Green functions on self-similar graphs and bounds for the spectrum of the Laplacian

Abstract

Combining the study of the simple random walk on graphs, generating functions (especially Green functions), complex dynamics and general complex analysis we introduce a new method of spectral analysis on self-similar graphs. We give an axiomatic definition of self-similar graphs which correspond to general nested but not necessarily finitely ramified fractals. For this class of graphs a graph theoretic analogue to the Banach fixed point theorem is proved. Functional equations and a decomposition algorithm for the Green functions of self-similar graphs with some more symmetric structure are obtained. Their analytic continuations are given by rapidly converging expressions. We study the dynamics of a certain complex rational Green function $d$ on finite directed subgraphs. If the Julia set $\cj$ of $d$ is a Cantor set, then the reciprocal spectrum $\spec^{-1}P=\{1/z\mid z\in\spec P\}$ of the Markov transition operator $P$ can be identified with the set of singularities of any Green function of the whole graph. Finally we get explicit upper and lower bounds for the reciprocal spectrum, where $\cd$ is a countable set of the $d$-backwards iterates of a certain finite set of real numbers.