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

Annales de l'Institut Fourier

Volumn 52, Issue 6, 2002, Pages

  Krön, Bernhard ^a  

^a Erwin Schrodinger International Institute for Mathematical Physics   (Austria)

Author keywords

Green functions; Self similar graphs

Indexed keywords

EID: 0037963469     PISSN: 03730956     EISSN: None     Source Type: Journal    
DOI: 10.5802/aif.1937     Document Type: Article

References (35)

1. M.T. BARLOW, J. KIGAMI, Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets, J. London Math. Soc., 56-2 (1997), 320-332.
2. M.T. BARLOW, E.A. PERKINS, Brownian motion on the Sierpiński gasket, Prob. Theory Related Fields, 79-4 (1988), 543-623.
3. L. BARTHOLDI, Croissance de groupes agissant sur des arbres. Ph. D. thesis, Université de Genéve, 2000.
4. L. BARTHOLDI, R.I. GRIGORCHUK, On the spectrum of Hecke type operators related to some fractal groups, Tr. Mat. Inst. Steklova (Din. Sist., Avtom. i Beskon. Gruppy), 231 (2000), 5-45.
5. L. BARTHOLDI. R.I. GRIGORCHUK, V. NEKRASHEVYCH, From fractal groups to fractal sets, in 'Fractals in Graz 2001', P.J. Grabner and W. Woess, eds., Birkhäuser, 2002.
6. A.F. BEARDON, Iteration of rational functions, Springer-Verlag, New York, 1991.
7. L. CARLESON, T.W. GAMELTN, Complex dynamics. Springer-Verlag, New York, 1993.
8. P.G. DOYLE, J.L. SNELL, Random walks and electric networks, Math. Association of America, Washington, DC, 1984.
9. N. DUNFORD, J.T. SCHWARTZ, Linear Operators I-II, Interscience, New York, 1963.
10. I.P. GOULDEN, D.M. JACKSON, Combinatorial enumeration, John Wiley & Sons, New York, 1983.
11. P.J. GRABNER, Functional iterations and stopping times for Brownian motion on the Sierpiński gasket. Mathematika, 44-2 (1997), 374-400.
12. P.J. GRABNER, W. WOESS, Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph. Stochastic Process. Appl., 69-1 (1997), 127-138.
13. B.M. HAMBLY, On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets, Prob. Theory Related Fields, 117-2 (2000), 221-247.
14. B.M. HAMBLY, V. METZ, The homogenization problem for the Vicsek set, Stochastic Process. Appl., 76-2 (1998), 167-190.
15. J.E. HUTCHINSON, Fractals and self-similarity, Indiana Univ. Math. J., 30-5 (1981), 713-747.
16. C. INNINGER, Rational iteration, Dissertation, Universitä̈tsverlag Rudolf Trauner, University of Linz, 2001.
17. O.D. JONES, Transition probabilities for the simple random walk on the Sierpinski graph, Stochastic Process. Appl., 61-1 (1996), 45-69.
18. H. KESTEN, Symmetric random walks on groups, Trans. Amer. Math. Soc., 1959.
19. J. KIGAMI, Harmonic calculus on p.c.f. self-similar sets., Trans. Amer. Math. Soc., 1993.
20. B. KRÖN, Spectral and structural theory of infinite graphs, PhD. thesis, Graz University of Technology, 2001.
21. B. KRÖN, Growth of self-similar graphs, preprint, 2002.
22. B. KRÖN, E. TEUFL, Asymptotics of the transition probabilities of the simple random walk on self-similar graphs, preprint, 2002.
23. T. LINDSTØRM, Brownian motion on nested fractals, Mem. Amer. Math. Soc., 83 (1990), 420.
24. L. MALOZEMOV, The integrated density of states for the difference Laplacian on the modified Koch graph, Comm. Math. Phys., 156-2 (1993), 387-397.
25. L. MALOZEMOV, Random walk and chaos of the spectrum. Solvable model, Chaos Solitons Fractals, 5-6 (1995), 895-907.
26. L. MALOZEMOV, A. TEPLYAEV, Pure point spectrum of the Laplacians on fractal graphs, J. Funct. Anal., 129-2 (1995), 390-405.
27. L. MALOZEMOV, A. TEPLYAEV, Self-similarity, operators and dynamics, preprint, 2001.
28. V. METZ, How many diffusions exist on the Vicsek snowflake?, Acta Appl. Math., 32-3 (1993). 227-241.
29. C.S.J.A. NASH-WILLIAMS, Random walk and electric currents in networks. Proc. Cambridge Phil. Soc., 55 (1959), 181-194.
30. R. RAMMAL, Random walk statistics on fractal structures, J. Stat. Phys., 36-5-6 (1984). 547-560.
31. R. RAMMAL, Spectrum of harmonic excitations on fractals, J. Physique, 45-2 (1984), 191-206.
32. R. RAMMAL, TOULOUSE. Random walks on fractal structures and percolation clusters. J. Physique - Lettres 44, 36-L13-L22 (1983).
33. C. SABOT, Pure point spectrum for the Laplacian on unbounded nested fractals, J. Funct. Anal., 173-2 (2000), 497-524.
34. A. TEPLYAEV., Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159-2 (1998), 537-567.
35. W. WOESS, Random Walks on Infinite Graphs and Groups, Cambridge University Press, Cambridge, 2000.