Boundary values of the Thurston pullback map

Lodge, Russell

doi:10.1090/S1088-4173-2013-00255-5

Boundary values of the Thurston pullback map
HTML articles powered by AMS MathViewer

by Russell Lodge

Conform. Geom. Dyn. 17 (2013), 77-118

DOI: https://doi.org/10.1090/S1088-4173-2013-00255-5

Published electronically: June 6, 2013

PDF | Request permission

Abstract:

For any Thurston map with exactly four postcritical points, we present an algorithm to compute the Weil-Petersson boundary values of the corresponding Thurston pullback map. This procedure is carried out for the Thurston map $f(z)=\frac {3z^2}{2z^3+1}$ originally studied by Buff, et al. The dynamics of this boundary map are investigated and used to solve the analogue of Hubbard’s Twisted Rabbit problem for $f$.

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
Laurent Bartholdi and Volodymyr V. Nekrashevych, Iterated monodromy groups of quadratic polynomials. I, Groups Geom. Dyn. 2 (2008), no. 3, 309–336. MR 2415302, DOI 10.4171/GGD/42
Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
Oleg Bogopolski, Introduction to group theory, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008. Translated, revised and expanded from the 2002 Russian original. MR 2396717, DOI 10.4171/041
Henk Bruin, Alexandra Kaffl, and Dierk Schleicher, Existence of quadratic Hubbard trees, Fund. Math. 202 (2009), no. 3, 251–279. MR 2476617, DOI 10.4064/fm202-3-4
Xavier Buff, Adam Epstein, Sarah Koch, and Kevin Pilgrim, On Thurston’s pullback map, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 561–583. MR 2508269, DOI 10.1201/b10617-20
J. W. Cannon, W. J. Floyd, and W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153–196. MR 1875951, DOI 10.1090/S1088-4173-01-00055-8
J. W. Cannon, W. J. Floyd, W. R. Parry, and K. M. Pilgrim, Nearly Euclidean Thurston maps, Conform. Geom. Dyn. 16 (2012), 209–255. MR 2958932, DOI 10.1090/S1088-4173-2012-00248-2
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
Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362, DOI 10.1007/978-1-4684-9449-5
John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, 2006. Teichmüller theory; With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra; With forewords by William Thurston and Clifford Earle. MR 2245223
G. Kelsey. Mega-bimodules of topological polynomials: Sub-hyperbolicity and Thurston obstructions. http://www.math.uiuc.edu/~gkelsey2/files/Papers/GAKThesis.pdf, 2011.
S. Koch. Teichmüller theory and critically finite endomorphisms. Submitted.
S. Koch, K. Pilgrim, and N. Selinger. Pullback invariants of Thurston maps. Preprint, 2012.
Serge Lang, Introduction to modular forms, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 222, Springer-Verlag, Berlin, 1995. With appendixes by D. Zagier and Walter Feit; Corrected reprint of the 1976 original. MR 1363488
R. Lodge. Boundary values of the Thurston pullback map. 2012. PhD Thesis.
Curt McMullen, Families of rational maps and iterative root-finding algorithms, Ann. of Math. (2) 125 (1987), no. 3, 467–493. MR 890160, DOI 10.2307/1971408
D. Meyer. Unmating of rational maps, sufficient criteria and examples. arXiv:1110.6784v1, 2011.
John Milnor, Pasting together Julia sets: a worked out example of mating, Experiment. Math. 13 (2004), no. 1, 55–92. MR 2065568, DOI 10.1080/10586458.2004.10504523
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
Volodymyr Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs, vol. 117, American Mathematical Society, Providence, RI, 2005. MR 2162164, DOI 10.1090/surv/117
Volodymyr Nekrashevych, Combinatorics of polynomial iterations, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 169–214. MR 2508257, DOI 10.1201/b10617-5
K. Pilgrim. An algebraic formulation of Thurston’s characterization of rational functions. To appear, special issue of Annales de la Faculte des Sciences de Toulouse, 2010.
N. Selinger. Thurston’s pullback map on the augmented Teichmüller space and applications. Inventiones mathematicae, pages 1–32, 2011.
Scott A. Wolpert, The Weil-Petersson metric geometry, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, Eur. Math. Soc., Zürich, 2009, pp. 47–64. MR 2497791, DOI 10.4171/055-1/2