Distribution of periodic torus orbits on homogeneous spaces

Abstract

We prove results toward the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of ${\mathrm{PGL}}_n(\mathbb{R})$. After attaching to each periodic orbit an integral invariant (the discriminant), our results have the following flavor: certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most $O({\Delta }^{ϵ})$ orbits of discriminant at most $\Delta$. The proof relies on the well-separatedness of periodic orbits together with measure rigidity for torus actions. We give examples of sequences of periodic orbits of this action that fail to become equidistributed even in higher rank. We also give an application of our results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees