An avoidance principle and Margulis functions for expanding translates of unipotent orbits

TL;DR

This paper establishes an avoidance principle and quantitative results for unipotent orbit translates in semisimple homogeneous spaces, using Margulis functions and representation theory to analyze orbit distribution and closed orbit counts.

Contribution

It introduces a new avoidance principle and quantitative isolation results for unipotent orbits, along with bounds on the number of closed orbits, advancing understanding of orbit dynamics in homogeneous spaces.

Findings

Proved an avoidance principle for expanding translates of unipotent orbits.

Established a quantitative isolation result for closed orbits.

Provided an upper bound on the number of closed orbits of bounded volume.

Abstract

We prove an avoidance principle for expanding translates of unipotent orbits for some semisimple homogeneous spaces. In addition, we prove a quantitative isolation result of closed orbits and give an upper bound on the number of closed orbits of bounded volume. The proofs of our results rely on the construction of a Margulis function and the theory of finite dimensional representations of semisimple Lie groups.