Self-similarity in the Kepler–Heisenberg Problem

The Kepler–Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. The dynamics are at least partially integrable, possessing two first integrals as well as a dilational momentum which is conserved by orbits with zero energy. The system is known to admit closed orbits of any rational rotation number, which all lie within the fundamental zero-energy integrable subsystem. Here, we demonstrate that, under mild conditions, zero-energy orbits are self-similar. Consequently, we find that these zero-energy orbits stratify into three families: future collision, past collision, and quasi-periodicity without collision. If a collision occurs, it occurs in finite time.