Topological mixing of positive diagonal flows

• Published in: Israel journal of mathematics Vol. 260; no. 1; pp. 1 - 71
• Main Author: Dang, Nguyen-Thi
• Format: Journal Article
• Language: English
• Published: Jerusalem The Hebrew University Magnes Press 2024; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Group Theory and Generalizations; Lie groups; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Subgroups; Theoretical; Topology
• ISSN: 0021-2172; 1565-8511
• DOI: 10.1007/s11856-023-2561-1

Let G be a connected, real linear, semisimple Lie group without compact factors and Γ < G a Zariski dense, discrete subgroup. We study the topological dynamics of positive diagonal flows on Γ 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 Γ 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.