Attracting or parabolic cycles other than fixed points

Question

I am studying the following complex polynomial $$P(z) = \frac{2z^{4}-2z^{3}+2z^2-z}{2z^{3}-2z^{2}+3z-2}$$ and I would like to know, if there are attracting or parabolic cycles for $P(z)$ different from the attracting fixed points $0,1$ and $\infty$. By numerical computations one can see that the critical points can be associated with those fixed points. Does that mean that there are no such cycles? Or how would I find them?

Also, how one could determine the qualitative local nature of the fixed point at infinity? For example with a local phase diagram. How can I draw such thing at infinity?

Answer 1

The direct answer to your first question, as originally answered in the comments is no - any attracting or parabolic cycle must contain a critical point in its immediate basin. This is stated as theorems 9.3.1 and 9.3.2 of Beardon's Iteration of Rational Functions.

Your function has six critical points, which you find by solving $P'(z)=0$. You can determine the forward orbit of these easily enough by direct computation:

• $ 1 \to 1$ (the super-attractive fixed point)
• $0.477 \to 0$ (the attractive fixed point with multiplier $1/2$)
• $ 0.154441 \pm 0.590415 i \to 0$ (the attractive fixed point with multiplier $1/2$ again)
• $ 0.106856 \pm 1.67351 i \to \infty$ (the neutral fixed point with multiplier $1$)

The dynamical picture, with the fixed points marked red and critical points marked green looks like so:

Your second question, originally asked in the comments, is "how do we determine the qualitative local nature of the fixed point at infinity? how can I draw such thing at infinity?" The trick there is to conjugate the function $P$ by the reciprocal function $\varphi(z) = 1/z$. That is, we investigate $$p(z) = 1/P(1/z) = \frac{2 z^4-3 z^3+2 z^2-2 z}{z^3-2 z^2+2 z-2}.$$ Note that $\varphi$ swaps zero and infinity. Thus, where $\infty$ is a fixed point of $P$, we have that zero is a fixed point of $p$. Furthermore, the zero is a neutral fixed point of $p$ since $p'(0)=1$. Thus, we say that infinity is a neutral fixed point of $P$. We can also draw the Julia set of $p$ to have something like a picture of the Julia set of $P$ near infinity:

Better yet, we could draw the image on the Riemann sphere:

Note that there is a higher quality interactive animation with the implementation in this Observable notebook.