Centralizers of derived-from-Anosov systems on $\mathbb{T}^3$: rigidity versus triviality

TL;DR

This paper investigates the structure of centralizers of certain partially hyperbolic diffeomorphisms on the 3-torus, revealing conditions under which they are either trivial or smoothly conjugate to linear automorphisms.

Contribution

It establishes a rigidity result for centralizers of partially hyperbolic diffeomorphisms homotopic to Anosov automorphisms on -torus, showing they are either trivial or conjugate to linear maps.

Findings

Centralizer is either virtually trivial or conjugate to a linear automorphism.

Diffeomorphisms homotopic to Anosov automorphisms exhibit rigidity in their centralizer structure.

Abstract

In this paper, we study the centralizer of a partially hyperbolic diffeomorphism on $\mathbb{T}^3$ which is homotopic to an Anosov automorphism, and we show that either its centralizer is virtually trivial or such diffeomorphism is smoothly conjugate to its linear part.