Rotational subsets of the circle under \(z^d\) (Q2488855)

Let \(\mathbb T=\mathbb R/\mathbb Z\) be the unit circle oriented counterclockwise and let \(\sigma_d:\mathbb T\to\mathbb T\) be the map given by \(\sigma_d(t)=dt\pmod 1\), induced by the complex-valued map \(z\to z^d\) on the unit circle in the complex plane. If \(f:\mathbb T\to\mathbb T\) is a map and \(A\) is a closed subset of \(\mathbb T\), then \(f\) is said to be order-preserving on \(A\) if \(f|_A\) can be extended to an order-preserving map \(F:\mathbb T\to\mathbb T\) so that \(F|_A=f|_A\). Among other results obtained the authors prove that if \(A\) is a closed subset of \(\mathbb T\), then \(\sigma_d\) is order-preserving on \(A\) if and only if \(\mathbb T\setminus A\) contains \(d-1\) pairwise disjoint open intervals, each of length \(\frac{1}{d}\), called holes of \(A\).