Conformal Geometry and Dynamics

ISSN 1088-4173



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
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

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.