Centralizer Rigidity near Elements of the Weyl Chamber Flow

TL;DR

This paper establishes rigidity results for centralizers near Weyl chamber flow elements on semisimple Lie groups, showing they are either trivial, one-dimensional, or conjugate to the original flow, with broader implications for partially hyperbolic systems.

Contribution

It proves a new rigidity theorem for centralizers near Weyl chamber flow elements and provides conditions for centralizers of partially hyperbolic diffeomorphisms to be Lie groups.

Findings

Centralizer of perturbed Weyl chamber flow elements is either trivial, one-dimensional, or conjugate to the original flow.

Established a general condition for the centralizer of partially hyperbolic diffeomorphisms to be a Lie group.

Extended rigidity results to a broad class of semisimple Lie group quotients.

Abstract

In this paper, we prove centralizer rigidity near an element of the Weyl chamber flow on a semisimple Lie group. We show that a volume preserving perturbation of an element of the Weyl chamber flow on a quotient $G/\Gamma$ of an $\mathbb{R}$-split, simple Lie group $G$ either has centralizer of dimension $0$ or $1$, or is smoothly conjugate to an element of the Weyl chamber flow. We also acquire a general condition for the centralizer of a partially hyperbolic diffeomorphism to be a Lie group.