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# Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure

Published:2008-04-01
Printed: Apr 2008
• Alexander Teplyaev
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## Abstract

We define sets with finitely ramified cell structure, which are generalizations of post-crit8cally finite 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. In particular, we consider post-critically infinite fractals. 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.
 Keywords: fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric
 MSC Classifications: 28A80 - Fractals [See also 37Fxx] 31C25 - Dirichlet spaces 53B99 - None of the above, but in this section 58J65 - Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60J60 - Diffusion processes [See also 58J65] 60G18 - Self-similar processes