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

@article{AntonioGarijo2011,
abstract = {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.},
author = {Antonio Garijo, Xavier Jarque, Mónica Moreno Rocha},
journal = {Fundamenta Mathematicae},
keywords = {transcendental entire function; transcendental dynamics; Julia set; non-landing hairs; indecomposable continuum},
language = {eng},
number = {2},
pages = {135-160},
title = {Non-landing hairs in Sierpiński curve Julia sets of transcendental entire maps},
url = {http://eudml.org/doc/283388},
volume = {214},
year = {2011},
}

TY - JOUR
AU - Antonio Garijo
AU - Xavier Jarque
AU - Mónica Moreno Rocha
TI - Non-landing hairs in Sierpiński curve Julia sets of transcendental entire maps
JO - Fundamenta Mathematicae
PY - 2011
VL - 214
IS - 2
SP - 135
EP - 160
AB - 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.
LA - eng
KW - transcendental entire function; transcendental dynamics; Julia set; non-landing hairs; indecomposable continuum
UR - http://eudml.org/doc/283388
ER -