I am trying to determine the set $\omega(z)$ of the possible subsqequences of the iterated sequence $(f^{\circ n}(z))_{n\in\mathbb{N}}$, with $f$ being the function described in the title and $z$ a complex number. I write $a=e^{i\theta}$ with $\theta\in\mathbb{R}$, and I consider two cases :
If $\theta\in2\pi\mathbb{Q}$ : we have that, writing $\theta=2\pi\frac{p}{q}$ with $p\in\mathbb{Z}$ and $q\in\mathbb{N}^*$, $f^{\circ q}(z)=z$. Thus, as $\omega(z)\subset\overline{\{z,f(z),\dots\}}=\{z,\dots,f^{\circ q-1}(z)\}$, we have that $\omega(z)$ is a finite set.
If $\theta\notin 2\pi\mathbb{Q}$ : I want to prove that $\omega(z)$ is the circle of center $0$ and of radius $|z|$, $S(0,|z|)$. The subgroup $\theta\mathbb{Z}+2\pi\mathbb{Z}$ is dense in $\mathbb{R}$, so $e^{i\theta\mathbb{Z}}$ is dense in $S^1=S(0,1)$. But : I don't know if (and I would to prove that !) $e^{i\theta\mathbb{N}}$ is dense in $S^1$, and how to use efficiently this fact to show that $\omega(z)=S(0,|z|)$ (if I can approximate any point $s\in S^1$ by a sequence $e^{i\theta a_n}$, $a_n\in\mathbb{N}$, how can I be assured that I can approximate $s$ by a sequence $e^{i\theta \tilde{a}_n}$ where $\tilde{a}_n$ is a strictly increasing sequence of positive integers ?).
Note that for the second point, I have tried something which perhaps does not lead to nothing : each of your comments/advice will be helpful. Thank you !
You may proceed as follows for the irrational case: Let $d(x,y)=\min_k|x-y-2k\pi|$ be a metric on ${\Bbb R}/2\pi {\Bbb Z}$. The sequence $x_n=n \theta$ (mod $2\pi$) must accumulate somewhere. So given $\epsilon>0$ there are $m<n$ so that $0<d(x_n,x_m)<\epsilon$ (equality is not possible by irrationality). But then $0<d(0,x_{n-m})<\epsilon$. Now, the subsequence $x_{k(n-m)}=kx_{n-m}$ mod $2\pi$ will be $\epsilon$ dense in $[0,2\pi]$.