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

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.

Removable sets for homogeneous linear partial differential equations in Carnot groups

“…We make use of the following theorem on the existence of selfsimilar sets in Carnot groups. Theorem 3.1 is a special case of Proposition 4.14 in [5]. In case the group G is of step two with rational structure constants (for instance, if G is the Heisenberg group), explicit tilings of this type were constructed by Strichartz [38], [39]; the latter tilings were further studied in [3] and [42].…”

Global conformal Assouad dimension in the Heisenberg group

“…Such a set can be constructed as the invariant set for an iterated function system satisfying the strong open set condition with fixed points in P . See [8] or Example 6.1 for more details. The Hausdorff measure H α−1 restricted to S 0 (and suitably normalized) is an (α − 1)-regular measure.…”

“…Examples include the Heisenberg cube and Strichartz tiles; further examples can be found in [5] and [8]. If the maps {F 1 , .…”

“…Almost sure dimension formulas for horizontal self-similar and self-affine fractals in the first Heisenberg group were established in [6]; see [5] for further results. Fractal geometry in general Carnot groups, with application to Gromov's dimension comparison problem, has been studied in [7], [8].…”

Geometric inequalities in Carnot groups

“…In this context, Carnot groups play the role of modeling the tangent space (in a suitable generalized sense, which is related with the Gromov-Hausdorff convergence) of a subRiemannian manifold; see [33], [49]. For this and many other reasons, Carnot groups are an intriguing field of research; see [5], [6], [7], [11], [18], [21,23], [27,28,29,30], [36], [40,41], [43,44], [54].…”