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I have a question in Complex Analysis that
Let $z \in \mathbb{C} $ such that $|z|=1$. Find all limits of subsequences of the sequence $\{z^n\}$ in the cases:
(i) $\arg z \in \mathbb{Q}\pi$
(ii) $\arg z \notin \mathbb{Q}\pi$

Let $z=e^{i\theta}$ with $\theta = \frac{p}{q}\pi, p \in \mathbb{Z}, q \in \mathbb{Z_+}, (p,q)=1$. We have $$z^n=e^{\frac{inp\pi}{q}} = e^{\frac{i(n+2q)p\pi}{q}}=z^{n+2q}$$ and $$z^n \neq z^{n+l} \text{ for }l=0,1,...,2q-1$$ Thus all limits of subsequences are $$\{e^\frac{ik\pi}{2q}:k=0,1,...,2q-1\}$$

Geometrically, the limits are the vertices of the equilateral $2q-$polygonal inscribed in the unit circle.

However, I have no idea in the case $\theta \notin \mathbb{Q}\pi$. I think that the limits are aperiodic, so I cannot imagine how it works. Could you show me some suggestions? Thank you very much for helps.

Nal
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  • you are looking at the function $e^{it} \mapsto e^{in t}$ aka. $t \mapsto nt \bmod 2\pi$ – reuns Feb 27 '17 at 23:26
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    iterate of irrationnal rotations are dense in unit circle, there are many posts in SE on this subject. http://math.stackexchange.com/questions/1569152/dense-set-in-the-unit-circle-reference-needed?noredirect=1&lq=1 – zwim Feb 27 '17 at 23:33
  • Thank you so much. I get it. – Nal Mar 02 '17 at 21:09

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