Integrated density of states of self-similar Sturm–Liouville operators and holomorphic dynamics in higher dimension

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1. [1] M.T Barlow, Diffusions on Fractals, St Flour Lecture Notes 1995, 1998. Zbl0916.60069MR1668115
2. [2] M.T Barlow, J Kigami, Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets, J. Lond. Math. Soc.56 (2) (1997) 320-332. Zbl0904.35064MR1489140
3. [3] R Carmona, J Lacroix, Spectral Theory of Random Schrödinger Operators, Probabilities and Applications, Birkhaüser, Boston, 1990. Zbl0717.60074MR1102675
4. [4] R Forman, Functional determinants and geometry, Invent. Math.88 (1987) 447-493. Zbl0602.58044MR884797
5. [5] R Forman, Determinants, finite-difference operators and boundary value problems, Comm. Math. Phys.147 (1992) 485-526. Zbl0767.58043MR1175491
6. [6] J.E Fornæss, Dynamics in Several Complex Variables, CBMS, Regional Conference Series in Math., No. 87, Amer. Math. Soc, 1996. Zbl0840.58016MR1363948
7. [7] J.E Fornæss, N Sibony, Complex dynamics in higher dimension. I, in: Complex Analytic Methods in Dynamical Systems (Rio de Janeiro, 1992), Astérisque, 222, 1994, pp. 201-231. Zbl0813.58030MR1285389
8. [8] J.E Fornæss, N Sibony, Complex dynamics in higher dimension. II, in: Modern Methods in Complex Analysis, Ann. Math. Studies, 137, University Press, Princeton, NJ, 1995, pp. 135-182. Zbl0847.58059MR1369137
9. [9] M Fukushima, Y Oshima, M Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Stud. Math. 19, Walter de Gruyter, Berlin, New York, 1994. Zbl0838.31001MR1303354
10. [10] M Fukushima, Dirichlet forms, diffusion processes and spectral dimensions for nested fractals, in: Albevario S, (Eds.), Ideas and Methods in Mathematical Analysis, Stochastics and Applications, Proc. Conf. in Memory of Hoegh-Krohn, Vol. 1, Cambridge Univ. Press, Cambridge, 1993, pp. 151-161. Zbl0764.60081MR1190496
11. [11] M Fukushima, T Shima, On the spectral analysis for the Sierpinski gasket, Potential Analysis1 (1992) 1-35. Zbl1081.31501MR1245223
12. [12] M Fukushima, T Shima, On the discontinuity and tail behaviours of the integrated density of states for nested pre-fractals, Comm. Math. Phys.163 (1994) 461-471. Zbl0798.58049MR1284792
13. [13] L Hörmander, Notions of Convexity, Progress in Mathematics, 127, Birkhäuser, 1994. Zbl0835.32001MR1301332
14. [14] J Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Am. Math. Soc.335 (1993) 721-755. Zbl0773.31009MR1076617
15. [15] J Kigami, Distributions of localized eigenvalues on post critically finite self-similar sets, J. Funct. Anal.159 (1998) 170-198. Zbl0908.35087MR1632976
16. [16] J Kigami, M.L Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys.158 (1) (1993) 93-125. Zbl0806.35130MR1243717
17. [17] Kusuoka S., Lecture on diffusion processes on nested fractals, in: Lecture Notes in Math., Vol. 1567, Springer. 
18. [18] M.L Lapidus, Analysis on fractals, Laplacians on self-similar sets, non-commutative geometry and spectral dimensions, Topological Methods in Nonlinear Analysis4 (1) (1994) 137-195. Zbl0836.35108MR1321811
19. [19] Y Le Jan, Mesures associées à une forme de Dirichlet. Applications, Bull. Soc. Math. de France106 (1978) 61-112. Zbl0393.31008MR508949
20. [20] T Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc.420 (1990). Zbl0688.60065MR988082
21. [21] L Pastur, A Figotin, Spectra of Random and Almost-Periodic Operators, Grundlehren der mathematischen Wissenschaften, 297, Springer-Verlag, Berlin, Heidelberg, 1992. Zbl0752.47002MR1223779
22. [22] R Rammal, Spectrum of harmonic excitations on fractals, J. de Physique45 (1984) 191-206. MR737523
23. [23] R Rammal, G Toulouse, J. Phys. Lett.44 (1983) L-13. 
24. [24] X Rudin, Real and Complex Analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill, NY, 1966. Zbl0142.01701MR210528
25. [25] C Sabot, Existence and uniqueness of diffusions on finitely ramified self-similar fractals, Ann. Scient. Ec. Norm. Sup., 4ème série30 (1997) 605-673. Zbl0924.60064MR1474807
26. [26] Sabot C., Density of states of diffusions on self-similar sets and holomorphic dynamics in Pk: the example of the interval [0,1], Comptes Rendus à l'Académie des Sciences, to appear. Zbl0922.60040
27. [27] Sibony N., Dynamique des applications rationnelles de Pk, preprint. MR1760844
28. [28] N Steinmetz, Rational Iteration, de Gruyter Studies in Mathematics, 16, W. de Gruyter, Berlin, 1993. Zbl0773.58010MR1224235