# Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

**Dichotomy**

**Theorem.**

**Quadchotomy**

**Theorem.**

#### Structure of the Paper

## 2. Hyperbolic Numbers

## 3. The Hyperbolic Mandelbrot Set

**Definition**

**3.**

**Definition**

**4.**

**Remark**

**1.**

**Lemma**

**1.**

- (i)
- a Cantor set not containing zero if $c<-2$,
- (ii)
- the interval $-{\rho}_{+}(c)\le x\le {\rho}_{+}(c)$ if $-2<c<\frac{1}{4}$,
- (iii)
- empty if $\frac{1}{4}<c$.

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

## 4. Hyperbolic Julia Sets

**Proposition**

**1.**

**Proof.**

**Theorem**

**2**

**(Quadchotomy).**

- (i)
- if $c\in S$, then ${\mathcal{K}}_{\mathbb{H}}({f}_{c})$ is nonempty and connected;
- (ii)
- if one of ${c}_{X},{c}_{Y}$ is in $[-2,\frac{1}{4}]$ and the other is less than or equal to $-2$, then ${\mathcal{K}}_{\mathbb{C}}({f}_{c})$ is disconnected;
- (iii)
- if ${c}_{X},{c}_{Y}<-2$, then ${\mathcal{K}}_{\mathbb{C}}({f}_{c})$ is totally disconnected;
- (iv)
- otherwise, ${\mathcal{K}}_{\mathbb{H}}({f}_{c})$ is empty.

**Proof.**

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Brooks, R.; Matelski, P. The dynamics of 2-generator subgroups of PSL(2,C). In Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference; Kra, I., Ed.; Princeton University Press: Princeton, NJ, USA, 1981; pp. 65–71. [Google Scholar]
- Devaney, R.L. A First Course in Chaotic Dynamical Systems: Theory and Experiment; Westview Press: Boulder, CO, USA, 1992. [Google Scholar]
- Harkin, A.A.; Harkin, J.B. Geometry of the Complex Numbers. Math. Mag.
**2004**, 77, 118–129. [Google Scholar] [CrossRef] - Catoni, F.; Boccaletti, D.; Cannata, R.; Catoni, V.; Nichelatti, E.; Zampetti, P. The Mathematics of Minkowski Space-Time with an Introduction to Commutative Hypercomplex Numbers; Birkhäuser: Basel, Switzerland, 2008. [Google Scholar]
- Catoni, F.; Boccaletti, D.; Cannata, R.; Catoni, V.; Zampetti, P. Geometry of Minkowski Space-Time; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Senn, P. The Mandelbrot set for binary numbers. Am. J. Phys.
**1990**, 58, 1018. [Google Scholar] [CrossRef] - Artzy, R. Dynamics of Quadratic Functions in Cycle Planes. J. Geom.
**1992**, 44, 26–32. [Google Scholar] [CrossRef] - Metzler, W. The “mystery” of the quadratic Mandelbrot set. Am. J. Phys.
**1994**, 62, 813–814. [Google Scholar] [CrossRef] - Fishback, P.E. Quadratic dynamics in binary number systems. J. Differ. Equ. Appl.
**2005**, 11, 597–603. [Google Scholar] [CrossRef] - Shipman, B.A.; Shipman, P.D.; Shipman, S.P. Lorentz-conformal transformations in the plane. Expos. Math.
**2017**, 35, 54–85. [Google Scholar] [CrossRef] [Green Version] - Fishback, P.E.; Horton, M.D. Quadratic dynamics in matrix rings: Tales of ternary numbers systems. Fractals
**2005**, 13, 147–156. [Google Scholar] [CrossRef]

**Figure 1.**The Mandelbrot and examples of Julia sets. Left panel: The Mandelbrot set is shown as a subset of parameter space $\mathbb{C}$. Center right panels: The filled Julia set for ${f}_{c}(z)={z}^{2}+c$ with $c=0.2$ (center panel) and $c=-1+0.5i$ (right panel). Colors represent the number of iterations before reaching the divergence criteria as described in [2]. That is, the colors represent the iterations performed before the norm of the iterate grew larger than a chosen bound (chosen to be four for these simulations). Red represents the quickest growth beyond our divergence criterion, whereas blue represents an initial condition whose orbit did not grow beyond the bound in the number of iterations we performed (200).

**Figure 2.**The hyperbolic Mandelbrot set is shown as a subset of parameter space $\mathbb{H}$ in the center panel. The four points labeled A–D in the center panel taken as parameter values c in ${f}_{c}(z)={z}^{2}+c$ give rise to the four types of Julia sets shown on the side panels: A, totally disconnected; B, disconnected; C, connected; and D, empty. As with Figure 1, the colors represent the iterations performed before the norm of the iterate grew larger than the bound (chosen to be four for these simulations). Red represents the quickest growth beyond our divergence criterion, whereas blue represents an initial condition whose orbit did not grow beyond the bound in the number of iterations we performed (200).

**Figure 3.**The wall-and-chamber decomposition of the quadchotomy theorem, Theorem 2. The hyperbolic plane in characteristic coordinates is divided into regions, as labeled, in which parameter values $({c}_{X},{c}_{Y})$ yield Julia sets for ${f}_{c}(z)={z}^{2}+c$, $c=\frac{1}{2}({c}_{X}+{c}_{Y})+\tau \frac{1}{2}({c}_{Y}-{c}_{X})$, ${c}_{X},{c}_{Y}\ne 0$, which are connected and nonempty, disconnected, but not totally disconnected, totally disconnected, or empty.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Blankers, V.; Rendfrey, T.; Shukert, A.; Shipman, P.D.
Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers. *Fractal Fract.* **2019**, *3*, 6.
https://doi.org/10.3390/fractalfract3010006

**AMA Style**

Blankers V, Rendfrey T, Shukert A, Shipman PD.
Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers. *Fractal and Fractional*. 2019; 3(1):6.
https://doi.org/10.3390/fractalfract3010006

**Chicago/Turabian Style**

Blankers, Vance, Tristan Rendfrey, Aaron Shukert, and Patrick D. Shipman.
2019. "Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers" *Fractal and Fractional* 3, no. 1: 6.
https://doi.org/10.3390/fractalfract3010006