Spectral properties of self-similar lattices and iteration of rational maps
Mémoires de la Société Mathématique de France, no. 92 (2003) , 110 p.

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.

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.

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, Serie 2, no. 92 (2003), 110 p. doi : 10.24033/msmf.405. http://numdam.org/item/MSMF_2003_2_92__1_0/

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

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

[3] M.T. Barlow, J. KigamiLocalized 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. BellissardRenormalization 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. BerezinThe 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. LascouxVariétés kähleriennes compactes. (French) Lecture Notes in Mathematics, vol. 154, Springer-Verlag, Berlin-New York, 1970. | MR | Zbl

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

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

[9] Y. Colin De VerdièreRéseaux 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. DemaillyMonge-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. FavreDynamics of bimeromorphic maps of surfaces, Preprint | MR | Zbl

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

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

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

[16] J.E. Fornaess, N. SibonyComplex 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. TakedaDirichlet forms and symmetric Markov processes, de Gruyter Stud. Math. vol. 19, Walter de Gruyter, Berlin, New-york, 1994. | MR

[18] M. FukushimaDirichlet 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. ShimaOn 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. GuedjRepresentation theorems for positive closed (1,1)-currents on flag manifolds of GL m (), Preprint.

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

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

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

[25] J. KigamiHarmonic 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. LapidusWeyl’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. KleinExtended states in the Anderson model on the Bethe lattice. Adv. Math., 133 (1998), no 1, 163–184. | MR | Zbl

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

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

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

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

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

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

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

[36] C. SabotExistence 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. VanchinathanSmall resolutions of Schubert varieties in symplectic and orthogonal Grassmannians. Publ. Res. Inst. Math. Sci., 30 (1994), no 3, 443–458 | MR | Zbl

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

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

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

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

[47] J. Sjöstrand, W.M. WangExponential 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. StrichartzFractals in the large, Canad. Math. J., 50 (1998), no 3, 638–657. | MR | Zbl

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

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

[51] W.M. WangLocalization 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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