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Constructing subdivision rules from rational maps
Authors:
J. W. Cannon, W. J. Floyd and W. R. Parry
Journal:
Conform. Geom. Dyn. 11 (2007), 128-136
MSC (2000):
Primary 37F10, 52C20; Secondary 57M12
Posted:
August 14, 2007
MathSciNet review:
2329140
Full-text PDF Free Access
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Additional Information
Abstract: This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if  is a critically finite rational map with no periodic critical points, then for any sufficiently large integer  the iterate  is the subdivision map of a finite subdivision rule. This enables one to give essentially combinatorial models for the dynamics of such iterates.
- 1.
M. Bonk and D. Meyer, Topological rational maps and subdivisions, in preparation.
- 2.
J.
W. Cannon, W.
J. Floyd, and W.
R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153–196
(electronic). MR
1875951 (2002j:52021), http://dx.doi.org/10.1090/S1088-4173-01-00055-8
- 3.
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
(2004f:37062), http://dx.doi.org/10.1090/S1088-4173-03-00082-1
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J.
W. Cannon, W.
J. Floyd, and W.
R. Parry, Expansion complexes for finite
subdivision rules. I, Conform. Geom. Dyn.
10 (2006), 63–99
(electronic). MR
2218641 (2007c:30048), http://dx.doi.org/10.1090/S1088-4173-06-00126-3
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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
(98i:57023), http://dx.doi.org/10.1090/S0002-9947-98-02107-2
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(2002i:37057)
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(2001m:57004), http://dx.doi.org/10.1006/aima.2000.1971
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Notes in Mathematics, vol. 1827, Springer-Verlag, Berlin, 2003. MR 2020454
(2004m:37087)
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W. P. Thurston, Lecture notes, CBMS Conference, University of Minnesota at Duluth, 1983.
- 1.
- M. Bonk and D. Meyer, Topological rational maps and subdivisions, in preparation.
- 2.
- J. W. Cannon, W. J. Floyd, and W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153-196 (electronic). MR 1875951 (2002j:52021)
- 3.
- 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 (electronic). MR 1992038 (2004f:37062)
- 4.
- J. W. Cannon, W. J. Floyd, and W. R. Parry, Expansion complexes for finite subdivision rules I, Conform. Geom. Dyn. 10 (2006), 63-99 (electronic). MR 2218641 (2007c:30048)
- 5.
- J. W. Cannon and E. L. Swenson, Recognizing constant curvature groups in dimension 3, Trans. Amer. Math. Soc. 350 (1998), 809-849. MR 1458317 (98i:57023)
- 6.
- A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993), 263-297. MR 1251582 (94j:58143)
- 7.
- J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, Vieweg & Sohn, Braunschweig/Wiesbaden, 1999. MR 1721240 (2002i:37057)
- 8.
- K. M. Pilgrim, Canonical Thurston obstructions, Adv. Math. 158 (2001), 154-168. MR 1822682 (2001m:57004)
- 9.
- K. M. Pilgrim, Combinations of Complex Dynamical Systems, Springer Lecture Notes in Mathematics 1827, Springer-Verlag, Berlin Heidelberg New York, 2003. MR 2020454 (2004m:37087)
- 10.
- D. Sullivan, Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains, Ann. Math. 122 (1985), 401-418. MR 819553 (87i:58103)
- 11.
- W. P. Thurston, Lecture notes, CBMS Conference, University of Minnesota at Duluth, 1983.
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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
Email:
floyd@math.vt.edu
W. R. Parry
Affiliation:
Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197
Email:
walter.parry@emich.edu
DOI:
http://dx.doi.org/10.1090/S1088-4173-07-00167-1
PII:
S 1088-4173(07)00167-1
Keywords:
Finite subdivision rule,
rational map,
conformality
Received by editor(s):
March 15, 2007
Posted:
August 14, 2007
Additional Notes:
This work was supported in part by NSF research grants DMS-0104030 and DMS-0203902.
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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