Koopman eigenfunction

A Koopman eigenfunction of an invertible probability-preserving transformation is a nonzero $f\in L^2(X,\mu)$ satisfying

U_Tf=\lambda f,\qquad\text{that is, }f(Tx)=\lambda f(x)\text{ almost everywhere}.

Unitarity forces $|\lambda|=1$ , and $f(T^nx)=\lambda^nf(x)$ .

Ergodic systems

In an ergodic system, $|f|$ is invariant and hence constant almost everywhere. Normalize it to one. Such an eigenfunction is a measurable circle-valued observable that extracts a rotation from the dynamics. Any two normalized eigenfunctions with the same eigenvalue have invariant product $f\overline h$ , so the eigenspace is one-dimensional.

Products and conjugates of normalized eigenfunctions are eigenfunctions with eigenvalues $\lambda\eta$ and $\overline\lambda$ . Thus the eigenvalues of an ergodic system form a subgroup of the circle.

References

Asgar Jamneshan and Henrik Kreidler, Ergodic Structure Theory and Applications, ISEM 28 lecture notes. §5.1.