Rational maps with half symmetries, Julia sets, and Multibrot sets in parameter planes

In this paper, we first work out formulas for nontrivial rational maps of minimal degree that are invariant under precompositions by the elements of finite Kleinian groups. In fact, up to precompositions and post-compositions by Mobius transformations, all formulas can be written as real-coefficient rational maps. Then using computer-generated pictures we explore the Julia sets of such maps in some one-parameter families and the Multibrot sets in the parameter planes, and we observe that the classifications of the Julia sets of the maps in these families have many similarities with the Julia sets of singularly perturbed rational maps studied by McMullen, and more extensively by Devaney and his collaborators.