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arXiv · 2012.03156

Quadratic Dynamics Over Hyperbolic Numbers

Sandra Hayes
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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

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Sandra Hayes. 2020-12-06. Quadratic Dynamics Over Hyperbolic Numbers. https://arxiv.org/abs/2012.03156

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