A Study of Dynamics of the Tricomplex Polynomial $\eta^p+c$

TL;DR

This paper explores the geometric properties of Mandelbric and Multibrot sets generated by polynomial functions in complex, hyperbolic, and tricomplex numbers, revealing their shapes and slice structures.

Contribution

It provides exact interval descriptions of Mandelbric cross sections and characterizes the shape and slice structure of Multibrot sets in tricomplex numbers, including the discovery of an octahedral slice.

Findings

Mandelbric cross section intervals are exactly determined.

Mandelbric on hyperbolic numbers forms a square centered at origin.

Tricomplex Mandelbric has four principal slices, one of which is an octahedron.

Abstract

In this article, we give the exact interval of the cross section of the so called Mandelbric set generated by the polynomial $z^3+c$ where $z$ and $c$ are complex numbers. Following that result, we show that the Mandelbric defined on the hyperbolic numbers $\mathbb{D}$ is a square with its center at the origin. Moreover, we define the Multibrot sets generated by a polynomial of the form $Q_{p,c}(\eta )=\eta^p+c$ ($p \in \mathbb{N}$ and $p \geq 2$) for tricomplex numbers. More precisely, we prove that the tricomplex Mandelbric has four principal slices instead of eight principal 3D slices that arise for the case of the tricomplex Mandelbrot set. Finally, we prove that one of these four slices is an octahedron.