Spectral Properties of Self-Similar Lattices and Iteration of Rational Maps
The method of a renormalization group, widely used in physics, produces some considerable mathematical difficulties, when applied to systems on usual, regular lattices. Hierarchical, self-similar lattices are much better suited for a detailed mathematical development of this method. The present book gives a detailed treatment of the problems, which were first studied by Rammal and Toulouse on the Sierpiński gasket, and puts them in a more general perspective. The goal of the book is to study systematically, in a mathematically rigorous way, the spectral properties of Laplace operators on self-similar sets. A new renormalization map, which is a rational map defined on a smooth projective variety, is introduced. Then, the characteristics of the spectrum of these operators are related with the characteristics of the dynamics of iterates of such a renormalization map. An explicit formula for the density of states is given, and it is shown that the spectral properties of the operator depend substantially on the asymptotic degree of the renormalization map. The contents (a slightly shortened list of the chapters of the book) are: Definitions and basic results (self-similar Laplacian, density of states); Preliminaries (Grassmann algebra, trace of a Dirichlet form); The renormalization map (construction, the main theorem in the lattice case); Analysis of the psh function G (the dichotomy theorem, asymptotic degree, regularity of the density of states, some related rational maps); Examples; Remarks, questions and conjecture; and Appendix (plurisubharmonic functions and positive currents, dynamics of rational maps on projective space, iteration of meromorphic maps on compact manifolds, G - Lagrangian Grassmanian).