Stationary probability measures on projective spaces for block-Lyapunov dominated systems

TL;DR

This paper investigates the existence and uniqueness of stationary probability measures on projective spaces for systems with block-Lyapunov contraction or expansion, refining and extending classical results in the theory of random matrix products.

Contribution

It establishes new conditions for lifting stationary measures in block-Lyapunov dominated systems, extending classical results by Furstenberg, Kifer, Hennion, and others.

Findings

Proves existence and uniqueness of lifts under contraction assumptions.

Characterizes when stationary measures on quotients have lifts under expansion.

Refines classical descriptions of stationary measures on projective spaces.

Abstract

Given a finite-dimensional real vector space $V$, a probability measure $\mu$ on $\operatorname{PGL}(V)$ and a $\mu$-invariant subspace $W$, under a block-Lyapunov contraction assumption, we prove existence and uniqueness of lifts to $P(V)\setminus P(W)$ of stationary probability measures on the quotient $P(V/W)$. In the other direction, i.e. under block-Lyapunov expansion, we prove that stationary measures on $P(V/W)$ have lifts if any only if the group generated by the support of $\mu$ stabilizes a subspace $W'$ not contained in $W$ and exhibiting a faster growth than on $W \cap W'$. These refine the description of stationary probability measures on projective spaces as given by Furstenberg, Kifer and Hennion, and under the same assumptions, extend corresponding results by Aoun, Benoist, Bru\`{e}re, Guivarc'h, and others.