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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sets of minimal Hausdorff dimension for quasiconformal maps


Author: Jeremy T. Tyson
Journal: Proc. Amer. Math. Soc. 128 (2000), 3361-3367
MSC (2000): Primary 30C65; Secondary 28A78
Published electronically: May 18, 2000
MathSciNet review: 1676353
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Abstract: For any $1\le\alpha\le n$, there is a compact set $E\subset\mathbb{R}^n$ of (Hausdorff) dimension $\alpha$ whose dimension cannot be lowered by any quasiconformal map $f:\mathbb{R}^n\to\mathbb{R}^n$. We conjecture that no such set exists in the case $\alpha<1$. More generally, we identify a broad class of metric spaces whose Hausdorff dimension is minimal among quasisymmetric images.


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Additional Information

Jeremy T. Tyson
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651
Email: jttyson@math.lsa.umich.edu, tyson@math.sunysb.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05433-2
PII: S 0002-9939(00)05433-2
Keywords: Hausdorff dimension, quasiconformal maps, generalized modulus
Received by editor(s): October 15, 1998
Received by editor(s) in revised form: January 15, 1999
Published electronically: May 18, 2000
Additional Notes: The results of this paper form part of the author’s doctoral dissertation at the University of Michigan. The author was supported by a National Science Foundation Graduate Research Fellowship and a Sloan Doctoral Dissertation Fellowship.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society