{
  "id": "1408.4289",
  "version": "v1",
  "published": "2014-08-19T10:47:26.000Z",
  "updated": "2014-08-19T10:47:26.000Z",
  "title": "Renormalization of three dimensional Hénon map I : Reduction of ambient space",
  "authors": [
    "Young Woo Nam"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Three dimensional analytic H\\'enon-like map $$ F(x,y,z) = (f(x) - \\epsilon(x,y,z),\\, x,\\, \\delta(x,y,z)) $$ and its {\\em period doubling} renormalization is defined. If $ F $ is infinitely renormalizable map, Jacobian determinant of $ n^{th} $ renormalized map, $ R^nF $ has asymptotically universal expression $$ Jac R^nF = b_F^{2^n}a(x)(1 + O(\\rho^n)) $$ where $ b_F $ is the average Jacobian of $ F $. The toy model map, $ F_{mod} $ is defined as the map satisfying $ \\partial_z \\epsilon \\equiv 0 $. The set of toy model map is invariant under renormalizaton. Moreover, if $ \\| \\partial_z \\delta \\| \\ll \\| \\partial_y \\epsilon \\| $, then there exists the continuous invariant plane field over $ \\mathcal O_F $ with dominated splitting. Under this condition, three dimensional H\\'enon-like map %with the dominated splitting is dynamically decomposed into two dimensional map with contraction along the strong stable direction. Any invariant line field on this plane filed over $ \\mathcal O_{F_{mod}} $ cannot be continuous.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-19T10:47:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dimensional hénon map",
      "ambient space",
      "toy model map",
      "renormalization",
      "continuous invariant plane field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.4289N"
    }
  }
}