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Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers

Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
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Fractal Fract 2019, 3(1), 6; https://doi.org/10.3390/fractalfract3010006
Received: 31 December 2018 / Revised: 18 February 2019 / Accepted: 18 February 2019 / Published: 20 February 2019
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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 + τ y for x , y R , and τ 2 = 1 but τ ± 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets. View Full-Text
Keywords: hyperbolic numbers; Julia sets; Mandelbrot set hyperbolic numbers; Julia sets; Mandelbrot set
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Blankers, V.; Rendfrey, T.; Shukert, A.; Shipman, P.D. Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers. Fractal Fract 2019, 3, 6.

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