Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## Topological dynamics of the Weyl Chamber flow

Vortrag von Dr. Nguyen-Thi Dang

Sprecher eingeladen von: Prof. Dr. Alexander Gorodnik

**Datum:** 11.11.19 **Zeit:** 14.00 - 15.00 **Raum:** ETH HG G 43

Let G be a connected, real linear, semisimple Lie group of non-compact type. Consider a Cartan subgroup A, a closed Weyl chamber A^+ in A and a maximal compact subgroup K for which the Cartan decomposition G = KA^+K holds. Denote by M the centralizer subgroup of A in K. Let D be a discrete subgroup of G. In this talk, we will assume that the real rank of G is higher than 2 and that D is Zariski dense (not necessarily a lattice). I am interested in dynamical systems of the form D\G/M x a_t, where a_t is a Weyl chamber flow. One can ask: when do they have non-diverging orbits? orbits that are dense in their A−orbit? is there topological mixing in a suitable subset of D\G/M? For G = SL(2, R), they identify with the action of the geodesic flow on the unit tangent bundle of D\H^2. The latter's topological dynamics are rather well understood due to the works of many people among which Hopf, Hedlund, Eberlein, Dal'bo... In particular, it is topologically mixing on its non-wandering set. In this talk, I will first state the main results on Weyl chamber flows of my thesis. Then I will sketch a proof of a joint work with O. Glorieux, a necessary and sufficient condition for topological mixing for regular Weyl chamber flows. Time permitting, I will present a generalization of this criteria for actions on D\G for SL(n, R) or SL(n, C).