Singular quasisymmetric mappings in dimensions two and greater

TL;DR

This paper constructs a counterexample in metric space theory showing that quasisymmetric mappings in dimensions two and higher can have inverses that are not absolutely continuous, challenging previous assumptions.

Contribution

It provides the first known example of a quasisymmetric map with a non-absolutely continuous inverse in higher dimensions, answering a long-standing question.

Findings

Existence of a quasisymmetric map with a non-absolutely continuous inverse.

Construction of a metric space with finite Hausdorff measure where this phenomenon occurs.

The inverse map can map a set of positive measure to a measure-zero set.

Abstract

For all $n \geq 2$, we construct a metric space $(X,d)$ and a quasisymmetric mapping $f\colon [0,1]^n \rightarrow X$ with the property that $f^{-1}$ is not absolutely continuous with respect to the Hausdorff $n$-measure on $X$. That is, there exists a Borel set $E \subset [0,1]^n$ with Lebesgue measure $|E|>0$ such that $f(E)$ has Hausdorff $n$-measure zero. The construction may be carried out so that $X$ has finite Hausdorff $n$-measure and $|E|$ is arbitrarily close to 1, or so that $|E| = 1$. This gives a negative answer to a question of Heinonen and Semmes.