# Julia and Mandelbrot sets for dynamics over the hyperbolic numbers

## TL;DR

This paper explores the hyperbolic analogs of Julia and Mandelbrot sets, revealing their structure and connection to hyperbolic numbers and Minkowski space, extending fractal dynamics beyond complex numbers.

## Contribution

It introduces the concept of hyperbolic Julia and Mandelbrot sets, analyzing their properties and the parameterization of connectedness in hyperbolic dynamics.

## Key findings

- Hyperbolic Mandelbrot set parameterizes connectedness of hyperbolic Julia sets.
- A wall-and-chamber decomposition of the hyperbolic plane is established.
- Hyperbolic sets exhibit fractal-like properties similar to complex Julia sets.

## Abstract

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form $x+\tau y$ for $x,y \in \mathbb{R}$, and $\tau^2 = 1$ but $\tau \neq \pm 1$, are the natural number system in which to encode geometric properties of the Minkowski space $\mathbb{R}^{1,1}$. We show that the hyperbolic analog of the Mandelbrot set parameterizes connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.