The Existence and Structure of Rotational Systems in the Circle

Abstract EN

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.