  Dynamical response functions are standard tools for probing local physics
near the equilibrium. They provide information about relaxation properties
after the equilibrium state is weakly perturbed. In this paper we focus on
systems which break the assumption of thermalization by exhibiting persistent
temporal oscillations. We provide rigorous bounds on the Fourier components of
dynamical response functions in terms of extensive or local dynamical
symmetries, i.e. extensive or local operators with periodic time dependence.
Additionally, we discuss the effects of spatially inhomogeneous dynamical
symmetries. The bounds are explicitly implemented on the example of an
interacting Floquet system, specifically in the integrable Trotterization of
the Heisenberg XXZ model.
