Open sets of diffeomorphisms with trivial centralizer in the $C^1$ topology

TL;DR

This paper demonstrates that in the $C^1$ topology on tori of dimensions 2, 3, and 4, there exist open sets of diffeomorphisms with trivial centralizer, using fixed point and periodic point methods near hyperbolic automorphisms.

Contribution

It introduces two approaches to establish the existence of open sets of diffeomorphisms with trivial centralizer in the $C^1$ topology on low-dimensional tori.

Findings

Open sets of diffeomorphisms with trivial centralizer exist in the $C^1$ topology.

Methods developed work near certain hyperbolic toral automorphisms.

Results apply to tori of dimensions 2, 3, and 4.

Abstract

On the torus of dimension $2$, $3$, or $4$, we show that the subset of diffeomorphisms with trivial centralizer in the $C^1$ topology has nonempty interior. We do this by developing two approaches, the fixed point and the odd prime periodic point, to obtain trivial centralizer for an open neighbourhood of Anosov diffeomorphisms arbitrarily near certain irreducible hyperbolic toral automorphism.