{
  "id": "2303.10609",
  "version": "v1",
  "published": "2023-03-19T09:10:55.000Z",
  "updated": "2023-03-19T09:10:55.000Z",
  "title": "Some measure rigidity and equidistribution results for $β$-maps",
  "authors": [
    "Nevo Fishbein"
  ],
  "comment": "17 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We prove $\\times a$ $\\times b$ measure rigidity for multiplicatively independent pairs when $a\\in\\mathbb{N}$ and $b>1$ is a ``specified'' real number (the $b$-expansion of $1$ has a tail or bounded runs of $0$'s) under a positive entropy condition. This is done by proving a mean decay of the Fourier series of the point masses average along $\\times b$ orbits. We also prove a quantitative version of this decay under stronger conditions on the $\\times a$ invariant measure. The quantitative version together with the $\\times b$ invariance of the limit measure is a step toward a general Host-type pointwise equidistribution theorem in which the equidistribution is for Parry measure instead of Lebesgue. We show that finite memory length measures on the $a$-shift meet the mentioned conditions for mean convergence. Our main proof relies on techniques of Hochman.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-19T09:10:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28D05",
      "11A63"
    ],
    "keywords": [
      "measure rigidity",
      "equidistribution results",
      "general host-type pointwise equidistribution theorem",
      "finite memory length measures",
      "point masses average"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}