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A characterization of the uniquely ergodic endomorphisms of the circle. (English) Zbl 0773.34036
A continuous endomorphism of the circle \(f\) is said to be uniquely ergodic if there exists a unique \(f\)-invariant probability measure on the circle. It is known that every homeomorphism of the circle with irrational rotation number is uniquely ergodic. This paper is to extend this result to the endomorphism of the circle. The main result of the paper is the following Theorem: A circle endomorphism is uniquely ergodic if and only if it has at most one periodic orbit.

37-XX Dynamical systems and ergodic theory
54H20 Topological dynamics (MSC2010)
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