Generalized baby Mandelbrot sets adorned with halos in families of rational maps

摘要

We consider the family of rational maps given by F-lambda(z) = z(n) + lambda/z(d) where n, d is an element of N with 1/n + 1/d < 1, the variable z is an element of <(C)over cap> and the parameter lambda is an element of C. It is known [1] that when n = d >= 3 there are n - 1 small copies of the Mandelbrot set symmetrically located around the origin in the parameter lambda-plane. These baby Mandelbrot sets have 'antennas' attached to the boundaries of Sierpinski holes. Sierpinski holes are open simply connected subsets of the parameter space for which the Julia sets of F-lambda are Sierpinski curves. In this paper we generalize the symmetry properties of F-lambda and the existence of the n - 1 baby Mandelbrot sets to the case when 1/n + 1/d < 1 where n is not necessarily equal to d.