Recurrence on affine Grassmannians

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$\unicode[STIX]{x1D707}$on$G$, we show that$X_{k,d}$supports a$\unicode[STIX]{x1D707}$-stationary measure$\unicode[STIX]{x1D708}$if and only if the$(k+1)\text{th}$Lyapunov exponent of$\unicode[STIX]{x1D707}$is strictly negative. In particular, when$\unicode[STIX]{x1D707}$is symmetric,$\unicode[STIX]{x1D708}$exists if and only if$2k\geq d$.