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: $$\ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j).$$ 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 $-\alpha_j$: $$V_j(x)=|x|^{-\alpha_j}V_j\left(\frac{x}{|x|}\right).$$ 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.