Periodic point free homeomorphisms and irrational rotation factors

We provide a complete characterization of periodic point free homeomorphisms of the $2$-torus admitting irrational circle rotations as topological factors. Given a homeomorphism of the $2$-torus without periodic points and exhibiting uniformly bounded rotational deviations with respect to a rational direction, we show that annularity and the geometry of its non-wandering set are the only possible obstructions for the existence of an irrational circle rotation as topological factor. Through a very precise study of the dynamics of the induced $\rho$-centralized skew-product, we extend and generalize considerably previous results of T. J\"ager [Inventiones Mathematicae, 176 (2009), n. 3, 601-616].