The Boundedness Locus and baby Mandelbrot sets for some generalized McMullen maps

Abstract

In this paper we study rational functions of the form $R_{n,a,c}(z)=z^n+\frac{a}{z^n}+c,$ with $n$ fixed and at least $3$, and hold either $a$ or $c$ fixed while the other varies. We locate some homeomorphic copies of the Mandelbrot set in the $c$-parameter plane for certain ranges of $a$, as well as in the $a$-plane for some $c$-ranges. We use techniques first introduced by Douady and Hubbard, that were applied for the subfamily $R_{n,a,0}$ by Robert Devaney. These techniques involve polynomial-like maps of degree two.

36 pages, 22 figures; in this version made minor editorial changes and improved some figures

Cited by in corpus (1)

• Baby Mandelbrot sets and Spines in some one-dimensional subspaces of the parameter space for generalized McMullen Maps