  The structure of the automorphism group of a simple TAI algebra is studied.
In particular, we show that $\frac{\bar{\mrm{Inn}} (A)}{\bar{\mrm{Inn}}_{0}
(A)}$ is isomorphic (as a topological group) to an inverse limit of discrete
abelian groups for a unital, simple, AH algebra with bounded dimension growth.
Consequently, $\frac{\bar{\mrm{Inn}} (A)}{\bar{\mrm{Inn}}_{0} (A)}$ is totally
disconnected.
  Another consequence of our results is the following: Suppose $A$ is the
transformation group \cstar-algebra of a minimal Furstenberg transformation
$(\mbb{T}^{n}, h_{n})$ with a unique $h_{n}$-invariant probability measure on
$\mbb{T}^{n}$. Then the automorphism group of $A$ is an extension of a simple
topological group by the discrete group $\mrm{Aut} (\totalk(A))_{+,1}$.
