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

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 $a\ge 3$. We also study the relation between non-landing hairs and the immediate basin of attraction of $z=-a$. Even as each non-landing hair accumulates onto the boundary of the immediate basin at a single point, its closure, nonetheless, becomes an indecomposable subcontinuum of the Julia set.

arXiv commentSome definitions have been introduced and clarifications have been made in several proofs. Substantial changes in Proposition 2.9. New references have been included. 31 pages, 5 figures. This article has been submitted for possible publication to Fundamenta Mathematicae