Spectral properties of self-similar lattices and iteration of rational maps

Abstract

Abstract: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 Grassmannians, 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 pcf selfsimilar sets of Kigami, cf [25]). 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 (cf [35], [34]).Résumé: Dans ce texte, nous considérons le Laplacien discret, défini sur un réseau(1) E-mail address: sabot@ccr.jussieu.fr 1 2 construità 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 trouveêtre une application rationnelle définie sur une variété projective lisse (plus précisément, cette variété est un produit de Grassmaniennes de trois types: Grassmaniennes classiques, Grassmaniennes Lagrangiennes, Grassmaniennes 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 propresà 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'appliqueà la classe des ensembles auto-similaires finiment ramifiés (ou autrement dità la classe des "pcf self-similar sets" de Kigami, cf [25]

Spectral analysis of a self-similar Sturm-Liouville operator

“…This seems to be the natural counterpart of the statistical invariance by translation for random Schrödinger operators (cf [12] for details). Thanks to the hypothesis (H), we can easily check that the operators H ± <n> (ω) are isomorphic for different ω.…”

“…By the previous remark, ν ± <n> does not depend on ω. We define the density of states as the limit µ = lim n→∞ 1 2 n ν ± <n> which exists and does not depend on the boundary condition ± (cf [3], [8] or [12]). …”

“…In [12] we gave similar formulas in the general case of finitely ramified self-similar sets for the density of states in terms of the Green function of a certain renormalization map. In [11] we deduced from the analysis of the dynamics of f that supp(µ) is a Cantor subset of R − for δ = 1 2 , and for certain values of the parameter δ we proved the local Hölder regularity of the Lyapounov exponent.…”

“…This study is motivated by the problem of understanding the nature of the spectrum for the class of self-similar Laplacians on finitely ramified selfsimilar sets. In [11], [12], [14], we proved that the spectral properties of these operators are related to the dynamics of a certain renormalization map, which is a rational map of a smooth complex projective variety. The main question emerging from these works is the nature of the spectrum of these operators in the non-degenerate case d ∞ = N (cf [12], d ∞ is the asymptotic degree of the renormalization map, N is the number of subcells of the set at level 1).…”

“…The strategy we adopt here to obtain the renormalization equation is a bit different than in [12], and much more straightforward: we write a renormalization equation directly on the propagator of the differential equation associed with the spectral problem. This simplification comes from the 1-dimensional nature of the problem in this example.…”