{
  "id": "1203.5145",
  "version": "v3",
  "published": "2012-03-22T23:05:16.000Z",
  "updated": "2015-12-06T00:56:51.000Z",
  "title": "On the mixing properties of piecewise expanding maps under composition with permutations",
  "authors": [
    "Nigel P. Byott",
    "Mark Holland",
    "Yiwei Zhang"
  ],
  "categories": [
    "math.DS",
    "math.CO",
    "math.GR"
  ],
  "abstract": "We consider the effect on the mixing properties of a piecewise smooth interval map $f$ when its domain is divided into $N$ equal subintervals and $f$ is composed with a permutation of these. The case of the stretch-and-fold map $f(x)=mx \\bmod 1$ for integers $m \\geq 2$ is examined in detail. We give a combinatorial description of those permutations $\\sigma$ for which $\\sigma \\circ f$ is still (topologically) mixing, and show that the proportion of such permutations tends to $1$ as $N \\to \\infty$. We then investigate the mixing rate of $\\sigma \\circ f$ (as measured by the modulus of the second largest eigenvalue of the transfer operator). In contrast to the situation for continuous time diffusive systems, we show that composition with a permutation cannot improve the mixing rate of $f$, but typically makes it worse. Under some mild assumptions on $m$ and $N$, we obtain a precise value for the worst mixing rate as $\\sigma$ ranges through all permutations; this can be made arbitrarily close to $1$ as $N \\to \\infty$ (with $m$ fixed). We illustrate the geometric distribution of the second largest eigenvalues in the complex plane for small $m$ and $N$, and propose a conjecture concerning their location in general. Finally, we give examples of other interval maps $f$ for which composition with permutations produces different behaviour than that obtained from the stretch-and-fold map.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-05-02T20:59:26.000Z",
      "abstract": "For a mixing and uniformly expanding interval map $f:I\\to I$ we pose the following questions. For which permutation transformations $\\sigma:I\\to I$ is the composition $\\sigma\\circ f$ again mixing? When $\\sigma\\circ f$ is mixing, how does the mixing rate of $\\sigma\\circ f$ typically compare with that of $f$? As a case study, we focus on the family of maps$f(x)=mx\\operatorname{mod}1$ for $2\\leq m\\in\\mathbb{N}$. We split $[0,1)$ into $N$ equal subintervals, and take $\\sigma$ to be a permutation of these. We analyse those $\\sigma \\in S_N$ for which $\\sigma \\circ f$ is mixing, and show that, for large $N$, typical permutations will preserve the mixing property. In contrast to the situation for continuous time diffusive systems, we find that composition with a permutation cannot improve the mixing rate, but may make it worse. We obtain a precise result on the worst mixing rate which can occur as $\\sigma$ varies, with $m$, $N$ fixed and$\\gcd(m,N)=1$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-12-06T00:56:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "piecewise expanding maps",
      "mixing property",
      "composition",
      "worst mixing rate",
      "case study"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.5145B"
    }
  }
}