# 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. doi: 10.1007/BF02392534. [4] F. R. Gantmacher, "Teoriya Matrits,", Second supplemented edition, (1966). [5] J. H. Hubbard, "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 1. Teichmüller Theory,", With contributions by Adrien Douady, (2006). [6] Y. Imayoshi and M. Taniguchi, "An Introduction to Teichmüller Spaces,", Translated and revised from the Japanese by the authors, (1992). doi: 10.1007/978-4-431-68174-8. [7] O. Lehto and K. I. Virtanen, "Quasiconformal Mappings in the Plane,", Second edition, (1973). [8] C. T. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Studies, 135 (1994). [9] J. Milnor, "Dynamics in One Complex Variable,", Third edition, 160 (2006). [10] _____, On Lattès maps,, in, (2006), 9. [11] K. M. Pilgrim, Canonical Thurston obstructions,, Adv. Math., 158 (2001), 154. doi: 10.1006/aima.2000.1971. [12] _____, "Combinations of Complex Dynamical Systems,", Lecture Notes in Mathematics, 1827 (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. [15] S. A. Wolpert, Geometry of the Weil-Petersson completion of Teichmüller space,, in, VIII (2003), 357. [16] _____, The Weil-Petersson metric geometry,, in, 13 (2009), 47.

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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. doi: 10.1007/BF02392534. [4] F. R. Gantmacher, "Teoriya Matrits,", Second supplemented edition, (1966). [5] J. H. Hubbard, "Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 1. Teichmüller Theory,", With contributions by Adrien Douady, (2006). [6] Y. Imayoshi and M. Taniguchi, "An Introduction to Teichmüller Spaces,", Translated and revised from the Japanese by the authors, (1992). doi: 10.1007/978-4-431-68174-8. [7] O. Lehto and K. I. Virtanen, "Quasiconformal Mappings in the Plane,", Second edition, (1973). [8] C. T. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Studies, 135 (1994). [9] J. Milnor, "Dynamics in One Complex Variable,", Third edition, 160 (2006). [10] _____, On Lattès maps,, in, (2006), 9. [11] K. M. Pilgrim, Canonical Thurston obstructions,, Adv. Math., 158 (2001), 154. doi: 10.1006/aima.2000.1971. [12] _____, "Combinations of Complex Dynamical Systems,", Lecture Notes in Mathematics, 1827 (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. [15] S. A. Wolpert, Geometry of the Weil-Petersson completion of Teichmüller space,, in, VIII (2003), 357. [16] _____, The Weil-Petersson metric geometry,, in, 13 (2009), 47.
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