The Rokhlin property and the tracial topological rank

Let A be a unital separable simple C∗-algebra with TR(A) ≤ 1 and α be an automorphism. We show that if α satisfies the tracially cyclic Rokhlin property then TR(A ⋊α Z) ≤ 1. We also show that whenever A has a unique tracial state and αm is uniformly outer for each m and αr is approximately inner for some r > 0, α satisfies the tracial cyclic Rokhlin property. By applying the classification theory of nuclear C∗algebras, we use the above result to prove a conjecture of Kishimoto: if A is a unital simple AT-algebra of real rank zero and α ∈ Aut(A) which is approximately inner and if α satisfies a Rokhlin property, then the crossed product A ⋊α Z is again an AT -algebra of real rank zero. As a by-product, we find that one can construct a large class of simple C∗-algebras with tracial rank one (and zero) from crossed products.