Quasi-Symmetric Embeddings in Euclidean Spaces

We consider quasi-symmetric embeddings $f: G \rightarrow R^n, G$ 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...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 1981-03, Vol.264 (1), p.191-204
Main Author: Vaisala, Jussi
Format: Article
Language:English
Subjects:
Cantor set
Cubes
Embeddings
Hausdorff dimensions
Hausdorff measures
Homeomorphism
Homomorphisms
Integers
Mathematics
Topology
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Summary:We consider quasi-symmetric embeddings $f: G \rightarrow R^n, G$ 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 Cech cohomology.
ISSN:0002-9947
DOI:10.2307/1998419