Symbolic template iterations of complex quadratic maps

The behavior of orbits for iterated polynomials has been widely studied since the dawn of discrete dynamics as a research field, in particular in the context of the complex quadratic family f:C→C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f :\mathbb {C} \rightarrow \mathbb {C}$$\end{document}, parametrized as fc(z)=z2+c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_c(z) = z^2 + c$$\end{document}, with c∈C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c \in \mathbb {C}$$\end{document}. While more recent research has been studying the orbit behavior when the map changes along with the iterations, many aspects of non-autonomous discrete dynamics remain largely unexplored. Our work is focused on studying the behavior of pairs of quadratic maps (1) when iterated according to a rule prescribed by a binary template and (2) when the maps are organized as nodes in a network, and interact in a time-dependent fashion. We investigate how the traditional theory changes in these cases, illustrating in particular how the hardwired structure (the symbolic template, and respectively the adjacency graph) can affect dynamics (behavior of orbits, topology of Julia and Mandelbrot sets). Our current manuscript addresses the first topic, while the second topic is the subject of a subsequent paper. This is of potential interest to a variety of applications (including genetic and neural coding), since (1) it investigates how an occasional or a reoccurring error in a replication or learning algorithm may affect the outcome and (2) it relates to algorithms of synaptic restructuring and neural dynamics in brain networks.