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

• Autores: Zoltán M. Balogh, Jeremy T. Tyson, Ben Warhurst
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 220, Nº 2, 2009, págs. 560-619
• Idioma: inglés
• DOI: 10.1016/j.aim.2008.09.018
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• Resumen

  • 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.