Harmonic coordinates on fractals with finitely ramified cell structure

Details Harmonic coordinates on fractals with finitely ramified cell structure Harmonic coordinates on fractals with finitely ramified cell structure

2005-06-14

arxiv.org/abs/math/0506261v4

Mathematics

Probability

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

Scientific paper

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.

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