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.
Highlights
The Mandelbrot set, arising from the study of dynamical systems on the complex plane, has been an object of interest ever since its introduction by Robert W
The Mandelbrot set MC is the set of complex numbers c ∈ C for which there exists some
Julia sets associated with the complex quadratic polynomial f c that defines the Mandelbrot set are shown in the center and right panels of Figure 1
Summary
The Mandelbrot set, arising from the study of dynamical systems on the complex plane, has been an object of interest ever since its introduction by Robert W. Julia sets associated with the complex quadratic polynomial f c that defines the Mandelbrot set are shown in the center and right panels of Figure 1. These examples illustrate a surprising connection of a topological nature between Mandelbrot and Julia sets given by the dichotomy theorem. Our main result is a wall-and-chamber decomposition of the hyperbolic plane, which gives the regions of the parameter space that give rise to each of these types of Julia sets This clarifies the results and statements of [7,9] by providing an explicit statement and proof of the hyperbolic-number analog to the dichotomy theorem: Quadchotomy Theorem.
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