As a particular problem within the field of non-autonomous discrete systems, we consider iterations of two quadratic maps f c0 = z 2 + c 0 and f c1 = z 2 + c 1 , according to a prescribed binary sequence, which we call template. We study the asymptotic behavior of the critical orbits, and define the Mandelbrot set in this case as the locus for which these orbits are bounded. However, unlike in the case of single maps, this concept can be understood in several ways. For a fixed template, one may consider this locus as a subset of the parameter space in (c 0 , c 1 ) ∈ C 2 ; for fixed quadratic parameters, one may consider the set of templates which produce a bounded critical orbit. In this paper, we consider both situations, as well as hybrid combinations of them, we study basic topological properties of these sets and propose applications.Using random iterations of discrete functions, we build a mathematical framework that can be used to study the effect of errors in copying mechanisms (such as DNA replication). In our theoretical setup -in which one of the functions is the correct one, and the other one is the erroneous perturbation -we consider problems that a sustainable replication system may have to solve when facing the potential for errors. We find that it is possible to tell which specific errors are more likely to affect the system's dynamics, in absence of prior knowledge of their timing. Moreover, within an optimal locus for the correct function, almost no errors can affect the sustainability of the system. Mathematically, our work complements broader existing results in nonautonomous dynamics with more specific detail for the case of two random iterated functions, which is a valuable context for applications.
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