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March  2020, 40(3): 1799-1811. doi: 10.3934/dcds.2020094

Topological cubic polynomials with one periodic ramification point

 1 Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile 2 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile

Received  July 2019 Published  December 2019

Fund Project: First author is supported by "Fondecyt Iniciación 11170276".
Second author is supported by CONICYT PIA ACT172001 and "Fondecyt 1160550".
Both authors partially supported by MathAmsud 18-Math-02 HidiParHodyn.

For $n \ge 1$, consider the space of affine conjugacy classes of topological cubic polynomials $f: \mathbb{C} \to \mathbb{C}$ with a period $n$ ramification point. It is shown that this space is a connected topological space.

Citation: Matthieu Arfeux, Jan Kiwi. Topological cubic polynomials with one periodic ramification point. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1799-1811. doi: 10.3934/dcds.2020094
References:

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References:
Illustration of Lemma 3.1 for a topological polynomial $f$ where $[(f, c, c')]\in{\mathcal{E}}({\mathcal{F}}_4)$ has kneading word $1000$
Illustration of the construction of the twisting loop corresponding to $m = 3$ and kneading word $1000$. The exterior curve in black is the level curve $g_{f_0} = g_{f_0}(c_0')$. The set $f_0^{-1}(Y)$ is drawn in gray
Illustration of the annulus $A$ around the twisting loop $\tau$ (left) and its preimage (right)
On both pictures, the doted curve represents the outer boundary of $\partial A_{ext}'$. The lightest gray regions are ${V'_0\cup V'_1}$. The other gray regions are the complement of $V'_0\cup V'_1$ in $f_0^{-1}(D)$ (left) and in $f_1^{-1}(D)$ (right).The lines in the darker gray regions represent the preimages of $\gamma$ by $f_0$ (left) and $f_1$ (right) where $\gamma$ is as in Section 4.2
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