On annular maps of the torus and sublinear diffusion

There is a classification by Misiurewicz and Ziemian of elements in Homeo$_0(\mathbf{T}^2)$ by their rotation set $\rho$, according to wether $\rho$ is a point, a segment or a set with nonempty interior. A recent classification of nonwandering elements in Homeo$_0(\mathbf{T}^2)$ by Koropecki and Tal has been given, according to the itrinsic underlying ambient where the dynamics takes place: planar, annular and strictly toral maps. We study the link between these two classifications, showing that, even abroad the nonwandering setting, annular maps are characterized by rotation sets which are \textit{rational segments}. Also, we obtain information on the \textit{sublinear diffusion} of orbits in the -not very well understood- case that $\rho$ has nonempty interior.