{
  "id": "1208.1473",
  "version": "v3",
  "published": "2012-08-07T17:35:13.000Z",
  "updated": "2014-04-21T18:21:53.000Z",
  "title": "Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior",
  "authors": [
    "Salvador Addas-Zanata"
  ],
  "comment": "to be published in Erg. Th. & Dyn. Sys",
  "categories": [
    "math.DS"
  ],
  "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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-04-21T18:21:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rotation set",
      "area-preserving diffeomorphisms",
      "non-empty interior",
      "periodic point",
      "interior point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.1473A"
    }
  }
}