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: x¨(t)=∑j=1N∇Vj(x(t)-cj).In this paper we prove symbolic dynamics at slightly negative energy for an N-centre problem where the potentials Vj are positive, anisotropic and homogeneous of degree - αj: Vj(x)=|x|-αjVj(x|x|).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.