  We study the planar motion of test particles in gravitational fields produced
by an external material halo, of the type found in many astrophysical systems,
such as elliptical galaxies and globular clusters. Both the Newtonian and the
general-relativistic dynamics are examined, and in the relativistic case the
dynamics of both massive and massless particles are investigated. The halo
field is given in general by a multipole expansion; we restrict ourselves to
multipole fields of pure order, whose Newtonian potentials are homogeneous
polynomials in cartesian coordinates. A pure (n)-pole field has (n) different
escapes, one of which is chosen by the particle according to its initial
conditions. We find that the escape has a fractal dependency on the initial
conditions for (n>2) both in the Newtonian and the relativistic cases for
massive test particles, but with important differences between them. The
relativistic motion of massless particles, however, was found to be regular for
all the fields we could study. The box-counting dimension was used in each case
to quantify the sensitivity to initial conditions which arises from the
fractality of the escape route.
