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

TL;DR

This paper investigates the conditions under which structurally stable diffeomorphisms exhibit sensitivity to initial conditions and have unique conjugacies, highlighting the special case of Anosov diffeomorphisms.

Contribution

It provides a necessary and sufficient condition for dense expansiveness in structurally stable diffeomorphisms and characterizes the space of conjugacies in relation to the centralizer.

Findings

Characterizes when structurally stable diffeomorphisms have dense points with sensitivity.

Shows the conjugacy space is homeomorphic to the centralizer.

Establishes that only Anosov diffeomorphisms have unique $C^0$-close conjugacies.

Abstract

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.