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Zbl 1038.37036
Sabot, Christophe
Integrated density of states of self-similar Sturm-Liouville operators and holomorphic dynamics in higher dimension. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 37, No. 3, 275-311 (2001). ISSN 0246-0203

Summary: We investigate the integrated density of states of a Sturm-Liouville operator $\frac {d}{dm} \frac {d}{dx}$ 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 $\bbfP^2$, and we give an explicit formula for the integrated density of states in terms of the Green function of this map. This allows us 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 {\it R. Rammal} and {\it G. Toulouse} [Physique Lettres 44, L13--L22 (1983)] and {\it R. Rammal} [Spectrum of harmonic excitations on fractals. J. Physique 45, 191--206 (1984)].

MSC 2000:: *37F10 Polynomials; rational maps; entire and meromorphic functions
34B24 Sturm-Liouville theory
32H50 Iteration problems for holomorphic maps on analytic spaces
34L40 Particular ordinary differential operators
28A80 Fractals

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