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Thurston's algorithm and rational maps from quadratic polynomial matings

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  • Topological mating is a combination that takes two same-degree polynomials and produces a new map with dynamics inherited from this initial pair. This process frequently yields a map that is Thurston-equivalent to a rational map $ F $ on the Riemann sphere. Given a pair of polynomials of the form $ z^2+c $ that are postcritically finite, there is a fast test on the constant parameters to determine whether this map $ F $ exists-but this test does not give a construction of $ F $. We present an iterative method that utilizes finite subdivision rules and Thurston's algorithm to approximate this rational map, $ F $. This manuscript expands upon results given by the Medusa algorithm in [9]. We provide a proof of the algorithm's efficacy, details on its implementation, the settings in which it is most successful, and examples generated with the algorithm.

    Mathematics Subject Classification: Primary: 37F20; Secondary: 37F10.

    Citation:

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  • Figure 1.  The conformal isomorphism $ \phi $ which determines external rays for $ z\mapsto z^2+i $. Shown on the right are external rays landing at points on the critical orbit of this polynomial.

    Figure 2.  Steps in the formation of the formal mating.

    Figure 3.  The Medusa and pseudo-equator algorithms are based upon Thurston's algorithm, highlighted in the commutative diagram above.

    Figure 4.  A rudimentary finite subdivision rule on $ \hat{\mathbb{C}} $.

    Figure 5.  The Julia set and Hubbard trees for $ f_{1/4} $.

    Figure 6.  The preimage of a Hubbard tree under its associated polynomial.

    Figure 7.  On the left, $ T_{1/4} $. On the right, the subdivision complex $ S_\mathcal{R} $ for the essential self-mating of $ f_{1/4} $.

    Figure 8.  On the left, the expected pullback of $ S_\mathcal{R} $ by the essential mating as based on local behavior of Hubbard trees. The essential mating is locally homeomorphic everywhere except on the critical set, so we complete the pullback as shown on the right.

    Figure 9.  The finite subdivision rule associated with $ f_{1/4}\;╨_e\;f_{1/4} $, along with marked pseudo-equator curves. $ C_0 $ is marked in blue on the left and its pullback $ C_1 $ is marked in blue on the right.

    Figure 10.  Pullbacks of the equator by a rational map that is Thurston-equivalent to the topological self-mating of $ f_{1/4} $. These pullbacks approximate the Julia set of the rational map, $ \hat{\mathbb{C}} $. (Image generated in Mathematica.)

    Figure 11.  Top: The Julia sets of $ f_{1/4} $ and $ f_{1/8} $, with external angles marked at postcritical points for reference. Middle: The Hubbard trees associated with these polynomials. Bottom: the finite subdivision rule associated with the essential mating $ f_{1/4}\;╨_e\;f_{1/8} $.

    Figure 12.  The critical orbit portrait and finite subdivision rule associated with $ f_{1/4}\;╨_e\;f_{1/8} $, along with marked pseudo-equator curves. $ C_0 $ is marked in blue above and its pullback $ C_1 $ is marked in blue below. We have relabeled the marked points to emphasize angle markings given by the parameterizations of $ C_0 $ and $ C_1 $.

    Figure 13.  Pullbacks of the equator by a sequence of rational maps which approximate the geometric mating of $ f_{1/4} $ and $ f_{1/8} $. (Image generated in Mathematica.)

    Figure 14.  The problem with using the canonical branch of the square root for pullbacks of $ C_n $: orientation is important, but harder to keep record of when our pullback curve is cut into several pieces.

    Figure 15.  The "pseudo-equator" is pinched by $ \sim_e $ into a non-Jordan curve.

  • [1] L. Bartholdi and V. Nekrashevych, Thurston equivalence of topological polynomials, Acta Math, 197 (2006), 1-51.  doi: 10.1007/s11511-006-0007-3.
    [2] H. Bruin and D. Schleicher, Symbolic dynamics of quadratic polynomials, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences, 7.
    [3] X. Buff, A. Epstein and S. Koch, Twisted matings and equipotential gluings, Annales de la Faculté des Sciences de Toulouse Mathématiques, 21 (2012), 995-1031. doi: 10.5802/afst.1360.
    [4] X. Buff, A. Epstein, S. Koch, D. Meyer, K. Pilgrim, M. Rees and L. Tan, Questions about polynomial matings, Annales de la Faculté des Sciences de Toulouse Mathématiques, 21 (2012), 1149-1176. doi: 10.5802/afst.1365.
    [5] J. Cannon, W. Floyd and W. Parry, Subdivision programs, https://www.math.vt.edu/people/floyd/research/software/subdiv.html.
    [6] J. CannonW. Floyd and W. Parry, Finite subdivision rules, Conform. Geom. Dyn., 5 (2001), 153-196.  doi: 10.1090/S1088-4173-01-00055-8.
    [7] A. Douady and J. H. Hubbard, Exploring the Mandelbrot set. The Orsay notes, Publ. Math. Orsay.
    [8] A. Douady and J. H. Hubbard, A proof of thurston's topological characterization of rational functions, Acta Mathematica, 171 (1993), 263-297.  doi: 10.1007/BF02392534.
    [9] S. Hruska Boyd and C. Henriksen, The Medusa algorithm for polynomial matings, Conform. Geom. Dyn., 16 (2012), 161-183.  doi: 10.1090/S1088-4173-2012-00245-7.
    [10] J. H. Hubbard and D. Schleicher, The spider algorithm, Complex Dynamical Systems, RL Devaney ed., Proc. Symp. Appl. Math, 49 (1994), 155-180.  doi: 10.1090/psapm/049/1315537.
    [11] W. Jung, Mandel version 5.11, http://www.mndynamics.com, 2014.
    [12] W. Jung, The Thurston algorithm for quadratic matings.,
    [13] D. Meyer, Unmating of rational maps, sufficient criteria and examples, in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday (ed. S. S. A. Bonifant M. Lyubich), Princeton University Press, 51 (2014), 197-233.
    [14] J. Milnor, Pasting together Julia sets: A worked out example of mating, Experiment. Math., 13 (2004), 55-92.  doi: 10.1080/10586458.2004.10504523.
    [15] C. Petersen and D. Meyer, On the notions of mating, Annales de la faculté des sciences de Toulouse Mathématiques, 21 (2012), 839-876. doi: 10.5802/afst.1355.
    [16] M. Rees, A partial description of the parameter space of rational maps of degree two: Part 1, Acta Math., 168 (1992), 11-87.  doi: 10.1007/BF02392976.
    [17] N. Selinger, Thurston's pullback map on the augmented Teichmüller space and applications, Inventiones Mathematicae, 189 (2012), 111-142.  doi: 10.1007/s00222-011-0362-3.
    [18] M. Shishikura, On a theorem of M. Rees for matings of polynomials, in The Mandelbrot Set, Theme and Variations (ed. Tan, L.), vol. London Mathematical Society Lecture Notes, 274, Cambridge University Press, 2000,289-305.
    [19] L. Tan, Matings of quadratic polynomials, Ergodic Theory Dynam. Systems, 12 (1992), 589-620.  doi: 10.1017/S0143385700006957.
    [20] M. Wilkerson, Finite Subdivision Rules from Matings of Quadratic Functions: Existence and Constructions, PhD thesis, Virginia Polytechnic Institute and State University, 2012.
    [21] M. Wilkerson, Subdivision rule constructions on critically preperiodic quadratic matings, New York J. Math., 22 (2016), 1055-1084. 
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