The W. Thurston algorithm applied to real polynomial maps

Bonifant, Araceli; Milnor, John; Sutherland, Scott

doi:10.1090/ecgd/365

The W. Thurston algorithm applied to real polynomial maps
HTML articles powered by AMS MathViewer

by Araceli Bonifant, John Milnor and Scott Sutherland

Conform. Geom. Dyn. 25 (2021), 179-199

DOI: https://doi.org/10.1090/ecgd/365

Published electronically: October 29, 2021

PDF | Request permission

Abstract:

This note will describe an effective procedure for constructing critically finite real polynomial maps with specified combinatorics.

References

Laurent Bartholdi and Volodymyr Nekrashevych, Thurston equivalence of topological polynomials, Acta Math. 197 (2006), no. 1, 1–51. MR 2285317, DOI 10.1007/s11511-006-0007-3
L. Bartholdi, IMG, Computations with iterated monodromy groups, a GAP package, version 0.1.1. laurentbartholdi.github.io/img/
Ben Bielefeld, Yuval Fisher, and John Hubbard, The classification of critically preperiodic polynomials as dynamical systems, J. Amer. Math. Soc. 5 (1992), no. 4, 721–762. MR 1149891, DOI 10.1090/S0894-0347-1992-1149891-3
Araceli Bonifant, Jan Kiwi, and John Milnor, Cubic polynomial maps with periodic critical orbit. II. Escape regions, Conform. Geom. Dyn. 14 (2010), 68–112. MR 2600536, DOI 10.1090/S1088-4173-10-00204-3
A. Bonifant, J. Milnor, and S. Sutherland, The W. Thurston Algorithm for Real Quadratic Rational Maps, arXiv:2009.10147 [math.DS], 2020.
Henk Bruin and Dierk Schleicher, Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials, J. Lond. Math. Soc. (2) 78 (2008), no. 2, 502–522. MR 2439637, DOI 10.1112/jlms/jdn033
Arnaud Chéritat, Tan Lei and Shishikura’s example of non-mateable degree 3 polynomials without a Levy cycle, Ann. Fac. Sci. Toulouse Math. (6) 21 (2012), no. 5, 935–980 (English, with English and French summaries). MR 3088263, DOI 10.5802/afst.1358
Peter Deuflhard, Newton methods for nonlinear problems, Springer Series in Computational Mathematics, vol. 35, Springer, Heidelberg, 2011. Affine invariance and adaptive algorithms; First softcover printing of the 2006 corrected printing. MR 2893875, DOI 10.1007/978-3-642-23899-4
Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263–297. MR 1251582, DOI 10.1007/BF02392534
H. B. Fine, On Newton’s method of approximation, Proc. Nat. Acad. Sci. USA 2 (1916) 546–552.
Suzanne Hruska Boyd and Christian Henriksen, The Medusa algorithm for polynomial matings, Conform. Geom. Dyn. 16 (2012), 161–183. MR 2943594, DOI 10.1090/S1088-4173-2012-00245-7
John Hamal Hubbard and Barbara Burke Hubbard, Vector calculus, linear algebra, and differential forms, Prentice Hall, Inc., Upper Saddle River, NJ, 1999. A unified approach. MR 1657732
John H. Hubbard and Dierk Schleicher, The spider algorithm, Complex dynamical systems (Cincinnati, OH, 1994) Proc. Sympos. Appl. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1994, pp. 155–180. MR 1315537, DOI 10.1090/psapm/049/1315537
A. Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), no. 1, 1–60 (German). MR 1510692, DOI 10.1007/BF01199469
W. Jung, The Thurston algorithm for quadratic matings. arXiv:1706.04177, 2017.
L. V. Kantorovič, On Newton’s method, Trudy Mat. Inst. Steklov. 28 (1949), 104–144 (Russian). MR 0038560
John Milnor, On Lattès maps, Dynamics on the Riemann sphere, Eur. Math. Soc., Zürich, 2006, pp. 9–43. MR 2348953, DOI 10.4171/011-1/1
John Milnor and Charles Tresser, On entropy and monotonicity for real cubic maps, Comm. Math. Phys. 209 (2000), no. 1, 123–178. With an appendix by Adrien Douady and Pierrette Sentenac. MR 1736945, DOI 10.1007/s002200050018
Alfredo Poirier, Critical portraits for postcritically finite polynomials, Fund. Math. 203 (2009), no. 2, 107–163. MR 2496235, DOI 10.4064/fm203-2-2
Alfredo Poirier, Hubbard trees, Fund. Math. 208 (2010), no. 3, 193–248. MR 2650982, DOI 10.4064/fm208-3-1
Nikita Selinger, Thurston’s pullback map on the augmented Teichmüller space and applications, Invent. Math. 189 (2012), no. 1, 111–142. MR 2929084, DOI 10.1007/s00222-011-0362-3
Dylan P. Thurston, From rubber bands to rational maps: a research report, Res. Math. Sci. 3 (2016), Paper No. 15, 49. MR 3500499, DOI 10.1186/s40687-015-0039-4