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Expansion complexes for finite subdivision rules. II
Author(s):
J.
W.
Cannon;
W.
J.
Floyd;
W.
R.
Parry
Journal:
Conform. Geom. Dyn.
10
(2006),
326-354.
MSC (2000):
Primary 30F45, 52C20;
Secondary 20F67, 52C20
Posted:
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.
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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 Tech, Blacksburg, Virginia 24061
Email:
floyd@math.vt.edu
W.
R.
Parry
Affiliation:
Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197
Email:
walter.parry@emich.edu
DOI:
10.1090/S1088-4173-06-00127-5
PII:
S 1088-4173(06)00127-5
Keywords:
Conformality,
expansion complex,
finite subdivision rule
Received by editor(s):
November 22, 2004
Posted:
December 6, 2006
Additional Notes:
This research was supported in part by NSF grants DMS-9803868, DMS-9971783, DMS-10104030, and DMS-0203902.
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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