How many geodesics join two points on a contact sub-Riemannian manifold?

We investigate the structure and the topology of the set of geodesics (critical points for the energy functional) between two points on a contact Carnot group G (or, more generally, corank-one Carnot groups). Denoting by (x, z) ∈ R2n × R exponential coordinates on G, we find constants C1, C2> 0 and R1, R2 such that the number ν̂(p) of geodesics joining the origin with a generic point p = (x, z) satisfies