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 Jäger.

We show that if there exists a counter example for the rational case of the Franks–Misiurewicz conjecture, then it must exhibit unbounded deviations in the complementary direction of its rotation set.