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Conformal Geometry and Dynamics

ISSN 1088-4173



Finite subdivision rules

Authors: J. W. Cannon, W. J. Floyd and W. R. Parry
Journal: Conform. Geom. Dyn. 5 (2001), 153-196
MSC (2000): Primary 20F65, 52C20; Secondary 05B45
Published electronically: December 18, 2001
MathSciNet review: 1875951
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Abstract: We introduce and study finite subdivision rules. A finite subdivision rule $\mathcal{R}$ consists of a finite 2-dimensional CW complex $S_{\mathcal{R}}$, a subdivision $\mathcal{R}(S_{\mathcal{R}})$ of $S_{\mathcal{R}}$, and a continuous cellular map $\varphi_{\mathcal{R}}\colon\thinspace \mathcal{R}(S_{\mathcal{R}}) \to S_{\mathcal{R}}$ whose restriction to each open cell is a homeomorphism. If $\mathcal{R}$ is a finite subdivision rule, $X$ is a 2-dimensional CW complex, and $f\colon\thinspace X\to S_{\mathcal{R}}$ is a continuous cellular map whose restriction to each open cell is a homeomorphism, then we can recursively subdivide $X$ to obtain an infinite sequence of tilings. We wish to determine when this sequence of tilings is conformal in the sense of Cannon's combinatorial Riemann mapping theorem. In this setting, it is proved that the two axioms of conformality can be replaced by a single axiom which is implied by either of them, and that it suffices to check conformality for finitely many test annuli. Theorems are given which show how to exploit symmetry, and many examples are computed.

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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 Polytechnic Institute & State University, Blacksburg, Virginia 24061

W. R. Parry
Affiliation: Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197

Keywords: Finite subdivision rule, conformality
Received by editor(s): September 20, 1999
Received by editor(s) in revised form: July 2, 2001
Published electronically: December 18, 2001
Additional Notes: This work was supported in part by NSF research grants and by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc.
Article copyright: © Copyright 2001 American Mathematical Society