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Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure


Published:20080401
Printed: Apr 2008
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Abstract
We define sets with finitely ramified cell structure, which are
generalizations of postcrit8cally finite selfsimilar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local selfsimilarity, and allow countably many cells
connected at each junction point.
In particular, we consider postcritically 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, selfsimilarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric
fractals, selfsimilarity, 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] Selfsimilar 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  Selfsimilar processes
