Circle skew-product minimality criterion ($f(Tx)=e^{2\pi im\rho (x)}f(x)$)

Let $T$ be a minimal homeomorphism of a compact metric space $X$, and let $\rho :X\to \mathbb{R}/\mathbb{Z}$ be continuous. The circle skew productis a minimal dynamical system precisely when there is no continuous circle-valued function $f$ satisfyingfor any nonzero integer $m$.

A solution gives an explicit obstruction: $e^{2\pi imy}/f(x)$ is a nonconstant continuous function invariant under $S$. Its level sets are proper closed invariant sets, so the extension is not minimal.

Conversely, take a minimal subsystem of a nonminimal extension. Its vertical translation stabilizer is a proper closed subgroup of the circle, and its fibers are cosets of that subgroup. A nontrivial circle character trivial on the subgroup descends to a continuous circle-valued function of the base, giving the displayed forbidden equation. Compactness and the isometric circle action supply this coset description; arbitrary nonisometric fiber dynamics need not have it.

For a base consisting of one point and constant $\rho =\alpha$, the equation reduces to $e^{2\pi im\alpha }=1$. Its absence for every nonzero $m$ says exactly that $\alpha$ is irrational, recovering minimality of an irrational rotation of the circle.