Topological mixing of positive diagonal flows (Q6561658)

Let \(G\) be a connected, real linear, semisimple Lie group without compact factors. Let \(A\) be a maximal \(\mathbb{R}\)-split torus, whose Lie algebra \(\mathfrak{a}\) is a Cartan subspace. Denote by \(\mathfrak{a}^+ \subset \mathfrak{a}\) a choice of a closed positive Weyl chamber with \(A^+=\text{exp}(\mathfrak{a}^+)\). Let \(K\) be a maximal compact subgroup of \(G\) for which the Cartan decomposition \(KA^+K\) of elements of \(G\) holds. Denote by \(M_0\) the connected component of the identity of \(Z_K(A)\). Let \(\Gamma\) be a Zariski dense, discrete subgroup of \(G\).\N\NThe author partitions the preimage in \(\Gamma\backslash G\) of the non-wandering set of mixing regular Weyl chamber flows, into finitely many \(A\)-invariant subsets that are dynamically conjugated to each other for positive diagonal flows. Then the author gives a necessary condition for the topological mixing properties of positive diagonal flows on \(\Gamma\backslash G\) and proves that it is sufficient when \(M_0\) is abelian. The key idea of the author is to construct Bruhat-Hopf coordinates that extend Hopf coordinates, which gives the framework to estimate the elliptic part of generic products of loxodromic elements.