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Thurston’s pullback map on the augmented Teichmüller space and applications

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Abstract

Let f be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map σ f of a finite-dimensional Teichmüller space. We prove that this map extends continuously to the augmented Teichmüller space and give an explicit construction for this extension. This allows us to characterize the dynamics of Thurston’s pullback map near invariant strata of the boundary of the augmented Teichmüller space. The resulting classification of invariant boundary strata is used to prove a conjecture by Pilgrim and to infer further properties of Thurston’s pullback map. Our approach also yields new proofs of Thurston’s theorem and Pilgrim’s Canonical Obstruction theorem.

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References

  1. Abikoff, W.: Degenerating families of Riemann surfaces. Ann. Math. (2) 105(1), 29–44 (1977). 0442293 (56 #679)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahlfors, L., Bers, L.: Riemann’s mapping theorem for variable metrics. Ann. Math. (2) 72, 385–404 (1960). 0115006 (22 #5813)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bers, L.: On spaces of Riemann surfaces with nodes. Bull. Am. Math. Soc. 80, 1219–1222 (1974). 0361165 (50 #13611)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bers, L.: Spaces of degenerating Riemann surfaces. In: Discontinuous Groups and Riemann Surfaces, Proc. Conf., Univ. Maryland, College Park, Md., 1973. Ann. of Math. Studies, vol. 79, pp. 43–55. Princeton Univ. Press, Princeton (1974). 0361051 (50 #13497)

    Google Scholar 

  5. Buff, X., Epstein, A., Koch, S., Pilgrim, K.: On Thurston’s pullback map. In: Complex Dynamics, pp. 561–583. AK Peters, Wellesley (2009). 2508269 (2010g:37071)

    Chapter  Google Scholar 

  6. Buff, X., Guizhen, C., Lei, T.: Teichüller spaces and holomorphic dynamics. In: Handbook of Teichmüller Theory, Vol. III (to appear). http://www.math.univ-angers.fr/tanlei/papers/teich.pdf

  7. Douady, A., Hubbard, J.H.: A proof of Thurston’s topological characterization of rational functions. Acta Math. 171(2), 263–297 (1993). 1251582 (94j:58143)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hubbard, J.H.: Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, vol. 1. Matrix Editions, Ithaca (2006). Teichmüller theory, With contributions by A. Douady, W. Dunbar, R. Roeder, S. Bonnot, D. Brown, A. Hatcher, C. Hruska and S. Mitra, With forewords by W. Thurston and C. Earle. 2245223 (2008k:30055)

    MATH  Google Scholar 

  9. Imayoshi, Y., Taniguchi, M.: An Introduction to Teichmüller Spaces. Springer, Tokyo (1992). Translated and revised from the Japanese by the authors. 1215481 (94b:32031)

    Book  MATH  Google Scholar 

  10. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications, vol. 54. Cambridge University Press, Cambridge (1995). With a supplementary chapter by A. Katok and L. Mendoza. 1326374 (96c:58055)

    MATH  Google Scholar 

  11. Masur, H.: Extension of the Weil-Petersson metric to the boundary of Teichmüller space. Duke Math. J. 43(3), 623–635 (1976). 0417456 (54 #5506)

    Article  MathSciNet  MATH  Google Scholar 

  12. McMullen, C.T.: The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. Math. (2) 151(1), 327–357 (2000). 1745010 (2001m:32032)

    Article  MathSciNet  MATH  Google Scholar 

  13. Milnor, J.: Dynamics in One Complex Variable, 3rd edn. Annals of Mathematics Studies, vol. 160. Princeton University Press, Princeton (2006). 2193309 (2006g:37070)

    MATH  Google Scholar 

  14. Pilgrim, K.M.: Canonical Thurston obstructions. Adv. Math. 158(2), 154–168 (2001). 1822682 (2001m:57004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pilgrim, K.M.: Combinations of Complex Dynamical Systems. Lecture Notes in Mathematics, vol. 1827. Springer, Berlin (2003). 2020454 (2004m:37087)

    Book  MATH  Google Scholar 

  16. Selinger, N.: On the boundary behavior of Thurston’s pullback map. In: Complex Dynamics, pp. 585–595. AK Peters, Wellesley (2009). 2508270 (2010g:37068)

    Chapter  Google Scholar 

  17. Thurston, W.P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull., New Ser., Am. Math. Soc. 19(2), 417–431 (1988). 956596 (89k:57023)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wolpert, S.A.: Geometry of the Weil-Petersson completion of Teichmüller space. In: Surveys in Differential Geometry, Boston, MA, 2002. Surv. Differ. Geom., vol. VIII, pp. 357–393. Int. Press, Somerville (2003). 2039996 (2005h:32032)

    Google Scholar 

  19. Wolpert, S.A.: The Weil-Petersson metric geometry. In: Handbook of Teichmüller Theory, vol. II. IRMA Lect. Math. Theor. Phys., vol. 13, pp. 47–64. Eur. Math. Soc., Zürich (2009). 2497791 (2010i:32012)

    Chapter  Google Scholar 

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  1. Jacobs University Bremen, Campus Ring 1, 28759, Bremen, Germany

    Nikita Selinger

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  1. Nikita Selinger
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Selinger, N. Thurston’s pullback map on the augmented Teichmüller space and applications. Invent. math. 189, 111–142 (2012). https://doi.org/10.1007/s00222-011-0362-3

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