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.
[1]
Mitchell J. Feigenbaum,et al.
The onset spectrum of turbulence
,
1979
.
[2]
Mitchell J. Feigenbaum,et al.
The transition to aperiodic behavior in turbulent systems
,
1980
.
[3]
S. Shenker,et al.
Quasiperiodicity in dissipative systems: A renormalization group analysis
,
1983
.
[4]
Jensen,et al.
Erratum: Fractal measures and their singularities: The characterization of strange sets
,
1986,
Physical review. A, General physics.
[5]
Jonathan D. Victor,et al.
Local structure theory for cellular automata
,
1987
.
[6]
Mitchell J. Feigenbaum.
Some characterizations of strange sets
,
1987
.
[7]
Mitchell J. Feigenbaum,et al.
Scaling spectra and return times of dynamical systems
,
1987
.
[8]
Procaccia,et al.
Scaling properties of multifractals as an eigenvalue problem.
,
1989,
Physical review. A, General physics.