We consider the family of transcendental entire maps given by 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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-2-3,
author = {Antonio Garijo and Xavier Jarque and M\'onica Moreno Rocha},
title = {Non-landing hairs in Sierpi\'nski curve Julia sets of transcendental entire maps},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {135-160},
zbl = {1245.37009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-2-3}
}
Antonio Garijo; Xavier Jarque; Mónica Moreno Rocha. Non-landing hairs in Sierpiński curve Julia sets of transcendental entire maps. Fundamenta Mathematicae, Tome 215 (2011) pp. 135-160. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-2-3/