How Many Geodesics Are There between Two Close Points on a Sub-Riemannian Manifold?

Given a point $q$ on a Riemannian manifold and a small enough neighborhood $U$ of this point, then for every other point $p\in U$ there will be only one geodesic joining these two points entirely contained in $U$.
Moving to the sub-Riemannian case, the situation dramatically changes.
Consider for example, the standard Heisenberg group $\mathbb{R}^{3}$ with coordinates $(x, y)$ (here $y$ is the “vertical” coordinate). Then the number $\hat{\nu}(p)$ of geodesics joining the origin with the point $p=(x,y)$ is given by:
$$ \tag{1} \hat{\nu}(p)=\frac{8 \|y\|}{\pi\|x\|^2}+O(1) $$
One should notice, for instance, that when the point is “vertical” ($x=0$) there are infinitely many geodesics and when the point is “horizontal” ($y=0$) there are finitely many (in fact just one).
On a general sub-Riemannian manifold, given a point $q$ and privileged coordinates on a neighborhood $U$ of $q$, one can consider the associated family of dilations:
$$\delta_{\epsilon}:U\to U,\quad \delta_{\epsilon}(q)=q.$$
When $\epsilon$ is very small, the geometry of this family approaches a limit geometry: the sub-Riemannian tangent space at $q$ (a Carnot group).
Given another point $p\in U$, it is natural to ask for the number $\nu(\delta_\epsilon(p))$ of geodesics between $q$ and $\delta_\epsilon(p)$ (i.e. when the two points get closer and closer, in the sub-Riemannian sense).
In this talk I will show how to relate the asymptotic for $\nu(\delta_\epsilon(p))$ to the count on the associated Carnot group (as performed in formula (1) above). I will show, for instance, that for the generic $p\in U$:
$$\lim_{\epsilon \to 0}\nu(\delta_\epsilon(p))=\hat{\nu}(p)$$
and discuss related questions and applications.
This is joint work with L. Rizzi