{
  "id": "1804.01837",
  "version": "v2",
  "published": "2018-04-05T13:21:27.000Z",
  "updated": "2019-07-10T08:04:19.000Z",
  "title": "Isentropes and Lyapunov exponents",
  "authors": [
    "Zoltán Buczolich",
    "Gabriella Keszthelyi"
  ],
  "comment": "This is the revised version after the referee's report",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "We consider skew tent maps $T_{{\\alpha}, {\\beta}}(x)$ such that $( {\\alpha}, {\\beta})\\in[0,1]^{2}$ is the turning point of $T {_ { {\\alpha}, {\\beta}}}$, that is, $T_{{\\alpha}, {\\beta}}=\\frac{{\\beta}}{{\\alpha}}x$ for $0\\leq x \\leq {\\alpha}$ and $T_{{\\alpha}, {\\beta}}(x)=\\frac{{\\beta}}{1- {\\alpha}}(1-x)$ for $ {\\alpha}<x\\leq 1$. We denote by $ {\\underline {M}}=K( {\\alpha}, {\\beta})$ the kneading sequence of $T {_ { {\\alpha}, {\\beta}}}$, by $h( {\\alpha}, {\\beta})$ its topological entropy and $\\Lambda=\\Lambda_{\\alpha,\\beta}$ denotes its Lyapunov exponent. For a given kneading squence $ {\\underline {M}}$ we consider isentropes (or equi-topological entropy, or equi-kneading curves), $( {\\alpha},\\Psi_{{\\underline {M}}}( {\\alpha}))$ such that $K( {\\alpha},\\Psi_{{\\underline {M}}}( {\\alpha}))= {\\underline {M}}$. On these curves the topological entropy $h( {\\alpha},\\Psi_{{\\underline {M}}}( {\\alpha}))$ is constant. We show that $\\Psi_{{\\underline {M}}}'( {\\alpha})$ exists and the Lyapunov exponent $\\Lambda_{\\alpha,\\beta}$ can be expressed by using the slope of the tangent to the isentrope. Since this latter can be computed by considering partial derivatives of an auxiliary function $ { \\Theta}_{{\\underline {M}}}$, a series depending on the kneading sequence which converges at an exponential rate, this provides an efficient new method of finding the value of the Lyapunov exponent of these maps.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2019-07-10T08:04:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B25",
      "28D20",
      "37E05",
      "37B40"
    ],
    "keywords": [
      "lyapunov exponent",
      "skew tent maps",
      "kneading sequence",
      "topological entropy",
      "considering partial derivatives"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}