Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups |
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Authors: | Zoltá n M. Balogh,Jeremy T. Tyson,Ben Warhurst |
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Affiliation: | a Department of Mathematics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland b Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801, USA c School of Mathematics, University of New South Wales, Sydney 2052, Australia |
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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-Riemannian 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-Carathéodory 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. |
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Keywords: | Carnot group Hausdorff dimension Iterated function system Self-similar fractal |
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