{
  "id": "1905.13256",
  "version": "v1",
  "published": "2019-05-30T18:37:41.000Z",
  "updated": "2019-05-30T18:37:41.000Z",
  "title": "Rigidity in dynamics and Möbius disjointness",
  "authors": [
    "Adam Kanigowski",
    "Mariusz Lemańczyk",
    "Maksym Radziwiłł"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $(X, T)$ be a topological dynamical system. We show that if all invariant measures of $(X, T)$ give rise to measure theoretic dynamical system that are rigid then $(X, T)$ satisfies Sarnak's conjecture on M\\\"obius disjointness. We show that the same conclusion also holds if there are countably many invariant ergodic measures, and they all give rise to rigid measure theoretic dynamical systems. This recovers several earlier results and immediately implies Sarnak's conjecture in the following new cases: for almost every interval exchange map of $d$ intervals with $d \\geq 2$ and almost all translation flows, for all $3$-interval exchange maps, and for absolutely continuous skew products over rotations. The latter two are improvement of earlier results of respectively Chaika-Eskin, Wang and Huang-Wang-Ye. We also discuss some purely arithmetic consequences for the Liouville function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-30T18:37:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "11N64"
    ],
    "keywords": [
      "möbius disjointness",
      "interval exchange map",
      "rigid measure theoretic dynamical systems",
      "earlier results",
      "immediately implies sarnaks conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}