Halmos–von Neumann theorem

Statement

An invertible ergodic probability-preserving system on a standard probability space has discrete spectrum if and only if it is measurably conjugate modulo null sets to

R_a:K\to K,\qquad R_a(x)=x+a,

where $K$ is a compact metrizable abelian group with normalized Haar measure and $\overline{\{na:n\in\mathbb Z\}}=K$ .

Spectral classification

Two such ergodic systems are measurably conjugate exactly when their groups of Koopman eigenvalues agree as subgroups of $\mathbb S^1$ . Standardness makes this group countable. Its Pontryagin dual supplies the compact rotation model.

With the forward convention $Uf=f\circ T$ , the rotation point is evaluation $a(\lambda)=\lambda$ . Inverse pullback conjugates these eigenvalues; the two conventions must not be mixed when writing the model.

Scope

The compact group can be finite, a torus, or an infinite totally disconnected group. The theorem does not classify systems with continuous spectral components from their Koopman representation alone.

References

Asgar Jamneshan and Henrik Kreidler, Ergodic Structure Theory and Applications, ISEM 28 lecture notes. Theorems 6.2.6–6.2.8.