Green functions on self-similar graphs and bounds for the spectrum of the Laplacian
| dc.creator | Krön, Bernhard | |
| dc.date | 2002-02-18 | |
| dc.date | 2002-09-12 | |
| dc.date.accessioned | 2026-07-07T04:46:32Z | |
| dc.date.available | 2026-07-07T04:46:32Z | |
| dc.description | 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. | |
| dc.description | 18 pages, 2 figures, other comments | |
| dc.identifier | https://arxiv.org/abs/math/0202172 | |
| dc.identifier | http://arxiv.org/abs/math/0202172 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63366 | |
| dc.subject | Combinatorics | |
| dc.subject | Dynamical Systems | |
| dc.subject | Probability | |
| dc.subject | Spectral Theory | |
| dc.title | Green functions on self-similar graphs and bounds for the spectrum of the Laplacian | |
| dc.type | text |