On the Dynamics of the Rational Family $f_{t}(z)=- t/4{(z^{2}- 2)^{2}/(z^{2}- 1)}$

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 $${f_{t}(z)}= - {t\over 4} {(z^{2}- 2)^{2}\over {z^{2}- 1}}$$ with free critical orbit $$\pm\sqrt{2}\mathop \rightarrow \limits^{(2)}0 \mathop \rightarrow \limits^{(4)}t \mathop\rightarrow \limits^{(1)}\cdots.$$ In particular we show 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 Sierpinski curve.