Definition
Discrete spectrum of a dynamical system
The property that Koopman eigenfunctions span the full space of square-integrable observables.
An invertible probability-preserving transformation has discrete spectrum, or pure point spectrum, if the closed linear span of its Koopman eigenfunctions is all of . Equivalently, has an orthonormal basis of Koopman eigenfunctions.
Examples
Finite permutations and translations on compact metrizable abelian groups have discrete spectrum. A nontrivial weakly mixing probability-preserving system cannot have discrete spectrum: after removing constants it has no eigenfunctions.
Terminology
“Discrete” here describes spectral measures and the existence of an eigenfunction basis. It does not say that the eigenvalues form a discrete subset of the circle. The eigenvalues of an irrational rotation are dense there.
References
- Asgar Jamneshan and Henrik Kreidler, Ergodic Structure Theory and Applications, ISEM 28 lecture notes. §6.1.