The Existence and Structure of Rotational Systems in the Circle

Abstract

By a rotational system, we mean a closed subset $X$ of the circle, $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, together with a continuous transformation $f:X\to X$ with the requirements that the dynamical system $(X,f)$ be minimal and that $f$ respect the standard orientation of $\mathbb{T}$. 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, $F:\mathbb{T}\to \mathbb{T}$. Because our main result makes no explicit mention of a global transformation on $\mathbb{T}$, we show that such a structure theorem holds for rotational systems that arise as invariant sets of any continuous transformation $F:\mathbb{T}\to \mathbb{T}$ 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 $F\left(\theta \right)=d·\theta \mathrm{mod}\mathrm{1}$ for an integer $d>\mathrm{1}$. 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.