Spectral theory of dynamical systems

Summary

The spectral theory of dynamical systems studies a measure-preserving transformation through the spectrum of its Koopman operator on $L^2$. Two systems with unitarily equivalent Koopman operators are called spectrally isomorphic, and properties preserved by this relation are spectral invariants. Ergodicity, weak mixing, mixing and discrete spectrum are spectral invariants. The theory is complete for ergodic systems with discrete spectrum, by the Halmos–von Neumann theorem. In general, spectral data do not determine a system up to isomorphism.