On ergodic decompositions related to the Kantorovich problem

D. A. Zaev

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXVI. Representation theory, dynamical systems, combinatorial methods, Tome 437 (2015), pp. 100-130

Résumé

Let $X$ be a Polish space, $\mathcal P(X)$ be the set of Borel probability measures on $X$, and $T\colon X\to X$ be a homeomorphism. We prove that for the simplex $\mathrm{Dom}\subseteq\mathcal P(X)$ of all $T$-invariant measures, the Kantorovich metric on $\mathrm{Dom}$ can be reconstructed from its values on the set of extreme points. This fact is closely related to the following result: the invariant optimal transportation plan is a mixture of invariant optimal transportation plans between extreme points of the simplex. The latter result can be generalized to the case of the Kantorovich problem with additional linear constraints and the class of ergodic decomposable simplices.