Harmonic calculus on p.c.f. self-similar sets

Kigami, Jun

doi:10.1090/S0002-9947-1993-1076617-1

Harmonic calculus on p.c.f. self-similar sets
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by Jun Kigami

Trans. Amer. Math. Soc. 335 (1993), 721-755

DOI: https://doi.org/10.1090/S0002-9947-1993-1076617-1

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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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