{
  "id": "2312.13165",
  "version": "v1",
  "published": "2023-12-20T16:25:54.000Z",
  "updated": "2023-12-20T16:25:54.000Z",
  "title": "Ergodic measures for periodic type $\\mathbb{Z}^m$-skew-products over Interval Exchange Transformations",
  "authors": [
    "Yuriy Tumarkin"
  ],
  "comment": "27 pages, 10 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider a special case of the question of classification of invariant Radon measures of $\\mathbb{Z}^m$-valued skew-products over interval exchange transformations, which arise as Poincar\\'e sections of the linear flow on periodic infinite translation surfaces. In the case of periodic type skew-products, we obtain a full classification of ergodic invariant Radon measures, showing them to be precisely the Maharam measures, a family of measures parametrised by $\\mathbb{R}^m$. For the proof we translate Rauzy-Veech renormalisation for skew-products into the symbolic language of the adic coding, and apply a symbolic result of Aaronson, Nakada, Sarig and Solomyak. Further, we use this language and a new extension of the Rauzy-Veech cocycle to find an explicit form for the Maharam measures and deduce the weak*-continuity of the measures depending on the parameter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-20T16:25:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "interval exchange transformations",
      "ergodic measures",
      "maharam measures",
      "periodic infinite translation surfaces",
      "ergodic invariant radon measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}