Mandelbulb, Mandelbrot, Mandelring and Hopfbrot

Abstract

A topological ring R, an escape set B in R and a family of maps z^d+c defines the degree d Mandelstuff as the set of parameters for which the closure of the orbit of 0 does not intersect R. If B is the complement of a ball of radius 2 in C or H it is the complex or quaternionic Mandelbrot set; in a vector space with polar decomposition x=|x| U(t) like R=R^m, the map z^d+c is defined as the map z=|z| U(t) to z^d=|z|^d U(d t). We review some Jacobi Mandelstuff of quadratic maps on almost periodic Jacobi matrices which have the spectrum on Julia sets. In a Banach algebra R, one can define A^d=|A|^d U^d with A=|A| U written as the product of a self-adjoint |A| and unitary element U. In R^4, the Hopf parametrization leads to the Hopfbrot, which has White-Nylander Mandelbulbs in R=R^3 as traces and the standard Mandelbrot sets in C as codimension 2 traces. It is an open problem of White whether Mandelbulbs in higher dimensions are connected. The document contains an appendix with a proof of the Douady-Hubbard theorem.