Sets of minimal Hausdorff dimension for quasiconformal maps

Tyson, Jeremy

doi:10.1090/S0002-9939-00-05433-2

Sets of minimal Hausdorff dimension for quasiconformal maps
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by Jeremy T. Tyson

Proc. Amer. Math. Soc. 128 (2000), 3361-3367

DOI: https://doi.org/10.1090/S0002-9939-00-05433-2

Published electronically: May 18, 2000

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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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