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Effective resistances for harmonic structures on p.c.f. self-similar sets

Published online by Cambridge University Press:  24 October 2008

Jun Kigami
Jun Kigami
Affiliation:
Department of Mathematics, College of General Education, Osaka University, Toyonaka 560, Japan
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In mathematics, analysis on fractals was originated by the works of Kusuoka [17] and Goldstein[8]. They constructed the ‘Brownian motion on the Sierpinski gasket’ as a scaling limit of random walks on the pre-gaskets. Since then, analytical structures such as diffusion processes, Laplacians and Dirichlet forms on self-similar sets have been studied from both probabilistic and analytical viewpoints by many authors, see [4], [20], [10], [22] and [7]. As far as finitely ramified fractals, represented by the Sierpinski gasket, are concerned, we now know how to construct analytical structures on them due to the results in [20], [18] and [11]. In particular, for the nested fractals introduced by Lindstrøm [20], one can study detailed features of analytical structures such as the spectral dimensions and various exponents of heat kernels by virtue of the strong symmetry of nested fractals, cf. [6] and [15]. Furthermore in [11], Kigami proposed a notion of post critically finite (p.c.f. for short) self-similar sets, which was a pure topological description of finitely ramified self-similar sets. Also it was shown that we can construct Dirichlet forms and Laplacians on a p.c.f. self-similar set if there exists a difference operator that is invariant under a kind of renormalization. This invariant difference operator was called a harmonic structure. In Section 2, we will give a review of the results in [11].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 2 , March 1994 , pp. 291 - 303
Copyright
Copyright © Cambridge Philosophical Society 1994

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