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Recurrence on affine Grassmannians

Published online by Cambridge University Press:  05 April 2018

YVES BENOIST  and
CAROLINE BRUÈRE
YVES BENOIST
Affiliation:
Université Paris-Sud, Orsay 91405, France email [email protected], [email protected]
CAROLINE BRUÈRE
Affiliation:
Université Paris-Sud, Orsay 91405, France email [email protected], [email protected]
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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 $\unicode[STIX]{x1D707}$ 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 $2k\geq d$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 12 , December 2019 , pp. 3207 - 3223
Copyright
© Cambridge University Press, 2018 

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