Skip to main content
SpringerLink
Log in

On the Pullback Relation on Curves Induced by a Thurston Map

  • Chapter
  • First Online:
In the Tradition of Thurston II

Abstract

Via taking connected components of preimages, a Thurston map f : (S2, Pf) → (S2, Pf) induces a pullback relation on the set of isotopy classes of curves in the complement of its postcritical set Pf. We survey known results about the dynamics of this relation, and pose some questions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. L. Bartholdi, D. Dudko, Algorithmic aspects of branched coverings II/V: sphere bisets and their decompositions (2016). https://arxiv.org/pdf/1603.04059.pdf

  2. L. Bartholdi, D. Dudko, Algorithmic aspects of branched coverings IV/V. Expanding maps. Trans. Am. Math. Soc. 370, 7679–7714 (2018)

    Article  MathSciNet  Google Scholar 

  3. J. Belk, J. Lanier, D. Margalit, R.R.Winarski, Recognizing topological polynomials by lifting trees (2019). https://arxiv.org/pdf/1906.07680.pdf

  4. I. Berstein, A.L. Edmonds, On the construction of branched coverings of low-dimensional manifolds. Trans. Am. Math. Soc. 247, 87–124 (1979)

    Article  MathSciNet  Google Scholar 

  5. M. Bonk, D. Meyer, Expanding Thurston Maps. Mathematical Surveys and Monographs, vol. 225 (American Mathematical Society, Providence, 2017)

    Google Scholar 

  6. M. Bonk, M. Hlushchanka, A. Iseli, Eliminating Thurston obstructions and controlling dynamics on curves (2021). arXiv:2105.06938

    Google Scholar 

  7. X. Buff, T.Gauthier, Perturbations of flexible Lattès maps. Bull. Soc. Math. France 141, 603–614 (2013)

    Article  MathSciNet  Google Scholar 

  8. X. Buff, A. Epstein, S. Koch, K. Pilgrim, On Thurston’s pullback map, in Complex Dynamics (A K Peters, Wellesley, 2009), pp. 561–583

    Google Scholar 

  9. X. Buff, A.L. Epstein, S. Koch, Böttcher coordinates. Indiana Univ. Math. J. 61, 1765–1799 (2012)

    Article  MathSciNet  Google Scholar 

  10. J.W. Cannon, W.J. Floyd, W.R. Parry, Finite subdivision rules. Conform. Geom. Dyn. 5, 153–196 (2001)

    Article  MathSciNet  Google Scholar 

  11. J.W. Cannon, W.J. Floyd, W.R. Parry, K.M. Pilgrim, Nearly Euclidean Thurston maps. Conform. Geom. Dyn. 16, 209–255 (2012)

    Article  MathSciNet  Google Scholar 

  12. G. Cui, Rational maps with (constant) pullback map. http://www.math.ac.cn/kyry/cgz/201501/W020150121410977178324.pdf

  13. G. Cui, W. Peng, L. Tan, Renormalizations and wandering Jordan curves of rational maps. Commun. Math. Phys. 344, 67–115 (2016)

    Article  MathSciNet  Google Scholar 

  14. A. Douady, J. Hubbard, A proof of Thurston’s topological characterization of rational functions. Acta. Math. 171, 263–297 (1993)

    Article  MathSciNet  Google Scholar 

  15. A. Fathi, F. Laudenbach, V. Poénaru, Thurston’s Work on Surfaces. Mathematical Notes,vol. 48 (Princeton University Press, Princeton, 2012). Translated from the 1979 French original by Djun M. Kim and Dan Margalit

    Google Scholar 

  16. W. Floyd, G. Kelsey, S. Koch, R. Lodge, W. Parry, K.M. Pilgrim, E. Saenz, Origami, affine maps, and complex dynamics. Arnold Math. J. 3, 365–395 (2016)

    Article  MathSciNet  Google Scholar 

  17. W. Floyd, W. Parry, K.M. Pilgrim, Rationality is decidable for nearly Euclidean Thurston maps. Geom. Dedicata 213, 487–512 (2021). https://doi.org/10.1007/s10711-020-00593-9. https://mathscinet.ams.org/mathscinet/search/publications.html?fmt=bibtex&pg1=MR&s1=4278340

  18. W.J. Harvey, Boundary structure of the modular group, in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, 1978). Annals of Mathematics Studies, vol. 97 (Princeton University Press, Princeton, 1981), pp. 245–251

    Google Scholar 

  19. M. Hlushchanka, Tischler graphs of critically fixed rational maps and their applications (2019). https://arxiv.org/pdf/1904.04759.pdf

  20. G. Kelsey, R. Lodge, Quadratic Thurston maps with few postcritical points (2017). https://arxiv.org/abs/1704.03929

  21. S. Koch, K.M. Pilgrim, N. Selinger, Pullback invariants of Thurston maps. Trans. Am. Math. Soc. 368, 4621–4655 (2016)

    Article  MathSciNet  Google Scholar 

  22. R. Lodge, Boundary values of the Thurston pullback map. Conform. Geom. Dyn. 17, 77–118 (2013)

    Article  MathSciNet  Google Scholar 

  23. E.A.S. Maldonado, On nearly Euclidean thurston maps. PhD. Thesis, Virginia Polytechnic University (2012)

    Google Scholar 

  24. H. Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space. Duke Math. J. 43, 623–635 (1976)

    Article  MathSciNet  Google Scholar 

  25. V. Nekrashevych, Combinatorial models of expanding dynamical systems. Ergodic Theory Dyn. Syst. 34, 938–985 (2014)

    Article  MathSciNet  Google Scholar 

  26. W. Parry, NET map slope functions (2018). arXiv:1811.01274

    Google Scholar 

  27. K.M. Pilgrim, Canonical Thurston obstructions. Adv. Math. 158, 154–168 (2001)

    Article  MathSciNet  Google Scholar 

  28. K.M. Pilgrim, Combinations of Complex Dynamical Systems. Lecture Notes in Mathematics, vol. 1827 (Springer, Berlin, 2003)

    Google Scholar 

  29. K.M. Pilgrim, An algebraic formulation of Thurston’s characterization of rational functions. Ann. Fac. Sci. Toulouse Math. 21, 1033–1068 (2012)

    Article  MathSciNet  Google Scholar 

  30. K.M. Pilgrim, L. Tan, Combining rational maps and controlling obstructions. Ergodic Theory Dyn. Syst. 18, 221–245 (1998)

    Article  MathSciNet  Google Scholar 

  31. R. Ramadas, Algebraic stability of meromorphic maps descended from Thurston’s pullback maps. Trans. Am. Math. Soc. 374, 565–587 (2021)

    Article  MathSciNet  Google Scholar 

  32. N. Selinger, Thurston’s pullback map on the augmented Teichmüller space and applications. Invent. Math. 189, 111–142 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially supported by Simons Foundation collaboration grants 4429419 and 615022.

Author information

Authors and Affiliations

  1. Indiana University, Bloomington, IN, USA

    Kevin M. Pilgrim

Authors
  1. Kevin M. Pilgrim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kevin M. Pilgrim .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Gakushuin University, Tokyo, Japan

    Ken’ichi Ohshika

  2. Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, Strasbourg, France

    Athanase Papadopoulos

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Pilgrim, K.M. (2022). On the Pullback Relation on Curves Induced by a Thurston Map. In: Ohshika, K., Papadopoulos, A. (eds) In the Tradition of Thurston II. Springer, Cham. https://doi.org/10.1007/978-3-030-97560-9_11

Download citation

Publish with us

Policies and ethics

Search

Navigation