Construction of Laplacians on P. C. F. Self-Similar Structures

Abstract

In this chapter, we will construct the analysis associated with Laplacians on connected post critically finite self-similar structures. In this chapter, L = ( K, S , { F i } i ∊ S ) is a post critically finite (p. c. f. for short) self-similar structure and K is assumed to be connected. (Also in this chapter, we always set S = {1,2, …, N }.) Recall that a condition for K being connected was given in 1.6. The key idea of constructing a Laplacian (or a Dirichlet form) on K is finding a “self-similar” compatible sequence of r-networks on { V m } m ≥0, where V m = V m ( L ) was defined in Lemma 1.3.11. Note that { V m } m ≥0 is a monotone increasing sequence of finite sets. We will formulate such a self-similar compatible sequence in 3.1. Once we get such a sequence, we can use the general theory in the last chapter and construct a resistance form (ℇ, F ) and a resistance metric R on V ∗ , where V >∗ = U m ≥0 V m . If the closure of V * with respect to the metric R were always identified with K , then we could apply Theorem 2.4.2 and see that (ℇ, F ) is a regular local Dirichlet form on L 2 ( K , μ) for any self-similar measure μ on K . Consequently, we could immediately obtain a Laplacian associated with the Dirichlet form (ℇ, F ) on L 2 ( K , μ).