Resistance Forms on Self-Similar Sets with Finite Ramification of Finite Type

In this paper, we introduce the finite neighboring type and the finite chain length conditions for a connected self-similar set K . We show that with these two conditions, K is a finitely ramified graph directed (f.r.g.d.) fractal defined by Hambly and Nyberg (Proc. Edinb. Math. Soc. (2) 46 (1), 1–34 2003 ). We give some nontrivial examples and compute the harmonic structures on them explicitly. Furthermore, for a f.r.g.d. self-similar set K , we provide an equivalent description, the finitely ramified of finite type (f.r.f.t.) cell structure of K , and investigate the relationship of harmonic structures associated with different f.r.f.t. cell structures of K .