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(2τπm/n) as the parameter value is changed. Each choice of m defines a different sequence of rc-tuplings, for which we construct a period ft-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.