Non-landing hairs in Sierpińnski 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 fa 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. © Instytut Matematyczny PAN, 2011.