Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure
Printed: Apr 2008
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
fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric
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