Analysis on Limits of Networks

Abstract

In this chapter, we will discuss limits of discrete Laplacians (or equivalently Dirichlet forms) on a increasing sequence of finite sets. The results in this chapter will play a fundamental role in constructing a Laplacian (or equivalently a Dirichlet form) on certain self-similar sets in the next chapter, where we will approximate a self-similar set by an increasing sequence of finite sets and then construct a Laplacian on the self-similar set by taking a limit of the Laplacians on the finite sets. More precisely, we will define a Dirichlet form and a Laplacian on a finite set in 2.1. The key idea is that every Dirichlet form on a finite set can be associated with an electrical network consisting of resistors. From such a point of view, we will introduce the important notion of effective resistance. In 2.2, we will study a limit of a “compatible” sequence of Dirichlet forms on increasing finite sets. Roughly speaking, the word “compatible” means that the Dirichlet forms appearing in the sequence induce the same effective resistance on the union of the increasing finite sets. In 2.3 and 2.4, we will present further properties of limits of compatible sequences of Dirichlet forms.