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The structure of the automorphism group of a simple TAI algebra is studied. In particular, we show that $${\frac{\overline{ \rm{Inn}}(A) }{ \overline{\rm{Inn}}_{0} ( A ) }}$$ 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, $${\frac{\overline{\rm{Inn}}(A)}{\overline{\rm{Inn}}_{0}(A) }}$$ is totally disconnected. Another consequence of our results is the following: Suppose A is the transformation group C*-algebra of a minimal Furstenberg transformation with a unique invariant probability measure. Then the automorphism group of A is an extension of a simple topological group by the discrete group $${\rm{Aut}(\underline{K}(A))_{+,1}}$$ . [ABSTRACT FROM AUTHOR]