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 . Their analytic continuations are given by rapidly converging expressions. We study the dynamics of a probability generating function associated with a random walk on a certain finite subgraph (“cell-graph”). The reciprocal spectrum coincides with the set of points in such that there is Green function which cannot be continued analytically from both half spheres in to . The Julia set of is an interval or a Cantor set. In the latter case is the set of singularities of all Green functions. Finally, we get explicit inner and outer bounds, where is the set of the -backward iterates of a finite set of real numbers.
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.
Keywords: self-similar graphs, Green functions
Mot clés : graphes auto-similaires, fonctions de Green
Krön, Bernhard 1
@article{AIF_2002__52_6_1875_0, author = {Kr\"on, Bernhard}, title = {Green functions on self-similar graphs and bounds for the spectrum of the laplacian}, journal = {Annales de l'Institut Fourier}, pages = {1875--1900}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {52}, number = {6}, year = {2002}, doi = {10.5802/aif.1937}, zbl = {1012.60063}, mrnumber = {1954327}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1937/} }
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 SP - 1875 EP - 1900 VL - 52 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1937/ DO - 10.5802/aif.1937 LA - en ID - AIF_2002__52_6_1875_0 ER -
%0 Journal Article %A Krön, Bernhard %T Green functions on self-similar graphs and bounds for the spectrum of the laplacian %J Annales de l'Institut Fourier %D 2002 %P 1875-1900 %V 52 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1937/ %R 10.5802/aif.1937 %G en %F AIF_2002__52_6_1875_0
Krön, Bernhard. Green functions on self-similar graphs and bounds for the spectrum of the laplacian. Annales de l'Institut Fourier, Volume 52 (2002) no. 6, pp. 1875-1900. doi : 10.5802/aif.1937. https://aif.centre-mersenne.org/articles/10.5802/aif.1937/
[1] Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets, J. London Math. Soc., Volume 56 (1997) no. 2, pp. 320-332 | DOI | MR | Zbl
[2] Brownian motion on the Sierpiński gasket, Prob. Theory Related Fields, Volume 79 (1988) no. 4, pp. 543-623 | DOI | MR | Zbl
[3] Croissance de groupes agissant sur des arbres (2000) (Ph. D. thesis, Université de Genève)
[4] On the spectrum of Hecke type operators related to some fractal groups, Tr. Mat. Inst. Steklova (Din. Sist., Avtom. i Beskon. Gruppy), Volume 231 (2000), pp. 5-45 | MR | Zbl
[5] From fractal groups to fractal sets, Fractals in Graz 2001 (2002) | Zbl
[6] Iteration of rational functions, Springer-Verlag, New York, 1991 | MR | Zbl
[7] Complex dynamics, Springer-Verlag, New York, 1993 | MR | Zbl
[8] Random walks and electric networks, Math. Association of America, Washington, DC, 1984 | MR | Zbl
[9] Linear Operators I-II, Interscience, New York, 1963 | Zbl
[10] Combinatorial enumeration, John Wiley \& Sons, New York, 1983 | MR | Zbl
[11] Functional iterations and stopping times for Brownian motion on the Sierpiński gasket, Mathematika, Volume 44 (1997) no. 2, pp. 374-400 | DOI | MR | Zbl
[12] Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph, Stochastic Process. Appl., Volume 69 (1997) no. 1, pp. 127-138 | DOI | MR | Zbl
[13] On the asymptotics of the eigenvalue counting function for random recursive Sierpiński gaskets, Prob. Theory Related Fields, Volume 117 (2000) no. 2, pp. 221-247 | DOI | MR | Zbl
[14] The homogenization problem for the Vicsek set, Stochastic Process. Appl., Volume 76 (1998) no. 2, pp. 167-190 | DOI | MR | Zbl
[15] Fractals and self-similarity, Indiana Univ. Math. J., Volume 30 (1981) no. 5, pp. 713-747 | DOI | MR | Zbl
[16] Rational iteration (2001) (Dissertation, Universitätsverlag Rudolf Trauner, University of Linz) | MR | Zbl
[17] Transition probabilities for the simple random walk on the Sierpiński graph, Stochastic Process. Appl., Volume 61 (1996) no. 1, pp. 45-69 | DOI | MR | Zbl
[18] Symmetric random walks on groups, Trans. Amer. Math. Soc., Volume 92 (1959), pp. 336-354 | DOI | MR | Zbl
[19] Harmonic calculus on p.c.f. self-similar sets., Trans. Amer. Math. Soc., Volume 335 (1993) no. 2, pp. 721-755 | DOI | MR | Zbl
[20] Spectral and structural theory of infinite graphs (2001) (PhD. thesis, Graz University of Technology)
[21] Growth of self-similar graphs (2002) (Preprint) | MR
[22] Asymptotics of the transition probabilities of the simple random walk on self-similar graphs (2002) (Preprint) | MR | Zbl
[23] Brownian motion on nested fractals, Mem. Amer. Math. Soc., Volume 83 (1990), pp. 420 | MR | Zbl
[24] The integrated density of states for the difference Laplacian on the modified Koch graph, Comm. Math. Phys., Volume 156 (1993) no. 2, pp. 387-397 | DOI | MR | Zbl
[25] Random walk and chaos of the spectrum. Solvable model, Chaos Solitons Fractals, Volume 5 (1995) no. 6, pp. 895-907 | DOI | MR | Zbl
[26] Pure point spectrum of the Laplacians on fractal graphs, J. Funct. Anal., Volume 129 (1995) no. 2, pp. 390-405 | DOI | MR | Zbl
[27] Self-similarity, operators and dynamics (2001) (Preprint) | Zbl
[28] How many diffusions exist on the Vicsek snowflake?, Acta Appl. Math., Volume 32 (1993) no. 3, pp. 227-241 | DOI | MR | Zbl
[29] Random walk and electric currents in networks, Proc. Cambridge Phil. Soc., Volume 55 (1959), pp. 181-194 | DOI | MR | Zbl
[30] Random walk statistics on fractal structures, J. Stat. Phys., Volume 36 (1984) no. 5-6, pp. 547-560 | DOI | MR | Zbl
[31] Spectrum of harmonic excitations on fractals, J. Physique, Volume 45 (1984) no. 2, pp. 191-206 | DOI | MR
[32] Random walks on fractal structures and percolation clusters, J. Physique - Lettres 44, Volume 36 (1983) no. L13-L22
[33] Pure point spectrum for the Laplacian on unbounded nested fractals, J. Funct. Anal., Volume 173 (2000) no. 2, pp. 497-524 | DOI | MR | Zbl
[34] Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., Volume 159 (1998) no. 2, pp. 537-567 | DOI | MR | Zbl
[35] Random Walks on Infinite Graphs and Groups, Cambridge University Press, Cambridge, 2000 | MR | Zbl
Cited by Sources: