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⋊αZ)⩽1. We also show that whenever A has a unique tracial state and αm is uniformly outer for each m(≠0) 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 some Rokhlin property, then the crossed product A⋊αZ 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.