Topological structures in complex dynamics

Abstract

[en] Many sets in the field of planar topology are considered "exotic" and not frequently encountered in everyday life. However, these sets possess unique and intriguing topological properties, and quite often also a visually appealing aesthetic. In recent years, thanks to the resurgence of complex dynamics, many of these exotic sets have been found to be Julia sets for complex analytic functions. In this work, we delve into the world of planar topology, provide an overview of the basics of complex dynamics, and present four examples of such sets: den- drites, Cantor sets, Sierpiński curves, and Cantor bouquets. To conclude, we also explain how these sets arise through specific families of complex maps, such as the quadratic family, the complex exponential family, and a certain type of singularly perturbed rational maps.