We can apply a well known theorem of

*Chaotic Dynamical Systems *(Jacobi's theorem) :

**Theorem**
Let $\displaystyle \lambda\in\mathbb{R}$ and consider:

$\displaystyle T_{\lambda}:S^1\rightarrow S^1,\quad T_{\lambda}(\theta)=\theta +2\pi\lambda$

Then, each orbit of $\displaystyle T_{\lambda}$ is dense in $\displaystyle S^1$ if $\displaystyle \lambda$ is irrational

**Particular case**
In our case we choose $\displaystyle \lambda=1/2\pi$ (irrational) , so:

$\displaystyle O^+=\{T^k(0):k\in\mathbb{N}\}=\{e^{ki}:k\in\mathbb {N}\}$

is dense in $\displaystyle S^1$ .

Fernando Revilla