Appears in collection : Homogeneous Dynamics and Geometry in Higher-Rank Lie Groups

Bowen and Margulis independently proved in the 70s that closed geodesics on compact hyperbolic surfaces equidistribute towards the measure of maximal entropy. From a homogeneous dynamics point of view, this measure is the quotient of the Haar measure on $\mathrm{PSL}(2,\mathbb{R})$ modulo some discrete cocompact sugroup.

In a joint work with Jialun Li, we investigate the higher rank setting of this problem by taking a higher rank Lie group (like $\mathrm{SL}(d,\mathbb{R})$ for $d\geq 3$) and by studying the dynamical properties of geodesic flows in higher rank: the so-called Weyl chamber flows and their induced diagonal action. We obtain an equidistribution formula of periodic tori (instead of closed orbits of the geodesic flow).