Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior

Abstract

AbstractIn 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.