The existence and structure of rotational systems in the circle

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Abstract and Applied Analysis


By a rotational system, we mean a closed subset X of the circle, T=R/Z, together with a continuous transformation f:X→X with the requirements that the dynamical system (X,f) be minimal and that f respect the standard orientation of T. We show that infinite rotational systems (X,f), with the property that map f 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, F:T→T. Because our main result makes no explicit mention of a global transformation on T, we show that such a structure theorem holds for rotational systems that arise as invariant sets of any continuous transformation F:T→T with finite preimages. In particular, there are no explicit conditions on the degree of F. We then give a development of known results in the case where Fθ=d·θmod -1 for an integer d>1. 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.

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