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

DOI:
https://doi.org/10.1090/S0002-9947-03-03171-4

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 , which is -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:
https://doi.org/10.1090/S0002-9947-03-03171-4

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