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: June 30, 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

[1]	J. W. Alexander, On the deformation of an n-cell, Proc. Nat. Acad. Sci., 9 (1923), 406-407.
[2]	A. Bonifant, J. Kiwi and J. Milnor, Cubic polynomial maps with periodic critical orbit. Ⅱ. Escape regions, Conform. Geom. Dyn., 14 (2010), 68-112. doi: 10.1090/S1088-4173-10-00204-3.
[3]	B. Branner, Cubic polynomials: Turning around the connectedness locus, Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, (1993), 391–427.
[4]	B. Branner and J. H. Hubbard, The iteration of cubic polynomials. Ⅰ. The global topology of parameter space, Acta Math., 160 (1988), 143-206. doi: 10.1007/BF02392275.
[5]	B. Branner and J. H. Hubbard, The iteration of cubic polynomials. Ⅱ. Patterns and parapatterns, Acta Math., 169 (1992), 229-325. doi: 10.1007/BF02392761.
[6]	G. Z. Cui and L. Tan., A characterization of hyperbolic rational maps, Invent. Math., 183 (2011), 451-516. doi: 10.1007/s00222-010-0281-8.
[7]	S. V. F. Levy, Critically Finite Rational Maps, PhD thesis, Princeton University, 1986.
[8]	J. Milnor, Dynamics in One Complex Variable, Third edition, Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006.
[9]	J. Milnor, Cubic polynomial maps with periodic critical orbit. Ⅰ, Complex Dynamics, A K Peters, Wellesley, MA, (2009), 333–411. doi: 10.1201/b10617-13.
[10]	M. Rees, Views of parameter space: Topographer and Resident, Astérisque, (2003).