Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Spectral properties of hierarchical lattices and iteration of rational maps


Code(s) de Classification MSC:

Résumé: In this text we are interested in spectral properties of discrete Laplace operators defined on lattices based on finitely-ramified self-similar sets. The basic example is the lattice based on the Sierpinski gasket. We introduce a new renormalization map which appears to be a rational self-map of a compact complex manifolds. We relate some characteristics of its dynamics with some characteristics of the spectrum of our operator. More specifically, we give an explicite 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 Neuman-Dirichlet eigenvalues which lead to eigenfunctions with compact support on the unbounded lattice. Depending on the asymptotic degree of the map we can prove drastic different spectral properties of the operator. Hence, this work aims at a generalization and a better understanding of the initial work of physisits Rammal and Toulouse on the Sierpinski gasket.

Mots Clés: Spectral theory of Schrödinger operators ; pluricomplex dynamics ; analysis on self-similar sets

Date: 2001-06-18

Prépublication numéro: PMA-670

Postscript file : PMA-670.ps