{
  "id": "1404.0456",
  "version": "v2",
  "published": "2014-04-02T04:38:20.000Z",
  "updated": "2015-08-25T20:35:06.000Z",
  "title": "On density of ergodic measures and generic points",
  "authors": [
    "Katrin Gelfert",
    "Dominik Kwietniak"
  ],
  "comment": "32 pages, 6 figures. This version replaces an earlier preprint entitled \"The (Poulsen) simplex of invariant measures\"",
  "categories": [
    "math.DS"
  ],
  "abstract": "We provide conditions which guarantee that ergodic measures are dense in the simplex of invariant probability measures of a dynamical system given by a continuous map acting on a Polish space. Using them we study generic properties of invariant measures and prove that every invariant measure has a generic point. In the compact case, density of ergodic measures means that the simplex of invariant measures is either a singleton of a measure concentrated on a single periodic orbit or the Poulsen simplex. Our properties focus on the set of periodic points and we introduce two concepts: close\\-ability with respect to a set of periodic points and linkability of a set of periodic points. Examples are provided to show that these are independent properties. They hold, for example, for systems having the periodic specification property. But they hold also for a much wider class of systems which contains, for example, irreducible Markov chains over a countable alphabet, all $\\beta$-shifts, all $S$-gap shifts, ${C}^1$-generic diffeomorphisms of a compact manifold $M$, and certain geodesic flows of a complete connected negatively curved manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-02T04:38:20.000Z",
      "title": "The (Poulsen) simplex of invariant measures",
      "abstract": "Two new concepts, closeability with respect to a set of periodic points and linkability of a set of periodic points of a dynamical system are introduced. Examples are provided to show that closeability and linkability are independent properties. Both properties together imply that the set of invariant measures is either a single periodic orbit or the Poulsen simplex - the unique non-trivial Choquet simplex in which extreme points are dense. Moreover, under these conditions every invariant measure has a generic point and an extension of Sigmund's theorem about generic properties of invariant measures still holds. The periodic specification property implies closeability and linkability for the set of periodic points. The methods apply beyond systems with specification, because all beta-shifts, all $S$-gap shifts, and many other dynamical systems are closeable with respect to some linkable sets of periodic points.",
      "comment": "27 pages, 5 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-08-25T20:35:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "37B10",
      "37A99",
      "37D25",
      "37C20"
    ],
    "keywords": [
      "invariant measure",
      "periodic points",
      "periodic specification property implies closeability",
      "unique non-trivial choquet simplex",
      "linkability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.0456G"
    }
  }
}