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). Google Scholar

[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). Google Scholar

[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.  Google Scholar

[4]

F. R. Gantmacher, "Teoriya Matrits," Second supplemented edition, With an appendix by V. B. Lidskiĭ, Izdat. "Nauka," Moscow, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. T. McMullen, "Complex Dynamics and Renormalization," Annals of Mathematics Studies, 135, Princeton University Press, Princeton, NJ, 1994.  Google Scholar

[9]

J. Milnor, "Dynamics in One Complex Variable," Third edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[10]

_____, On Lattès maps, in "Dynamics on the Riemann Sphere," Eur. Math. Soc., Zürich, (2006), 9-43.  Google Scholar

[11]

K. M. Pilgrim, Canonical Thurston obstructions, Adv. Math., 158 (2001), 154-168. doi: 10.1006/aima.2000.1971.  Google Scholar

[12]

_____, "Combinations of Complex Dynamical Systems," Lecture Notes in Mathematics, 1827, Springer-Verlag, Berlin, 2003.  Google Scholar

[13]

N. Selinger, "On Thurston's Characterization Theorem for Branched Covers," Ph.D Thesis, 2011. Google Scholar

[14]

_____, Thurston's pullback map on the augmented Teichmüller space and applications, Inventiones Mathematicae, 189 (2012), 111-142.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. Bonnot, M. Braverman and M. Yampolsky, Thurston equivalence to a rational map is decidable, to appear in Moscow Math. J., (2010). Google Scholar

[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). Google Scholar

[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.  Google Scholar

[4]

F. R. Gantmacher, "Teoriya Matrits," Second supplemented edition, With an appendix by V. B. Lidskiĭ, Izdat. "Nauka," Moscow, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. T. McMullen, "Complex Dynamics and Renormalization," Annals of Mathematics Studies, 135, Princeton University Press, Princeton, NJ, 1994.  Google Scholar

[9]

J. Milnor, "Dynamics in One Complex Variable," Third edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[10]

_____, On Lattès maps, in "Dynamics on the Riemann Sphere," Eur. Math. Soc., Zürich, (2006), 9-43.  Google Scholar

[11]

K. M. Pilgrim, Canonical Thurston obstructions, Adv. Math., 158 (2001), 154-168. doi: 10.1006/aima.2000.1971.  Google Scholar

[12]

_____, "Combinations of Complex Dynamical Systems," Lecture Notes in Mathematics, 1827, Springer-Verlag, Berlin, 2003.  Google Scholar

[13]

N. Selinger, "On Thurston's Characterization Theorem for Branched Covers," Ph.D Thesis, 2011. Google Scholar

[14]

_____, Thurston's pullback map on the augmented Teichmüller space and applications, Inventiones Mathematicae, 189 (2012), 111-142.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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