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Topological cubic polynomials with one periodic ramification point

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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  • 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.

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

    Citation:

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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 $

    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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    [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.
    [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.
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    [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.
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    [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).
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