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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Fractafolds based on the Sierpinski gasket and their spectra


Author: Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 355 (2003), 4019-4043
MSC (2000): Primary 28A80
Published electronically: June 18, 2003
MathSciNet review: 1990573
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Abstract: We introduce the notion of ``fractafold'', which is to a fractal what a manifold is to a Euclidean half-space. We specialize to the case when the fractal is the Sierpinski gasket SG. We show that each such compact fractafold can be given by a cellular construction based on a finite cell graph $G$, which is $3$-regular in the case that the fractafold has no boundary. We show explicitly how to obtain the spectrum of the fractafold from the spectrum of the graph, using the spectral decimation method of Fukushima and Shima. This enables us to obtain isospectral pairs of nonhomeomorphic fractafolds. We also show that although SG is topologically rigid, there are fractafolds based on SG that are not topologically rigid.


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Additional Information

Robert S. Strichartz
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853
Email: str@math.cornell.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03171-4
PII: S 0002-9947(03)03171-4
Received by editor(s): May 30, 2002
Published electronically: June 18, 2003
Additional Notes: This research was supported in part by the National Science Foundation, grant DMS-0140194
Article copyright: © Copyright 2003 American Mathematical Society