Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups
type: Item
instance of: scholarly article
title: Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups
zbMATH Open document ID: 1155.22011
author: Zoltán M. Balogh
author: Jeremy T. Tyson
author: Ben Warhurst
published in: Advances in Mathematics
publication date: 2008-12-22T00:00:00Z
review text: Carnot groups are simply connected nilpotent Lie groups with graded Lie algebra equipped with a left invariant metric of sub-Riemannian type. In the paper under review Gromov's dimension comparison problem is solved for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating the optimal mutual covering of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. It is shown, that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type are obtained. The main results of the paper are illustrated by jet spaces Carnot groups.
reviewed by: Victor Sharapov
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 53C17
zbMATH DE Number: 5480737
zbMATH Keywords: Carnot group
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: iterated function system
zbMATH Keywords: self-similar fractal
Wikidata QID: Q115362116
MaRDI profile type: Publication
full work available at URL: j.aim.2008.09.018
OpenAlex ID: W1993961717
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Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups (cites work)
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Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups (DOI)
DOI: 10.1016/J.AIM.2008.09.018

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