{
  "id": "1512.05146",
  "version": "v1",
  "published": "2015-12-16T12:32:51.000Z",
  "updated": "2015-12-16T12:32:51.000Z",
  "title": "Equi-topological entropy curves for skew tent maps in the square",
  "authors": [
    "Zoltan Buczolich",
    "Gabriella Keszthelyi"
  ],
  "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 $T_ {{\\alpha}, {\\beta}}$ and by $h({\\alpha}, {\\beta})$ its topological entropy. For a given kneading squence $ {\\underline{M}}$ we consider equi-kneading, (or equi-topological entropy, or isentrope) curves $({\\alpha}, \\varphi_{{\\underline{M}}}({\\alpha}))$ such that $K({\\alpha}, {\\varphi}_{{\\underline{M}}}({\\alpha}))= {\\underline{M}}$. To study the behavior of these curves an auxiliary function $ {\\Theta}_{{\\underline{M}}}({\\alpha}, {\\beta})$ is introduced. For this function $ {\\Theta}_{{\\underline{M}}}({\\alpha}, \\varphi_{{\\underline{M}}}({\\alpha}))=0$, but it may happen that for some kneading sequences $\\Theta_{{\\underline{M}}}({\\alpha}, {\\beta})=0$ for some $ {\\beta}< \\varphi_{{\\underline{M}}}({\\alpha})$ with $({\\alpha}, {\\beta})$ still in the interesting region. Using $ {\\Theta}_{{\\underline{M}}}$ we show that the curves $({\\alpha},\\varphi_{{\\underline{M}}}({\\alpha}))$ hit the diagonal $\\{({\\beta}, {\\beta}): 0.5< {\\beta}<1 \\}$ almost perpendicularly if $({\\beta}, {\\beta})$ is close to $(1,1)$. Answering a question asked by M. Misiurewicz at a conference we show that these curves are not necessarily exactly orthogonal to the diagonal, for example for $ {\\underline{M}}=RLLRC$ the curve $(\\alpha, {\\varphi}_{{\\underline{M}}}({\\alpha}))$ is not orthogonal to the diagonal. On the other hand, for $ {\\underline{M}}=RLC$ it is. With different parametrization properties of equi-kneading maps for skew tent maps were considered by J.C. Marcuard, M. Misiurewicz and E. Visinescu.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-16T12:32:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B40",
      "28D20",
      "37E05",
      "37E20"
    ],
    "keywords": [
      "skew tent maps",
      "equi-topological entropy curves",
      "kneading sequence",
      "auxiliary function",
      "misiurewicz"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151205146B"
    }
  }
}