Local Centralizer Rigidity near Elements of the Weyl Chamber Flow

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 $C^1$ 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 a centralizer of dimension $0$ or $1$, or is smoothly conjugate to an element of the Weyl chamber flow. We also obtain a general condition for the center-fixing centralizer of a partially hyperbolic diffeomorphism to be a Lie group.

arXiv comment49 pages, 2 figures