A Sierpinski Mandelbrot spiral for rational maps of the form Zᴺ + λ / Zᴰ

Abstract

We identify three structures that lie in the parameter plane of the rational map F(z) = zⁿ + λ / zᵈ, for which z is a complex number, λ a complex parameter, n ≥ 4 is even, and d ≥ 3 is odd.

There exists a Sierpindelbrot arc, an infinite sequence of pairs of Mandelbrot sets and Sierpinski holes, that limits to the parameter at the end of the arc.

There exists as well a qualitatively different Sierpindelbrot arc, an infinite sequence of pairs of Mandelbrot sets and Sierpinski holes, that limits to the parameter at the center of the arc.

Furthermore, there exist infinitely many arcs of each type. A parameter can travel along a continuous path from the Cantor set locus, along infinitely many arcs of the first type in a successively smaller region of the parameter plane, while passing through an arc of the second type, to the parameter at the center of the latter arc. This infinite sequence of Sierpindelbrot arcs is a Sierpinski Mandelbrot spiral.