*Submitted Paper*

**Inserted:** 21 dec 2018

**Last Updated:** 21 dec 2018

**Year:** 2018

**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.

**Tags:**
GeoMeG

**Keywords:**
quasisymmetric mapping, absolute continuity