{
  "id": "2004.02078",
  "version": "v1",
  "published": "2020-04-05T02:49:51.000Z",
  "updated": "2020-04-05T02:49:51.000Z",
  "title": "Global Behaviors of weak KAM Solutions for exact symplectic Twist Maps",
  "authors": [
    "Jianlu Zhang"
  ],
  "comment": "19 pages,1 figure",
  "categories": [
    "math.DS"
  ],
  "abstract": "We investigated several global behaviors of the weak KAM solutions $u_c(x,t)$ parametrized by $c\\in H^1(\\mathbb T,\\mathbb R)$. For the suspended Hamiltonian $H(x,p,t)$ of the exact symplectic twist map, we could find a family of weak KAM solutions $u_c(x,t)$ parametrized by $c(\\sigma)\\in H^1(\\mathbb T,\\mathbb R)$ with $c(\\sigma)$ continuous and monotonic and \\[ \\partial_tu_c+H(x,\\partial_x u_c+c,t)=\\alpha(c),\\quad \\text{a.e.\\ } (x,t)\\in\\mathbb T^2, \\] such that sequence of weak KAM solutions $\\{u_c\\}_{c\\in H^1(\\mathbb T,\\mathbb R)}$ is $1/2-$H\\\"older continuity of parameter $\\sigma\\in \\mathbb{R}$. Moreover, for each generalized characteristic (no matter regular or singular) solving \\[ \\left\\{ \\begin{aligned} &\\dot{x}(s)\\in \\text{co} \\Big[\\partial_pH\\Big(x(s),c+D^+u_c\\big(x(s),s+t\\big),s+t\\Big)\\Big], & \\\\ &x(0)=x_0,\\quad (x_0,t)\\in\\mathbb T^2,& \\end{aligned} \\right. \\] we evaluate it by a uniquely identified rotational number $\\omega(c)\\in H_1(\\mathbb T,\\mathbb R)$. This property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories. Besides, we discussed this applies to high-dimensional cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-05T02:49:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E40",
      "37E45",
      "37J40",
      "37J45",
      "37J50",
      "49L25"
    ],
    "keywords": [
      "weak kam solutions",
      "exact symplectic twist map",
      "global behaviors",
      "matter regular",
      "uniquely identified rotational number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}