2002/02/18 by Bernhard Krön, Krön, Bernhard
Mathematics · #Advanced Operator Algebra Research #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Graph theory and applications #Probability (math.PR) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.CO #math.DS #math.PR #math.SP
paper · pdf · doi:10.48550/arxiv.math/0202172
18 pages, 2 figures, other comments
openalex publication_date 2002/02/18 · arxiv created 2002/09/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-1P=\1/z| z∈\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.