  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.
