In this paper we consider ${C}^{1+ \epsilon } $ area-preserving diffeomorphisms of the torus $f$, either homotopic to the identity or to Dehn twists. We suppose that $f$ has a lift $\widetilde {f} $ to the plane such that its rotation set has interior and prove, among other things, that if zero is an interior point of the rotation set, then there exists a hyperbolic $\widetilde {f} $-periodic point $\widetilde {Q} \in { \mathbb{R} }^{2} $ such that ${W}^{u} (\widetilde {Q} )$ intersects ${W}^{s} (\widetilde {Q} + (a, b))$ for all integers $(a, b)$, which implies that $ \overline{{W}^{u} (\widetilde {Q} )} $ is invariant under integer translations. Moreover, $ \overline{{W}^{u} (\widetilde {Q} )} = \overline{{W}^{s} (\widetilde {Q} )} $ and $\widetilde {f} $ restricted to $ \overline{{W}^{u} (\widetilde {Q} )} $ is invariant and topologically mixing. Each connected component of the complement of $ \overline{{W}^{u} (\widetilde {Q} )} $ is a disk with diameter uniformly bounded from above. If $f$ is transitive, then $ \overline{{W}^{u} (\widetilde {Q} )} = { \mathbb{R} }^{2} $ and $\widetilde {f} $ is topologically mixing in the whole plane.