Existence of the Mandelbrot set in parameter planes for some generalized McMullen maps

Abstract

In this paper we study rational functions of the form Rn,a,c(z) = z n + a z n + c and hold either a or c fixed while the other varies. We show that for certain ranges of a, the c-parameter plane contains homeomorphic copies of the Mandelbrot set, as well as the a-plane for certain ranges of c.We use techniques first introduced by Douady and Hubbard in [1] that were applied for the subfamily Rn,a,0 by Robert Devaney in [2]. These techniques involve polynomial-like maps of degree two.