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Conformal Geometry and Dynamics

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Global conformal Assouad dimension in the Heisenberg group


Author: Jeremy T. Tyson
Journal: Conform. Geom. Dyn. 12 (2008), 32-57
MSC (2000): Primary 30C65; Secondary 28A78, 43A80
DOI: https://doi.org/10.1090/S1088-4173-08-00177-X
Published electronically: March 6, 2008
MathSciNet review: 2385407
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Abstract: We study global conformal Assouad dimension in the Heisenberg group $\mathbb {H}^n$. For each $\alpha \in \{0\}\cup [1,2n+2]$, there is a bounded set in $\mathbb {H}^n$ with Assouad dimension $\alpha$ whose Assouad dimension cannot be lowered by any quasiconformal map of $\mathbb {H}^n$. On the other hand, for any set $S$ in $\mathbb {H}^n$ with Assouad dimension strictly less than one, the infimum of the Assouad dimensions of sets $F(S)$, taken over all quasiconformal maps $F$ of $\mathbb {H}^n$, equals zero. We also consider dilatation-dependent bounds for quasiconformal distortion of Assouad dimension. The proofs use recent advances in self-similar fractal geometry and tilings in $\mathbb {H}^n$ and regularity of the Carnot–Carathéodory distance from smooth hypersurfaces.


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Additional Information

Jeremy T. Tyson
Affiliation: Department of Mathematics, University of Illinois, West Green Street, Urbana, Illinois 61801
MR Author ID: 625886
Email: tyson@math.uiuc.edu

Keywords: Quasiconformal map, conformal dimension, Assouad dimension, Heisenberg group, self-affine tiling.
Received by editor(s): August 27, 2007
Published electronically: March 6, 2008
Additional Notes: Research supported by NSF grant DMS 0555869
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.