Singular quasisymmetric mappings in dimensions two and greater

For all n≥2, we construct a metric space (X,d) and a quasisymmetric mapping f:[0,1] ⁿ →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⊂[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.