Julia sets on ℝℙ² and dianalytic dynamics

We study analytic maps of the sphere that project to well-defined maps on the nonorientable real surface R P 2 \mathbb {RP}^2 . We parametrize all maps with two critical points on the Riemann sphere C ∞ \mathbb {C}_\infty , and study the moduli space associated to these maps. These maps are also called quasi-real maps and are characterized by being conformally conjugate to a complex conjugate version of themselves. We study dynamics and Julia sets on R P 2 \mathbb {RP}^2 of a subset of these maps coming from bicritical analytic maps of the sphere.