Stationary probability measures on projective spaces 1: block-Lyapunov dominated systems

Given a finite-dimensional real vector space V, a probability measure $$\mu $$ μ on $${\text {PGL}}(V)$$ PGL ( V ) and a $$\mu $$ μ -invariant subspace W, under a block-Lyapunov contraction assumption, we prove existence and uniqueness of lifts to $$P(V){\setminus } P(W)$$ P ( V ) \ P ( W ) of stationary probability measures on the quotient P(V/W). In the other direction, i.e. 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 $$\mu $$ μ stabilizes a subspace $$W'$$ W ′ not contained in W and exhibiting a faster growth than on $$W \cap W'$$ W ∩ 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, Bruère, Guivarc’h, and others.