On The Dynamics Of The Rational Family ${\mathbf{f}}_{\mathbf{t}}(\mathbf{z})=-\frac{\mathbf{t}}{4}\frac{({\mathbf{z}}^2-2{)}^2}{{\mathbf{z}}^2-1}$

Abstract

In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps $\ds f_t(z)=-\frac{t}{4}\frac{(z^{2}-2)^{2}}{z^{2}-1}$ with free critical orbit $\pm \sqrt{2}\stackrel{(2)}{\to }0\stackrel{(4)}{\to }t\stackrel{(1)}{\to }...$. In particular it is shown that for any escape parameter $t$ the boundary of the basin at infinity $\A_t$ is either a Cantor set, a curve with infinitely many complementary components, or else a Jordan curve. In the latter case the Julia set is a Sierpiński curve.