Effects of local mutations in quadratic iterations

We introduce mutations in replication systems in which the intact copying mechanism is performed by discrete iterations of a complex quadratic map in the family $f_c(z) = z^2+c$. More specifically, we consider a "correct" function $f_{c_1}$ acting on the complex plane (representing the RNA to be copied). A "mutation" $f_{c_0}$ is a different ("erroneous") map acting on a locus of given radius $r$ around a mutation focal point $\xi^*$. The effect of the mutation is interpolated radially to eventually recover the original map $f_{c_1}$ 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 (replication error) affects the temporal evolution of the system, in the context of cellular differentiation. 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). In the context of genetics, this framework may increase our understanding of the factors and mechanisms that shape the genetic expression, in a specialized cell, in the process of differentiation from a stem cell.