Relatively almost periodic observable (${\min }_j\Vert U_T^{n}f-g_j{\Vert }_y<\epsilon$)

For every positive $\epsilon$, require finitely many global $L^2$ centers $g_j$ such that the displayed inequality holds for every integer $n$ on almost every fiber of the factor map. A center can depend on the fiber and time; its finite list is fixed. This is weaker than almost periodic observable in global norm. For noninvertible systems use nonnegative times.

Multiplying a relatively almost periodic observable by a base-set indicator preserves relative almost periodicity: on each fiber an orbit value is either unchanged or zero, so add zero to the finite list of centers. Conditional norm covariance under a factor map also proves closure under convergence in the uniform conditional L2 norm.