= Circle skew-product minimality criterion
{title2=$f(Tx)=e^{2\pi i m\rho(x)}f(x)$}
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 $e^{2\pi i m y}/f(x)$ a nonconstant continuous invariant function.
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