We describe the quadratic dynamics in certain two-component number systems, which like the complex numbers, can be expressed as rings of two by two real matrices. This description is accomplished using the properties of the real quadratic family and its derivative. We also demonstrate that the Mandelbrot set for any of these systems may be defined in two equivalent ways that are analogous to the two characterizations of the usual complex Mandelbrot set.
We describe the quadratic dynamics in certain three-component number systems, which like the complex numbers, can be expressed as rings of real matrices. This description is accomplished using the properties of the real quadratic family and its various first-and second-order phase and parameter derivatives. We demonstrate that the fundamental dichotomy of defining the Mandelbrot set either in terms of filled Julia sets or in terms of the orbit of the origin extends to these ternary number systems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.