Topological cubic polynomials with one periodic ramification point

Matthieu Arfeux; Jan Kiwi

doi:10.3934/dcds.2020094

Article Contents

2020, Volume 40, Issue 3: 1799-1811. Doi: 10.3934/dcds.2020094

This issue Previous Article Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy Next Article An epiperimetric inequality approach to the parabolic Signorini problem

Topological cubic polynomials with one periodic ramification point

Matthieu Arfeux^1, and
Jan Kiwi^2,

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

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..

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Abstract

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.

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Mathematics Subject Classification: Primary: 37F10, 37F20, 37F30.

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Figure 1. Illustration of Lemma 3.1 for a topological polynomial $ f $ where $ [(f, c, c')]\in{\mathcal{E}}({\mathcal{F}}_4) $ has kneading word $ 1000 $

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Figure 2. 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

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Figure 3. Illustration of the annulus $ A $ around the twisting loop $ \tau $ (left) and its preimage (right)

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Figure 4. 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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References

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