{
  "id": "2106.06177",
  "version": "v1",
  "published": "2021-06-11T05:42:40.000Z",
  "updated": "2021-06-11T05:42:40.000Z",
  "title": "Existence of quasiconformal maps with maximal stretching on any given countable set",
  "authors": [
    "Tyler Bongers",
    "James T. Gill"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "Quasiconformal maps are homeomorphisms with useful local distortion inequalities; infinitesimally, they map balls to ellipsoids with bounded eccentricity. This leads to a number of useful regularity properties, including quantitative H\\\"older continuity estimates; on the other hand, one can use the radial stretches to characterize the extremizers for H\\\"older continuity. In this work, given any bounded countable set in $\\mathbb{R}^d$, we will construct an example of a $K$-quasiconformal map which exhibits the maximum stretching at each point of the set. This will provide an example of a quasiconformal map that exhibits the worst-case regularity on a surprisingly large set, and generalizes constructions from the planar setting into $\\mathbb{R}^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-11T05:42:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C65"
    ],
    "keywords": [
      "quasiconformal map",
      "countable set",
      "maximal stretching",
      "useful local distortion inequalities",
      "surprisingly large set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}