Unique ergodicity of continuous self-maps of the circle (Q1191761)

The author shows that for \(f:S^ 1\to S^ 1\) a continuous self map of the circle, \(f\) is uniquely ergodic if and only if \(f\) has at most one periodic orbit. This generalizes a well known result of H. Furstenberg which says that if \(f:S^ 1\to S^ 1\) is a homeomorphism with no periodic points, then \(f\) is uniquely ergodic. The proof uses ideas from the proof of Furstenberg's theorem [\textit{H. Furstenberg}, Am. J. Math. 83, 573-601 (1961; Zbl 0178.384)] and also a result of \textit{J. Auslander} and \textit{Y. Katznelson} [Isr. J. Math. 32, 375-381 (1979; Zbl 0442.54011)].