  Let $X$ be a finite set and $\Omega=\{1,...,d\}^{\mathbb{N}}$ be the
Bernoulli space. Denote by $\sigma$ the shift map acting on $\Omega$. For a
fixed probability $\mu$ on $X$ with supp($\mu$)$=X$, define $\Pi(\mu,\sigma)$
as the set of all Borel probabilities $\pi \in P(X\times \Omega)$ such that the
$x$-marginal of $\pi$ is $\mu $ and the $y$-marginal of $\pi$ is
$\sigma$-invariant. We consider a fixed Lipschitz cost function $c: X \times
\Omega \to \mathbb{R}$ and an associated Ruelle operator. We introduce the
concept of Gibbs plan, which is a probability on $X \times \Omega$. Moreover,
we define entropy, pressure and equilibrium plans. The study of equilibrium
plans can be seen as a generalization of the optimal cost problem where the
concept of entropy is introduced. We show that an equilibrium plan is a Gibbs
plan. Our main result is a Kantorovich duality Theorem on this setting. The
pressure plays an important role in the establishment of the notion of
admissible pair. Finally, given a parameter $\beta$, which plays the role of
the inverse of temperature, we consider equilibrium plans for $\beta c$ and its
limit $\pi_\infty$, when $\beta \to \infty$, which is also known as ground
state. We compare this with other previous results on Ergodic Transport in
temperature zero.
