{
  "id": "2012.07481",
  "version": "v2",
  "published": "2020-12-14T12:58:50.000Z",
  "updated": "2022-08-17T21:54:59.000Z",
  "title": "Logarithmic bounds for ergodic sums of certain flows on the torus: a short proof",
  "authors": [
    "Jérôme Carrand"
  ],
  "comment": "Version v2 is the electronic copy of the version published in Qualitative Theory of Dynamical Systems",
  "journal": "Qualitative Theory of Dynamical Systems (2022)",
  "doi": "10.1007/s12346-022-00632-8",
  "categories": [
    "math.DS"
  ],
  "abstract": "We give a short proof that the ergodic sums of $\\mathcal{C}^1$ observables for a $\\mathcal{C}^1$ flow on $\\mathbb{T}^2$ admitting a closed transversal curve whose Poincar\\'e map has constant type rotation number have growth deviating at most logarithmically from a linear one. For this, we relate the latter integral to the Birkhoff sum of a well-chosen observable on the circle and use the Denjoy-Koksma inequality. We also give an example of a nonminimal flow satisfying the above assumptions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-08-17T21:54:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A25",
      "37D20",
      "37E35"
    ],
    "keywords": [
      "ergodic sums",
      "short proof",
      "logarithmic bounds",
      "constant type rotation number",
      "nonminimal flow"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}