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.
- Previous Article
- DCDS-S Home
- This Issue
-
Next Article
Hereditarily non uniformly perfect sets
December
2019, 12(8): 2403-2433.
doi: 10.3934/dcdss.2019151
Thurston's algorithm and rational maps from quadratic polynomial matings
Department of Mathematics and Statistics, Coastal Carolina University, PO Box 261954, Conway, SC 29528-6054, USA |
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 [
Keywords:
Mating,
rational maps,
Thurston's algorithm,
Medusa,
finite subdivision rule,
pseudo-equator.
Mathematics Subject Classification:
Primary: 37F20; Secondary: 37F10.
Citation:
Mary Wilkerson. Thurston's algorithm and rational maps from quadratic polynomial matings. Discrete and Continuous Dynamical Systems - S,
2019, 12
(8)
: 2403-2433.
doi: 10.3934/dcdss.2019151
![]()
References:
| [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. Cannon, W. 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.
|
show all references
References:
| [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. Cannon, W. 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.
|
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] |
Peter Haïssinsky, Kevin M. Pilgrim. An algebraic characterization of expanding Thurston maps. Journal of Modern Dynamics, 2012, 6 (4) : 451-476. doi: 10.3934/jmd.2012.6.451 |
| [2] |
Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005 |
| [3] |
Pedro A. S. Salomão. The Thurston operator for semi-finite combinatorics. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 883-896. doi: 10.3934/dcds.2006.16.883 |
| [4] |
S. R. Bullett and W. J. Harvey. Mating quadratic maps with Kleinian groups via quasiconformal surgery. Electronic Research Announcements, 2000, 6: 21-30. |
| [5] |
Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1321-1329. doi: 10.3934/dcds.2016.36.1321 |
| [6] |
David Julitz. Numerical approximation of atmospheric-ocean models with subdivision algorithm. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 429-447. doi: 10.3934/dcds.2007.18.429 |
| [7] |
Cezar Joiţa, William O. Nowell, Pantelimon Stănică. Chaotic dynamics of some rational maps. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 363-375. doi: 10.3934/dcds.2005.12.363 |
| [8] |
Eriko Hironaka, Sarah Koch. A disconnected deformation space of rational maps. Journal of Modern Dynamics, 2017, 11: 409-423. doi: 10.3934/jmd.2017016 |
| [9] |
Richard Sharp, Anastasios Stylianou. Statistics of multipliers for hyperbolic rational maps. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1225-1241. doi: 10.3934/dcds.2021153 |
| [10] |
Russell Lodge, Sabyasachi Mukherjee. Invisible tricorns in real slices of rational maps. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1755-1797. doi: 10.3934/dcds.2020340 |
| [11] |
Yan Gao, Jinsong Zeng, Suo Zhao. A characterization of Sierpiński carpet rational maps. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5049-5063. doi: 10.3934/dcds.2017218 |
| [12] |
Jeffrey Diller, Han Liu, Roland K. W. Roeder. Typical dynamics of plane rational maps with equal degrees. Journal of Modern Dynamics, 2016, 10: 353-377. doi: 10.3934/jmd.2016.10.353 |
| [13] |
Huaibin Li. An equivalent characterization of the summability condition for rational maps. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4567-4578. doi: 10.3934/dcds.2013.33.4567 |
| [14] |
Yan Gao, Luxian Yang, Jinsong Zeng. Subhyperbolic rational maps on boundaries of hyperbolic components. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 319-326. doi: 10.3934/dcds.2021118 |
| [15] |
Weihua Liu, Andrew Klapper. AFSRs synthesis with the extended Euclidean rational approximation algorithm. Advances in Mathematics of Communications, 2017, 11 (1) : 139-150. doi: 10.3934/amc.2017008 |
| [16] |
Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35 |
| [17] |
Ayla Sayli, Ayse Oncu Sarihan. Statistical query-based rule derivation system by backward elimination algorithm. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1341-1356. doi: 10.3934/dcdss.2015.8.1341 |
| [18] |
Rui Gao, Weixiao Shen. Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2013-2036. doi: 10.3934/dcds.2014.34.2013 |
| [19] |
Piotr Pokora, Tomasz Szemberg. Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone. Electronic Research Announcements, 2014, 21: 126-131. doi: 10.3934/era.2014.21.126 |
| [20] |
Weiyuan Qiu, Fei Yang, Yongcheng Yin. Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3375-3416. doi: 10.3934/dcds.2016.36.3375 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]