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arXiv · 1703.10816

Recurrence on Affine Grassmannians

Yves Benoist·Caroline Bruère
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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 $2k\geq d$.

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Yves Benoist, Caroline Bruère. 2017-03-31. Recurrence on Affine Grassmannians. https://doi.org/10.1017/etds.2018.18

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