= Circle skew-product minimality criterion
{title2=$f(Tx)=e^{2\pi i m\rho(x)}f(x)$}
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 product>
$$
S(x,y)=(Tx,y+\rho(x))
$$
is a <minimal dynamical system> precisely when there is no continuous circle-valued function $f$ satisfying
$$
f(Tx)=e^{2\pi i m\rho(x)}f(x)
$$
for any nonzero <integer> $m$.
A solution gives an explicit obstruction: $e^{2\pi i my}/f(x)$ 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 $e^{2\pi i m\alpha}=1$. Its absence for every nonzero $m$ says exactly that $\alpha$ is irrational, recovering minimality of an <irrational rotation of the circle>.
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