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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 82B44 Disordered systems (random Ising models, random Schrodinger operators, etc.)
- 32H50 Iteration problems
- 28A80 Fractals, See also {58Fxx}

**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