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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
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tom 214
|
nr 2
135-160
EN
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.
2
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On connectivity of Julia sets of transcendental meromorphic maps and weakly repelling fixed points II

100%
Fagella N., Jarque X., Taixés J.
Fundamenta Mathematicae
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2011
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tom 215
|
nr 2
177-202
EN
Following the attracting and preperiodic cases, in this paper we prove the existence of weakly repelling fixed points for transcendental meromorphic maps, provided that their Fatou set contains a multiply connected parabolic basin. We use quasi-conformal surgery and virtually repelling fixed point techniques.
3
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Hyperbolic components of the complex exponential family

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Devaney R., Fagella N., Jarque X.
Fundamenta Mathematicae
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2002
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tom 174
|
nr 3
193-215
EN
We describe the structure of the hyperbolic components of the parameter plane of the complex exponential family, as started in [1]. More precisely, we label each component with a parameter plane kneading sequence, and we prove the existence of a hyperbolic component for any given such sequence. We also compare these sequences with the more commonly used dynamical kneading sequences.
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