Constructing subdivision rules from rational maps
HTML articles powered by AMS MathViewer
- by J. W. Cannon, W. J. Floyd and W. R. Parry PDF
- Conform. Geom. Dyn. 11 (2007), 128-136 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.
Additional Information
- J. W. Cannon
- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- Email: cannon@math.byu.edu
- W. J. Floyd
- Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061
- MR Author ID: 67750
- Email: floyd@math.vt.edu
- W. R. Parry
- Affiliation: Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197
- MR Author ID: 136390
- Email: walter.parry@emich.edu
- Received by editor(s): March 15, 2007
- Published electronically: August 14, 2007
- Additional Notes: This work was supported in part by NSF research grants DMS-0104030 and DMS-0203902.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn. 11 (2007), 128-136
- MSC (2000): Primary 37F10, 52C20; Secondary 57M12
- DOI: https://doi.org/10.1090/S1088-4173-07-00167-1
- MathSciNet review: 2329140