ABSTRACT
The role of functional equations to describe the exact local structure of highly bifurcated attractors of xn + 1 = λf(xn ) independent of a specific f is formally developed. A hierarchy of universal functions gr (x) exists, each descriptive of the same local structure but at levels of a cluster of 2r points. The hierarchy obeys gr-1(x) = − agr (gr (x/α)), with g = lim r →∞ gr existing and obeying g(x) = −αg(g(x/α)), an equation whose solution determines both g and α. For r asymptotic
g r ∼ g − δ − r h https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203734636/af41cd04-f159-4387-917e-fec41ddcf0d6/content/equ18_1a.tif"/>where δ > 1 and h are determined as the associated eigenvalue and eigenvector of the operator L:
L [ ψ ] = − α [ ψ ( g ( x / α ) ) + g ′ ( g ( x / α ) ) ψ ( − x / α ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203734636/af41cd04-f159-4387-917e-fec41ddcf0d6/content/unequ18_207_1.tif"/>We conjecture that L possesses a unique eigenvalue in excess of 1, and show that this δ is the λ-convergence rate. The form (*) is then continued to all λ rather than just discrete λr and bifurcation values Λr and dynamics at such λ is determined. These results hold for the high bifurcations of any fundamental cycle. We proceed to analyze the approach to the asymptotic regime and show, granted L's spectral conjecture, the stability of the gr limit of highly iterated λf's, thus establishing our theory in a local sense. We show in the course of this that highly iterated λf's are conjugate to gr's, thereby providing some elementary approximation schemes for obtaining λ r for a chosen f.
