Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Constructing rational maps from subdivision rules

Authors: J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry
Journal: Conform. Geom. Dyn. 7 (2003), 76-102
MSC (2000): Primary 37F10, 52C20; Secondary 57M12
Published electronically: July 28, 2003
MathSciNet review: 1992038
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose $\mathcal{R}$ is an orientation-preserving finite subdivision rule with an edge pairing. Then the subdivision map $\sigma_{\mathcal{R}}$ is either a homeomorphism, a covering of a torus, or a critically finite branched covering of a 2-sphere. If $\mathcal{R}$ has mesh approaching $0$ and $S_{\mathcal{R}}$ is a 2-sphere, it is proved in Theorem 3.1 that if $\mathcal{R}$ is conformal, then $\sigma_{\mathcal{R}}$ is realizable by a rational map. Furthermore, a general construction is given which, starting with a one-tile rotationally invariant finite subdivision rule, produces a finite subdivision rule $\mathcal{Q}$ with an edge pairing such that $\sigma_{\mathcal{Q}}$ is realizable by a rational map.

References [Enhancements On Off] (What's this?)

  • 1. P. L. Bowers and K. Stephenson, A ``regular'' pentagonal tiling of the plane, Conform. Geom. Dyn. 1 (1997), 58-68 (electronic). MR 99d:52016
  • 2. E. Brezin, R. Byrne, J. Levy, K. Pilgrim, and K. Plummer, A census of rational maps, Conform. Geom. Dyn. 4 (2000), 35-74 (electronic). MR 2001d:37052
  • 3. H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144. MR 33:2805
  • 4. J. W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), 155-234. MR 95k:30046
  • 5. J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension $3$, Trans. Amer. Math. Soc. 350 (1998), 809-849. MR 98i:57023
  • 6. J. W. Cannon, W. J. Floyd, and W. R. Parry, Squaring rectangles: the finite Riemann mapping theorem, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), Contemp. Math., 169, Amer. Math. Soc., Providence, RI, 1994, pp. 133-212. MR 95g:20045
  • 7. J. W. Cannon, W. J. Floyd, and W. R. Parry, Sufficiently rich families of planar rings, Ann. Acad. Sci. Fenn. Math. 24 (1999), 265-304. MR 2000k:20057
  • 8. J. W. Cannon, W. J. Floyd, and W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153-196 (electronic). MR 2002j:52021
  • 9. J. W. Cannon, W. J. Floyd, and W. R. Parry, Expansion complexes for finite subdivision rules I, preprint.
  • 10. J. W. Cannon, W. J. Floyd, and W. R. Parry, Expansion complexes for finite subdivision rules II, in preparation.
  • 11. A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993), 263-297. MR 94j:58143
  • 12. L. R. Ford, Automorphic Functions, Mc-Graw Hill, New York, 1929.
  • 13. T. Kuusalo, Verallgemeinerter Riemannscher Abbildungssatz und quasikonforme Mannigfaltgkeiten, Ann. Acad. Sci. Fenn. Ser. A I 409 (1967), 24 pages. MR 36:1645
  • 14. S. Lattès, Sur l'iteration des substitutions rationelles et les fonctions de Poincaré, C. R. Acad. Sci. Paris 16 (1918), 26-28.
  • 15. J. Lehner, Discontinuous Groups and Automorphic Functions, Mathematical Surveys, Number VIII, American Mathematical Society, Providence, RI, 1964. MR 29:1332
  • 16. C. T. McMullen, Complex Dynamics and Renormalization, Annals of Mathematics Studies 135, Princeton University Press, Princeton, NJ, 1994. MR 96b:58097
  • 17. D. Meyer, Quasisymmetric embedding of self similar surfaces and origami with rational maps, Ann. Acad. Sci. Fenn. Math. 27 (2002), 461-484. MR 2003g:52037
  • 18. J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, Vieweg, Braunschweig/Wiesbaden, 1999. MR 2002i:37057
  • 19. K. M. Pilgrim, Cylinders for iterated rational maps, Ph.D. thesis, University of California, Berkeley, 1994.
  • 20. K. M. Pilgrim, Canonical Thurston obstructions, Adv. Math. 158 (2001), 154-168. MR 2001m:57004
  • 21. K. Stephenson, CirclePack, software, available from
  • 22. E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. MR 35:1007
  • 23. D. Sullivan, Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains, Ann. Math. 122 (1985), 401-418. MR 87i:58103
  • 24. D. Sullivan, Conformal dynamical systems, Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Math. 1007, Springer, Berlin, 1983, pp. 725-752. MR 85m:58112
  • 25. W. P. Thurston, Lecture notes, CBMS Conference, University of Minnesota at Duluth, 1983.

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 37F10, 52C20, 57M12

Retrieve articles in all journals with MSC (2000): 37F10, 52C20, 57M12

Additional Information

J. W. Cannon
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

W. J. Floyd
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061

R. Kenyon
Affiliation: Laboratoire de Topologie, Université Paris-Sud, Bat. 425, 91405 Orsay Cedex-France

W. R. Parry
Affiliation: Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197

Keywords: Finite subdivision rule, rational map, conformality
Received by editor(s): September 5, 2001
Received by editor(s) in revised form: April 4, 2003
Published electronically: July 28, 2003
Additional Notes: This research was supported in part by NSF grants DMS-9803868, DMS-9971783, and DMS-10104030.
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society