{
  "id": "2405.02204",
  "version": "v1",
  "published": "2024-05-03T16:03:24.000Z",
  "updated": "2024-05-03T16:03:24.000Z",
  "title": "Pseudo-monodromy and the Mandelbrot set",
  "authors": [
    "Yutaka Ishii",
    "Thomas Richards"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We investigate the discontinuity of codings for the Julia set of a quadratic map. To each parameter ray, we associate a natural coding for Julia sets on the ray. Given a hyperbolic component $H$ of the Mandelbrot set, we consider the codings along the two parameter rays landing on the root point of $H$. Our main result describes the discontinuity of these two coding in terms of the kneading sequences of the hyperbolic components which are conspicuous to $H$. This result can be interpreted as a solution to the degenerated case of Lipa's conjecture on the monodromy problem of the horseshoe locus for the complex H\\'enon family.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-03T16:03:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F20",
      "37F46",
      "37F80"
    ],
    "keywords": [
      "mandelbrot set",
      "pseudo-monodromy",
      "julia set",
      "parameter ray",
      "hyperbolic component"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}