Topological obstructions for robustly transitive endomorphisms on surfaces

We address the problem of necessary conditions and topological obstructions for existence robustly transitive endomorphisms on surfaces. Concretely, we show that a weak form hyperbolicity (namely, partial hyperbolicity) is condition in order to have displaying critical points, only surfaces supporting this class systems are either torus or Klein bottle. Furthermore, also prove induced action by partially hyperbolic endomorphism first homology group has at least one eigenvalue with modulus larger than one.