Constructing subdivision rules from rational maps

Cannon, J.; Floyd, W.; Parry, W.

doi:10.1090/S1088-4173-07-00167-1

Constructing subdivision rules from rational maps
HTML articles powered by AMS MathViewer

by J. W. Cannon, W. J. Floyd and W. R. Parry

Conform. Geom. Dyn. 11 (2007), 128-136

DOI: https://doi.org/10.1090/S1088-4173-07-00167-1

Published electronically: August 14, 2007

PDF | Request permission

Abstract:

This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if $f$ is a critically finite rational map with no periodic critical points, then for any sufficiently large integer $n$ the iterate $f^{\circ n}$ is the subdivision map of a finite subdivision rule. This enables one to give essentially combinatorial models for the dynamics of such iterates.

References

M. Bonk and D. Meyer, Topological rational maps and subdivisions, in preparation.
J. W. Cannon, W. J. Floyd, and W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153–196. MR 1875951, DOI 10.1090/S1088-4173-01-00055-8
J. W. Cannon, W. J. Floyd, R. Kenyon, and W. R. Parry, Constructing rational maps from subdivision rules, Conform. Geom. Dyn. 7 (2003), 76–102. MR 1992038, DOI 10.1090/S1088-4173-03-00082-1
J. W. Cannon, W. J. Floyd, and W. R. Parry, Expansion complexes for finite subdivision rules. I, Conform. Geom. Dyn. 10 (2006), 63–99. MR 2218641, DOI 10.1090/S1088-4173-06-00126-3
J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension $3$, Trans. Amer. Math. Soc. 350 (1998), no. 2, 809–849. MR 1458317, DOI 10.1090/S0002-9947-98-02107-2
Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263–297. MR 1251582, DOI 10.1007/BF02392534
John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
Kevin M. Pilgrim, Canonical Thurston obstructions, Adv. Math. 158 (2001), no. 2, 154–168. MR 1822682, DOI 10.1006/aima.2000.1971
Kevin M. Pilgrim, Combinations of complex dynamical systems, Lecture Notes in Mathematics, vol. 1827, Springer-Verlag, Berlin, 2003. MR 2020454, DOI 10.1007/b14147
Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), no. 3, 401–418. MR 819553, DOI 10.2307/1971308
W. P. Thurston, Lecture notes, CBMS Conference, University of Minnesota at Duluth, 1983.