Advertisement

Journal of Statistical Physics

, Volume 27, Issue 1, pp 183–200 | Cite as

Asymptotic properties of sequences of iterates of nonlinear transformations

  • James B. McGuire
  • Colin J. Thompson
Articles

Abstract

By considering functions defined on the unit interval with a single zero minimum and a single unit maximum we are led to a version of the doubling or universal transformation. The fixed point functions of this doubling transformation have certain invariance properties under conjugacy. These invariance properties lead to a widening of the concept of universality to power law conjugacy classes in which the Feigenbaum divergence parameter δ is a function only of the product of the powers with which iterating functions approach unity at the maximum and zero at the minimum. We also construct an effective method for computing the divergence parameter from iterates, and derivatives of iterates, generated by the appropriate fixed point function.

Key words

Maps on an interval iteration conjugacy universal fixed point function functional equation doubling transformation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. Collet and J-P. Eckmann,Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Boston 1980).Google Scholar
  2. 2.
    R. M. May,Nature 261:459 (1976).Google Scholar
  3. 3.
    M. J. Feigenbaum,J. Stat. Phys. 19:25 (1978);21:669 (1979).Google Scholar
  4. 4.
    P. Collet, J-P. Eckmann, and O. E. Lanford III,Commun. Math. Phys. 76:211 (1980).Google Scholar
  5. 5.
    O. E. Lanford III, private communication.Google Scholar

Copyright information

© Plenum Publishing Corporation 1982

Authors and Affiliations

  • James B. McGuire
    • 1
  • Colin J. Thompson
    • 2
  1. 1.Physics DepartmentFlorida Atlantic UniversityBoca Raton
  2. 2.Institute for Advanced StudyPrinceton

Personalised recommendations