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

TL;DR

This paper constructs quasiconformal maps in any dimension that achieve maximal local stretching on any given countable set, demonstrating worst-case regularity behavior and extending planar results to higher dimensions.

Contribution

It introduces a method to create quasiconformal maps with maximal stretching on arbitrary countable sets in any dimension, generalizing planar extremizers to $\

Findings

Constructed quasiconformal maps with maximal stretching on countable sets.

Demonstrated worst-case regularity behavior of quasiconformal maps.

Extended planar extremizer constructions to higher dimensions.

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 $\mathbb{R}^d$.