  In this paper we study $C^1$-structurally stable diffeomorphisms, that is,
$C^1$ Axiom A diffeomorphisms with the strong transversality condition. In
contrast to the case of dynamics restricted to a hyperbolic basic piece,
structurally stable diffeomorphisms are in general not expansive and the
conjugacies between $C^1$-close structurally stable diffeomorphisms may be
non-unique, even if there are assumed $C^0$-close to the identity. Here we give
a necessary and sufficient condition for a structurally stable diffeomorphism
to admit a dense subset of points with expansiveness and sensitivity to initial
conditions. Morever, we prove that the set of conjugacies between elements in
the same conjugacy class is homeomorphic to the $C^0$-centralizer of the
dynamics. Finally, we use this fact to deduce that any two $C^1$-close
structurally stable diffeomorphismsare conjugated by a unique conjugacy
$C^0$-close to the identity if and only if these are Anosov.
