Bounded and unbounded behavior for area-preserving rational pseudo-rotations

A rational pseudo-rotation $f$ of the torus is a homeomorphism homotopic to the identity with a rotation set consisting of a single vector $v$ of rational coordinates. We give a classification for rational pseudo-rotations with an invariant measure of full support, in terms of the deviations from the constant rotation $x\mapsto x+v$ in the universal covering. For the simpler case that $v=(0,0)$, it states that either every orbit by the lifted dynamics is bounded, or the displacement in some rational direction is uniformly bounded (implying that the dynamics is annular) or the set of fixed points of $f$ contains a large continuum which is the complement of a disjoint union of disks (i.e. a fully essential continuum). In the analytic setting, the latter case is ruled out. In order to prove this classification, we introduce tools that are of independent interest and can be applied in a more general setting: in particular, a geometric result about the quasi-convexity and existence of asymptotic directions for certain chains of disks, and a Poincar\'e recurrence theorem on the universal covering for irrotational measures.