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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:
[1]

J. W. Alexander, On the deformation of an n-cell, Proc. Nat. Acad. Sci., 9 (1923), 406-407.   Google Scholar

[2]

A. BonifantJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. V. F. Levy, Critically Finite Rational Maps, PhD thesis, Princeton University, 1986.  Google Scholar

[8]

J. Milnor, Dynamics in One Complex Variable, Third edition, Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[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.  Google Scholar

[10]

M. Rees, Views of parameter space: Topographer and Resident, Astérisque, (2003).  Google Scholar

show all references

References:
[1]

J. W. Alexander, On the deformation of an n-cell, Proc. Nat. Acad. Sci., 9 (1923), 406-407.   Google Scholar

[2]

A. BonifantJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. V. F. Levy, Critically Finite Rational Maps, PhD thesis, Princeton University, 1986.  Google Scholar

[8]

J. Milnor, Dynamics in One Complex Variable, Third edition, Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[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.  Google Scholar

[10]

M. Rees, Views of parameter space: Topographer and Resident, Astérisque, (2003).  Google Scholar

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