PUB - Publikationen an der Universität Bielefeld

This paper introduces an equivalent condition for the existence of regular local irreducible conservative Dirichlet forms that are self-similar under a given array of weights on p.c.f. (post-critically finite) fractals, which is expressed by a uniform resistance estimate on an iterated sequence of electrical networks. The corresponding construction of Dirichlet forms through Gamma-convergence, which is inspired by recent works by Yang and Grigor'yan, plays a central role. It avoids solving the complicated renormalization equations for a resistance form which is necessary in Kigami's construction. Techniques for resistance networks are needed for our method to be realized on general p.c.f. fractals. We also point out how the resistance estimate is deduced if a self-similar diffusion exists, and show by examples how this equivalent condition helps to decide which arrays of weights admit self-similar Dirichlet forms.