The role of functional equations to describe the exact local structure of highly bifurcated attractors of x_{n + 1} = λf(x_n ) independent of a specific f is formally developed. A hierarchy of universal functions g_r (x) exists, each descriptive of the same local structure but at levels of a cluster of 2^r points. The hierarchy obeys g_r-1(x) = − ag_r (g_r (x/α)), with g = lim _{r →∞} g_r 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 λ_r and bifurcation values Λ_r 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 g_r 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 g_r's, thereby providing some elementary approximation schemes for obtaining λ _r for a chosen f.