Entropy, pressure and duality for Gibbs plans in Ergodic transport

Abstract

Let X be a finite set and Ω = {1, ..., d} N be the Bernoulli space. Denote by σ the shift map acting on Ω. We consider a fixed Lipschitz cost (or potential) function c : X × Ω → R and an associated Ruelle operator. We introduce the concept of Gibbs plan for c, which is a probability on X × Ω such that the y marginal is invariant. Moreover, we define entropy, pressure and equilibrium plans. The study of equilibrium plans can be seen as a generalization of the equilibrium probability problem where the concept of entropy for plans is introduced. We show that an equilibrium plan is a Gibbs plan.For a fixed probability µ on X with supp(µ)= X, define Π(µ, σ) as the set of all Borel probabilities π ∈ P (X ×Ω) such that the x-marginal of π is µ and the y-marginal of π is σ-invariant. We also investigate the pressure problem over Π(µ, σ), that is with constraint µ. Our main result is a Kantorovich 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 µ on X in a problem of pressure without constraint. Finally, given a parameter β, which plays the role of the inverse of temperature, we consider equilibrium plans for βc and its limit π ∞ , when β → ∞, which is also known as ground state. We compare this with other previous results on Ergodic Transport at temperature zero.

Duality results for iterated function systems with a general family of branches

“…Although this particular result seems to recall the main result in [12], we have a different situation. In order to illustrate the differences, we present an example of application in Thermodynamic Formalism.…”

“…Proof. The structure of this proof is close to [18] and [12]. In order to make the computations let E = C(X × Y × Z × W ).…”

“…Part of this section contains ideas previously developed in [10] and [12]. Given a normed linear space E and a convex function Θ : E → R∪{+∞}, the Legendre-Fenchel transform of Θ is the function Θ * : E * → R ∪ {+∞}, given by…”

“…Using ideas of transport theory (see [18], [10], [12]), we fix two probabilities µ ∈ P(X) and ν ∈ P(Y ) and denote by Π(µ, ν, τ ) the set of probabilities π ∈ Π(·, ·, τ ) satisfying…”

On the Thin Boundary of the Fat Attractor

“…Expressions of the kind (2) appear in Ergodic Transport (see [29] [30] [28] [32]). Equation (11) just after Theorem 6 describes relation (2) under certain general hypothesis: the Lipschitz setting (see section 9).…”