{
  "id": "1803.03130",
  "version": "v1",
  "published": "2018-03-08T14:50:25.000Z",
  "updated": "2018-03-08T14:50:25.000Z",
  "title": "From Cantor to Semi-hyperbolic Parameter along External Rays",
  "authors": [
    "Yi-Chiuan Chen",
    "Tomoki Kawahira"
  ],
  "comment": "34 pages, 5 figures",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "For the quadratic family $f_{c}(z) = z^2+c$ with $c$ in the exterior of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. Let $\\hat{c}$ be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this paper we prove that for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$ is uniformly $O(1/\\sqrt{|c-\\hat{c}|})$ when $c$ belongs to a parameter ray that lands on $\\hat{c}$. We also characterize the degeneration of the dynamics along the parameter ray.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-08T14:50:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F45",
      "37F99"
    ],
    "keywords": [
      "semi-hyperbolic parameter",
      "external rays",
      "parameter ray",
      "mandelbrot set",
      "julia set moves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}