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

Antonio Garijo¹ ; Xavier Jarque¹ ; Mónica Moreno Rocha²

¹ Dept. d'Enginyeria Informàtica i Mathemàtiques Universitat Rovira i Virgili Tarragona 43007, Spain
² Centro de Investigación en Matemáticas Guanajuato 36240, Mexico

Fundamenta Mathematicae, Tome 214 (2011) no. 2, pp. 135-160

Résumé

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.

Détail

DOI : 10.4064/fm214-2-3

Keywords: consider family transcendental entire maps given z a exp where complex parameter every map has superattracting fixed point a asymptotic value julia set known homeomorphic sierpi ski universal curve containing embedded copies one dimensional plane continuum paper study subcontinua julia set defined combinatorial manner particular existence non landing hairs prescribed combinatorics embedded julia set parameters geq study relation between non landing hairs immediate basin attraction a even though each non landing hair accumulates boundary immediate basin single point its closure indecomposable subcontinuum julia set