The centralizer of Komuro-expansive flows and expansive R^d-actions

Abstract

In this paper we study the centralizer of flows and ${\mathbb{R}}^d$-actions on compact Riemannian manifolds. We prove that the centralizer of every $C^{\infty }$ Komuro-expansive flow with non-ressonant singularities is trivial, meaning it is the smallest possible, and deduce there exists an open and dense subset of geometric Lorenz attractors with trivial centralizer. We show that ${\mathbb{R}}^d$-actions obtained as suspension of ${\mathbb{Z}}^d$-actions are expansive if and only if the same holds for the ${\mathbb{Z}}^d$-actions. We also show that homogeneous expansive ${\mathbb{R}}^d$-actions have quasi-trivial centralizers, meaning that it consists of orbit invariant, continuous linear reparametrizations of the ${\mathbb{R}}^d$-action. In particular, homogeneous Anosov ${\mathbb{R}}^d$-actions have quasi-trivial centralizer.

Final version, to appear at Mathematische Zeitschrift