Symbolic dynamics for the $N$-centre problem at negative energies

We consider the planar N-centre problem, with homogeneous potentials of degree -alpha < 0, alpha is an element of [1, 2). We prove the existence of in finitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the N centres in two non-empty sets.