Topological mixing of positive diagonal flows

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 $\mathrm{\Gamma }\mathrm{\backslash }G$. 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 $\mathrm{\Gamma }\mathrm{\backslash }G$ 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.