{
  "id": "1907.09434",
  "version": "v1",
  "published": "2019-07-22T17:10:27.000Z",
  "updated": "2019-07-22T17:10:27.000Z",
  "title": "On the topology of nearly-integrable Hamiltonians at simple resonances",
  "authors": [
    "L. Biasco",
    "L. Chierchia"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We show that, in general, averaging at simple resonances a real--analytic, nearly--integrable Hamiltonian, one obtains a one--dimensional system with a cosine--like potential; ``in general'' means for a generic class of holomorphic perturbations and apart from a finite number of simple resonances with small Fourier modes; ``cosine--like'' means that the potential depends only on the resonant angle, with respect to which it is a Morse function with one maximum and one minimum. \\\\ Furthermore, the (full) transformed Hamiltonian is the sum of an effective one--dimen\\-sio\\-nal Hamiltonian (which is, in turn, the sum of the unperturbed Hamiltonian plus the cosine--like potential) and a perturbation, which is exponentially small with respect to the oscillation of the potential. \\\\ As a corollary, under the above hypotheses, if the unperturbed Hamiltonian is also strictly convex, the effective Hamiltonian at {\\sl any simple resonance} (apart a finite number of low--mode resonances) has the phase portrait of a pendulum. \\\\ The results presented in this paper are an essential step in the proof (in the ``mechanical'' case) of a conjecture by Arnold--Kozlov--Neishdadt (\\cite[Remark~6.8, p. 285]{AKN}), claiming that the measure of the ``non--torus set'' in general nearly--integrable Hamiltonian systems has the same size of the perturbation; compare \\cite{BClin}, \\cite{BC}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-22T17:10:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple resonance",
      "finite number",
      "cosine-like potential",
      "small fourier modes",
      "general nearly-integrable hamiltonian systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}