Repeller structure in a hierarchical model. II. Metric properties View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

1991-10

AUTHORS

R. Livi, A. Politi, S. Ruffo

ABSTRACT

We study the renormalization dynamics deriving from a hierarchical tight-binding Schrödinger equation. In the Part I of this work we analyzed the topological structure of the recurrent set-a chaotic repeller- and its relation with the spectral problem. In this part we turn our attention to the metric properties of the repeller. We first study periodic orbits and their bifurcation unfolding, and we organize them on a binary tree. We then apply a thermodynamic formalism which provides a complete characterization of the scaling properties of the energy spectrum. The distributionf(α) of local dimensions is determined by computing both a generalized ζ-function through the periodic orbits and the bandwidths of periodic approximants of the Schrödinger operator. When the growth rateR of the potential is smaller than 1, we find evidence of a phase transition, implying that two different classes of states coexist in the spectrum. The asymptotic behavior of the Lebesgue measureμ of the spectrum is also studied. A linear scaling ofμ to 0 is observed forR → 1−, while forR > 1, the measure of the periodic approximants goes to 0 as R−h with the hierarchical orderh. Finally, we show that the localized state, present for R<1, is characterized by a superexponential scaling of the bandwidth. More... »

PAGES

73-95

References to SciGraph publications

  • 1987-09. The spectrum of a quasiperiodic Schrödinger operator in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1988-08. The spectrum of a one-dimensional hierarchical model in JOURNAL OF STATISTICAL PHYSICS
  • 1989-12. Cantor spectrum and singular continuity for a hierarchical Hamiltonian in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • Journal

    TITLE

    Journal of Statistical Physics

    ISSUE

    1-2

    VOLUME

    65

    Author Affiliations

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/bf01329851

    DOI

    http://dx.doi.org/10.1007/bf01329851

    DIMENSIONS

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