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

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

2011-02-16

arxiv.org/abs/1102.3401v3

Mathematics

Dynamical Systems

Scientific paper

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 $\pm\sqrt{2}\xrightarrow{(2)}0\xrightarrow{(4)}t\xrightarrow{(1)}...$. 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 Sierpi\'nski curve.

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