# Singular quasisymmetric mappings in dimensions two and greater

@article{Romney2018SingularQM,
title={Singular quasisymmetric mappings in dimensions two and greater},
author={Matthew Romney},
journal={arXiv: Metric Geometry},
year={2018}
}
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 $|E| = 1… Expand 3 Citations #### Figures from this paper Uniformization with Infinitesimally Metric Measures • Mathematics • 2019 We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces$X$homeomorphic to$\mathbb R^2$. Given a measure$\mu$on such a space, we introduceExpand On the inverse absolute continuity of quasiconformal mappings on hypersurfaces • Mathematics • 2018 We construct quasiconformal mappings$f\colon \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$for which there is a Borel set$E \subset \mathbb{R}^2 \times \{0\}$of positive Lebesgue$2$-measure whoseExpand The branch set of minimal disks in metric spaces • Mathematics • 2020 We study the structure of the branch set of solutions to Plateau's problem in metric spaces satisfying a quadratic isoperimetric inequality. In our first result, we give examples of spaces withExpand #### References SHOWING 1-10 OF 26 REFERENCES Plane with A∞-Weighted Metric not Bilipschitz Embeddable to Rn A planar set$G \subset {\bb R}^2$is constructed that is bilipschitz equivalent to ($G, d^z$), where ($G, d$) is not bilipschitz embeddable to any uniformly convex Banach space. Here,$z \inExpand
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