Counting and equidistribution of periodic diagonal orbits.

Let $G=SL_d(R)$, $\Gamma=SL_d(Z)$ and $M$ be the subgroup of signs. I will talk about the counting and equidistribution of compact periodic orbits of the diagonal group on $\Gamma\backslash G/M$. Counting periodic diagonal orbits is a generalization of the prime geodesic theorem on compact hyperbolic surfaces, dating back to Huber, Margulis and Bowen. For $SL_2(Z)$ case, it is considered in Sarnak’s thesis.

I will sketch a proof. We use Hopf coordinates, an idea of Roblin for hyperbolic cases, the angular distribution of lattices points  and a version of non-escape of mass of periodic diagonal orbits. The talk is based on a recent joint work with Thi Dang.