Symbolic Dynamics for the Anisotropic N-Centre Problem at Negative Energies

The planar N-centre problem describes the motion of a particle moving in the plane under the action of the force fields of N fixed attractive centres: $$\begin{aligned} \ddot{x}(t)=\sum _{j=1}^N\nabla V_j(x(t)-c_j). \end{aligned}$$ In this paper we prove symbolic dynamics at slightly negative energy for an N-centre problem where the potentials $$V_j$$ are positive, anisotropic and homogeneous of degree $$-\alpha _j$$ : $$\begin{aligned} V_j(x)=|x|^{-\alpha _j}V_j\left( \frac{x}{|x|}\right) . \end{aligned}$$ The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis’ functional. Compared with the classical N-centre problem with Kepler potentials, a major difficulty arises from the lack of a regularization of the singularities. We will consider both the collisional dynamics and the non collision one. Symbols describe geometric and topological features of the associated trajectory.