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

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Garijo A., Jarque X., Moreno Rocha M.

Fundamenta Mathematicae

2011

tom 214

nr 2

135-160

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 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.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Garijo A.

1 Jarque X.

1 Moreno Rocha M.

1 2011

Wyniki wyszukiwania

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