  Quasiperiodic systems in one dimension can host non-ergodic states, e.g.
localized in position or momentum. Periodic quenches within localized phases
yield Floquet eigenstates of the same nature, i.e. spatially localized or
ballistic. However, periodic quenches across these two non-ergodic phases were
thought to produce ergodic diffusive-like states even for non-interacting
particles. We show that this expectation is not met at the thermodynamic limit
where the system always attains a non-ergodic state. We find that ergodicity
may be recovered by scaling the Floquet quenching period with system size and
determine the corresponding scaling function. Our results suggest that while
the fraction of spatially localized or ballistic states depends on the model's
details, all Floquet eigenstates belong to one of these non-ergodic categories.
Our findings demonstrate that quasiperiodicity hinders ergodicity and
thermalization, even in driven systems where these phenomena are commonly
expected.
