Topological mixing of positive diagonal flows

TL;DR

This paper investigates the topological mixing properties of positive diagonal flows on certain Lie groups, extending coordinate systems and providing conditions for mixing based on the structure of the centralizer.

Contribution

It introduces Bruhat-Hopf coordinates for analyzing diagonal flows and establishes necessary and sufficient conditions for topological mixing in this setting.

Findings

Extended Hopf coordinates to Bruhat-Hopf coordinates.

Partitioned the non-wandering set into finitely many conjugated subsets.

Proved a necessary and, under certain conditions, sufficient condition for topological mixing.

Abstract

Let $G$ be a semi-simple real Lie group without compact factors and $ \Gamma < G$ a Zariski dense, discrete subgroup. We study the topological dynamics of positive diagonal flows on $\Gamma \backslash G$. We extend Hopf coordinates to Bruhat-Hopf coordinates of $G$, which gives the framework to estimate the elliptic part of products of large generic loxodromic elements. By rewriting results of Guivarc'h-Raugi into Bruhat-Hopf coordinates, we partition the preimage in $\Gamma \backslash G$ of the non-wandering set of mixing regular Weyl chamber flows, into finitely many dynamically conjugated subsets. We prove a necessary condition for topological mixing, and when the connected component of the identity of the centralizer of the Cartan subgroup is abelian, we prove it is sufficient.