The Existence and Structure of Rotational Systems in the Circle

By a rotational system, we mean a closed subset of the circle, , together with a continuous transformation with the requirements that the dynamical system be minimal and that respect the standard orientation of . We show that infinite rotational systems , with the property that map has finite preimages, are extensions of irrational rotations of the circle. Such systems have been studied when they arise as invariant subsets of certain specific mappings, . Because our main result makes no explicit mention of a global transformation on , we show that such a structure theorem holds for rotational systems that arise as invariant sets of any continuous transformation with finite preimages. In particular, there are no explicit conditions on the degree of . We then give a development of known results in the case where for an integer . The paper concludes with a construction of infinite rotational sets for mappings of the unit circle of degree larger than one whose lift to the universal cover is monotonic.