Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups
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.
scientific article; zbMATH DE number 5480737
scientific article