Quadratic dynamics over hyperbolic numbers: a brief survey

Abstract

Hyperbolic numbers, also called split complex or perplex numbers in the literature, are a variation of complex numbers established as a theory primarily by W. Clifford in the nineteenth century who applied them to mechanics. Today hyperbolic numbers are considered to be the basic mathematics of Einstein's theory of special relativity. The goal of this paper is to present concise, direct proofs of results contained in ite1,2,4,5,6,7 on the dynamics of hyperbolic numbers which is quite different from the dynamics of complex numbers. 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, namely it is connected, totally disconnected, or disconnected but not totally disconnected. This is in contrast to the complex case where it is always a non-empty totally disconnected set.

Keywords

• hyperbolic (split complex or perplex) numbers
• quadratic dynamics
• mandelbrot set
• filled julia sets