Complex universality

The theory of period doublings for one-parameter families of iterated real mappings is generalized to period n-tuplings for complex mappings. An n-tupling occurs when the eigenvalue of a stable periodic orbit passes through the value ω=exp(2 πim/n) as the parameter value is changed. Each choice of m defines a different sequence of n-tuplings, for which we construct a period n-tupling renormalization operator with a universal fixpoint function, a universal unstable manifold and universal scaling numbers. These scaling numbers can be organized by Farey trees. The present paper gives a general description and numerical support for the universality conjectured above. © 1989 Springer-Verlag.