Asymptotic properties of sequences of iterates of nonlinear transformations

Abstract

By considering functions defined on the unit interval with a single zero minimum and a single unit maximum we are led to a version of the doubling or universal transformation. The fixed point functions of this doubling transformation have certain invariance properties under conjugacy. These invariance properties lead to a widening of the concept of universality to power law conjugacy classes in which the Feigenbaum divergence parameter δ is a function only of the product of the powers with which iterating functions approach unity at the maximum and zero at the minimum. We also construct an effective method for computing the divergence parameter from iterates, and derivatives of iterates, generated by the appropriate fixed point function.