Abstract
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.
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Feigenbaum, M.J. Presentation functions, fixed points, and a theory of scaling function dynamics. J Stat Phys 52, 527–569 (1988). https://doi.org/10.1007/BF01019716
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DOI: https://doi.org/10.1007/BF01019716