Spectral properties of self-similar lattices and iteration of rational maps
[Propriétés spectrales des réseaux auto-similaires et itération d’applications rationelles]
Mémoires de la Société Mathématique de France, no. 92 (2003) , 110 p.

Dans ce texte, nous considérons le laplacien discret, défini sur un réseau construit à partir d’un ensemble auto-similaire finiment ramifié, et son analogue continu défini sur l’ensemble auto-similaire lui-même. Nous nous intéressons aux propriétés spectrales de ces opérateurs. L’exemple le plus classique est celui du triangle de Sierpinski (Sierpinski gasket) et du réseau discret associé. Nous introduisons une nouvelle application de renormalisation qui se trouve être une application rationnelle définie sur une variété projective lisse (plus précisément, cette variété est un produit de grassmanniennes de trois types : grassmanniennes classiques, grassmanniennes lagrangiennes, grassmanniennes orthogonales). Nous relions certaines propriétés spectrales de ces opérateurs avec la dynamique des itérés de cette application. En particulier, nous donnons une formule explicite de la densité d’états en termes du courant de Green de l’application, et nous caractérisons le spectre de Neumann-Dirichlet (qui correspond aux fonctions propres à support compact sur l’ensemble infini) à l’aide des points d’indétermination de l’application. Suivant le degré asymptotique de l’application nous pouvons prouver que les propriétés spectrales de l’opérateur sont très différentes. Notre formalisme s’applique à la classe des ensembles auto-similaires finiment ramifiés (ou autrement dit à la classe des « p.c.f. self-similar sets » de Kigami). Ainsi, ce travail généralise et donne une compréhension plus profonde des résulats obtenus initialement par Rammal et Toulouse dans le cas du triangle de Sierpinski.

In this text we consider discrete Laplace operators defined on lattices based on finitely-ramified self-similar sets, and their continuous analogous defined on the self-similar sets themselves. We are interested in the spectral properties of these operators. The basic example is the lattice based on the Sierpinski gasket. We introduce a new renormalization map which appears to be a rational map defined on a smooth projective variety (more precisely, this variety is isomorphic to a product of three types of Grassmannians: complex Grassmannians, Lagrangian Grassmannian, orthogonal Grassmannians). We relate some characteristics of the dynamics of its iterates with some characteristics of the spectrum of our operator. More specifically, we give an explicit formula for the density of states in terms of the Green current of the map, and we relate the indeterminacy points of the map with the so-called Neumann-Dirichlet eigenvalues which lead to eigenfunctions with compact support on the unbounded lattice. Depending on the asymptotic degree of the map we can prove drastically different spectral properties of the operators. Our formalism is valid for the general class of finitely ramified self-similar sets (i.e. for the class of p.c.f. self-similar sets of Kigami). Hence, this work aims at a generalization and a better understanding of the initial work of the physicists Rammal and Toulouse on the Sierpinski gasket.

DOI : 10.24033/msmf.405
Classification : 82B44, 32H50, 28A80
Keywords: Spectral theory of Schrödinger operators, pluricomplex analysis, dynamics in several complex variables, electrical networks, analysis on self-similar sets, fractal graphs
Mot clés : Théorie spectrale, analyse et dynamique à plusieurs variables complexes, réseaux électriques, analyse sur les ensembles auto-similaires, graphes fractals 
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Sabot, Christophe. Spectral properties of self-similar lattices and iteration of rational maps. Mémoires de la Société Mathématique de France, Série 2, no. 92 (2003), 110 p. doi : 10.24033/msmf.405. http://numdam.org/item/MSMF_2003_2_92__1_0/

[1] S. Alexander – Some properties of the spectrum of the Sierpinski gasket in a magnetic field, Phys. Rev., B29 (1984), 5504–5508. | MR

[2] M.T. Barlow, E. Perkins – Brownian motion on the Sierpiński gasket. Probab. Theory Related Fields, 79 (1988), no 4, 543–623. | MR | Zbl

[3] M.T. Barlow, J. Kigami – Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets, J. Lond. Math. Soc., 56 (1997), no 2, 320–332. | MR | Zbl

[4] J. Bellissard – Renormalization group analysis and quasicrystals. Ideas and methods in quantum and statistical physics (Oslo, 1988), Cambridge Univ. Press, Cambridge, 1992, 118–148. | MR | Zbl

[5] F.A. Berezin – The method of second quantization. Translated from the Russian by Nobumichi Mugibayashi and Alan Jeffrey. Pure and Applied Physics, vol. 24, Academic Press, New York-London, 1966. | MR | Zbl

[6] M. Berger, A. Lascoux – Variétés kähleriennes compactes. (French) Lecture Notes in Mathematics, vol. 154, Springer-Verlag, Berlin-New York, 1970. | MR | Zbl

[7] D. Carlson – What are Schur complements, anyway? Linear Algebra Appl., 74 (1986), 257–275. | MR | Zbl

[8] R. Carmona, J. Lacroix – Spectral Theory of Random Schrödinger Operators, Probabilities and applications, Birkhaüser, Boston, 1990. | MR | Zbl

[9] Y. Colin De Verdière – Réseaux électriques planaires I, Comment. Math. Helv., 69 (1994), 351–374. | MR | EuDML | Zbl

[10] —, Déterminants et intégrales de Fresnel, Ann. Inst. Fourier, 49 (1999), no 3, 861–881. | MR | EuDML | Zbl | Numdam

[11] J.-P. Demailly – Monge-Ampère operators, Lelong numbers and intersection theory, complex analysis and geometry, Univ. Ser. Math., Plenum Press, 1993, 115–193. | MR | Zbl

[12] J. Diller, C. Favre – Dynamics of bimeromorphic maps of surfaces, Preprint | MR | Zbl

[13] C. Favre – Dynamique des applications rationelles. PhD thesis, Université Paris-Sud-Orsay.

[14] C. Favre, V. Guedj – Dynamique des applications rationelles des espaces multi-projectifs, To appear in Indiana Math. J.

[15] R. Forman – Functional determinants and geometry, Invent. Math., 88 (1987), 447–493. | MR | EuDML | Zbl

[16] J.E. Fornaess, N. Sibony – Complex dynamics in higher dimension II. Modern methods in complex analysis (Princeton, NJ, 1992), Ann. of Math. Stud., vol. 137, Princeton Univ. Press, Princeton, NJ, 1995, 135–182. | MR | Zbl

[17] M. Fukushima, Y. Oshima, M. Takeda – Dirichlet forms and symmetric Markov processes, de Gruyter Stud. Math. vol. 19, Walter de Gruyter, Berlin, New-york, 1994. | MR

[18] M. Fukushima – Dirichlet forms, diffusion processes and spectral dimensions for nested fractals, in : Ideas and Methods in Mathematical analysis, Stochastics and Applications, Proc. Conf. in Memory of Hoegh-Krohn, vol. 1 (S. Albevario et al., eds.), Cambridge Univ. Press, Cambridge, 1993, 151–161. | MR

[19] M. Fukushima, T. Shima – On the spectral analysis for the Sierpinski gasket, Potential Analysis, 1 (1992), 1–35. | MR | Zbl

[20] —, On the discontinuity and tail behaviours of the integrated density of states for nested pre-fractals., Comm. Math. Phys., 163 (1994), 461–471. | MR | Zbl

[21] V. Guedj – Representation theorems for positive closed (1,1)-currents on flag manifolds of GL m (ℂ), Preprint.

[22] P.A. Griffiths, J. Harris – Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994. | MR | Zbl

[23] B. Hambly – Brownian motion on a homogeneous random fractal. Probab. Theory Related Fields, 94 (1992), no 1, 1–38. | MR | Zbl

[24] L. Hörmander – Notions of convexity. Progress in Mathematics, vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994. | MR | Zbl

[25] J. Kigami – Harmonic calculus on p.c.f. self-similar sets, Trans. Am. Math. Soc., 335 (1993), 721–755. | MR | Zbl

[26] —, Distribution of localized eigenvalues of Laplacians on post-critically finite self-similar sets, J. Funct. Anal., 159 (1998), no 1, 170–198. | Zbl

[27] J. Kigami, M.L. Lapidus – Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Commun. Math. Phys., 158 (1993), no 1, 93–125. | MR | Zbl

[28] A. Klein – Extended states in the Anderson model on the Bethe lattice. Adv. Math., 133 (1998), no 1, 163–184. | MR | Zbl

[29] A. Korányi – A Schwartz lemma for bounded symmetric domains, Proc. Am. Math. Soc., 17 (1966), 210–213. | MR | Zbl

[30] S. Kusuoka – Dirichlet forms on fractals and products of random matrices. Publ. Res. Inst. Math. Sci., 25 (1989), no 4, 659–680. | MR | Zbl

[31] T. Lindstrøm – Brownian motion on nested fractals. Mem. Amer. Math. Soc., 420, 1990. | Zbl

[32] V. Metz – Shorted operators: an application in potential theory. Linear Algebra Appl., 264 (1997), 439–455. | MR | Zbl

[33] L. Pastur, A. Figotin – Spectra of Random and Almost-Periodic Operators, Grundlehren der mathematischen Wissenschaften, vol. 297, Springer-Verlag, Berlin Heidelberg, 1992. | MR | Zbl

[34] R. Rammal – Spectrum of harmonic excitations on fractals, J. de Physique, 45 (1984), 191–206. | MR

[35] R. Rammal, G. Toulouse – J. Phys. Lett., 44 (1983), L-13.

[36] C. Sabot – Existence and uniqueness of diffusions on finitely ramified self-similar fractals, in Ann. Scient. Ec. Norm. Sup., 4ème série, t. 30 (1997), 605–673. | MR | EuDML | Zbl | Numdam

[37] —, Espaces de Dirichlet reliés par des points et application aux diffusions sur les fractals finiment ramifiés. Potential Analysis, 11 (1999), no 2, 183–212. | MR

[38] —, Integrated density of states of self-similar Sturm-Liouville operators and holomorphic dynamics in higher dimension, Ann. Inst. H. Poincaré Probab. Statist., 37 (2001), no 3, 275–311. | MR | EuDML | Zbl | Numdam

[39] —, Pure point spectrum for the Laplacian on unbounded nested fractals. J. Funct. Anal., 173 (2000), no 2, 497–524. | MR | Zbl

[40] —, Schrödinger operators on fractal lattices with random blow-up, Preprint, arXiv:math-ph/0201041, to appear in Potential Analysis.

[41] —, Spectral Analysis of a self-similar Sturm-Liouville operator, preprint. | Zbl

[42] P. Sankaran, P. Vanchinathan – Small resolutions of Schubert varieties in symplectic and orthogonal Grassmannians. Publ. Res. Inst. Math. Sci., 30 (1994), no 3, 443–458 | MR | Zbl

[43] E. Seneta – Nonnegative matrices and Markov chains. Second edition. Springer Series in Statistics. Springer-Verlag, New York, 1981. | MR | Zbl

[44] J.-P. Serre – Linear Representations of Finite Groups, Graduated Texts in Mathematics, Springer-Verlag. | MR

[45] N. Sibony – Dynamique des applications rationnelles de ℙ k . (French) Dynamique et géométrie complexes (Lyon, 1997). Panor. Synthèses, 8, Soc. Math. France, Paris, 1999, 97–185. | MR

[46] C.L. Siegel – Symplectic geometry. Amer. J. Math., 65 (1943), 1–86. | MR | Zbl

[47] J. Sjöstrand, W.M. Wang – Exponential decay of averaged Green functions for random Schršdinger operators. A direct approach. Ann. Sci. École Norm. Sup. (4), 32 (1999), no 3. | EuDML

[48] R.S. Strichartz – Fractals in the large, Canad. Math. J., 50 (1998), no 3, 638–657. | MR | Zbl

[49] A. Teplyaev – Spectral Analysis on infinite Sierpinski Gasket, J. Funct. Anal., 159 (1998), no 2, 537–567. | MR | Zbl

[50] A. Terras – Harmonic analysis on symmetric spaces and applications, I, II, Spinger-Verlag New-York Inc. | MR

[51] W.M. Wang – Localization and universality of Poisson statistics for the multidimensional Anderson model at weak disorder, Invent. Math., 146 (2001), 365–398. | MR | Zbl

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