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
DOI:
https://doi.org/10.1090/S0002-9939-00-05433-2
Published electronically:
May 18, 2000
MathSciNet review:
1676353
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
MR Author ID:
625886
Email:
jttyson@math.lsa.umich.edu, tyson@math.sunysb.edu
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
