Consider the following rational functions: $R\_1(z)=z^2+z-1$ and $R\_2(z)=z+\frac{1}{z}$. The first has fixed points at $z^2+z-1=z \Rightarrow z=\pm 1.$ The second has a single fixed point at $\infty.$ 

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> Show that neither of the two following propositions implies the other:
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> > (i) $R$ has all of it's fixed points at $\xi$  
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> >  (ii) $R^{-1}\{\xi\}=\xi$
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My reasoning that $(i)$ and $(ii)$ are not related is because of the following:

Notice that $R\_1^{-1}(-1)=\{0,-1\}$ since $z^2+z-1=-1 \Rightarrow z^2+z=0 \Rightarrow z=0,-1$ But $R\_1(0)=-1$. So $0$ is not a fixed point.

But considering $R\_2$, we have $R\_2(0)=R\_2(\infty)=\infty$. But $R^{-1}\{{\infty}\}=\{0, \infty\}$