## Expansion complexes for finite subdivision rules. II

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- by J. W. Cannon, W. J. Floyd and W. R. Parry
- Conform. Geom. Dyn.
**10**(2006), 326-354 - DOI: https://doi.org/10.1090/S1088-4173-06-00127-5
- Published electronically: December 6, 2006
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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.## References

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## Bibliographic 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 Tech, 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): 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: https://doi.org/10.1090/S1088-4173-06-00127-5
- MathSciNet review: 2268483