Equidistribution of divergent orbits of the diagonal group in the space of lattices

TL;DR

This paper investigates the distributional behavior of divergent orbits of diagonal groups in lattice spaces, establishing conditions for equidistribution and limitations on orbit confinement in higher dimensions.

Contribution

It introduces invariants for divergent orbits and proves equidistribution results using entropy and measure rigidity methods.

Findings

Virtually all divergent orbits of a specific type become equidistributed as discriminant increases.

In dimensions three and higher, only a few divergent orbits can remain in a compact set before diverging.

The study links orbit invariants to their distributional limits in the space of lattices.

Abstract

We consider divergent orbits of the group of diagonal matrices in the space of lattices in Euclidean space. We define two natural numerical invariants of such orbits: The discriminant - an integer - and the type - an integer vector. We then study the question of the limit distributional behaviour of these orbits as the discriminant goes to infinity. Using entropy methods we prove that for divergent orbits of a specific type, virtually any sequence of orbits equidistribute as the discriminant goes to infinity. Using measure rigidity for higher rank diagonal actions we complement this result and show that in dimension 3 or higher only very few of these divergent orbits can spend all of their life-span in a given compact set before they diverge.