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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Finite subdivision rules
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by J. W. Cannon, W. J. Floyd and W. R. Parry PDF
Conform. Geom. Dyn. 5 (2001), 153-196 Request permission

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
  • 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.
  • © Copyright 2001 American Mathematical Society
  • 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
  • MathSciNet review: 1875951