Locally unipotent invariant measures and limit distribution of a sequence of polynomial trajectories on homogeneous spaces

TL;DR

This paper introduces locally unipotent invariant measures on homogeneous spaces and proves their relevance to the equidistribution of polynomial trajectories, extending understanding of limit distributions in this setting.

Contribution

It defines locally unipotent invariant measures and proves their emergence as limit measures for polynomial trajectories on homogeneous spaces, addressing a problem posed by Shah.

Findings

Limit measures on polynomial trajectories are locally unipotent invariant.

Provides partial solution to Shah's equidistribution problem.

Utilizes Ratner's measure classification and linearization techniques.

Abstract

Let $G$ be a Lie group and $\Gamma$ be a lattice in $G$. We introduce the notion of locally unipotent invariant measures on $G/\Gamma$. We then prove that under some conditions, the limit measure supported on the image of polynomial trajectories on $G/\Gamma$ is locally unipotent invariant, thus give a partial answer to an equidistribution problem for higher dimensional polynomial trajectories on homogeneous spaces, which was raised by Shah in \cite{shah1994limit}. The proof relies on Ratner's measure classification theorem, linearization technique for polynomial trajectories near singular sets and a twisting technique of Shah.