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A combinatorial invariant for escape time Sierpiński rational maps

100%
Moreno Rocha M.
Fundamenta Mathematicae
|
2013
|
tom 222
|
nr 2
99-130
EN
An escape time Sierpiński map is a rational map drawn from the McMullen family z ↦ zⁿ + λ/zⁿ with escaping critical orbits and Julia set homeomorphic to the Sierpiński curve continuum. We address the problem of characterizing postcritically finite escape time Sierpiński maps in a combinatorial way. To accomplish this, we define a combinatorial model given by a planar tree whose vertices come with a pair of combinatorial data that encodes the dynamics of critical orbits. We show that each escape time Sierpiński map realizes a subgraph of the combinatorial tree and the combinatorial information is a complete conjugacy invariant.
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Non-landing hairs in Sierpiński curve Julia sets of transcendental entire maps

51%
Garijo A., Jarque X., Moreno Rocha M.
Fundamenta Mathematicae
|
2011
|
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.
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