Tricomplex dynamical systems generated by polynomials of even degree

TL;DR

This paper characterizes the geometric structure of Multibrot sets generated by even-degree polynomials in complex and hyperbolic numbers, revealing their interval bounds, shape properties, and 3D generalizations.

Contribution

It provides exact interval bounds for the Multibrot sets, shows they form squares in hyperbolic numbers, and introduces a 3D octahedral generalization for tricomplex polynomials.

Findings

Multibrot sets have explicitly determined cross section intervals.

Hyperbolic Multibrots are always squares.

3D generalization results in an octahedral shape.

Abstract

In this article, we give the exact interval of the cross section of the Multibrot sets generated by the polynomial $z^p+c$ where $z$ and $c$ are complex numbers and $p \geq 2$ is an even integer. Furthermore, we show that the same Multibrots defined on the hyperbolic numbers are always squares. Moreover, we give a generalized 3D version of the hyperbolic Multibrot set and prove that our generalization is an octahedron for a specific 3D slice of the tricomplex polynomial $\eta^p+c$ where $p \geq 2$ is an even integer.