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arXiv · 2305.02879

Stationary probability measures on projective spaces 2: the critical case

Richard Aoun·Cagri Sert
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Abstract

In a previous article, given a finite-dimensional real vector space $V$ and a probability measure $\mu$ on $\operatorname{PGL}(V)$ with finite first moment, we gave a description of all $\mu$-stationary probability measures on the projective space $\operatorname{P}(V)$ in the non-critical (or Lyapunov dominated) case. In the current article, we complete the analysis by providing a full description of the more subtle critical case. Our results demonstrate an algebraic rigidity in this situation. Combining our results with those of Furstenberg--Kifer ('83), Guivarch--Raugi ('07) $\&$ Benoist--Quint ('14), we deduce a classification of all stationary probability measures on the projective space for i.i.d random matrix products with finite first moment without any algebraic assumption.

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Richard Aoun, Cagri Sert. 2023-05-04. Stationary probability measures on projective spaces 2: the critical case. https://arxiv.org/abs/2305.02879

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