  The question of which costs admit unique optimizers in the Monge-Kantorovich
problem of optimal transportation between arbitrary probability densities is
investigated. For smooth costs and densities on compact manifolds, the only
known examples for which the optimal solution is always unique require at least
one of the two underlying spaces to be homeomorphic to a sphere. We introduce a
(multivalued) dynamics which the transportation cost induces between the target
and source space, for which the presence or absence of a sufficiently large set
of periodic trajectories plays a role in determining whether or not optimal
transport is necessarily unique. This insight allows us to construct smooth
costs on a pair of compact manifolds with arbitrary topology, so that the
optimal transportation between any pair of probility densities is unique.
