# American Institute of Mathematical Sciences

January  2013, 7(1): 99-117. doi: 10.3934/jmd.2013.7.99

## Topological characterization of canonical Thurston obstructions

 1 Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY 11794-3660, United States

Received  August 2012 Published  May 2013

Let $f$ be an obstructed Thurston map with canonical obstruction $\Gamma_f$. We prove the following generalization of Pilgrim's conjecture: if the first-return map $F$ of a periodic component $C$ of the topological surface obtained from the sphere by pinching the curves of $\Gamma_f$ is a Thurston map then the canonical obstruction of $F$ is empty. Using this result, we give a complete topological characterization of canonical Thurston obstructions.
Citation: Nikita Selinger. Topological characterization of canonical Thurston obstructions. Journal of Modern Dynamics, 2013, 7 (1) : 99-117. doi: 10.3934/jmd.2013.7.99
##### References:
 [1] S. Bonnot, M. Braverman and M. Yampolsky, Thurston equivalence to a rational map is decidable, to appear in Moscow Math. J., (2010). [2] A. Chéritat, Tan Lei and Shishikura's example of non-mateable degree 3 polynomials without a Levy cycle, to appear in Annales de la faculté des sciences de Toulouse, (2012). [3] A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297. doi: 10.1007/BF02392534. [4] F. R. Gantmacher, "Teoriya Matrits," Second supplemented edition, With an appendix by V. B. Lidskiĭ, Izdat. "Nauka," Moscow, 1966. [5] J. H. Hubbard, "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 1. 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, Matrix Editions, Ithaca, NY, 2006. [6] Y. Imayoshi and M. Taniguchi, "An Introduction to Teichmüller Spaces," Translated and revised from the Japanese by the authors, Springer-Verlag, Tokyo, 1992. doi: 10.1007/978-4-431-68174-8. [7] O. Lehto and K. I. Virtanen, "Quasiconformal Mappings in the Plane," Second edition, Translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York, 1973. [8] C. T. McMullen, "Complex Dynamics and Renormalization," Annals of Mathematics Studies, 135, Princeton University Press, Princeton, NJ, 1994. [9] J. Milnor, "Dynamics in One Complex Variable," Third edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006. [10] _____, On Lattès maps, in "Dynamics on the Riemann Sphere," Eur. Math. Soc., Zürich, (2006), 9-43. [11] K. M. Pilgrim, Canonical Thurston obstructions, Adv. Math., 158 (2001), 154-168. doi: 10.1006/aima.2000.1971. [12] _____, "Combinations of Complex Dynamical Systems," Lecture Notes in Mathematics, 1827, Springer-Verlag, Berlin, 2003. [13] N. Selinger, "On Thurston's Characterization Theorem for Branched Covers," Ph.D Thesis, 2011. [14] _____, Thurston's pullback map on the augmented Teichmüller space and applications, Inventiones Mathematicae, 189 (2012), 111-142. [15] S. A. Wolpert, Geometry of the Weil-Petersson completion of Teichmüller space, in "Surveys in Differential Geometry, Vol. VIII" (Boston, MA, 2002), Surv. Differ. Geom., VIII, Int. Press, Somerville, MA, (2003), 357-393. [16] _____, The Weil-Petersson metric geometry, in "Handbook of Teichmüller Theory, Vol. II," IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, (2009), 47-64.

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##### References:
 [1] S. Bonnot, M. Braverman and M. Yampolsky, Thurston equivalence to a rational map is decidable, to appear in Moscow Math. J., (2010). [2] A. Chéritat, Tan Lei and Shishikura's example of non-mateable degree 3 polynomials without a Levy cycle, to appear in Annales de la faculté des sciences de Toulouse, (2012). [3] A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297. doi: 10.1007/BF02392534. [4] F. R. Gantmacher, "Teoriya Matrits," Second supplemented edition, With an appendix by V. B. Lidskiĭ, Izdat. "Nauka," Moscow, 1966. [5] J. H. Hubbard, "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 1. 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, Matrix Editions, Ithaca, NY, 2006. [6] Y. Imayoshi and M. Taniguchi, "An Introduction to Teichmüller Spaces," Translated and revised from the Japanese by the authors, Springer-Verlag, Tokyo, 1992. doi: 10.1007/978-4-431-68174-8. [7] O. Lehto and K. I. Virtanen, "Quasiconformal Mappings in the Plane," Second edition, Translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York, 1973. [8] C. T. McMullen, "Complex Dynamics and Renormalization," Annals of Mathematics Studies, 135, Princeton University Press, Princeton, NJ, 1994. [9] J. Milnor, "Dynamics in One Complex Variable," Third edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006. [10] _____, On Lattès maps, in "Dynamics on the Riemann Sphere," Eur. Math. Soc., Zürich, (2006), 9-43. [11] K. M. Pilgrim, Canonical Thurston obstructions, Adv. Math., 158 (2001), 154-168. doi: 10.1006/aima.2000.1971. [12] _____, "Combinations of Complex Dynamical Systems," Lecture Notes in Mathematics, 1827, Springer-Verlag, Berlin, 2003. [13] N. Selinger, "On Thurston's Characterization Theorem for Branched Covers," Ph.D Thesis, 2011. [14] _____, Thurston's pullback map on the augmented Teichmüller space and applications, Inventiones Mathematicae, 189 (2012), 111-142. [15] S. A. Wolpert, Geometry of the Weil-Petersson completion of Teichmüller space, in "Surveys in Differential Geometry, Vol. VIII" (Boston, MA, 2002), Surv. Differ. Geom., VIII, Int. Press, Somerville, MA, (2003), 357-393. [16] _____, The Weil-Petersson metric geometry, in "Handbook of Teichmüller Theory, Vol. II," IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, (2009), 47-64.
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