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Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure

Published online by Cambridge University Press:  20 November 2018

Alexander Teplyaev
Alexander Teplyaev*
Affiliation:
Department of Mathematics, University of Connecticut, Storrs CT 06269-3009, U.S.A. e-mail: [email protected]
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Abstract

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We define sets with finitely ramified cell structure, which are generalizations of post-critically 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

28A8031C2553B9958J6560J6060G18fractalsself-similarityenergyresistanceDirichlet formsdiffusionsquantum graphsgeneralized Riemannian metric
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 2 , 01 April 2008 , pp. 457 - 480
Copyright
Copyright © Canadian Mathematical Society 2008

References

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