Scaling properties of multifractals as an eigenvalue problem

Mitchell J. Feigenbaum, Itamar Procaccia, Tams Tél

Research output: Contribution to journalArticle

81 Citations (Scopus)


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
Issue number10
Publication statusPublished - Jan 1 1989

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics

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