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Gromov's dimension comparison and fractal geometry in Carnot groups

ARGOMENTI: Convegni

SEMINARI DI EQUAZIONI DIFFERENZIALI E APPLICAZIONI
Giovedi' 10 aprile alle ore 12.15 in aula 2AB/40 TA il professor Zoltan Balogh dell'Universita' di Berna terra' un seminario dal titolo "Gromov's dimension comparison and fractal geometry in Carnot groups".

-Abstract
We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a
Carnot-Caratheodory 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.

Rif. int. M. Bardi, R. Monti, A. Cesaroni