2010/11/16 by Antonio Garijo, Garijo, Antonio, Xavier Jarque +3
Mathematics · #37F10 #37F20 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37F10 #msc:37F20
paper · pdf · doi:10.48550/arxiv.1011.3774
Some 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
arxiv created 2011/04/09 · arxiv updated 2015/03/17
We consider the family of transcendental entire maps given by fa(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 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.