{
  "id": "math/9905169",
  "version": "v1",
  "published": "1999-05-26T19:33:12.000Z",
  "updated": "1999-05-26T19:33:12.000Z",
  "title": "Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account",
  "authors": [
    "John W. Milnor"
  ],
  "comment": "51 pages, 25 PostScript figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "A key point in Douady and Hubbard's study of the Mandelbrot set $M$ is the theorem that every parabolic point $c\\ne 1/4$ in $M$ is the landing point for exactly two external rays with angle which are periodic under doubling. This note will try to provide a proof of this result and some of its consequences which relies as much as possible on elementary combinatorics, rather than on more difficult analysis. It was inspired by section 2 of the recent thesis of Schleicher (see also Stony Brook IMS preprint 1994/19, with E. Lau), which contains very substantial simplifications of the Douady-Hubbard proofs with a much more compact argument, and is highly recommended. The proofs given here are rather different from those of Schleicher, and are based on a combinatorial study of the angles of external rays for the Julia set which land on periodic orbits. The results in this paper are mostly well known; there is a particularly strong overlap with the work of Douady and Hubbard. The only claim to originality is in emphasis, and the organization of the proofs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1999-05-26T19:33:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "periodic orbits",
      "mandelbrot set",
      "externals rays",
      "expository account",
      "external rays"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math......5169M"
    }
  }
}