{
  "id": "2101.08045",
  "version": "v1",
  "published": "2021-01-20T09:59:37.000Z",
  "updated": "2021-01-20T09:59:37.000Z",
  "title": "A class of Newton maps with Julia sets of Lebesgue measure zero",
  "authors": [
    "Mareike Wolff"
  ],
  "comment": "44 pages, 5 figures",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "Let $g(z)=\\int_0^zp(t)\\exp(q(t))\\,dt+c$ where $p,q$ are polynomials and $c\\in\\mathbb{C}$, and let $f$ be the function from Newton's method for $g$. We show that under suitable assumptions the Julia set of $f$ has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $f^n(z)$ converges to zeros of $g$ almost everywhere in $\\mathbb{C}$ if this is the case for each zero of $g''$. In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-20T09:59:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "30D05"
    ],
    "keywords": [
      "lebesgue measure zero",
      "julia set",
      "newton maps",
      "newtons method",
      "result implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}