  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.
