Almost periodic observable ($\{U_T^{n}f:n\in \mathbb{Z}\}$)

An $L^2$ observable of an invertible probability measure-preserving system is almost periodic when its Koopman operator orbit is totally bounded in the global $L^2$ norm. A noninvertible system uses its forward orbit. This is the Hilbert-space dynamical definition, distinguished from uniform almost periodicity for continuous functions of a real variable.

If the forward orbit of $f$ under an isometry $U$ is totally bounded, choose a finite net whose centers are $U^{n_j}f$ and let $M={\max }_j{n}_j$. For every sufficiently large $n$, some center is within $\epsilon$ of $U^{n}f$, so cancellation gives $n-n_j\in R_{\epsilon }$. Thus $R_{\epsilon }$ is a syndetic set. Telescoping also gives $\Vert U^{kn}f-f{\Vert }_2\le k\Vert U^{n}f-f{\Vert }_2$.

A probability measure-preserving system is compact if every $L^2$ observable has a relatively compact Koopman operator orbit in the global $L^2$ norm. For an invertible system one may use all integer times; for a general transformation use nonnegative times. This is a measure-theoretic definition, stronger than compactness of the underlying topological space.