Global conformal assouad dimension in the heisenberg group

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Abstract

We study global conformal Assouad dimension in the Heisenberg group Hn. For each α ∈ (0) ∪ [1, 2n + 2], there is a bounded set in Hn with Assouad dimension α whose Assouad dimension cannot be lowered by any quasiconformal map of Hn. On the other hand, for any set S in Hn with Assouad dimension strictly less than one, the infimum of the Assouad dimensions of sets F(S), taken over all quasiconformal maps F of Hn, 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 Hn and regularity of the Carnot–Carathéodory distance from smooth hypersurfaces.

Original languageEnglish (US)
Pages (from-to)32-57
Number of pages26
JournalConformal Geometry and Dynamics
Volume12
Issue number4
DOIs
StatePublished - Mar 6 2008

Keywords

  • Assouad dimension
  • Conformal dimension
  • Heisenberg group
  • Quasiconformal map
  • Self-affine tiling

ASJC Scopus subject areas

  • Geometry and Topology

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