REPELLER STRUCTURE IN A HIERARCHICAL MODEL .2. METRIC PROPERTIES

R LIVI, A POLITI, S RUFFO

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

We study the renormalization dynamics deriving from a hierarchical tight-binding Schrodinger 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 distribution f (alpha) of local dimensions is determined by computing both a generalized zeta-function through the periodic orbits and the bandwidths of periodic approximants of the Schrodinger operator. When the growth rate R 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 measure-mu of the spectrum is also studied. A linear scaling of mu to 0 is observed for R --> 1-, while for R > 1, the measure of the periodic approximants goes to 0 as R(-h) with the hierarchical order h. Finally, we show that the localized state, present for R < 1, is characterized by a superexponential scaling of the bandwidth.

Original languageEnglish
Pages (from-to)73-95
Number of pages23
JournalJournal of Statistical Physics
Volume65
Issue number1-2
Publication statusPublished - Oct 1991

Keywords

  • HIERARCHICAL STRUCTURES
  • MULTIFRACTALITY
  • PHASE TRANSITIONS
  • STRANGE REPELLERS
  • LOCALIZATION
  • STRANGE SETS
  • SPECTRUM
  • MAPS

Cite this

REPELLER STRUCTURE IN A HIERARCHICAL MODEL .2. METRIC PROPERTIES. / LIVI, R ; POLITI, A ; RUFFO, S .

In: Journal of Statistical Physics, Vol. 65, No. 1-2, 10.1991, p. 73-95.

Research output: Contribution to journalArticle

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title = "REPELLER STRUCTURE IN A HIERARCHICAL MODEL .2. METRIC PROPERTIES",
abstract = "We study the renormalization dynamics deriving from a hierarchical tight-binding Schrodinger 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 distribution f (alpha) of local dimensions is determined by computing both a generalized zeta-function through the periodic orbits and the bandwidths of periodic approximants of the Schrodinger operator. When the growth rate R 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 measure-mu of the spectrum is also studied. A linear scaling of mu to 0 is observed for R --> 1-, while for R > 1, the measure of the periodic approximants goes to 0 as R(-h) with the hierarchical order h. Finally, we show that the localized state, present for R < 1, is characterized by a superexponential scaling of the bandwidth.",
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AU - POLITI, A

AU - RUFFO, S

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N2 - We study the renormalization dynamics deriving from a hierarchical tight-binding Schrodinger 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 distribution f (alpha) of local dimensions is determined by computing both a generalized zeta-function through the periodic orbits and the bandwidths of periodic approximants of the Schrodinger operator. When the growth rate R 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 measure-mu of the spectrum is also studied. A linear scaling of mu to 0 is observed for R --> 1-, while for R > 1, the measure of the periodic approximants goes to 0 as R(-h) with the hierarchical order h. Finally, we show that the localized state, present for R < 1, is characterized by a superexponential scaling of the bandwidth.

AB - We study the renormalization dynamics deriving from a hierarchical tight-binding Schrodinger 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 distribution f (alpha) of local dimensions is determined by computing both a generalized zeta-function through the periodic orbits and the bandwidths of periodic approximants of the Schrodinger operator. When the growth rate R 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 measure-mu of the spectrum is also studied. A linear scaling of mu to 0 is observed for R --> 1-, while for R > 1, the measure of the periodic approximants goes to 0 as R(-h) with the hierarchical order h. Finally, we show that the localized state, present for R < 1, is characterized by a superexponential scaling of the bandwidth.

KW - HIERARCHICAL STRUCTURES

KW - MULTIFRACTALITY

KW - PHASE TRANSITIONS

KW - STRANGE REPELLERS

KW - LOCALIZATION

KW - STRANGE SETS

KW - SPECTRUM

KW - MAPS

M3 - Article

VL - 65

SP - 73

EP - 95

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -