mark

Show that $\infty$ is a super-attractive fixed point for the polynomial

$$p(z) = z^4 + 2z^3 - z - 1.$$

An archived instance of discourse for discussion in undergraduate Complex Dynamics.

mark

$$p(z) = z^4 + 2z^3 - z - 1.$$

RedCrayon

Note that $\phi(z)=1/z$ maps $\infty$ to zero. Conjugating $p$ with $\phi$ gives $$f(z)=\phi\circ p\circ\phi^{-1}(z)=\frac{z^4}{1+2z-z^3-z^4}$$ Note that $f(0)=0$ and $f^\prime(0)=0$, so zero is a super-attractive fixed point of $f$. This implies that $\infty$ is a super-attractive fixed point of $p$.