Fractal measures and their singularities: The characterization of strange sets

Thomas C. Halsey, Mogens H. Jensen, Leo P. Kadanoff, Itamar Procaccia, and Boris I. Shraiman
Phys. Rev. A 33, 1141 – Published 1 February 1986; Erratum Phys. Rev. A 34, 1601 (1986)
PDFExport Citation

Abstract

We propose a description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory. We focus upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: α, which determines the strength of their singularities; and f, which describes how densely they are distributed. The spectrum of singularities is described by giving the possible range of α values and the function f(α). We apply this formalism to the 2 cycle of period doubling, to the devil’s staircase of mode locking, and to trajectories on 2-tori with golden-mean winding numbers. In all cases the new formalism allows an introduction of smooth functions to characterize the measures. We believe that this formalism is readily applicable to experiments and should result in new tests of global universality.

  • Received 26 August 1985

DOI:https://doi.org/10.1103/PhysRevA.33.1141

©1986 American Physical Society

Erratum

Erratum: Fractal measures and their singularities: The characterization of strange sets [Phys. Rev. A 33, 1141 (1986)]

Thomas C. Halsey, Mogens H. Jensen, Leo P. Kadanoff, Itamar Procaccia, and Boris I. Shraiman
Phys. Rev. A 34, 1601 (1986)

Authors & Affiliations

Thomas C. Halsey, Mogens H. Jensen, Leo P. Kadanoff, Itamar Procaccia, and Boris I. Shraiman

  • The James Franck Institute, The Enrico Fermi Institute for Nuclear Studies, and Department of Chemistry, The University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637

References (Subscription Required)

Click to Expand
Issue

Vol. 33, Iss. 2 — February 1986

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review A

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×