Calculus of Variations and Geometric Measure Theory
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Z. Balogh - J. Tyson - B. Warhurst

Sub-Riemanian vs. Euclidean dimension comparison and fractal geometry on Carnot groups

created by zoltan on 13 Aug 2007

[BibTeX]

Submitted Paper

Inserted: 13 aug 2007

Year: 2007

Abstract:

We solve Gromov's dimension comparison problem 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 optimal mutual coverings of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemann\-ian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Carathé\-odory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples.

Keywords: Carnot groups, Hausdorff dimension, Carnot-Carathéodory metric, Itereated functions systems


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