{
  "id": "0911.5411",
  "version": "v3",
  "published": "2009-11-28T19:08:41.000Z",
  "updated": "2011-07-17T19:34:48.000Z",
  "title": "Typical points for one-parameter families of piecewise expanding maps of the interval",
  "authors": [
    "Daniel Schnellmann"
  ],
  "comment": "33 pages, 3 figures; inclusion of a new section about almost sure typicality in transversal families of piecewise expanding unimodal maps; in the first part of the paper the conditions in order to obtain almost sure typicality are weakened; several other (small) improvements",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $I\\subset\\mathbb{R}$ be an interval and $T_a:[0,1]\\to[0,1]$, $a\\in I$, a one-parameter family of piecewise expanding maps such that for each $a\\in I$ the map $T_a$ admits a unique absolutely continuous invariant probability measure $\\mu_a$. We establish sufficient conditions on such a one-parameter family such that a given point $x\\in[0,1]$ is typical for $\\mu_a$ for a full Lebesgue measure set of parameters $a$, i.e. $$ \\frac{1}{n}\\sum_{i=0}^{n-1}\\delta_{T_a^i(x)} \\overset{\\text{weak-}*}{\\longrightarrow}\\mu_a,\\qquad\\text{as} n\\to\\infty, $$ for Lebesgue almost every $a\\in I$. In particular, we consider $C^{1,1}(L)$-versions of $\\beta$-transformations, skew tent maps, and Markov structure preserving one-parameter families. For the skew tent maps we show that the turning point is almost surely typical.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-07-17T19:34:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05",
      "37A05",
      "37D20"
    ],
    "keywords": [
      "one-parameter family",
      "piecewise expanding maps",
      "typical points",
      "continuous invariant probability measure",
      "absolutely continuous invariant probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.5411S"
    }
  }
}