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 $\mu$ on $\mathrm{PGL}(V)$ with finite first moment, we gave a description of all $\mu$-stationary probability measures on the projective space $\mathrm{P}(V)$ 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) $\mathrm{\&}$ 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.