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Existence of quasiconformal maps with maximal stretching on any given countable set

Rosemarie Bongers, James T. Gill

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 Hölder continuity estimates; on the other hand, one can use the radial stretches to characterize the extremizers for Hölder 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 $\mathbb{R}^d$.

Existence of quasiconformal maps with maximal stretching on any given countable set

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 Hölder continuity estimates; on the other hand, one can use the radial stretches to characterize the extremizers for Hölder continuity. In this work, given any bounded countable set in , we will construct an example of a -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 .

Paper Structure

This paper contains 5 sections, 1 theorem, 43 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theorem:existence']}
  3. Construction of $F_{\Lambda}$
  4. Establishing $K$-quasiconformality
  5. Stretching exponent for points in $\Lambda$

Key Result

Theorem 1.1

Fix a bounded countable set $\Lambda = \{\lambda_1, \lambda_2, ...\} \subseteq \mathbb{R}^d$ and a $K > 1$. There is a $K$-quasiconformal map $F_{\Lambda}$ which stretches with exponent $1/K$ at every point in $\Lambda$.

Theorems & Definitions (2)

  • Definition 1.1
  • Theorem 1.1