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Stretching and Rotation Sets of Quasiconformal Mappings

Rosemarie Bongers

Abstract

Quasiconformal maps in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. In this work, we study the singularities of these maps, in particular the sizes of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve results by Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and rotation sets with non-sigma-finite measure at the appropriate Hausdorff dimension. We also improve this to give examples with positive Riesz capacity at the critical homogeneity, as well as positivity for a broad class of gauged Hausdorff measures at that dimension.

Stretching and Rotation Sets of Quasiconformal Mappings

Abstract

Quasiconformal maps in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. In this work, we study the singularities of these maps, in particular the sizes of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve results by Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and rotation sets with non-sigma-finite measure at the appropriate Hausdorff dimension. We also improve this to give examples with positive Riesz capacity at the critical homogeneity, as well as positivity for a broad class of gauged Hausdorff measures at that dimension.

Paper Structure

This paper contains 4 sections, 15 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Dimension Zero
  4. Dimension Greater than Zero

Key Result

Theorem 1.1

If $f : \mathbb{C} \to \mathbb{C}$ is a $K$-quasiconformal mapping with $K > 1$, and $\alpha(1 + i \gamma) \in B_K$, then the Hausdorff dimension of the stretching and rotation set $E_f$ of $f$ is bounded by and this result is sharp at the level of dimension.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • ...and 15 more