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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 and Continuous Dynamical Systems, 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. [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).

show all references

##### 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).
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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