Effective equidistribution of translates of large submanifolds in semisimple homogeneous spaces

TL;DR

This paper proves that translates of certain curved submanifolds in semisimple homogeneous spaces become uniformly distributed over the space as they are iteratively transformed, with an effective rate of convergence.

Contribution

It establishes effective equidistribution results for translates of large submanifolds in semisimple homogeneous spaces, extending previous understanding to curved submanifolds of small codimension.

Findings

Effective convergence of translated submanifolds to the uniform measure

Quantitative rates of equidistribution for specific submanifolds

Application to horospherical subgroup dynamics

Abstract

Let $G=SL_2(\mathbb R)^d$ and $\Gamma=\Gamma_0^d$ with $\Gamma_0$ a lattice in $SL_2(\mathbb R)$. Let $S$ be any "curved" submanifold of small codimension of a maximal horospherical subgroup of $G$ relative to an $\mathbb R$-diagonalizable element $a$ in the diagonal of $G$. Then for $S$ compact our result can be described by saying that $a^n \text{vol}_S$ converges in an effective way to the volume measure of $G/\Gamma$ when $n\to \infty$, with $\text{vol}_S$ the volume measure on $S$.