Integrated density of states of self-similar Sturm–Liouville operators and holomorphic dynamics in higher dimension

Abstract

We investigate the integrated density of states of a Sturm–Liouville operator d dm d dx when the measure m is constructed from a self-similar measure on the interval [0,1]. We show that this involves the dynamics of a rational map on the complex projective plane P 2 , and we give an explicit formula for the integrated density of states in terms of the Green function of this map. This allows to deduce several results on the structure of the integrated density of states by a study of the dynamics of this map. This operator is a particular case of the so-called diffusions on self-similar sets and is relevant in this context. Indeed it is the first example, except for the sets of the Sierpinski gasket type (usually called decimable), where a connection is established between the spectrum of the operator and the dynamics of the iterates of a certain rational map. Therefore it is a new step toward a generalization of the initial work of Rammal and Toulouse (1983) and Rammal (1984).