  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}$$\in {\rm I}\negthinspace {\rm R^2}$ such that
$W^u(\widetilde{Q})$ intersects $W^s(\widetilde{Q}+(a,b))$ for all integers
$(a,b)$, which implies that $\bar{W^u(\widetilde{Q})}$ is invariant under
integer translations. Moreover,
$\bar{W^u(\widetilde{Q})}=\bar{W^s(\widetilde{Q})}$ and $\widetilde{f}$
restricted to $\bar{W^u(\widetilde{Q})}$ is invariant and topologically mixing.
Each connected component of the complement of $\bar{W^u(\widetilde{Q})}$ is a
disk with uniformly bounded diameter. If $f$ is transitive, then
$\bar{W^u(\widetilde{Q})}=$${\rm I}\negthinspace {\rm R^2}$ and $\widetilde{f}$
is topologically mixing in the whole plane.
