On Hausdorff Measures of Curves in Sub-Riemannian Geometry

In sub-Riemannian geometry, the length of a non-horizontal path is not defined (or is equal to +∞). However several other notions allow to measure a path, such as the Hausdorff measures, the class of k-dimensional lengths introduced in [2], or notions based on approximations by discrete sets, like the nonholonomic interpolation complexity and the entropy (see [8, 10]). The purpose of this paper is to compare these different notions and to extend our knowledge of the k-lengths, the entropy and the complexity to the Hausdorff measures. We show in particular that k-lengths coincide with the Hausdorff measures, thus providing an integral representation of these measures and the value of the Hausdorff dimension. We also give asymptotics of the entropy and of the nonholonomic interpolation complexity in function of the Hausdorff measures, which in turn allow to compute these measures in many cases.