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

• Autorzy: Antonio Garijo , Xavier Jarque , Mónica Moreno Rocha
• Języki publikacji: EN

Abstrakty

EN

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 ≥ 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.

Kategorie tematyczne

• 37F20: Combinatorics and topology
• 37F10: Polynomials; rational maps; entire and meromorphic functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2011
• Tom: 214
• Numer: 2
• Strony: 135-160

Daty

wydano

2011

Twórcy

autor

Antonio Garijo

• Dept. d'Enginyeria Informàtica, i Mathemàtiques, Universitat Rovira i Virgili, Tarragona 43007, Spain

autor

Xavier Jarque

• Dept. d'Enginyeria Informàtica, i Mathemàtiques, Universitat Rovira i Virgili, Tarragona 43007, Spain

autor

Mónica Moreno Rocha

• Centro de Investigación en Matemáticas, Guanajuato 36240, Mexico

Identyfikatory

DOI

10.4064/fm214-2-3