Scaling properties of multifractals as an eigenvalue problem

Mitchell J. Feigenbaum, Itamar Procaccia, T. Tél

Research output: Contribution to journalArticle

81 Citations (Scopus)

Abstract

The calculation of the scaling properties of multifractal sets is presented as an eigenvalue problem. The eigenvalue equation unifies the treatment of sets that can be organized on regular treesbe they complete or incomplete (pruned) trees. In particular, this approach unifies the multifractal analyses of sets at the borderline of chaos with those of chaotic sets. Phase transitions in the thermodynamic formalism of multifractals are identified as a crossing of the largest discrete eigenvalue with the continuous part of the spectrum of the relevant operator. An analysis of the eigenfunctions is presented and several examples are solved in detail. Of particular interest is the analysis of intermittent maps, which shows the existence of an infinite-order phase transitions, and of (Smale) incomplete maps of the interval with finite and infinite rules of pruning of the multifractal trees.

Original languageEnglish
Pages (from-to)5359-5372
Number of pages14
JournalPhysical Review A
Volume39
Issue number10
DOIs
Publication statusPublished - 1989

Fingerprint

eigenvalues
scaling
chaos
eigenvectors
formalism
intervals
operators
thermodynamics

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Atomic and Molecular Physics, and Optics

Cite this

Scaling properties of multifractals as an eigenvalue problem. / Feigenbaum, Mitchell J.; Procaccia, Itamar; Tél, T.

In: Physical Review A, Vol. 39, No. 10, 1989, p. 5359-5372.

Research output: Contribution to journalArticle

Feigenbaum, Mitchell J. ; Procaccia, Itamar ; Tél, T. / Scaling properties of multifractals as an eigenvalue problem. In: Physical Review A. 1989 ; Vol. 39, No. 10. pp. 5359-5372.
@article{085cc3ece3384094bfce10d529d9d272,
title = "Scaling properties of multifractals as an eigenvalue problem",
abstract = "The calculation of the scaling properties of multifractal sets is presented as an eigenvalue problem. The eigenvalue equation unifies the treatment of sets that can be organized on regular treesbe they complete or incomplete (pruned) trees. In particular, this approach unifies the multifractal analyses of sets at the borderline of chaos with those of chaotic sets. Phase transitions in the thermodynamic formalism of multifractals are identified as a crossing of the largest discrete eigenvalue with the continuous part of the spectrum of the relevant operator. An analysis of the eigenfunctions is presented and several examples are solved in detail. Of particular interest is the analysis of intermittent maps, which shows the existence of an infinite-order phase transitions, and of (Smale) incomplete maps of the interval with finite and infinite rules of pruning of the multifractal trees.",
author = "Feigenbaum, {Mitchell J.} and Itamar Procaccia and T. T{\'e}l",
year = "1989",
doi = "10.1103/PhysRevA.39.5359",
language = "English",
volume = "39",
pages = "5359--5372",
journal = "Physical Review A - Atomic, Molecular, and Optical Physics",
issn = "1050-2947",
publisher = "American Physical Society",
number = "10",

}

TY - JOUR

T1 - Scaling properties of multifractals as an eigenvalue problem

AU - Feigenbaum, Mitchell J.

AU - Procaccia, Itamar

AU - Tél, T.

PY - 1989

Y1 - 1989

N2 - The calculation of the scaling properties of multifractal sets is presented as an eigenvalue problem. The eigenvalue equation unifies the treatment of sets that can be organized on regular treesbe they complete or incomplete (pruned) trees. In particular, this approach unifies the multifractal analyses of sets at the borderline of chaos with those of chaotic sets. Phase transitions in the thermodynamic formalism of multifractals are identified as a crossing of the largest discrete eigenvalue with the continuous part of the spectrum of the relevant operator. An analysis of the eigenfunctions is presented and several examples are solved in detail. Of particular interest is the analysis of intermittent maps, which shows the existence of an infinite-order phase transitions, and of (Smale) incomplete maps of the interval with finite and infinite rules of pruning of the multifractal trees.

AB - The calculation of the scaling properties of multifractal sets is presented as an eigenvalue problem. The eigenvalue equation unifies the treatment of sets that can be organized on regular treesbe they complete or incomplete (pruned) trees. In particular, this approach unifies the multifractal analyses of sets at the borderline of chaos with those of chaotic sets. Phase transitions in the thermodynamic formalism of multifractals are identified as a crossing of the largest discrete eigenvalue with the continuous part of the spectrum of the relevant operator. An analysis of the eigenfunctions is presented and several examples are solved in detail. Of particular interest is the analysis of intermittent maps, which shows the existence of an infinite-order phase transitions, and of (Smale) incomplete maps of the interval with finite and infinite rules of pruning of the multifractal trees.

UR - http://www.scopus.com/inward/record.url?scp=0000253786&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000253786&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.39.5359

DO - 10.1103/PhysRevA.39.5359

M3 - Article

VL - 39

SP - 5359

EP - 5372

JO - Physical Review A - Atomic, Molecular, and Optical Physics

JF - Physical Review A - Atomic, Molecular, and Optical Physics

SN - 1050-2947

IS - 10

ER -