On the Dynamics of Rational Maps with Two Free Critical Points

TL;DR

This paper analyzes the dynamics of a family of rational maps with two free critical points, revealing invariant basins and inactive critical points, and provides a detailed description of their dynamical and parameter planes.

Contribution

It characterizes the invariant basins and activity of critical points in a specific rational family, offering a detailed dynamical and parameter plane structure.

Findings

Either of the basins is completely invariant for certain parameter ranges.

At least one free critical point remains inactive in these ranges.

The paper provides a detailed map of the dynamical and parameter plane structures.

Abstract

In this paper we discuss the dynamical structure of the rational family $(f_t)$ given by $$f_t(z)=tz^{m}\Big(\frac{1-z}{1+z}\Big)^{n}\quad(m\ge 2,~t\ne 0).$$ Each map $f_t$ has two super-attracting immediate basins and two free critical points. We prove that for $0<|t|\le 1$ and $|t|\ge 1$, either of these basins is completely invariant and at least one of the free critical points is inactive. Based on this separation we draw a detailed picture the structure of the dynamical and the parameter plane.