### 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 language | English |
---|---|

Pages (from-to) | 73-95 |

Number of pages | 23 |

Journal | Journal of Statistical Physics |

Volume | 65 |

Issue number | 1-2 |

Publication status | Published - Oct 1991 |

### Keywords

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

### Cite this

*Journal of Statistical Physics*,

*65*(1-2), 73-95.

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

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 65, no. 1-2, pp. 73-95.

}

TY - JOUR

T1 - REPELLER STRUCTURE IN A HIERARCHICAL MODEL .2. METRIC PROPERTIES

AU - LIVI, R

AU - POLITI, A

AU - RUFFO, S

PY - 1991/10

Y1 - 1991/10

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 -