Journal of Statistical Physics

, Volume 52, Issue 3–4, pp 527–569 | Cite as

Presentation functions, fixed points, and a theory of scaling function dynamics

  • Mitchell J. Feigenbaum


Presentation functions provide the time-ordered points of the forward dynamics of a system as successive inverse images. They generally determine objects constructed on trees, regular or otherwise, and immediately determine a functional form of the transfer matrix of these systems. Presentation functions for regular binary trees determine the associated forward dynamics to be that of a period doubling fixed point. They are generally parametrized by the trajectory scaling function of the dynamics in a natural way. The requirement that the forward dynamics be smooth with a critical point determines a complete set of equations whose solution is the scaling function. These equations are compatible with a dynamics in the space of scalings which is conjectured, with numerical and intuitive support, to possess its solution as a unique, globally attracting fixed point. It is argued that such dynamics is to be sought as a program for the solution of chaotic dynamics. In the course of the exposition new information pertaining to universal mode locking is presented.

Key words

Scaling thermodynamics period doubling mode locking dynamical systems chaos renormalization group 


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Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • Mitchell J. Feigenbaum
    • 1
  1. 1.Rockefeller UniversityNew York

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