Kishimotos Conjugacy Theorems in simple $C^*$-algebras of tracial rank one

Let $A$ be a unital separable simple amenable $C^*$-algebra with finite tracial rank which satisfies the Universal Coefficient Theorem (UCT). Suppose $af$ and $bt$ are two automorphisms with the Rokhlin property that {induce the 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 sequence of strongly asymptotically inner automorphisms $sigma_n$ such that $$ af={rm Ad}, u_ncirc sigma_ncirc btcirc sigma_n^{-1}andeqn lim_{ntoinfty}|u_n-1|=0, $$ and that the converse holds. {We then give a $K$-theoretic description as to exactly when $af$ and $bt$ are cocycle conjugate, at least under a mild restriction. Moreover, we show that given any $K$-theoretical data, there exists an automorphism $af$ with the Rokhlin property which has the same $K$-theoretical data.