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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Global conformal Assouad dimension in the Heisenberg group
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by Jeremy T. Tyson PDF
Conform. Geom. Dyn. 12 (2008), 32-57 Request permission

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
  • Received by editor(s): August 27, 2007
  • Published electronically: March 6, 2008
  • Additional Notes: Research supported by NSF grant DMS 0555869
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2385407