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Entropy, pressure and duality for Gibbs plans in ergodic transport. (English) Zbl 1352.37091

Summary: Let \(X\) be a finite set and \(\Omega=\{1,\dots,d\}^{\mathbb N}\) be the Bernoulli space. Denote by \(\sigma\) the shift map acting on \(\Omega\). We consider a fixed Lipschitz cost (or potential) function \(c:X\times\Omega\to\mathbb R\) and an associated Ruelle operator. We introduce the concept of Gibbs plan for \(c\), which is a probabilityon \(X\times\Omega\) such that themarginal in the second variable is \(\sigma\)-invariant. Moreover, we define entropy, pressure and equilibriumplans. The study of equilibriumplans can be seen as a generalization of the equilibriumprobability problem where the concept of entropy for plans is introduced.
We show that an equilibriumplan is a Gibbs plan.
For a fixed probability \(\mu\) on \(X\) with \(\operatorname{supp}(\mu)=X\), define \(\Pi(\mu,\sigma)\) as the set of all Borel probabilities \(\pi\) on \(X\times\Omega\) such that the marginal in the first variable is \(\mu\) and the marginal in the second variable is \(\sigma\)-invariant. We also investigate the pressure problem over \(\Pi(\mu,\sigma)\), that is with constraint \(\mu\). Our main result is a duality Theorem on this setting. The pressure without constraint plays an important role in the establishment of the notion of admissible pair. Basically we want to transform a problem of pressure with a constraint \(\mu\) on \(X\) in a problem of pressure without constraint. Finally, given a parameter \(\beta\), which plays the role of the inverse of temperature, we consider equilibrium plans for \(\beta c\) and an accumulation point \(\pi_\infty\), when \(\beta\to\infty\), which is also known as ground state. We compare this with other previous results on Ergodic Transport at temperature zero.

MSC:

37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37A60 Dynamical aspects of statistical mechanics
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References:

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