Classification of Principle 3D Slices of Filled-in Julia Sets in Multicomplex Spaces

TL;DR

This work extends filled-in Julia sets to multicomplex spaces and develops a concrete framework to classify their 3D slice representations. It introduces multicomplex algebra, an idempotent decomposition, and a visualization approach via principal 3D slices, augmented by an equivalence relation on slice spaces that respects the dynamics. The authors prove a precise count of principal 3D slices: $4$ slices for $p$ even, and $4/8/9$ slices for $p$ odd depending on $c$ and the ambient dimension $n$, highlighting intrinsic symmetries not present in the real or complex cases. The results refine and extend earlier work on multicomplex fractals, offering both theoretical classification and practical visualization guidance for higher-dimensional Julia sets.

Abstract

A generalization of the filled-in Julia set is presented using the multicomplex numbers and an algorithm is presented to visualize these sets in the tridimensional space. There are many ways to visualize these higher dimensional fractals sets on a computer. We therefore introduce an equivalence relation between 3D representations and show that, for the filled-in Julia sets associated to the polynomial $z^p + c$, there are nine 3D slices when $p$ is an odd integer and four when $p$ is even. These results differs from the recent characterization obtained by Brouillette and Rochon in 2019 and the proofs require different arguments in the context of the filled-in Julia sets.

Figures (2)

• Figure 1: Graphical results of principal 3D slices of the filled-in Julia set obtained using the algorithm
• Figure 2: Graphical results of principal 3D slices of the filled-in Julia set $\mathcal{K}_{3, 0.25}^3$

Theorems & Definitions (31)

• theorem 1
• theorem 2
• theorem 3
• definition 1
• theorem 4
• proof
• lemma 1
• proof
• theorem 5
• proof