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On quasiconformal dimension distortion for subsets of the real line

Petteri Nissinen, István Prause

Abstract

Optimal quasiconformal dimension distortions bounds for subsets of the complex plane have been established by Astala. We show that these estimates can be improved when one considers subsets of the real line of arbitrary Hausdorff dimension. We present some explicit numerical bounds.

On quasiconformal dimension distortion for subsets of the real line

Abstract

Optimal quasiconformal dimension distortions bounds for subsets of the complex plane have been established by Astala. We show that these estimates can be improved when one considers subsets of the real line of arbitrary Hausdorff dimension. We present some explicit numerical bounds.
Paper Structure (4 sections, 11 theorems, 62 equations)

This paper contains 4 sections, 11 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dimension distortion for antisymmetric maps
  4. Improved dimension estimates on the line

Key Result

Theorem 2.5

Let $f:{\mathbb{C}} \to {\mathbb{C}}$ be $K=\frac{1+k}{1-k}$-quasiconformal. Then there exists a holomorphic motion $\Phi: \mathbb{D}\times {\mathbb{C}} \to {\mathbb{C}}$ such that $f(z)=\Phi(k, z)=\Phi_k(z)$, $z\in {\mathbb{C}}$.

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Theorem 12.5.3 AIM
  • Definition 2.6
  • Lemma 2.7: Lemma 10.7 FRY
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 14 more