Let $\sigma$ be a homeomorphism of $S^1$. Then the following statements are equivalent;
(1) O(z) is dense in S for some z in S,
(2) O(z) is dense in $S^1$ for every z in $S^1$,
(3) $\sigma$ is conjugate to $R_{\theta}$ for some irrational number $\theta$, (0<$\theta$<1/2).
When $\sigma$ satisfies the condition (1) or (2), the rotation $R_{\theta}$ in (3) is uniquely determined.
(where for a real number $\theta$, we denote by $R_{\theta}$ the rotation: $R_{\theta}(e^{2\pi ix})=e^{2\pi i(x+\theta)}$ on $S^1$ and orbit O(z) :={$\sigma^n(z):n \in \mathbb{Z}$}).
In this theorem I have proved that (1) implies (3) and (2) implies (1) but unable to prove (3) implies (1) since to prove that these are equivalent, we need to prove that (1)=>(3), (3)=>(2) and (2)=>(1).
And the uniqueness of $R_{\theta}$ can be proved by using the set {$e^{2\pi in\theta}|n\in\mathbb{Z}$} of eigenvalues of $R_{\theta}$ but unable to get the result. Please help me in proving the result.
If I understand you correctly, you have proved (1) implies (3) and (2) implies (1), so what remains is (3) implies (2). For this, have a look at this answer.