Islands of stability and complex universality relations

For complex mappings of the type z→λz(1−z), universality constants α and δ can be defined along islands of stability lying on filamentary sequences in the complex λ plane. As the end of the filament is approached, asymptotic values α N ∼λ N−1 ∞, δ N /α2 N ∼1 are attained, where μ∞=λ∞(λ∞−2)/4, is associated with the limiting form of the universal function for that sequence, g(z)=1−μ∞ z 2. These results are complex generalizations of the real mapping case (applying to tangent bifurcations and windows of stability) where μ∞=2 and δ/α2→ (2)/(3) correspond to the filament running along the real axis.