Harmonic Calculus on Limits of Networks and Its Application to Dendrites

AbstractThe main purpose of this paper is to give a general framework to analysis on fractals including finitely ramified self-similar sets, random fractals, Canter sets, dendrites, and the recurrent case of infinitely ramified self-similar sets. We study a limit of a sequence of electrical networks satisfying certain compatibility, which is a generalization of the decimation invariance of random walks playing an important role in constructing diffusion processes on self-similar sets. As a limit, we propose notions of finite resistance forms and effective resistance metrics. Also it is shown that a finite resistance form induces an effective resistance metric and vice versa. A finite resistance form becomes a regular Dirichlet form if the domain is dense in L2-space. In such a case, the effective resistance metric is thought of as the intrinsic metric for the Dirichlet form. In the latter half of the paper we apply the general framework to dendrites, which are topological spaces containing no loops. A shortest path metric, where all arcs are geodesics, on a dendrite is shown to be an effective resistance metric and the corresponding finite resistance form turns out to be a local regular Dirichlet form. Furthermore, we establish a simple relation between the Hausdorff dimensions and the spectral dimensions of dendrites with shortest path metrics