{
  "id": "0806.1012",
  "version": "v6",
  "published": "2008-06-05T16:44:20.000Z",
  "updated": "2009-01-06T11:30:53.000Z",
  "title": "Negative Entropy, Zero temperature and stationary Markov Chains on the interval",
  "authors": [
    "Artur O. Lopes",
    "Joana Mohr",
    "Rafael R. Souza",
    "Philippe Thieullen"
  ],
  "categories": [
    "math.DS",
    "math.PR"
  ],
  "abstract": "We analyze some properties of maximizing stationary Markov probabilities on the Bernoulli space $[0,1]^\\mathbb{N}$, More precisely, we consider ergodic optimization for a continuous potential $A$, where $A: [0,1]^\\mathbb{N}\\to \\mathbb{R}$ which depends only on the two first coordinates. We are interested in finding stationary Markov probabilities $\\mu_\\infty$ on $ [0,1]^\\mathbb{N}$ that maximize the value $ \\int A d \\mu,$ among all stationary Markov probabilities $\\mu$ on $[0,1]^\\mathbb{N}$. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential $A$. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities $\\mu_\\beta$ which weakly converges to $\\mu_\\infty$. The probabilities $\\mu_\\beta$ are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure. Under the hypothesis of $A$ being $C^2$ and the twist condition, that is, $\\frac{\\partial^2 A}{\\partial_x \\partial_y} (x,y) \\neq 0$, for all $(x,y) \\in [0,1]^2$, we show the graph property.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2009-01-06T11:30:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A35",
      "37A50",
      "37A60",
      "60J10"
    ],
    "keywords": [
      "stationary markov chains",
      "negative entropy",
      "probability",
      "variational principle similar",
      "large deviation principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.1012L"
    }
  }
}