Harmonic coordinates on fractals with finitely ramified cell structure

Abstract

We define sets with finitely ramified cell structure, which are generalizations of p.c.f. self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. We prove that if Kigami's resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.

a better version of pictures can be found here: http://www.math.uconn.edu/~teplyaev/research/arxiv0506261.pdf

References in corpus (2)

• Quantum Graphs II: Some spectral properties of quantum and combinatorial graphs
• Stability of parabolic Harnack inequalities on metric measure spaces

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