Universal metric properties of nonlinear transformations

The role of functional equations to describe the exact local structure of highly bifurcated attractors of x/sub n/+1=lambdaf (X/sub n/) independent of a specific f is formally developed. A hierarchy of universal functions g/sub tau/(X) exists, each descriptive of the same local structure but at levels of a cluster of 2/sup tau/ points. The hierarchy obeys g/sub tau-1/(X) =-αg/sub tau/(g/sub tau/(X/α)), with g =lim/sub tauarrow-rightinfinity/g/sub tau/ existing and obeying g (X) =-αg (g (X/α)), an equation whose solution determines both g and α. For r asymptotic g/sub tau/approx.g-delta/sup -tau/h where delta>1 and h are determined as the associated eigenvalue and eigenvector of the operator L: L[psi]=-α[psi (g (X/α))+g' (g (X/α)) psi (-X/α)] We conjecture that L possesses a unique eigenvalue in excess of 1, and show that this delta is the lambda-convergence rate. The form (*) is then continued to all lambda rather than just discrete lambda/sub tau/ and bifurcation values Λ/sub tau/ and dynamics at such lambda 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/sub tau/ limit of highly iterated lambdaf's, thus establishing our theory in a local sense. We show in the course of this that highly iterated lambdaf's are conjugate to g/sub tau/'s, thereby providing some elementary approximation schemes for obtaining lambda/sub tau/ for a chosen f