{
  "id": "1412.8337",
  "version": "v1",
  "published": "2014-12-29T13:42:09.000Z",
  "updated": "2014-12-29T13:42:09.000Z",
  "title": "Renormalization of $C^r$ Hénon map : Two dimensional embedded map in three dimension",
  "authors": [
    "Young Woo Nam"
  ],
  "comment": "24 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study renormalization of highly dissipative analytic three dimensional H\\'enon maps $$ F(x,y,z) = (f(x) - \\varepsilon(x,y,z),\\ x,\\ \\delta(x,y,z)) $$ where $ \\varepsilon(x,y,z) $ is a sufficiently small perturbation of $ \\varepsilon_{2d}(x,y) $. Under certain conditions, $ C^r $ single invariant surfaces each of which is tangent to the invariant plane field over the critical Cantor set exist for $ 2 \\leq r < \\infty $. The $ C^r $ conjugation from an invariant surface to the $ xy- $plane defines renormalization two dimensional $ C^r $ H\\'enon-like map. It also defines two dimensional embedded $ C^r $ H\\'enon-like maps in three dimension. In this class, universality theorem is re-constructed by conjugation. Geometric properties on the critical Cantor set in invariant surfaces are the same as those of two dimensional maps --- non existence of the continuous line field and unbounded geometry. The set of embedded two dimensional H\\'enon-like maps is open and dense subset of the parameter space of average Jacobian, $ b_{F_{2d}} $ for any given smoothness, $ 2 \\leq r < \\infty $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-29T13:42:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37Cxx"
    ],
    "keywords": [
      "dimensional embedded map",
      "hénon map",
      "henon-like map",
      "critical cantor set",
      "dimensional henon maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.8337N"
    }
  }
}