Abstract
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·θmod1 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.
Highlights
In what follows, T = R/Z denotes the unit circle with the standard orientation
We point out that parts of the analysis of rotational systems with irrational rotation number can be done without explicit reference to an ambient transformation on the unit circle
This leads to a structure result for rotational subsystems of a wide class of continuous transformations F : T → T, those with finite preimages
Summary
The angle of the rotation of T in the preceding theorem is called the rotation number of (X, f) Such systems are of particular interest when they arise as invariant subsets of a continuous mapping on the whole circle, F : T → T. We point out that parts of the analysis of rotational systems with irrational rotation number can be done without explicit reference to an ambient transformation on the unit circle This leads to a structure result for rotational subsystems of a wide class of continuous transformations F : T → T, those with finite preimages. For a given continuous F : T → T, Theorem 4 and its corollary shed no light on how to determine which irrational numbers can be realized by rotational subsystems of (T, F). When the degree is larger than 2, there are examples for every irrational rotation number
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