A class of -rectifiable subsets of metric spaces (and Hausdorff measures of curves in SR geometry)
Joint work with R. Ghezzi.
1. Case of curves
Let distribution define a sub-Riemannian structure on . Let denote the distributions generated by brackets. If is a curve tangent to , then . We give a formula for Hausdorff measure. When , equals length.
With Falbel, I defined a few years ago the metric differential of a curve,
which allows to define
Question. In what generality does
An -chain over a curve is an ordered set of points with .
Definition 1 Let be a curve. Let be the minimal number of points in an -chain of . The complexity of is
Question. Compare complexity and Hausdorff measure.
Falbel-Jean: for in contact and quasi-contact case.
Gauthier-Zakalyukin: for co-rank and distributios, and a few other cases.
Say a curve is equiregular if have constant rank along .
Theorem 2 (Falbel-Jean) Let be a embedded curve in . Assume that is tangent to . Assume that is equiregular. Then
Furthermore, if , , then
2.1. Euclidean case
In Euclidean space, a set is -rectifiable if -almost all of it is the union of a countable collection of Lipschitz images of . One may replace Lipschitz by .
Theorem 3 (Federer, Preiss) Let have finite measure. Then is -rectifiable if and only if
exists and is -almost everywhere constant.
Rectifiability implies a.e. differentiability (existence of a tangent -plane).
The situation is pretty different in sub-Riemannian geometry. Let us concentrate on curves now. Lipschitz images of are horizontal curves, so -rectifiable curves of positive measure exist only if and they are horizontal.
This forces us to enlarge the class of model curves to be used in the definition of rectifiability.
Definition 4 Let be a metric space. Let be a continuous curve in . Say that admits a metric derivative of degree at if the limit
exists everywhere. Say is metrically of degree (abbreviated into ) if is continuous.
Such curves are -Hölder continuous, locally bi-Hölder at points where .
Proposition 5 1. In Euclidean (or Riemannian) geometry, -curves with positive are Lipschitz and .
2. Let be a and equiregular curve is a sub-Riemannian manifold. Let be the smallest integer such that . Then is and does not vanish everywhere.
Remark 1 Assume is absolutely continuous (in Riemannian sense) and , then is the smallest integer such that .
It turns out that Theorem 1 extends to arbitrary metric spaces.
Theorem 6 (Ghezzi-Jean) Let be a metric space. Let be an injective curve in . Then
Furthermore, if , then density exists at at , i.e.
I expect that -curves in sub-Riemannian manifolds even have a derivative (an element in the tangent Carnot group), not merely a numerical metric differential.
2.3. Rectifiability of curves in metric spaces
Definition 7 Let be a metric space. A set is -rectifiable if -almost all of it is the union of a countable collection of images of curves.
Theorem 8 (Ghezzi-Jean) Let be a metric space. Let be a -rectifiable set with finite -measure. Then
I am convinced that density exists and is equal to 2.
is not preserved by bi-Lipschitz homeomorphisms, so our definition of -rectifiability is not bi-Lipschitz invariant.
Question. Can one cook up a bi-Lipschitz invariant definition and have the same conclusion ?