Translates of homogeneous measures associated with observable subgroups on some homogeneous spaces

TL;DR

This paper investigates the limiting behavior of translated invariant measures on homogeneous spaces associated with observable subgroups, providing classification, non-divergence criteria, and extending previous foundational work.

Contribution

It offers a comprehensive classification of measure limits, criteria for non-divergence, and extends prior results in the dynamics of homogeneous spaces.

Findings

Complete classification of measure limits in non-divergent cases

Criteria for non-divergence of measures under translation

Extension of Eskin--Mozes--Shah and Shapira--Zheng results

Abstract

In the present article we study the following problem. Let G be a linear algebraic group over Q, $\Gamma$ be an arithmetic lattice and H be an observable Q-subgroup. There is a H-invariant measure $\mu_H$ supported on the closed submanifold $H\Gamma/\Gamma$. Given a sequence $g_n$ in G we study the limiting behavior of $(g_n)_*\mu_H$. In the non-divergent case we give a rather complete classification. We further supplement this by giving criterion of non-divergence and prove non-divergence for arbitrary sequence $g_n$ for certain H. This work can be viewed as a natural extension of the work of Eskin--Mozes--Shah and Shapira--Zheng.