  Let $G$ be an inductive limit of finite cyclic groups and let $A$ be a unital
simple projectionless C*-algebra with $K_1(A) \cong G$ and with a unique
tracial state, as constructed based on dimension drop algebras by Jiang and Su.
First, we show that any two aperiodic elements in $\Aut(A)/\WInn(A)$ are
conjugate, where $\WInn(A)$ means the subgroup of $\Aut(A)$ consisting of
automorphisms which are inner in the tracial representation.
  In the second part of this paper, we consider a class of unital simple
C*-algebras with a unique tracial state which contains the class of unital
simple AT-algebras of real rank zero with a unique tracial state. This class is
closed under inductive limits and under crossed products by actions of $\Z$
with the Rohlin property. Let $A$ be a TAF-algebra in this class. We show that
for any automorphism $\alpha$ of $A$ there exists an automorphism
$\widetilde{\alpha}$ of $A$ with the Rohlin property such that
$\widetilde{\alpha}$ and $\alpha$ are asymptotically unitarily equivalent. In
its proof we use an aperiodic automorphism of the Jiang-Su algebra.
