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Conformal Geometry and Dynamics

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Expansion complexes for finite subdivision rules. I

Authors: J. W. Cannon, W. J. Floyd and W. R. Parry
Journal: Conform. Geom. Dyn. 10 (2006), 63-99
MSC (2000): Primary 30F45, 52C20; Secondary 20F67, 52C26
Published electronically: March 22, 2006
MathSciNet review: 2218641
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Abstract: This paper develops the basic theory of conformal structures on finite subdivision rules. The work depends heavily on the use of expansion complexes, which are defined and discussed in detail. It is proved that a finite subdivision rule with bounded valence and mesh approaching $0$ is conformal (in the combinatorial sense) if there is a partial conformal structure on the model subdivision complex with respect to which the subdivision map is conformal. This gives a new approach to the difficult combinatorial problem of determining when a finite subdivision rule is conformal.

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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
MR Author ID: 67750

W. R. Parry
Affiliation: Department of Mathematics Eastern Michigan University, Ypsilanti, Michigan 48197
MR Author ID: 136390

Keywords: Conformality, expansion complex, finite subdivision rule
Received by editor(s): November 22, 2004
Published electronically: March 22, 2006
Additional Notes: This research was supported in part by NSF grants DMS-9803868, DMS-9971783, DMS-10104030, and DMS-0203902
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.