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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 fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric
MSC Classifications: 28A80, 31C25, 53B99, 58J65, 60J60, 60G18 show english descriptions Fractals [See also 37Fxx]
Dirichlet spaces
None of the above, but in this section
Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]
Diffusion processes [See also 58J65]
Self-similar processes
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
 

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