{
  "id": "1408.4619",
  "version": "v1",
  "published": "2014-08-20T12:02:29.000Z",
  "updated": "2014-08-20T12:02:29.000Z",
  "title": "Invariant space under Hénon renormalization : Intrinsic geometry of Cantor attractor",
  "authors": [
    "Young Woo Nam"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Three dimensional H\\'non-like map $$ F(x,y,z) = (f(x) - \\epsilon (x,y,z),\\ x,\\ \\delta (x,y,z)) $$ is defined on the cubic box $ B $. An invariant space under renormalization would appear only in higher dimension. Consider renormalizable maps each of which satisfies the condition $$ \\partial_y \\delta \\circ F(x,y,z) + \\partial_z \\delta \\circ F(x,y,z) \\cdot \\partial_x \\delta (x,y,z) \\equiv 0 $$ for $ (x,y,z) \\in B $. Denote the set of maps satisfying the above condition be $ \\mathcal N $. Then the set $ \\mathcal N \\cap \\mathcal I(\\bar \\epsilon) $ is invariant under the renormalization operator where $ \\mathcal I(\\bar \\epsilon) $ is the set of infinitely renormalizable maps. H\\'enon like diffeomorphism in $ \\mathcal N \\cap \\mathcal I(\\bar \\epsilon) $ has universal numbers, $ b_2 \\asymp | \\partial_z \\delta | $ and $ b_1 = b_F /b_2 $ where $ b_F $ is the average Jacobian of $ F $. The Cantor attractor of $ F \\in \\mathcal N \\cap \\mathcal I(\\bar \\epsilon) $, $ \\mathcal O_F $ has {\\em unbounded geometry} almost everywhere in the parameter space of $ b_1 $. If two maps in $ \\mathcal N $ has different universal numbers $ b_1 $ and $ \\widetilde b_1 $, then the homeomorphism between two Cantor attractor is at most H\\\"older continuous, which is called {\\em non rigidity}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-20T12:02:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cantor attractor",
      "invariant space",
      "hénon renormalization",
      "intrinsic geometry",
      "universal numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.4619N"
    }
  }
}