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

DOI:
https://doi.org/10.1090/S1088-4173-01-00055-8

Published electronically:
December 18, 2001

MathSciNet review:
1875951

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Email:
cannon@math.byu.edu

**W. J. Floyd**

Affiliation:
Department of Mathematics, Virginia Polytechnic Institute & State University, 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

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