The effective potential and transshipment in thermodynamic formalism at temperature zero

Denote the points in {1,2,..,r}^{Z}= {1,2,..,r}^{N} x {1,2,..,r}^{N} by ({y}^*, {x}). Given a Lipschitz continuous observable A: {1,2,..,r}^{Z} \to {R} , we define the map {G}^+: {H}\to {H} by {G}^+(\phi)({y}^*) = \sup_{\mu \in {M}_\sigma} [\int_{\{1,2,..,r\}^{N}} ( A({y}^*, {x}) + \phi({x})) d\mu({x}) + h_\mu(\sigma) ], where: \sigma is the left shift map acting on {1,2,..,r}^{N}; {M}_\sigma denotes the set of \sigma-invariant Borel probabilities; h_\mu(\sigma) indicates the Kolmogorov-Sinai entropy; {H } is the Banach space of Lipschitz real-valued functions on {1,2,..,r}^{N}. We show there exist a unique \phi^+ \in {H } and a unique \lambda^+\in {R} such that {G}^+ (\phi^+) = \phi^+ + \lambda^+. We say that \phi^+ is the effective potential associated to A. This also defines a family of $\sigma$-invariant Borel probabilities \mu_{{y}^*} on {1,2,..,r}^{N}, indexed by the points {y}^* \in {1,2,..,r}^{N}. Finally, for A fixed and for variable positive real values \beta, we consider the same problem for the Lipschitz observable \beta A. We investigate then the asymptotic limit when \beta\to \infty of the effective potential (which depends now on \beta) as well as the above family of probabilities. We relate the limit objects with an ergodic version of Kantorovich transshipment problem. In statistical mechanics \beta \propto 1/T, where T is the absolute temperature. In this way, we are also analyzing the problem related to the effective potential at temperature zero.