{
  "id": "math/0601644",
  "version": "v1",
  "published": "2006-01-26T15:48:18.000Z",
  "updated": "2006-01-26T15:48:18.000Z",
  "title": "Virtual Immediate Basins of Newton Maps and Asymptotic Values",
  "authors": [
    "Xavier Buff",
    "Johannes Rueckert"
  ],
  "comment": "15 pages, 1 figure",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "Newton's root finding method applied to a (transcendental) entire function f:C->C is the iteration of a meromorphic function N. It is well known that if for some starting value z, Newton's method converges to a point x in C, then f has a root at x. We show that in many cases, if an orbit converges to infinity for Newton's method, then f has a `virtual root' at infinity. More precisely, we show that if N has an invariant Baker domain that satisfies some mild assumptions, then 0 is an asymptotic value for f. Conversely, we show that if f has an asymptotic value of logarithmic type at 0, then the singularity over 0 is contained in an invariant Baker domain of N, which we call a virtual immediate basin. We show by way of counterexamples that this is not true for more general types of singularities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-01-26T15:48:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "30D05",
      "49M15"
    ],
    "keywords": [
      "virtual immediate basin",
      "asymptotic value",
      "newton maps",
      "invariant baker domain",
      "root finding method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1644B"
    }
  }
}