Trivial centralizers for Axiom A diffeomorphisms

TL;DR

The paper proves that a generic set of Axiom A diffeomorphisms, including non-Anosov and surface cases, have trivial centralizers, meaning they commute only with their iterates, highlighting rigidity in their symmetry structure.

Contribution

It establishes the generic triviality of centralizers for a broad class of Axiom A diffeomorphisms, extending previous results to non-Anosov and surface cases with no cycles.

Findings

Residual set of non-Anosov Axiom A diffeomorphisms have trivial centralizer.

Open and dense set of surface Axiom A diffeomorphisms have trivial centralizer.

Analysis of commuting diffeomorphisms on hyperbolic chain recurrent classes.

Abstract

We show there is a residual set of non-Anosov $C^{\infty}$ Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. If $M$ is a surface and $2\leq r\leq \infty$, then we will show there exists an open and dense set of of $C^r$ Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. Additionally, we examine commuting diffeomorphisms preserving a compact invariant set $\Lambda$ where $\Lambda$ is a hyperbolic chain recurrent class for one of the diffeomorphisms.