  We define the tracial Rokhlin property for actions of finite cyclic groups on
stably finite simple unital C*-algebras. We prove that when the algebra is in
addition simple and has tracial rank zero, then the crossed product again has
tracial rank zero. Under a kind of weak approximate innerness assumption and
one other technical condition, we prove that if the action has the the tracial
Rokhlin property and the crossed product has tracial rank zero, then the
original algebra has tracial rank zero. We give examples showing how the
tracial Rokhlin property differs from the Rokhlin property of Izumi.
  We use these results, together with work of Elliott-Evans and Kishimoto, to
give an inductive proof that every simple higher dimensional noncommutative
torus is an AT algebra. We further prove that the crossed product of every
simple higher dimensional noncommutative torus by the flip is an AF algebra,
and that the crossed products of irrational rotation algebras by the standard
actions of the cyclic groups of orders 3, 4, and 6 are simple AH algebras with
real rank zero.
