$p$-adic alternated Julia sets

TL;DR

This paper extends the concept of Julia sets to the $p$-adic setting, providing a mathematical framework, visualization techniques, and exploring their connectivity properties to aid in the study of arithmetic dynamics.

Contribution

It introduces the $p$-adic alternated Julia sets, extending prior complex dynamics concepts to $p$-adic spaces and offering new visualization tools.

Findings

Extended Julia sets to $p$-adic spaces

Developed visualization algorithms for $p$-adic Julia sets

Analyzed connectivity properties of these 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.