# Recurrence on Affine Grassmannians

## TL;DR

This paper investigates the dynamics of the affine group acting on affine subspaces, establishing conditions for the existence of stationary measures based on Lyapunov exponents, with implications for symmetric measures.

## Contribution

It provides a precise criterion linking Lyapunov exponents to the existence of stationary measures on affine Grassmannians.

## Key findings

- Stationary measure exists iff the (k+1)th Lyapunov exponent is negative.
- For symmetric measures, stationary measures exist if and only if 2k ≥ d.
- The result characterizes the action of affine groups on affine subspaces in terms of Lyapunov exponents.

## Abstract

We study the action of the affine group $G$ of $\mathbb{R}^d$ on the space $X_{k,\,d}$ of $k$-dimensional affine subspaces. Given a compactly-supported Zariski dense probability measure $\mu$ on $G$, we show that $X_{k,\,d}$ supports a $\mu$-stationary measure $\nu$ if and only if the $(k\!+\!1)^{\rm th}$-Lyapunov exponent of $\mu$ is strictly negative. In particular, when $\mu$ is symmetric, $\nu$ exists if and only if $2k\geq d$.