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

Krön, Bernhard

Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1875-1900.

Pour une classe de graphes auto-similaires, les prolongements analytiques de ses fonctions de Green peuvent être calculés explicitement. Si le spectre de l'opérateur de Markov n'est pas un intervalle, alors il coïncide avec l'ensemble des valeurs réciproques des singularités des fonctions de Green. Nous donnons des bornes intérieures et extérieures pour ce spectre.

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 ${\mathrm{spec}}^{-1}P=\{1/\lambda \mid \lambda \in \mathrm{spec}P\}$ coincides with the set of points $z$ in $\overline{ℝ}\setminus (-1,1)$ such that there is Green function which cannot be continued analytically from both half spheres in $\overline{ℂ}\setminus \overline{ℝ}$ to $z$. The Julia set $𝒥$ of $d$ is an interval or a Cantor set. In the latter case ${\mathrm{spec}}^{-1}P$ is the set of singularities of all Green functions. Finally, we get explicit inner and outer bounds, $𝒥\subset {\mathrm{spec}}^{-1}P\subset 𝒥\cup 𝒟,$ where $𝒟$ is the set of the $d$-backward iterates of a finite set of real numbers.

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