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Conformal Geometry and Dynamics

ISSN 1088-4173



Mapping schemes realizable by obstructed topological polynomials

Author: Gregory A. Kelsey
Journal: Conform. Geom. Dyn. 16 (2012), 44-80
MSC (2010): Primary 37F20; Secondary 20F65
Published electronically: March 13, 2012
MathSciNet review: 2893472
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Abstract: In 1985, Levy used a theorem of Berstein to prove that all hyperbolic topological polynomials are equivalent to complex polynomials. We prove a partial converse to the Berstein-Levy Theorem: given post-critical dynamics that are in a sense strongly non-hyperbolic, we prove the existence of topological polynomials which are not equivalent to any complex polynomial that realize these post-critical dynamics. This proof employs the theory of self-similar groups to demonstrate that a topological polynomial admits an obstruction and produces a wealth of examples of obstructed topological polynomials.

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  • [BG00] Laurent Bartholdi and Rostislav Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups. Proc. Steklov Inst. Math. 231 (2000), no. 4, 1-41. MR 1841750 (2002d:37017)
  • [BN06] Laurent Bartholdi and Volodymyr Nekrashevych, Thurston Equivalence of Topological Polynomials. Acta Math. 197 (2006), no. 1, 1-51. MR 2285317 (2008c:37072)
  • [BV05] Laurent Bartholdi and Bálint Virág, Amenability via random walks. Duke Math. J. 130 (2005), no. 1, 39-56. MR 2176547 (2006h:43001)
  • [Ber73] Claude Berge, Graphs and Hypergraphs. North-Holland Mathematical Library 6, North-Holland Publishing Company, Amsterdam, 1973. MR 0357172 (50:9640)
  • [BFH92] Benjamin 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 (93h:58128)
  • [Bla84] Paul Blanchard, Complex Analytic Dynamics. Bull. Amer. Math. Soc. 11 (1984), no. 1, 85-141. MR 0741725 (85h:58001)
  • [BK02] Mario Bonk and Bruce Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150 (2002), no. 1, 127-183. MR 1930885 (2004k:53057)
  • [BM] Mario Bonk and Daniel Meyer, Expanding Thurston maps. arXiv:1009.3647.
  • [B+00] E. Brezin, R. Bryne, J. Levy, K. Pilgrim, and K. Plummer, A Census of Rational Maps. Conform. Geom. Dyn. 4 (2000), 35-74. MR 1749249 (2001d:37052)
  • [BS02] Henk Bruin and Dierk Schleicher, ``Symbolic dynamics of quadratic polynomials''. Institut Mittag-Leffler, Report No. 7, 2001/2002.
  • [BP06] Kai-Uwe Bux and Rodrigo Pérez, On the growth of iterated monodromy groups. in Topological and asymptotic aspects of group theory, Contemp. Math. 394, American Mathematical Society, Providence, RI, 2006. MR 2216706 (2006m:20062)
  • [DH82] Adrien Douady and John Hubbard, Itération des polynômes quadratiques complexes. C. R. de Acad. Sci. Paris, Sér. I Math. 294 (1982), no. 3, 123-126. MR 0651802 (83m:58046)
  • [DH84] Adrien Douady and John Hubbard, Etude dynamique des polynomes complexes, I, II. Publications Mathematiques d'Orsay 84-2, 85-4, Université de Paris-Sud, Départment de Mathématiques, Orsay, 1984, 1985. MR 0762431 (87f:58072a); MR 0812271 (87f:58072b)
  • [DH93] Adrien Douady and John Hubbard, A Proof of Thurston's Characterization of Rational Functions. Acta Math. 171 (1993), no. 2, 263-297. MR 1251582 (94j:58143)
  • [FG91] Jacek Fabrykowski and Narain Gupta, On groups with sub-exponential growth functions, II. J. Indian Math. Soc. (N.S.) 56 (1991), no. 1-4, 217-228. MR 1153150 (93g:20053)
  • [Fat19] Pierre Fatou, Sur les equations fonctionnelles. Bull. Soc. Math. France 47 (1919), 161-271. MR 1504787.
  • [Fat20] Pierre Fatou, Sur les equations fonctionnelles. Bull. Soc. Math. France 48 (1920), 33-94, 208-314. MR 1504792; MR 1504797.
  • [Gri80] Rostislav Grigorchuk, On Burnside's problem on periodic groups. Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 53-54. MR 565099 (81m:20045)
  • [Gri84] Rostislav Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 5, 939-985. MR 0764305 (86h:20041)
  • [GŻ02a] Rostislav Grigorchuk and Andrzej Żuk, On a torsion-free weakly branch group defined by a three state automaton. Internat. J. Algebra Comput. 12 (2001), no. 1-2, 223-246. MR 1902367 (2003c:20048)
  • [GŻ02b] Rostislav Grigorchuk and Andrzej Żuk, Spectral properties of a torsion-free weakly branch group defined by a three state automaton. in Computational and statistical group theory, Contemp. Math. 298, Amer. Math. Soc., Providence, RI, 2002. MR 1929716 (2003h:60011)
  • [HP09] Peter Haïssinsky and Kevin M. Pilgrim. Coarse expanding conformal dynamics. Astérisque, 325, 2009. MR 2662902
  • [Jul18] Gaston Julia, Memoire sur l'itération des fonctions rationnelles. J. Math. Pures Appl. 8 (1918), 47-245.
  • [Kam01] Atsushi Kameyama, The Thurston equivalence for postcritically finite branched coverings. Osaka J. Math. 38 (2001), no. 3, 565-610. MR 1860841 (2002h:57004)
  • [Koc07] Sarah Koch, Teichmüller Theory and Endomorphisms of $ \mathbb{P}^n$. Ph.D. thesis, Université de Provence, 2007.
  • [Koch] Sarah Koch. Private communication, 7 August, 2009.
  • [Lev85] Silvio Levy, Critically Finite Rational Maps. Ph.D. thesis, Princeton University, 1985.
  • [Man82] Benoit Mandelbrot, The fractal geometry of nature. W.H. Freeman and Co., San Francisco, 1982. MR 0665254 (84h:00021)
  • [McM91] Curtis T. McMullen, Rational maps and Kleinian groups. in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991. MR 1159274 (93h:57024)
  • [McM94] Curtis T. McMullen, The classification of conformal dynamical systems. in Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994. MR 1474980 (98h:58162)
  • [Nek05] Volodymyr Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs, 117, Amer. Math. Soc., Providence, RI, 2005. MR 2162164 (2006e:20047)
  • [Nek09] Volodymyr Nekrashevych, Combinatorics of Polynomial Iterations in Complex Dynamics: Families and Friends, D. Schleicher, ed., A K Peters, Wellesley, MA, 2009. MR 2508257
  • [Pil00] Kevin M. Pilgrim, Dessins d'enfants and Hubbard trees. Ann. Sci. École Norm. Sup. 33 (2000), no. 5, 671-693. MR 1834499 (2002m:37062)
  • [Pil01] Kevin M. Pilgrim, Canonical Thurston obstructions. Adv. Math. 158 (2001), no. 2, 154-168. MR 1822682 (2001m:57004)
  • [Pil03a] Kevin M. Pilgrim, Combinations of Complex Dynamical Systems, Lecture Notes in Mathematics 1827, Springer, 2003. MR 2020454 (2004m:37087)
  • [Pil03b] Kevin M. Pilgrim, An algebraic formulation of Thurston's combinatorial equivalence Proc. Amer. Math. Soc. 131 (2003), no. 11, 3527-3534. MR 1991765 (2005g:37087)
  • [Poi09] Alfredo Poirier, Critical portraits for post-critically finite polynomials. Fund. Math. 203 (2009), no. 2, 107-163. MR 2496235 (2010c:37095)
  • [Ree86] Mary Rees, Realization of matings of polynomials as rational maps of degree $ 2$. 1986.
  • [Sel] Nikita Selinger, Thurston's pullback map on the augmented Teichmüller space and applications. arXiv:1010.1690.
  • [Shi00] Mitsuhiro Shishikura, On a theorem of M. Rees for mating of polynomials in The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser. 274, Cambridge University Press, Cambridge, 2000. MR 1765095 (2002d:37072)
  • [Tan92] Lei Tan, Matings of quadratic polynomials. Ergodic Theory Dynam. Systems 12 (1992), no. 3, 589-620. MR 1182664 (93h:58129)

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Additional Information

Gregory A. Kelsey
Affiliation: Department of Mathematics, Computing Sciences, and Physics, Immaculata University, P.O. Box 648, Immaculata, Pennsylvania 19345

Keywords: Combinatorics of complex dynamics, self-similar groups
Received by editor(s): January 27, 2011
Received by editor(s) in revised form: July 26, 2011
Published electronically: March 13, 2012
Additional Notes: The author acknowledges support from National Science Foundation grant DMS 08-38434 “EMSW21-MCTP: Research Experience for Graduate Students”.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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