{
  "id": "1311.7334",
  "version": "v1",
  "published": "2013-11-28T14:38:47.000Z",
  "updated": "2013-11-28T14:38:47.000Z",
  "title": "Around the stability of KAM-tori",
  "authors": [
    "Hakan Eliasson",
    "Bassam Fayad",
    "Raphaël Krikorian"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We show that an analytic invariant torus $\\cT_0$ with Diophantine frequency $\\o_0$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at $\\cT_0$ satisfies a R\\\"ussmann transversality condition, the torus $\\cT_0$ is accumulated by KAM tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least $d+1$ that is foliated by analytic invariant tori with frequency $\\o_0$. For frequency vectors $\\o_0$ having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), we show that if the Hamiltonian $H$ satisfies a Kolmogorov non degeneracy condition at $\\cT_0$, then $\\cT_0$ is accumulated by KAM tori of positive total measure. In $4$ degrees of freedom or more, we construct for any $\\o_0 \\in \\R^d$, $C^\\infty$ (Gevrey) Hamiltonians $H$ with a smooth invariant torus $\\cT_0$ with frequency $\\o_0$ that is not accumulated by a positive measure of invariant tori.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-28T14:38:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "analytic invariant torus",
      "birkhoff normal form",
      "positive total measure",
      "kolmogorov non degeneracy condition",
      "finite uniform diophantine exponent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.7334E"
    }
  }
}