Science China Mathematics

, Volume 61, Issue 12, pp 2237–2266 | Cite as

Expansion properties for finite subdivision rules I

  • William J. Floyd
  • Walter R. Parry
  • Kevin M. Pilgrim


Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate relationships between various notions of expansion—combinatorial, dynamical, algebraic, and coarse-geometric.


finite subdivision rule expanding map postcritically finite Thurston map 


37F10 52C20 57M12 



Kevin M. Pilgrim was supported by Simons Foundation Collaboration Grants for Mathematicians (Grant No. 245269). The authors express heartfelt gratitude to Jim Cannon, who helped them in many ways. Without him this paper would not exist.


  1. 1.
    Bielefeld B, Fisher Y, Hubbard J. The classification of critically preperiodic polynomials as dynamical systems. J Amer Math Soc, 1992, 5: 721–762MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bonk M, Kleiner B. Quasisymmetric parametrizations of two-dimensional metric spheres. Invent Math, 2002, 150: 127–183MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonk M, Meyer D. Expanding Thurston Maps. Mathematical Surveys and Monographs, vol. 225. Providence: Amer Math Soc, 2017CrossRefzbMATHGoogle Scholar
  4. 4.
    Cannon J W. The combinatorial Riemann mapping theorem. Acta Math, 1994, 173: 155–234MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cannon J W, Floyd W J, Kenyon R, et al. Constructing rational maps from subdivision rules. Conform Geom Dyn, 2003, 7: 76–102MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cannon J W, Floyd W J, Parry W R. Conformal modulus: The graph paper invariant or the conformal shape of an algorithm. In: Geometric Group Theory Down Under. Berlin: de Gruyter, 1999, 71–102Google Scholar
  7. 7.
    Cannon J W, Floyd W J, Parry W R. Finite subdivision rules. Conform Geom Dyn, 2001, 5: 153–196MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cannon J W, Floyd W J, Parry W R. Expansion complexes for finite subdivision rules I. Conform Geom Dyn, 2006, 10: 63–99MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cannon J W, Floyd W J, Parry W R. Expansion complexes for finite subdivision rules II. Conform Geom Dyn, 2006, 10: 326–354MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cannon J W, Floyd W J, Parry W R. Constructing subdivision rules from rational maps. Conform Geom Dyn, 2007, 11: 128–136MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cannon J W, Floyd W J, Parry W R, et al. Subdivision rules and virtual endomorphisms. Geom Dedicata, 2009, 141: 181–195MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Douady A, Hubbard J H. A proof of Thurston's topological characterization of rational functions. Acta Math, 1993, 171: 263–297MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ghys E, de la Harpe P. Sur les groupes hyperboliques d'après Mikhael Gromov. Progress in Mathematics, vol. 83. Boston: Birkhäuser, 1990Google Scholar
  14. 14.
    Haïssinsky P, Pilgrim K M. Coarse Expanding Conformal Dynamics. Paris: Soc Math France, 2009zbMATHGoogle Scholar
  15. 15.
    Haïssinsky P, Pilgrim K M. An algebraic characterization of expanding Thurston maps. J Mod Dyn, 2012, 6: 451–476MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kaimanovich V. Random walks on Sierpiński graphs. In: Fractals in Graz 2001: Analysis Dynamics Geometry Stochastics. Basel-Boston-Berlin: Birkhäuser, 2003, 145–183CrossRefGoogle Scholar
  17. 17.
    Levy S. Critically finite rational maps. PhD Thesis. Princeton: Princeton University, 1985Google Scholar
  18. 18.
    Lyndon R C, Schupp P E. Combinatorial Group Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 89. Berlin: Springer, 1977Google Scholar
  19. 19.
    Milnor J, Thurston W. On iterated maps of the interval. In: Dynamical Systems. Lecture Notes in Mathematics, vol. 1342. Berlin-Heidelberg: Springer, 1988, 465–563MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nekrashevych V. Self-Similar Groups. Mathematical Surveys and Monographs, vol. 117. Providence: Amer Math Soc, 2005Google Scholar
  21. 21.
    Pilgrim K M. Julia sets as Gromov boundaries following V. Nekrashevych. Topology Proc, 2005, 29: 293–316MathSciNetzbMATHGoogle Scholar
  22. 22.
    Rushton B. Classification of subdivision rules for geometric groups of low dimension. Conform Geom Dyn, 2014, 18: 171–191MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Stephenson K. Circlepack, software. Http://
  24. 24.
    Thurston W P. On the geometry and dynamics of iterated rational maps. In: Complex Dynamics. Wellesley: A K Peters, 2009, 3–137CrossRefGoogle Scholar
  25. 25.
    Whyburn G T. Topological characterization of the Sierpiński curve. Fund Math, 1958, 45: 320–324MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science in China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • William J. Floyd
    • 1
  • Walter R. Parry
    • 2
  • Kevin M. Pilgrim
    • 3
  1. 1.Department of MathematicsVirginia TechBlacksburgUSA
  2. 2.Department of Mathematics and StatisticsEastern Michigan UniversityYpsilantiUSA
  3. 3.Department of MathematicsIndiana UniversityBloomingtonUSA

Personalised recommendations