{
  "id": "0806.2922",
  "version": "v1",
  "published": "2008-06-18T08:31:46.000Z",
  "updated": "2008-06-18T08:31:46.000Z",
  "title": "A Point is Normal for Almost All Maps $βx + α\\mod 1$ or Generalized $β$-Maps",
  "authors": [
    "B. Faller",
    "C. -E. Pfister"
  ],
  "comment": "Latex, 16 pages",
  "journal": "Ergod. Th & Dynam. Sys. 29 (2009) 1529-1547",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider the map $T_{\\alpha,\\beta}(x):= \\beta x + \\alpha \\mod 1$, which admits a unique probability measure of maximal entropy $\\mu_{\\alpha,\\beta}$. For $x \\in [0,1]$, we show that the orbit of $x$ is $\\mu_{\\alpha,\\beta}$-normal for almost all $(\\alpha,\\beta)\\in[0,1)\\times(1,\\infty)$ (Lebesgue measure). Nevertheless we construct analytic curves in $[0,1)\\times(1,\\infty)$ along them the orbit of $x=0$ is at most at one point $\\mu_{\\alpha,\\beta}$-normal. These curves are disjoint and they fill the set $[0,1)\\times(1,\\infty)$. We also study the generalized $\\beta$-maps (in particular the tent map). We show that the critical orbit $x=1$ is normal with respect to the measure of maximal entropy for almost all $\\beta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-18T08:31:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K16",
      "37E05",
      "37B10"
    ],
    "keywords": [
      "maximal entropy",
      "unique probability measure",
      "construct analytic curves",
      "lebesgue measure"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.2922F"
    }
  }
}