Sub-Riemannian
vs Euclidean dimension comparison
and
fractal geometry on Carnot groups
Abstract: A Carnot group G is naturally equipped with equivalent
Euclidean and subriemannian metrics. Gromov has asked the following question for submanifolds of G:
Determine all possible pairs (α, β) of real numbers such that there exists
a submanifold M
of G with Euclidean Hausdorff dimension α
and sub-Riemannian Hausdorff dimension β.
To answer this question in general is difficult because the structure of
the underlying Lie algebra is significant. If we consider Gromov’s
questions for subsets of G, then a complete answer can be formulated. The
solution uses elements of sub-Riemannian fractal geometry associated to
horizontal self-similar iterated function systems on Carnot
groups.
An interesting bi-product of this work is a relatively simple method for
calculating dimensions of nonlinear iterated function systems. This is the
result of joint work with Zoltan Balogh
(