  We study a general Kishimoto's problem for automorphisms on simple
C*-algebras with tracial rank zero. Let $A$ be a unital separable simple
C*-algebra with tracial rank zero and let $\alpha$ be an automorphism. Under
the assumption that $\alpha$ has certain Rokhlin property, we present a proof
that $A\rtimes_{\alpha}\Z$ has tracial rank zero. We also show that if the
induced map $\alpha_{*0}$ on $K_0(A)$ fixes a "dense" subgroup of $K_0(A)$ then
the tracial Rokhlin property implies a stronger Rokhlin property. Consequently,
the induced crossed product C*-algebras have tracial rank zero.
