Paths in sub-Riemannian geometry

In sub-Riemannian geometry only horizontal paths - i.e. tangent to the distribution - can have finite length. The aim of this talk is to study non-horizontal paths, in particular to measure them and give their metric dimension. For that we introduce two metric invariants, the entropy and the complexity, and corresponding measures of the paths depending on a small parameter epsilon. We give estimates for the entropy and the complexity, and a condition for these quantities to be equivalent. The estimates depend on a epsilon -norm on the tangent space, which tends to the sub-Riemannian metric as epsilon goes to zero. The results are based on an estimation of sub-Riemannian balls depending uniformly of their radius.