Ergodic theory and the duality principle on homogeneous spaces

TL;DR

This paper establishes quantitative ergodic theorems for lattice actions on infinite volume homogeneous varieties, providing explicit convergence rates and illustrating their application in equidistribution problems.

Contribution

It introduces new quantitative ergodic theorems for lattice actions on homogeneous spaces, with explicit convergence rates and optimal forms of the duality principle.

Findings

Established mean and pointwise ergodic theorems with explicit rates

Provided optimal quantitative forms of the duality principle

Applied results to various equidistribution problems

Abstract

We prove mean and pointwise ergodic theorems for the action of a discrete lattice subgroup in a connected algebraic Lie group, on infinite volume homogeneous algebraic varieties. Under suitable necessary conditions, our results are quantitative, namely we establish rates of convergence in the mean and pointwise ergodic theorems, which can be estimated explicitly. Our results give a precise and in most cases optimal quantitative form to the duality principle governing dynamics on homogeneous spaces. We illustrate their scope in a variety of equidistribution problems.