Global conformal Assouad dimension in the Heisenberg group

Tyson, Jeremy

doi:10.1090/S1088-4173-08-00177-X

Global conformal Assouad dimension in the Heisenberg group
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by Jeremy T. Tyson

Conform. Geom. Dyn. 12 (2008), 32-57

DOI: https://doi.org/10.1090/S1088-4173-08-00177-X

Published electronically: March 6, 2008

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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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