}\|u_n-1\|=0, $$ and that the converse holds. {We then give a$K$-theoretic description as to exactly when $\af$ and $\bt$ are cocycleconjugate, at least under a mild restriction. Moreover, we show that given any$K$-theoretical data, there exists an automorphism $\af$ with the Rokhlinproperty which has the same $K$-theoretical data.

Kishimoto's Conjugacy Theorems in simple $C^*$-algebras of tracial rank one