  We study a class of singular dynamical systems which generalise the classical
N-centre problem of Celestial Mechanics to the case in which the configuration
space is a Riemannian surface. We investigate the existence of topological
conjugation with the archetypal chaotic dynamical system, the Bernoulli shift.
After providing infinitely many geometrically distinct and collision-less
periodic solutions, we encode them in bi-infinite sequences of symbols.
Solutions are obtained as minimisers of the Maupertuis functional in suitable
free homotopy classes of the punctured surface, without any collision
regularisation. For any sufficiently large value of the energy, we prove that
the generalised N-centre problem admits a symbolic dynamics. Moreover, when the
Jacobi-Maupertuis metric curvature is negative, we construct chaotic invariant
subsets.
