Effects of local mutations in quadratic iterations.

Abstract

We introduce mutations in the process of discrete iterations of complex quadratic maps in the family fc(z)=z2+c. More specifically, we consider a "correct" function fc1 acting on the complex plane. A "mutation" fc0 is a different ("erroneous") map acting on a locus of given radius r around a mutation focal point ξ∗. The effect of the mutation is interpolated radially to eventually recover the original map fc1 when reaching an outer radius R. We call the resulting map a "mutated" map. In the theoretical framework of mutated iterations, we study how a mutation affects the temporal evolution of the system and the asymptotic behavior of its orbits. We use the prisoner set of the system to quantify simultaneously the long-term behavior of the entire space under mutated maps. We analyze how the position, timing, and size of the mutation can alter the system's long-term evolution (as encoded in the topology of its prisoner set). The framework is then discussed as a metaphoric model for studying the impact of copying errors in natural replication systems.