The Boundedness Locus and Baby Mandelbrot Sets for Some Generalized McMullen Maps

Abstract

In this paper, we study rational functions of the form [Formula: see text] with [Formula: see text] fixed and at least [Formula: see text], and hold either [Formula: see text] or [Formula: see text] fixed while the other varies. We locate some homeomorphic copies of the Mandelbrot set in the [Formula: see text]-parameter plane for certain ranges of [Formula: see text], as well as in the [Formula: see text]-plane for some [Formula: see text]-ranges. We use techniques first introduced by Douady and Hubbard [1985] that were applied for the subfamily [Formula: see text] by Devaney [2006]. These techniques involve polynomial-like maps of degree two.