On the dynamics of rational maps with two free critical points

Abstract

In this paper we discuss the dynamical structure of the rational family $(f_t)$ given by

$$\begin{aligned} f_t(z)=tz^{m}\left( \frac{1-z}{1+z}\right) ^{n}\quad (m\ge 2,~n\in \mathbb N,~t\in \mathbb C{\setminus }\{0\}). \end{aligned}$$

Each map $f_t$ has super-attracting immediate basins $\mathscr {A}_t$ and $\mathscr {B}_t$ about $z=0$ and $z=\infty$, respectively, and two free critical points. We prove that $\mathscr {A}_t$ (for $0<|t|\le 1$) and $\mathscr {B}_t$ (for $|t|\ge 1$) are completely invariant, and at least one of the free critical points is inactive. Based on this separation we draw a detailed picture of the structure of the dynamical and the parameter plane.