Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure

Alexander Teplyaev
2008 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
We define sets with finitely ramified cell structure, which are generalizations of p.c.f. 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. 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
more » ... uared 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.
doi:10.4153/cjm-2008-022-3 fatcat:3snbuk4dpjc6rirmrr6pane3ce