Non-landing hairs in Sierpiński curve Julia sets of transcendental entire maps

Tom 214 / 2011

Antonio Garijo, Xavier Jarque, Mónica Moreno Rocha Fundamenta Mathematicae 214 (2011), 135-160 MSC: Primary 37F10; Secondary 37F20. DOI: 10.4064/fm214-2-3

Streszczenie

We consider the family of transcendental entire maps given by $f_a(z)=a(z-(1-a))\exp(z+a)$ where $a$ is a complex parameter. Every map has a superattracting fixed point at $z=-a$ and an asymptotic value at $z=0$. For $a>1$ the Julia set of $f_a$ is known to be homeomorphic to the Sierpiński universal curve, thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of non-landing hairs with prescribed combinatorics embedded in the Julia set for all parameters $a\geq 3$. We also study the relation between non-landing hairs and the immediate basin of attraction of $z=-a$. Even though each non-landing hair accumulates on the boundary of the immediate basin at a single point, its closure is an indecomposable subcontinuum of the Julia set.

Autorzy

  • Antonio GarijoDept. d'Enginyeria Informàtica
    i Mathemàtiques
    Universitat Rovira i Virgili
    Tarragona 43007, Spain
    e-mail
  • Xavier JarqueDept. d'Enginyeria Informàtica
    i Mathemàtiques
    Universitat Rovira i Virgili
    Tarragona 43007, Spain
    e-mail
  • Mónica Moreno RochaCentro de Investigación en Matemáticas
    Guanajuato 36240, Mexico
    e-mail

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