Polynomial dynamic and lattice orbits in S-arithmetic homogeneous spaces

TL;DR

This paper investigates the distribution of dense orbits in S-arithmetic homogeneous spaces, extending known results from Lie groups to p-adic and S-arithmetic contexts using dynamical methods.

Contribution

It introduces a framework for analyzing orbit distribution in S-arithmetic homogeneous spaces, generalizing equidistribution results beyond Lie groups.

Findings

Established equidistribution of dense orbits in S-arithmetic spaces

Extended polynomial dynamics techniques to p-adic settings

Provided new insights into orbit repartition in non-Archimedean groups

Abstract

Consider an homogeneous space under a locally compact group G and a lattice in G. Then the lattice naturally acts on the homogeneous space. Looking at a dense orbit, one may wonder how to describe its repartition. One then adopt a dynamical point of view and compare the asymptotic distribution of points in the orbits with the natural measure on the space. In the setting of Lie groups and their homogeneous spaces, several results showed an equidistribution of points in the orbits using Ratner's rigidity of polynomial dynamics in homogeneous spaces. We address here this problem in the setting of p-adic and S-arithmetic groups.