# Non-landing hairs in Sierpinski curve Julia sets of transcendental entire maps - Mathematics > Dynamical Systems

Abstract: We consider the family of transcendental entire maps given by\$f az=az-1-a\expz+a\$ where \$a\$ is a complex parameter. Every map has asuperattracting 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\-nskiuniversal curve, thus containing embedded copies of any one-dimensional planecontinuum. In this paper we study subcontinua of the Julia set that can bedefined in a combinatorial manner. In particular, we show the existence ofnon-landing hairs with prescribed combinatorics embedded in the Julia set forall parameters \$a\geq 3\$. We also study the relation between non-landing hairsand the immediate basin of attraction of \$z=-a\$. Even as each non-landing hairaccumulates onto the boundary of the immediate basin at a single point, itsclosure, nonetheless, becomes an indecomposable subcontinuum of the Julia set.

Author: Antonio Garijo, Xavier Jarque, Monica Moreno Rocha

Source: https://arxiv.org/