If $|w|=1$ and $w^n$≠1 for any $n \in \mathbb{N}_+$, show that $ \{w^n ; n \in \mathbb{N}_+\}$ is dense in ∂$\Bbb D$.

Question

The problem is that in the title:

If $|w|=1$ and $w^n\not=1$ for any $n\in \mathbb{N}_+$, show that $\{w^n ; n \in \mathbb{N}_+\}$ is dense in $\partial \mathbb D$, where $\mathbb D$ is the open unit disk in $\mathbb C$.

I found this proposition in Composition operator theory by Xuxianming (page $47$), which is a book in Chinese.

Any hint will be appreciated, thanks.

Does this answer your question? Prove that the orbit of an iterated rotation of 0 (by (A)(Pi), A irrational) around a circle centered at the origin is dense in the circle. — Anne Bauval, Oct 30 '22 at 03:18
Thanks a lot. I think they are almost the same. But the answer in this question is kind of "geometric". And I prefer a proof which is more algebraic. — xdyy, Oct 30 '22 at 09:21

score 0 · Answer 1 · answered Oct 30 '22 at 04:03

0

Hint

Prove that $a_n=an\mod 2\pi$ is dense in $(0,2\pi)$ when $\frac{a}{2\pi}\notin \Bbb Q$.

answered Oct 30 '22 at 04:03

Mostafa Ayaz

31,924

Thanks. I think I have solved it by proving that ${m+n\alpha | m,n \in \mathbb{Z}, \alpha \notin \mathbb{Q} }$ is dense in $\mathbb{R}$. – xdyy Oct 30 '22 at 09:23
1

@AnneBauval Yes. Thanks for the complements. – xdyy Oct 30 '22 at 09:27
You're welcome. Good luck! – Mostafa Ayaz Oct 30 '22 at 16:59

If $|w|=1$ and $w^n$≠1 for any $n \in \mathbb{N}_+$, show that $ \{w^n ; n \in \mathbb{N}_+\}$ is dense in ∂$\Bbb D$.

1 Answers1