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Random walks on projective spaces
Published online by Cambridge University Press: 17 July 2014
Abstract
Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a connected real semisimple Lie group,
$V$ be a finite-dimensional representation of
$G$ and
$\mu $ be a probability measure on
$G$ whose support spans a Zariski-dense subgroup. We prove that the set of ergodic
$\mu $-stationary probability measures on the projective space
$\mathbb{P}(V)$ is in one-to-one correspondence with the set of compact
$G$-orbits in
$\mathbb{P}(V)$. When
$V$ is strongly irreducible, we prove the existence of limits for the empirical measures. We prove related results over local fields as the finiteness of the set of ergodic
$\mu $-stationary measures on the flag variety of
$G$.
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- © The Author(s) 2014
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