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Journal Article

Harmonic Calculus on P.C.F. Self-Similar Sets

Jun Kigami
Transactions of the American Mathematical Society
Vol. 335, No. 2 (Feb., 1993), pp. 721-755
DOI: 10.2307/2154402
Stable URL: https://www.jstor.org/stable/2154402
Page Count: 35
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Harmonic Calculus on P.C.F. Self-Similar Sets
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Abstract

The main object of this paper is the Laplace operator on a class of fractals. First, we establish the concept of the renormalization of difference operators on post critically finite (p.c.f. for short) self-similar sets, which are large enough to include finitely ramified self-similar sets, and extend the results for Sierpinski gasket given in [10] to this class. Under each invariant operator for renormalization, the Laplace operator, Green function, Dirichlet form, and Neumann derivatives are explicitly constructed as the natural limits of those on finite pre-self-similar sets which approximate the p.c.f. self-similar sets. Also harmonic functions are shown to be finite dimensional, and they are characterized by the solution of an infinite system of finite difference equations.

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