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:**

- 37F10 Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]
- 32H50 Iteration problems
- 53D10 Contact manifolds, general
- 94C99 None of the above, but in this section
- 28A80 Fractals [See also 37Fxx]

**Résumé:** We present the renormalization map we introduced in [23] in order
to describe the spectral properties of self-similar lattices, from the
point of view of symplectic geometry.
We show that this map comes from a
symplectic reduction and that its key-properties
come from general properties of symplectic reductions, that we
prove in this text.
In particular, the singularities of the symplectic reduction,
considered as a rational map, are explicitly described and play an important
role. We also present new examples, where we can compute the renormalization map.

**Mots Clés:** *Spectral theory ; Symplectic geometry ; Symplectic reductions ; Lagrangian Grassmannians ;
Fractal geometry ; Laplace operators on fractals ; Dirichlet forms*

**Date:** 2003-03-12

**Prépublication numéro:** *PMA-806*

**Postscript file : **PMA-806.ps