Journal of Statistical Physics

, Volume 85, Issue 1–2, pp 261–276 | Cite as

Fractal properties of critical invariant curves

  • Brian R. Hunt
  • Konstantin M. Khanin
  • Yakov G. Sinai
  • James A. Yorke
Short Communications

Abstract

We examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers, through both the thermodynamic formalism and numerical computation. The maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed. Our results indicate that the dimension is universal for a given type of singularity and rotation number, and that among all rotation numbers, the golden mean produces the largest dimension.

Key Words

Thermodynamic formalism fractal dimension invariant measure circle homeomorphism rotation number twist map critical curve renormalization 

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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Brian R. Hunt
    • 1
  • Konstantin M. Khanin
    • 2
    • 3
  • Yakov G. Sinai
    • 2
    • 3
  • James A. Yorke
    • 1
  1. 1.Institute for Physical Science and TechnologyUniversity of MarylandCollege Park
  2. 2.Department of MathematicsPrinceton UniversityPrinceton
  3. 3.Landau Institute of Theoretical PhysicsRussian Academy of SciencesMoscowRussia

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