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Spectral properties of self-similar lattices and iteration of rational maps. (English) Zbl 1036.82013
This monograph considers in a very general framework the spectral properties of the discrete Laplacian on lattices based on finitely-ramified self-similar sets as well as the continuous analogue on self-similar sets. In particular the paper generalizes the work of R. Rammal and G. Toulouse [J. Physique Lettres 44, L13–L19 (1983), ibid. 49, 1194–1197 (1982)] on the Sierpinski gasket; this set serves also as the basic example to illustrate the techniques used in the work.
The key tool is a new renormalization map, eventually rational, on some compact Kähler manifold. It is used to prove the existence of the (integrated) density and its relation to the Green function. In this way the spectral properties of the Laplace operator are related to the dynamical properties of the renormalization map. As a consequence a natural dichotomy is proven: Either eigenfunctions of compact support generically do not exist or if they do the (integrated) density of states is determined completely by the related eigenvalues. Furthermore, regularity properties of the (integrated) density of states are shown.
The article is self-contained, the necessary tools, e.g. Grassmann algebras, facts about pluriharmonic functions, currents, Green functions, iterations of meromorphic maps on compact complex manifolds are provided in the main text or in an extensive appendix. Several examples apart from the Sierpinski gasket are discussed.

MSC:
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
28A80 Fractals
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
39A70 Difference operators
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