Quadratic Dynamics Over Hyperbolic Numbers

TL;DR

This paper explores the unique dynamics of quadratic functions over hyperbolic numbers, revealing distinct geometric structures of Mandelbrot and Julia sets compared to complex numbers.

Contribution

It provides a comprehensive review of the properties of hyperbolic Mandelbrot and Julia sets, highlighting their differences from classical complex dynamics.

Findings

Hyperbolic Mandelbrot set is a filled square.

Filled Julia sets are rectangles or have 3 topological types.

Distinct topological behaviors outside the hyperbolic Mandelbrot set.

Abstract

Hyperbolic numbers are a variation of complex numbers, but their dynamics is quite different. The hyperbolic Mandelbrot set for quadratic functions over hyperbolic numbers is simply a filled square, and the filled Julia set for hyperbolic parameters inside the hyperbolic Mandelbrot set is a filled rectangle. For hyperbolic parameters outside the hyperbolic Mandelbrot set, the filled Julia set has 3 possible topological descriptions, if it is not empty, in contrast to the complex case where it is always a non-empty totally disconnected set. These results were proved in [1,2,4,5,6,7] and are reviewed here