Roots of Unity of a specific argument

Question

I am asked to find an unstable period 5 point for $f(z)=z^2$ with an argument which lies between -0.74 and -0.44.

I can solve to get all the roots of unity, but how can I narrow it down the the one that will have an argument between -0.74 and -0.44?

Answer 1

The equation to solve is thus $f^5(z)-z=z^{32}-z=0$. Which gives $z=0$ and the unit roots of degree $31$. So you have to find all $k$ such that $$\frac{2\pi}{31}k\in[-0.74 , -0.44].$$

This can be transformed into an interval for $k$.

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For this point to be unstable you need $1<|f'(z)|=32\cdot |z^{31}|$, which obviously is satisfied for all unit roots.