Spectral properties of self-similar lattices and iteration of rational maps

TL;DR

This paper investigates the spectral properties of Laplace operators on self-similar lattices, introducing a rational renormalization map on complex varieties and relating its dynamics to the spectrum, including density of states and eigenvalues.

Contribution

It introduces a new rational renormalization map on complex Grassmannian varieties and links its dynamics to spectral characteristics of self-similar lattice operators, generalizing previous models.

Findings

Explicit formula for density of states via Green current

Relation between map indeterminacy points and Neuman-Dirichlet eigenvalues

Different spectral behaviors depending on the map's asymptotic degree

Abstract

In this text we consider discrete Laplace operators defined on lattices based on finitely-ramified self-similar sets, and their continuous analogous defined on the self-similar sets themselves. We are interested in the spectral properties of these operators. The basic example is the lattice based on the Sierpinski gasket. We introduce a new renormalization map which appears to be a rational map defined on a smooth projective variety (more precisely, this variety is isomorphic to a product of three types of Grassmannians: complex Grassmannians, Lagrangian Grassmannians, orthogonal Grassmannians). We relate some characteristics of the dynamics of its iterates with some characteristics of the spectrum of our operator. More specifically, we give an explicit 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 drastically different spectral properties of the operators. Our formalism is valid for the general class of finitely ramified self-similar sets (i.e. for the class of pcf self-similar sets of Kigami). Hence, this work aims at a generalization and a better understanding of the initial work of the physicists Rammal and Toulouse on the Sierpinski gasket.