{
  "id": "1405.0481",
  "version": "v2",
  "published": "2014-05-02T19:29:23.000Z",
  "updated": "2015-06-09T00:34:25.000Z",
  "title": "On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation",
  "authors": [
    "Nigel P. Byott",
    "Congping Lin",
    "Yiwei Zhang"
  ],
  "comment": "27 pages 6 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "For an integer $m \\geq 2$, let $\\mathcal{P}_m$ be the partition of the unit interval $I$ into $m$ equal subintervals, and let $\\mathcal{F}_m$ be the class of piecewise linear maps on $I$ with constant slope $\\pm m$ on each element of $\\mathcal{P}_m$. We investigate the effect on mixing properties when $f \\in \\mathcal{F}_m$ is composed with the interval exchange map given by a permutation $\\sigma \\in S_N$ interchanging the $N$ subintervals of $\\mathcal{P}_N$. This extends the work in a previous paper [N.P. Byott, M. Holland and Y. Zhang, DCDS, {\\bf 33}, (2013) 3365--3390], where we considered only the \"stretch-and-fold\" map $f_{sf}(x)=mx \\bmod 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-02T19:29:23.000Z",
      "abstract": "For an integer $m \\geq 2$, we consider the class $\\mathcal{F}_m$ of maps on the unit interval $I$ with constant slope $\\pm m$ on each subinterval of the partition $\\mathcal{P}_m$ of $I$ into $m$ equal subintervals. We investigate the effect on mixing properties when $f \\in \\mathcal{F}_m$ is composed with the interval exchange map given by a permutation $\\sigma \\in S_N$ interchanging the $N$ subintervals of the partition $\\mathcal{P}_N$. This extends the work in a previous paper [N.P. Byott, M. Holland and Y. Zhang, DCDS, {\\bf 33}, (2013) 3365--3390], where we considered only the \"stretch-and-fold\" map $f_{sf}(x)=mx \\bmod 1$. We give necessary and sufficient conditions for the existence of a permutation $\\sigma$ for which $\\sigma \\circ f$ fails to be topologically mixing. We also consider the worst mixing rate $\\tau^{(m,N)}(f)$ which occurs as $\\sigma$ varies (where a mixing rate close to $1$ indicates slow mixing). We obtain the exact value of $\\tau^{(m,N)}(f_{zz})$ where $f_{zz}$ is the \"zigzag\" map, with the maximal number of changes of slope. We show that, for odd $m$, no $f \\in \\mathcal{F}_m$ can have worse asymptotic mixing behaviour than $f_{zz}$ in the following sense: there is an explicit constant $c(m)$, depending only on $m$, such that $\\liminf\\limits_{N\\to\\infty} (1-\\tau^{(m,N)}(f))/(1-\\tau^{(m,N)}(f_{zz})) \\geq c(m)$ with $\\gcd(m,N)=1$. When $m=2$, $f_{zz}$ is the familiar tent map. In this case, there are always permutations $\\sigma$ for which $\\sigma \\circ f_{zz}$ is not mixing. We give a partial result on the worst mixing rate which occurs for the tent map when such permutations are excluded.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-09T00:34:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "piecewise expanding maps",
      "mixing properties",
      "non-constant orientation",
      "permutation",
      "worst mixing rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0481B"
    }
  }
}