{
  "id": "1805.02607",
  "version": "v1",
  "published": "2018-05-07T16:35:57.000Z",
  "updated": "2018-05-07T16:35:57.000Z",
  "title": "Pointwise ergodic theorem for locally countable quasi-pmp graphs",
  "authors": [
    "Anush Tserunyan"
  ],
  "categories": [
    "math.DS",
    "math.LO"
  ],
  "abstract": "We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, analogous to pointwise ergodic theorems for group actions, replacing the group with a Schreier graph of the action. Our theorem equates the global condition of ergodicity to locally approximating the means of $L^1$-functions via an increasing sequence of Borel subgraphs with finite connected components. Equivalently, it states that any (not necessarily pmp) locally countable Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is descriptive set theoretic and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant, a method of producing finite equivalence subrelations with large domain, and a simple method of exploiting nonamenability of a measured graph. The non-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-07T16:35:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A30",
      "03E15",
      "05C63",
      "37A20",
      "37A25"
    ],
    "keywords": [
      "pointwise ergodic theorem",
      "locally countable quasi-pmp graphs",
      "standard probability space contains",
      "ergodic hyperfinite subgraph",
      "producing finite equivalence subrelations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}