Pullback invariants of Thurston maps
HTML articles powered by AMS MathViewer
- by Sarah Koch, Kevin M. Pilgrim and Nikita Selinger PDF
- Trans. Amer. Math. Soc. 368 (2016), 4621-4655 Request permission
Abstract:
Associated to a Thurston map $f: S^2 \to S^2$ with postcritical set $P$ are several different invariants obtained via pullback: a relation $\mathcal {S}_{P} {\stackrel {f}{\longleftarrow }} \mathcal {S}_{P}$ on the set $\mathcal {S}_{P}$ of free homotopy classes of curves in $S^2\setminus P$, a linear operator $\lambda _f: \mathbb {R}[\mathcal {S}_{P}]\to \mathbb {R}[\mathcal {S}_{P}]$ on the free $\mathbb {R}$-module generated by $\mathcal {S}_{P}$, a virtual endomorphism $\phi _f: \mathrm {PMod}(S^2, P) \dashrightarrow \mathrm {PMod}(S^2, P)$ on the pure mapping class group, an analytic self-map $\sigma _f: \mathcal {T}(S^2, P) \to \mathcal {T}(S^2, P)$ of an associated Teichmüller space, and an analytic self-correspondence $X\circ Y^{-1}: \mathcal {M}(S^2, P) \rightrightarrows \mathcal {M}(S^2, P)$ of an associated moduli space. Viewing these associated maps as invariants of $f$, we investigate relationships between their properties.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
- Robert W. Bell and Dan Margalit, Injections of Artin groups, Comment. Math. Helv. 82 (2007), no. 4, 725–751. MR 2341838, DOI 10.4171/CMH/108
- Joan S. Birman, Alex Lubotzky, and John McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983), no. 4, 1107–1120. MR 726319, DOI 10.1215/S0012-7094-83-05046-9
- 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, Subdivision rules and virtual endomorphisms, Geom. Dedicata 141 (2009), 181–195. MR 2520071, DOI 10.1007/s10711-009-9352-7
- 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
- Albert Fathi, François Laudenbach, and Valentin Poénaru, Thurston’s work on surfaces, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ, 2012. Translated from the 1979 French original by Djun M. Kim and Dan Margalit. MR 3053012
- Peter Haïssinsky and Kevin M. Pilgrim, Finite type coarse expanding conformal dynamics, Groups Geom. Dyn. 5 (2011), no. 3, 603–661. MR 2813529, DOI 10.4171/GGD/141
- Peter Haïssinsky and Kevin M. Pilgrim, An algebraic characterization of expanding Thurston maps, J. Mod. Dyn. 6 (2012), no. 4, 451–476. MR 3008406, DOI 10.3934/jmd.2012.6.451
- John H. Hubbard and Sarah Koch, An analytic construction of the Deligne-Mumford compactification of the moduli space of curves, J. Differential Geom. 98 (2014), no. 2, 261–313. MR 3263519
- 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
- Nikolai V. Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, vol. 115, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by E. J. F. Primrose and revised by the author. MR 1195787, DOI 10.1090/mmono/115
- Nikolai V. Ivanov, Mapping class groups, http://www.math.msu.edu/~ivanov/m99.ps, 1998.
- Atsushi Kameyama, The Thurston equivalence for postcritically finite branched coverings, Osaka J. Math. 38 (2001), no. 3, 565–610. MR 1860841
- Gregory A. Kelsey, Mapping schemes realizable by obstructed topological polynomials, Conform. Geom. Dyn. 16 (2012), 44–80. MR 2893472, DOI 10.1090/S1088-4173-2012-00239-1
- Sarah Koch, Teichmüller theory and endomorphisms of $\mathbb {P}^n$, PhD thesis, University of Provence, 2007.
- Sarah Koch, Teichmüller theory and critically finite endomorphisms, Adv. Math. 248 (2013), 573–617. MR 3107522, DOI 10.1016/j.aim.2013.08.019
- Sarah Koch, Kevin M. Pilgrim, and Nikita Selinger, Limit sets of Thurston pullback semigroups, Manuscript, 2014.
- Russell Lodge, Boundary values of the Thurston pullback map, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)–Indiana University. MR 3054977
- Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623–635. MR 417456
- Volodymyr Nekrashevych, Combinatorics of polynomial iterations, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 169–214. MR 2508257, DOI 10.1201/b10617-5
- Volodymyr Nekrashevych, Combinatorial models of expanding dynamical systems, Ergodic Theory Dynam. Systems 34 (2014), no. 3, 938–985. MR 3199801, DOI 10.1017/etds.2012.163
- Kevin M. Pilgrim, Canonical Thurston obstructions, Adv. Math. 158 (2001), no. 2, 154–168. MR 1822682, DOI 10.1006/aima.2000.1971
- Kevin M. Pilgrim, Combination, decomposition, and structure theory for postcritically finite branched coverings of the two-sphere to itself, to appear, Springer Lecture Notes in Math., 2003.
- Kevin M. Pilgrim, An algebraic formulation of Thurston’s characterization of rational functions, Ann. Fac. Sci. Toulouse Math. (6) 21 (2012), no. 5, 1033–1068 (English, with English and French summaries). MR 3088266, DOI 10.5802/afst.1361
- 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
- Nikita Selinger, Topological characterization of canonical Thurston obstructions, J. Mod. Dyn. 7 (2013), no. 1, 99–117. MR 3071467, DOI 10.3934/jmd.2013.7.99
- 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
Additional Information
- Sarah Koch
- Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
- MR Author ID: 874529
- Kevin M. Pilgrim
- Affiliation: Department of Mathematics, Indiana University, 831 E. Third Street, Bloomington, Indiana 47405
- MR Author ID: 614176
- Nikita Selinger
- Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660
- MR Author ID: 874467
- Received by editor(s): April 2, 2013
- Received by editor(s) in revised form: May 10, 2014
- Published electronically: September 24, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 4621-4655
- MSC (2010): Primary 30F60, 32G15, 37F20
- DOI: https://doi.org/10.1090/tran/6482
- MathSciNet review: 3456156