Trivial centralizers for codimension-one attractors

TL;DR

This paper proves that for certain hyperbolic attractors in dynamical systems, most nearby systems have trivial centralizers, meaning they have no nontrivial symmetries on the basin of attraction.

Contribution

It establishes that for codimension-one hyperbolic attractors, trivial centralizers are generic among nearby diffeomorphisms, extending understanding of symmetries in dynamical systems.

Findings

Trivial centralizers are generic near certain hyperbolic attractors.

Most nearby systems have no nontrivial symmetries on the basin.

The result applies to non-Anosov systems with codimension-one attractors.

Abstract

We show that if $\Lambda$ is a codimension-one hyperbolic attractor for a $C^r$ diffeomorphism $f$, where $2\leq r\leq \infty$, and $f$ is not Anosov, then there is a neighborhood $\mathcal{U}$ of $f$ in $\mathrm{Diff}^r(M)$ and an open and dense set $\mathcal{V}$ of $\mathcal{U}$ such that any $g\in\mathcal{V}$ has a trivial centralizer on the basin of attraction for $\Lambda$.