Equidistribution and counting of periodic tori in the space of Weyl chambers

Abstract

Let $G$ be a semisimple Lie group without compact factor and $\Gamma <G$ a torsion-free, cocompact, irreducible lattice. According to Selberg, periodic orbits of regular Weyl chamber flows live on tori. We prove that these periodic tori equidistribute exponentially fast towards the quotient of the Haar measure. From the equidistribution formula, we deduce a higher rank prime geodesic theorem. As a corollary, we obtain an upper bound of the growth of conjugacy classes with non-trivial polynomial term.