Singular quasisymmetric mappings in dimensions two and greater

Abstract

For all $n\ge 2$, we construct a metric space $(X,d)$ and a quasisymmetric mapping $f:[0,1{]}^n\to X$ 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.

arXiv comment13 pages, 2 figures, to appear in Adv. Math