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/

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