{
  "id": "1903.08703",
  "version": "v1",
  "published": "2019-03-20T19:04:40.000Z",
  "updated": "2019-03-20T19:04:40.000Z",
  "title": "On Stable and Unstable Behaviours of Certain Rotation Segments",
  "authors": [
    "Salvador Addas-Zanata",
    "Xiao-Chuan Liu"
  ],
  "comment": "5 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper, we study non-wandering homeomorphisms of the two torus in the identity homotopy class, whose rotation sets are non-trivial line segments from $(0,0)$ to some totally irrational vector $(\\alpha,\\beta)$. We show this rotation set is in fact a non-generic phenomenon for any $C^r$ diffeomorphisms, with $r \\geq 1$. When such a rotation set does happen, assuming several natural conditions that are generically satisfied in the area-preserving world, we give a clearer description of its rotational behavior. More precisely, the dynamics admits bounded deviation along the direction $-(\\alpha,\\beta)$ in the lift, and the rotation set is locked inside an arbitrarily small cone with respect to small $C^0$-perturbations of the dynamics. On the other hand, for any non-wandering homeomorphism $f$ with this kind of rotation set, we also present a perturbation scheme in order for the rotation set to be eaten by rotation sets of nearby dynamics, in the sense that the later set has non-empty interior and contains the former one. These two flavors interplay and share the common goal of understanding the stability/instability properties of this kind of rotation set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-20T19:04:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rotation set",
      "rotation segments",
      "unstable behaviours",
      "identity homotopy class",
      "non-wandering homeomorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}