Julia and Mandelbrot sets for dynamics over the hyperbolic numbers

2019/01/07 by Blankers, Vance, Rendfrey, Tristan, Shukert, Aaron +1
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.02085

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+τy for x,y ∈ ℝ, and τ² = 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 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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