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

TL;DR

This paper investigates area-preserving diffeomorphisms of the torus with rotation sets having interior, establishing the existence of hyperbolic periodic points with invariant, topologically mixing stable and unstable manifolds, and characterizing their global dynamics.

Contribution

It proves the existence of hyperbolic periodic points with invariant manifolds under certain rotation set conditions, extending understanding of the dynamics of area-preserving diffeomorphisms on the torus.

Findings

Existence of hyperbolic periodic points with invariant stable and unstable manifolds.

Invariant and topologically mixing behavior on the closure of unstable manifolds.

Global dynamics characterized by the entire plane being the closure of unstable manifolds in transitive cases.

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 $\widetilde{f}$ 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.