By Mitchell J. Feigenbaum (auth.), Roberto Artuso, Predrag Cvitanović, Giulio Casati (eds.)
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O(X1. l) F ( ) F ( ) Cn +1 ... Co - eo Xen+l ... el - (52) to Xen+l ... et But now notice that the final n c's are exactly the same in every argument of the F's. If the final n c's are the same, that means the same linear restriction of Fo. Taking a difference eliminates the constant value, the slope cancels out in the fraction, so (52) can be rewritten as (53) 20 This is the final result. The statement is that if Fo is constructed of 2" linear restrictions, then the scaling function to all orders is just the 2,,+1 values determined by the nth level set of scalings.
In particular, notice that however many € 's I have, it is only the first n +1 of themthe refinement information, corresponding to higher powers of 2- that ()' depends upon. At this level of 2" linear restrictions, the fall-off is not exponential, it is absolutely strict, there is no memory after n + 1 levels, so this is a replacement of the continuum problem by a problem with a finite amount of memory. This statement essentially finishes the computation of the scaling function. If you understand well what is in this statement you can immediately write down the equations that determine the scaling function, and you no longer need the renormalization group theory.
Feigenbaum. "Universal metric properties of non-linear transformations". J. Stat. Phys. 21,669-706 (1979). [MY] W de Melo and S. Van Strien. "Schwarzian derivative and beyond". Bull Amer Math Soc 18,159-162 (1988). [S 1] D. Sullivan D. "Bounds, quadratic, differentials, and renormalization conjectures ". To appear in AMS volume (2) (1991) celebrating the Centennial of the American Mathematical Society. [S2] D. Sullivan. "Quasiconformal homeomorphisms III dynamics, topology, and geometry". ICM Berkeley 1986.