Holder stability of diffeomorphisms

TL;DR

This paper establishes that certain structurally stable diffeomorphisms on compact manifolds are characterized by their Hölder stability, meaning small perturbations result in conjugate systems via Hölder continuous homeomorphisms.

Contribution

It proves an equivalence between Axiom A with strong transversality and Hölder stability for $C^2$ diffeomorphisms on compact manifolds, linking dynamical stability to Hölder conjugacy.

Findings

$C^2$ diffeomorphisms satisfying Axiom A and transversality are Hölder stable.

Small $C^1$ perturbations lead to conjugate systems via Hölder homeomorphisms.

Hölder stability characterizes a class of structurally stable diffeomorphisms.

Abstract

We prove that a $C^2$ diffeomorphism $f$ of a compact manifold $M$ satisfies Axiom A and the strong transversality condition if and only if it is H\"{o}lder stable, that is, any $C^1$ diffeomorphism $g$ of $M$ sufficiently $C^1$ close to $f$ is conjugate to $f$ by a homeomorphism which is H\"{o}lder on the whole manifold.