On sensitivity to initial conditions and uniqueness of conjugacies for structurally stable diffeomorphisms

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.