*Published Paper*

**Inserted:** 20 may 2014

**Last Updated:** 15 may 2017

**Journal:** Journal of symplectic geometry

**Volume:** Volume 15

**Number:** 1

**Pages:** 247 – 305

**Year:** 2017

**Doi:** 10.4310/JSG.2017.v15.n1.a7

**Abstract:**

We investigate the number of geodesics between two points $p$ and $q$ on a
contact sub-Riemannian manifold M. We show that the count of geodesics on $M$
is controlled by the count on its nilpotent approximation at $p$ (a contact
Carnot group). For contact Carnot groups we make the count explicit in
exponential coordinates $(x,z) \in \mathbb{R}^{2n} \times \mathbb{R}$ centered
at $p$. In this case we prove that for the generic $q$ the number of geodesics
$\nu(q)$ between $p$ and $q=(x,z)$ satisfies: \[ C_1\frac{

z

}{\

x\

^2} + R_1
\leq \nu(q) \leq C_2\frac{

z

}{\

x\

^2} + R_2\] for some constants $C_1,C_2$
and $R_1,R_2$. We recover exact values for Heisenberg groups, where $C_1=C_2 =
\frac{8}{\pi}$.
Removing the genericity condition for $q$, geodesics might appear in families
and we prove a similar statement for their topology. We study these families,
and in particular we focus on the unexpected appearance of isometrically
non-equivalent geodesics: families on which the action of isometries is not
transitive.
We apply the previous study to contact sub-Riemannian manifolds: we prove
that for any given point $p \in M$ there is a sequence of points $p_n$ such
that $p_n \to p$ and that the number of geodesics between $p$ and $p_n$ grows
unbounded (moreover these geodesics have the property of being contained in a
small neighborhood of $p$).

**Keywords:**
Carnot groups, Geodesics, sub-Riemannian, contact

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