  We study if the interplay between dynamical localization and interactions in
periodically driven quantum systems can give rise to anomalous thermalization
behavior. Specifically, we consider one-dimensional models with interacting
spinless fermions with nearest-neighbor hopping and density-density
interactions, and a periodically driven on-site potential with spatial
periodicity $m=2$ and $m=4$. At a dynamical localization point, these models
evade thermalization either due to the presence of an extensive number of
conserved quantities (for weak interactions) or due to the kinetic constraints
caused by drive-induced resonances (for strong interactions). Our models
therefore illustrate interesting mechanisms for generating constrained dynamics
in Floquet systems which are difficult to realize in an undriven system.
