{
  "id": "0811.2588",
  "version": "v2",
  "published": "2008-11-16T16:32:43.000Z",
  "updated": "2009-06-04T13:18:08.000Z",
  "title": "Period Doubling in Area-Preserving Maps: An Associated One Dimensional Problem",
  "authors": [
    "Denis Gaidashev",
    "Hans Koch"
  ],
  "comment": "Typos removed. An argument about sequential compactness replaced by one based on Tikhonov-Schauder theorem",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of $\\field{R}^2$. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods by J.-P. Eckmann, H. Koch and P. Wittwer (1982 and 1984). As it is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period doubling universality exists to date. We argue that the period doubling renormalization fixed point for area-preserving maps is almost one dimensional, in the sense that it is close to the following Henon-like map: $$H^*(x,u)=(\\phi(x)-u,x-\\phi(\\phi(x)-u)),$$ where $\\phi$ solves $$\\phi(x)={2 \\over \\lambda} \\phi(\\phi(\\lambda x))-x.$$ We then give a ``proof'' of existence of solutions of small analytic perturbations of this one dimensional problem, and describe some of the properties of this solution. The ``proof'' consists of an analytic argument for factorized inverse branches of $\\phi$ together with verification of several inequalities and inclusions of subsets of $\\field{C}$ numerically. Finally, we suggest an analytic approach to the full period doubling problem for area-preserving maps based on its proximity to the one dimensional. In this respect, the paper is an exploration of a possible analytic machinery for a non-trivial renormalization problem in a conservative two-dimensional system.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-06-04T13:18:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E20",
      "37F25",
      "28D05",
      "37E30",
      "65P30"
    ],
    "keywords": [
      "area-preserving map",
      "dimensional problem",
      "feigenbaum-coullet-tresser period doubling universality",
      "full period doubling problem",
      "small analytic perturbations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.2588G"
    }
  }
}