An Ergodic Theorem on Ergodic Transport

Here we present an ergodic theorem which adapts a Theorem by J. Elton to the classical thermodynamical formalism and to ergodic transport. First, we discuss how Elton's theorem can be used to characterise Gibbs measures for expanding maps. Such characterisation will be done by constructing a stochastic process, defined by a iterated function system (IFS), whose empirical measure converges to the Gibbs measure, in the sense that the mean of any test function evaluated in the outcomes of this stochastic process converges to the integral of such test function with respect to the Gibbs measure. In this way we present a stochastic algorithm that compute integrals of functions. After this, we turn our attention to ergodic transport: given two sets $X$ and $\Omega$, a measure $\mu$ on $X$ and a dynamics $T$ on $\Omega$, we consider the set of probability measures on $X \times \Omega$ whose projections on the second coordinate are $T$-invariant, while the projections on the first coordinate are $\mu$. Such measures are called transport plans. We call Gibbs plan any transport plan that maximizes a pressure functional that is defined by a potential function added to an entropy term. As in the classical thermodynamical formalism case, we adapt Elton's theorem to define a stochastic process (using a IFS) whose empirical measures converges to the Gibbs plan. We provide examples and show explicitly calculations in the case where $X$ has two elements and the cost function depends on the two first coordinates of $\Omega$.