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

For a minimal dynamical system on a compact metric space, the continuous circle extension $(x,y)\mapsto (Tx,y+\rho (x))$ is minimal precisely when the displayed equation has no continuous circle-valued solution $f$ for any nonzero integer $m$. A minimal subsystem of the extension has fibers that are cosets of its closed vertical translation stabilizer. If that stabilizer is proper, a nontrivial circle character trivial on it gives the forbidden equation. Conversely a solution makes $e^{2\pi imy}/f(x)$ a nonconstant continuous invariant function.