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  "body": "\n## Summary\n\nThe Koopman operator of a [[measure-preserving transformation|556WBDAHNP]] $T$ acts on functions by $f \\mapsto f \\circ T$.\nIt is the [[composition operator|CFASPSF7BB]] with symbol $T$ on the space $L^2$ of the invariant measure.\nThere it is an isometry, and a [[unitary operator|MM0EJ2GDE4]] when $T$ is invertible.\nIt turns a nonlinear dynamical system into a linear operator on a Hilbert space.\nErgodicity, weak mixing and mixing all become properties of this operator, which is the starting point of the [[spectral theory of dynamical systems|Z7GZ6SPFA5]].\n\n## Explanations\n\n### L1 human\n\nInstead of following points, follow measurements.\nA measurement is a function $f$ on the space, and after one step of the system the new measurement is $f \\circ T$: measure where the point has moved to.\nThis rule is linear in $f$ even when $T$ is very nonlinear.\nFor a rotation $T(x) = x + \\alpha$ of the circle, the function $e^{2\\pi i x}$ becomes $e^{2\\pi i \\alpha} e^{2\\pi i x}$, so it is an eigenfunction with eigenvalue $e^{2\\pi i \\alpha}$.\n",
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