{
  "id": "1407.2563",
  "version": "v1",
  "published": "2014-07-09T17:01:00.000Z",
  "updated": "2014-07-09T17:01:00.000Z",
  "title": "Connectedness locus for pairs of affine maps and zeros of power series",
  "authors": [
    "Boris Solomyak"
  ],
  "comment": "23 pages, 5 figures; comments welcome",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "We study the connectedness locus N for the family of iterated function systems of pairs of affine-linear maps in the plane (the non-self-similar case). First results on the set N were obtained in joint work with P. Shmerkin (2006). Here we establish rigorous bounds for the set N based on the study of power series of special form. We also derive some bounds for the region of \"*-transversality\" which have applications to the computation of Hausdorff measure of the self-affine attractor. We prove that a large portion of the set N is connected and locally connected, and conjecture that the entire connectedness locus is connected. We also prove that the set N has many zero angle \"cusp corners,\" at certain points with algebraic coordinates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-09T17:01:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "30B10"
    ],
    "keywords": [
      "power series",
      "affine maps",
      "entire connectedness locus",
      "non-self-similar case",
      "algebraic coordinates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.2563S"
    }
  }
}