Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers
Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
Author to whom correspondence should be addressed.
Received: 31 December 2018 / Revised: 18 February 2019 / Accepted: 18 February 2019 / Published: 20 February 2019
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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
, are the natural number system in which to encode geometric properties of the Minkowski space
. 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.
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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.
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.
Blankers, Vance; Rendfrey, Tristan; Shukert, Aaron; Shipman, Patrick D. 2019. "Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers." Fractal Fract 3, no. 1: 6.
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