Symbolic template iterations of complex quadratic maps

TL;DR

This paper explores how changing iteration patterns, or symbolic templates, influence the dynamics of complex quadratic maps, extending classical theory and suggesting applications in genetic and neural coding.

Contribution

It introduces the concept of symbolic template iterations for complex quadratic maps and analyzes their impact on Julia and Mandelbrot set topology, extending traditional dynamical systems theory.

Findings

Iteration patterns affect Julia and Mandelbrot set structures.

Classical Fatou-Julia theory partially extends to symbolic templates.

Potential applications in genetic and neural coding are discussed.

Abstract

The behavior of orbits for iterated logistic maps has been widely studied since the dawn of discrete dynamics as a research field, in particular in the context of the complex quadratic family. However, little is is known about orbit behavior if the map changes along with the iterations. We investigate in which ways the traditional theory of Fatou-Julia may still apply in this case, illustrating how the iteration pattern (symbolic template) can affect the topology of the Julia and Mandelbrot sets. We briefly discuss the potential of this extension towards a variety of applications in genetic and neural coding, since it investigates how an occasional or a reoccurring error in a replication or learning algorithm may affect the dynamic outcome.