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

Bernhard Krön[1]

  • [1] Erwin Schrödinger Institute (ESI), Boltzmanngasse 9, 1090 Wien (Autriche)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 6, page 1875-1900
  • ISSN: 0373-0956

Abstract

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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 for spectral analysis on self-similar graphs.First, for a rather general, axiomatically defined class of self-similar graphs a graph theoretic analogue to the Banach fixed point theorem is proved. The subsequent results hold for a subclass consisting of “symmetrically” self-similar graphs which however is still more general then other axiomatically defined classes of self-similar graphs studied in this context before: we obtain functional equations and a decomposition algorithm for the Green functions of the simple random walk Markov transition operator P . Their analytic continuations are given by rapidly converging expressions. We study the dynamics of a probability generating function d associated with a random walk on a certain finite subgraph (“cell-graph”). The reciprocal spectrum spec - 1 P = { 1 / λ ∣ λ ∈ spec P } coincides with the set of points z in ℝ ¯ ∖ ( - 1 , 1 ) such that there is Green function which cannot be continued analytically from both half spheres in ℂ ¯ ∖ ℝ ¯ to z . The Julia set 𝒥 of d is an interval or a Cantor set. In the latter case spec - 1 P is the set of singularities of all Green functions. Finally, we get explicit inner and outer bounds, 𝒥 ⊂ spec - 1 P ⊂ 𝒥 ∪ 𝒟 , where 𝒟 is the set of the d -backward iterates of a finite set of real numbers.

How to cite

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Krön, Bernhard. "Green functions on self-similar graphs and bounds for the spectrum of the laplacian." Annales de l’institut Fourier 52.6 (2002): 1875-1900. <http://eudml.org/doc/116030>.

@article{Krön2002,
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 for spectral analysis on self-similar graphs.First, for a rather general, axiomatically defined class of self-similar graphs a graph theoretic analogue to the Banach fixed point theorem is proved. The subsequent results hold for a subclass consisting of “symmetrically” self-similar graphs which however is still more general then other axiomatically defined classes of self-similar graphs studied in this context before: we obtain functional equations and a decomposition algorithm for the Green functions of the simple random walk Markov transition operator $P$. Their analytic continuations are given by rapidly converging expressions. We study the dynamics of a probability generating function $d$ associated with a random walk on a certain finite subgraph (“cell-graph”). The reciprocal spectrum $\{\rm spec\}^\{-1\}\!P=\lbrace 1/\lambda \mid \lambda \in \{\rm spec\}\, P\rbrace $ coincides with the set of points $z$ in $\bar\{\mathbb \{R\}\}\setminus (-1,1)$ such that there is Green function which cannot be continued analytically from both half spheres in $\bar\{\mathbb \{C\}\}\setminus \bar\{\mathbb \{R\}\}$ to $z$. The Julia set $\{\mathcal \{J\}\}$ of $d$ is an interval or a Cantor set. In the latter case $\{\rm spec\}^\{-1\}\!P$ is the set of singularities of all Green functions. Finally, we get explicit inner and outer bounds, $\{\mathcal \{J\}\}\subset \{\rm spec\}^\{-1\}\!P\subset \{\mathcal \{J\}\}\cup \{\mathcal \{D\}\},$ where $\{\mathcal \{D\}\}$ is the set of the $d$-backward iterates of a finite set of real numbers.},
affiliation = {Erwin Schrödinger Institute (ESI), Boltzmanngasse 9, 1090 Wien (Autriche)},
author = {Krön, Bernhard},
journal = {Annales de l’institut Fourier},
keywords = {self-similar graphs; Green functions; self-similar graph; simple random walk; Green function; Julia set},
language = {eng},
number = {6},
pages = {1875-1900},
publisher = {Association des Annales de l'Institut Fourier},
title = {Green functions on self-similar graphs and bounds for the spectrum of the laplacian},
url = {http://eudml.org/doc/116030},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Krön, Bernhard
TI - Green functions on self-similar graphs and bounds for the spectrum of the laplacian
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 6
SP - 1875
EP - 1900
AB - 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 for spectral analysis on self-similar graphs.First, for a rather general, axiomatically defined class of self-similar graphs a graph theoretic analogue to the Banach fixed point theorem is proved. The subsequent results hold for a subclass consisting of “symmetrically” self-similar graphs which however is still more general then other axiomatically defined classes of self-similar graphs studied in this context before: we obtain functional equations and a decomposition algorithm for the Green functions of the simple random walk Markov transition operator $P$. Their analytic continuations are given by rapidly converging expressions. We study the dynamics of a probability generating function $d$ associated with a random walk on a certain finite subgraph (“cell-graph”). The reciprocal spectrum ${\rm spec}^{-1}\!P=\lbrace 1/\lambda \mid \lambda \in {\rm spec}\, P\rbrace $ coincides with the set of points $z$ in $\bar{\mathbb {R}}\setminus (-1,1)$ such that there is Green function which cannot be continued analytically from both half spheres in $\bar{\mathbb {C}}\setminus \bar{\mathbb {R}}$ to $z$. The Julia set ${\mathcal {J}}$ of $d$ is an interval or a Cantor set. In the latter case ${\rm spec}^{-1}\!P$ is the set of singularities of all Green functions. Finally, we get explicit inner and outer bounds, ${\mathcal {J}}\subset {\rm spec}^{-1}\!P\subset {\mathcal {J}}\cup {\mathcal {D}},$ where ${\mathcal {D}}$ is the set of the $d$-backward iterates of a finite set of real numbers.
LA - eng
KW - self-similar graphs; Green functions; self-similar graph; simple random walk; Green function; Julia set
UR - http://eudml.org/doc/116030
ER -

References

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