$p$-adic alternated Julia sets

Abstract

The study of dynamical systems involves analyzing how functions behave under iteration in different mathematical spaces. In the context of complex dynamics, tools such as Julia sets and filled Julia sets are used to understand the long-term behavior of functions in the complex Euclidean field. In this paper, we will present a review of Julia sets and filled Julia sets, provide an overview of the mathematical formulation of the alternated Julia sets introduced in the work of Danca-Romera-Pastor, extend it to the $p$-adic setting, and propose a tool that can potentially be used to study the arithmetic dynamics of various types of functions. Additionally, we will summarize key results on connectivity properties and visualization techniques as discussed in the work of Danca-Bourke-Romera and provide a visualization algorithm and pseudocode that enable the visualization of alternated Julia sets with various connectivity properties.