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

In this paper we study C¹-structurally stable diffeomorphisms, that is, C¹ 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 C¹-close structurally stable diffeomorphisms may be non-unique, even if there are assumed C⁰-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 C⁰-centralizer of the dynamics. Finally, we use this fact to deduce that any two C¹-close structurally stable diffeomorphisms are conjugated by a unique conjugacy C⁰-close to the identity if and only if these are Anosov.