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This article deals with directional rotational deviations for non-wanderingperiodic point free homeomorphisms of the 2-torus which are homotopic to theidentity. We prove that under mild assumptions, such a homeomorphism exhibitsuniformly bounded rotational deviations in some direction if and only it leavesinvariant a pseudo-foliation, a notion which is a slight generalization ofclassical one-dimensional foliations. To get these results, we introduce anovel object called $\tilde\rho$-centralized skew-product and their associatedstable sets at infinity.

Rotational deviations and invariant pseudo-foliations for periodic point free torus homeomorphisms