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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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Expansion complexes for finite subdivision rules. II
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by J. W. Cannon, W. J. Floyd and W. R. Parry PDF
Conform. Geom. Dyn. 10 (2006), 326-354 Request permission


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
  • Email:
  • W. J. Floyd
  • Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061
  • MR Author ID: 67750
  • Email:
  • W. R. Parry
  • Affiliation: Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197
  • MR Author ID: 136390
  • Email:
  • 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.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Conform. Geom. Dyn. 10 (2006), 326-354
  • MSC (2000): Primary 30F45, 52C20; Secondary 20F67, 52C20
  • DOI:
  • MathSciNet review: 2268483