{
  "id": "1203.6432",
  "version": "v13",
  "published": "2012-03-29T05:53:43.000Z",
  "updated": "2014-11-09T12:05:33.000Z",
  "title": "Equilibrium states and invariant measures for random dynamical systems",
  "authors": [
    "Ivan Werner"
  ],
  "comment": "The article is published in DCDS-A, but without the 3rd paragraph on page 4 (the complete removal of the paragraph became the condition for the publication in the DCDS-A after the reviewer run out of the citation suggestions collected in the paragraph)",
  "journal": "DCDS-A 35 (3) (March 2015) 1285-1326",
  "doi": "10.3934/dcds.2015.35.1285",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Random dynamical systems with countably many maps which admit countable Markov partitions on complete metric spaces such that the resulting Markov systems are uniformly continuous and contractive are considered. A non-degeneracy and a consistency conditions for such systems, which admit some proper Markov partitions of connected spaces, are introduced, and further sufficient conditions for them are provided. It is shown that every uniformly continuous Markov system associated with a continuous random dynamical system is consistent if it has a dominating Markov chain. A necessary and sufficient condition for the existence of an invariant Borel probability measure for such a non-degenerate system with a dominating Markov chain and a finite (16) is given. The condition is also sufficient if the non-degeneracy is weakened with the consistency condition. A further sufficient condition for the existence of an invariant measure for such a consistent system which involves only the properties of the dominating Markov chain is provided. In particular, it implies that every such a consistent system with a finite Markov partition and a finite (16) has an invariant Borel probability measure. A bijective map between these measures and equilibrium states associated with such a system is established in the non-degenerate case. Some properties of the map and the measures are given.",
  "revisions": [
    {
      "version": "v12",
      "updated": "2014-06-10T09:44:58.000Z",
      "comment": "Weakened the dominating Markov chain condition and made other minor improvements and corrections. The article is accepted for the publication in the journal Discrete and Continuous Dynamical System - A under the condition that the anonymous reviewer and the Editorial Office will be satisfied with the citations in the article",
      "journal": null,
      "doi": null
    },
    {
      "version": "v13",
      "updated": "2014-11-09T12:05:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37Hxx",
      "82B26",
      "82C05",
      "60G10",
      "37H99"
    ],
    "keywords": [
      "random dynamical system",
      "equilibrium states",
      "invariant measure",
      "invariant borel probability measure",
      "dominating markov chain"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.6432W"
    }
  }
}