Non-landing hairs in Sierpinski curve Julia sets of transcendental entire maps - Mathematics > Dynamical SystemsReport as inadecuate




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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/







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