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

TL;DR

This paper investigates the structure of Julia sets for a family of transcendental entire maps, revealing the existence of non-landing hairs with specific combinatorics and their relation to the immediate basin of attraction.

Contribution

It demonstrates the existence of non-landing hairs with prescribed combinatorics in Julia sets of certain transcendental maps for all parameters a ≥ 3.

Findings

Non-landing hairs with prescribed combinatorics exist in Julia sets for all a ≥ 3.

Each non-landing hair accumulates on the basin boundary but forms an indecomposable continuum.

The structure of Julia sets includes embedded copies of complex continua, like the Sierpiński curve.

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 Sierpi\'nski 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 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.