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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quasisymmetric embeddings in Euclidean spaces


Author: Jussi Väisälä
Journal: Trans. Amer. Math. Soc. 264 (1981), 191-204
MSC: Primary 30C60; Secondary 28A75, 54C25, 54E40, 57N45
MathSciNet review: 597876
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Abstract: We consider quasi-symmetric embeddings $ f:G \to {R^n}$, $ G$ open in $ {R^p}$, $ p \leqslant n$. If $ p = n$, quasi-symmetry implies quasi-conformality. The converse is true if $ G$ has a sufficiently smooth boundary. If $ p < n$, the Hausdorff dimension of $ fG$ is less than $ n$. If $ fG$ has a finite $ p$-measure, $ f$ preserves the property of being of $ p$-measure zero. If $ p < n$ and $ n \geqslant 3$, $ {R^n}$ contains a quasi-symmetric $ p$-cell which is topologically wild.

We also prove auxiliary results on the relations between Hausdorff measure and Čech cohomology.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0597876-7
PII: S 0002-9947(1981)0597876-7
Article copyright: © Copyright 1981 American Mathematical Society