{
  "id": "1405.3604",
  "version": "v3",
  "published": "2014-05-14T18:10:58.000Z",
  "updated": "2015-01-15T13:06:25.000Z",
  "title": "Krieger's finite generator theorem for ergodic actions of countable groups I",
  "authors": [
    "Brandon Seward"
  ],
  "comment": "The introduction was rewritten, the last section and appendix were moved to Part II, and two new sections were added",
  "categories": [
    "math.DS"
  ],
  "abstract": "For an ergodic probability-measure-preserving action $G \\curvearrowright (X, \\mu)$ of a countable group $G$, we define the Rokhlin entropy $h_G^{\\mathrm{Rok}}(X, \\mu)$ to be the infimum of the Shannon entropies of countable generating partitions. It is known that for free ergodic actions of amenable groups this notion coincides with classical Kolmogorov--Sinai entropy. It is thus natural to view Rokhlin entropy as a close analogue to classical entropy. Under this analogy we prove that Krieger's finite generator theorem holds for all countably infinite groups. Specifically, if $h_G^{\\mathrm{Rok}}(X, \\mu) < \\log(k)$ then there exists a generating partition consisting of $k$ sets. We actually obtain a notably stronger result which is new even in the case of actions by the integers. Our proofs are entirely self-contained and do not rely on the original Krieger theorem.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-03T20:57:00.000Z",
      "abstract": "For an ergodic probability-measure-preserving action $G \\curvearrowright (X, \\mu)$ of a countable group $G$, we consider the infimum $h_G^{\\mathrm{Rok}}(X, \\mu)$ of the Shannon entropies of countable generating partitions. In the case of free ergodic actions of amenable groups it is known that this quantity is equal to the entropy of the action. It is thus natural to view $h_G^{\\mathrm{Rok}}(X, \\mu)$ as a close analogue to entropy. Under this analogy we prove that Krieger's finite generator theorem holds for all countably infinite groups. Specifically, if $h_G^{\\mathrm{Rok}}(X, \\mu) < \\log(k)$ then there exists a generating partition consisting of $k$ sets. We actually obtain a notably stronger result which is new even in the case of actions by the integers. Furthermore, we develop formulas for $h_G^{\\mathrm{Rok}}(X, \\mu)$ which are analogues to the classical Kolmogorov and Kolmogorov--Sinai theorems from entropy theory. Our proofs are entirely self-contained and do not rely on the original Krieger theorem.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-01-15T13:06:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A15",
      "37A35"
    ],
    "keywords": [
      "countable group",
      "kriegers finite generator theorem holds",
      "generating partition",
      "free ergodic actions",
      "original krieger theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.3604S"
    }
  }
}