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

Abstract

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

arXiv commentto be published in Erg. Th. & Dyn. Sys