Stationary probability measures on projective spaces 2: The critical case

In a previous article, given a finite-dimensional real vector space V and a probability measure μ on PGL(V) with finite first moment, we gave a description of all μ-stationary probability measures on the projective space P(V) in the noncritical (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 and Kifer, Guivarch and Raugi, and Benoist and Quint, we deduce a classification of all stationary probability measures on the projective space for independent and identically distributed random matrix products with finite first moment without any algebraic assumption.