Entropy, pressure and duality for Gibbs plans in Ergodic transport

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• 作者：A.O. Lopes ; J.K. Mengue ; J. Mohr ; R.R. Souza
• 关键词：entropy ; pressure ; Thermodynamic Formalism ; Gibbs plans ; ergodic transport ; Kantorovich Duality ; Fenchel ; Rockafellar duality ; Ruelle Operator ; zero temperature ; subaction ; 37D35 ; 37A35 ; 37C30 ; 90B06 ; 37A60
• 刊名：Bulletin of the Brazilian Mathematical Society
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：46
• 期：3
• 页码：353-389
• 全文大小：287 KB
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• 作者单位：A.O. Lopes (1)
  J.K. Mengue (1)
  J. Mohr (1)
  R.R. Souza (1)
  
  1. Instituto de Matemática, UFRGS, Porto Alegre, Brazil
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1678-7714

文摘

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 probabilityon X×Ωsuch that themarginal in the second variable is σ-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 μ on X with supp(μ)= X, define Π(μ, σ) as the set of all Borel probabilities π on X × Ω such that the marginal in the first variable is μ and the marginal in the second variable is σ-invariant. We also investigate the pressure problem over Π(μ, σ), that is with constraint μ. 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 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 β c and an accumulation point π _{?/sub>, when β ?? which is also known as ground state. We compare this with other previous results on Ergodic Transport at temperature zero. Keywords entropy pressure Thermodynamic Formalism Gibbs plans ergodic transport Kantorovich Duality Fenchel-Rockafellar duality Ruelle Operator zero temperature subaction}