{
  "id": "1809.02372",
  "version": "v1",
  "published": "2018-09-07T09:38:24.000Z",
  "updated": "2018-09-07T09:38:24.000Z",
  "title": "On the transversal dependence of weak K.A.M. solutions for symplectic twist maps",
  "authors": [
    "Marie-Claude Arnaud",
    "Maxime Zavidovique"
  ],
  "comment": "43 pages, 3 figures",
  "categories": [
    "math.DS",
    "math.SG"
  ],
  "abstract": "For a symplectic twist map, we prove that there is a choice of weak K.A.M. solutions that depend in a continuous way on the cohomology class. We thus obtain a continuous function $u(\\theta, c)$ in two variables: the angle $\\theta$ and the cohomology class $c$. As a result, we prove that the Aubry-Mather sets are contained in pseudographs that are vertically ordered by their rotation numbers. Then we characterize the $C^0$ integrable twist maps in terms of regularity of $u$ that allows to see $u$ as a generating function. We also obtain some results for the Lipschitz integrable twist maps. With an example, we show that our choice is not the so-called discounted one (see \\cite{DFIZ2}), that is sometimes discontinuous. We also provide examples of `strange' continuous foliations that cannot be straightened by a symplectic homeomorphism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-07T09:38:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E40",
      "37J50",
      "37J30",
      "37J35"
    ],
    "keywords": [
      "symplectic twist map",
      "transversal dependence",
      "cohomology class",
      "lipschitz integrable twist maps",
      "symplectic homeomorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}