{
  "id": "1111.5561",
  "version": "v2",
  "published": "2011-11-23T17:27:18.000Z",
  "updated": "2012-08-07T17:20:13.000Z",
  "title": "Dynamics of homeomorphisms of the torus homotopic to Dehn twists",
  "authors": [
    "Braulio Garcia",
    "Fabio Armando Tal",
    "Salvador Addas-Zanata"
  ],
  "journal": "final version, as in Ergodic Theory and Dynamical Systems 2012",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential \"horizontal\" set K, invariant under $f$. In other words, if we consider the lift $\\hat{f}$ of $f$ to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of $\\hat{f}$. Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland's conjecture to this setting: If $f$ is area preserving and has a lift $\\hat{f}$ to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under $\\hat{f}$, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-08-07T17:20:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dehn twists",
      "torus homotopic",
      "vertical rotation set",
      "homeomorphisms",
      "zero lebesgue measure vertical rotation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.5561G"
    }
  }
}