Iterated Maps Using Complex, Dual, and Split-Complex Numbers

The classical iterated complex number mapping is a quadratic function of the form with the seed and a complex constant . A complex number is a member of a filled-in Julia set if it is a seed for a mapping that remains bounded after a large number of iterations. Different values of will produce different Julia sets, and any constant associated with a non-empty Julia set is a member of the Mandelbrot set. This Demonstration offers images of Julia and Mandelbrot sets on complex, dual, or split-complex planes. The left panel shows the Mandelbrot set containing values of constant that are associated with non-empty iterated quadratic Julia sets for the selected number type. You can select different values of the constant with the mouse, and the corresponding Julia set, if one exists, appears on the right.

Binary (two-part) numbers have the form , where and are real and is defined by the property for complex numbers, for dual numbers, and for split-complex numbers (sometimes called perplex numbers). This Demonstration extends to dual and split-complex numbers the iterated quadratic mappings of complex numbers that yield Julia and Mandelbrot sets.