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Hereditarily non uniformly perfect sets
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 [
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] | |
[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] | |
[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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