A short proof of a theorem concerning homeomorphisms of the unit circle (Q762406)

In this note the author gives a short proof of a theorem concerning homeomorphisms \(\sigma\) of the unit circle \(S^ 1\) in which the rotation number of \(\sigma\) can be neglected. He proved an equivalence relation as the following Theorem. Let \(\sigma\) be a homeomorphism of \(S^ 1\). Then the following statements are equivalent: (1) \(\theta\) (z) is dense in \(S^ 1\) for some \(z\in S^ 1\), (2) \(\theta\) (z) is dense in \(S^ 1\) for every \(z\in S^ 1\), (3) \(\sigma\) is conjugate to R for some irrational number \(\theta\) \((0<\theta <)\). When \(\sigma\) satisfies (1) or (2), the rotation \(R_{\theta}\) in (3) is uniquely determined.