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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Analysis on products of fractals


Author: Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 357 (2005), 571-615
MSC (2000): Primary 31C45, 28A80
Published electronically: September 23, 2004
MathSciNet review: 2095624
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Abstract: For a class of post-critically finite (p.c.f.) fractals, which includes the Sierpinski gasket (SG), there is a satisfactory theory of analysis due to Kigami, including energy, harmonic functions and Laplacians. In particular, the Laplacian coincides with the generator of a stochastic process constructed independently by probabilistic methods. The probabilistic method is also available for non-p.c.f. fractals such as the Sierpinski carpet. In this paper we show how to extend Kigami's construction to products of p.c.f. fractals. Since the products are not themselves p.c.f., this gives the first glimpse of what the analytic theory could accomplish in the non-p.c.f. setting. There are some important differences that arise in this setting. It is no longer true that points have positive capacity, so functions of finite energy are not necessarily continuous. Also the boundary of the fractal is no longer finite, so boundary conditions need to be dealt with in a more involved manner. All in all, the theory resembles PDE theory while in the p.c.f. case it is much closer to ODE theory.


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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/S0002-9947-04-03685-2
Received by editor(s): July 8, 2003
Published electronically: September 23, 2004
Additional Notes: The author’s research was supported in part by the National Science Foundation, grant DMS–0140194
Article copyright: © Copyright 2004 American Mathematical Society