  We consider the multidimensional Monge-Kantrovich transport problem in an
abstract setting. Our main results state that if a cost function and marginal
measures are invariant by a family of transformations, then a solution of the
Kantrovich relaxation problem and a solution of its dual can be chosen so that
they are invariant under the same family of transformations. This provides a
new tool to study and analyze the support of optimal transport plans and
consequently to scrutinize the Monge problem. Birkhoff's Ergodic theorem is an
essential tool in our analysis.
