Unique ergodicity
= Unique ergodicity
= Uniquely ergodic
{synonym}
A continuous self-map of a compact <metric space> is uniquely ergodic when it has exactly one invariant <Borel probability measure>. An <irrational rotation of the circle> is uniquely ergodic: invariance multiplies its nonzero-index <Fourier coefficients> by nontrivial phases, forcing those coefficients to vanish. The <Stone-Weierstrass theorem> then identifies the invariant measure as normalized <Lebesgue measure>.