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Conformal Geometry and Dynamics

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Expansion complexes for finite subdivision rules. II

Authors: J. W. Cannon, W. J. Floyd and W. R. Parry
Journal: Conform. Geom. Dyn. 10 (2006), 326-354
MSC (2000): Primary 30F45, 52C20; Secondary 20F67, 52C20
Published electronically: December 6, 2006
MathSciNet review: 2268483
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Abstract: This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a one-tile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching $0$) has an invariant partial conformal structure, and hence is conformal. The paper next considers one-tile single valence finite subdivision rules. It is shown that an expansion map for such a finite subdivision rule can be conjugated to a linear map, and that the finite subdivision rule is conformal exactly when this linear map is either a dilation or has eigenvalues that are not real. Finally, an example is given of an irreducible finite subdivision rule that has a parabolic expansion complex and a hyperbolic expansion complex.

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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: December 6, 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.