  We explore how interactions can facilitate classical like dynamics in models
with sequentially activated hopping. Specifically, we add local and short range
interaction terms to the Hamiltonian, and ask for conditions ensuring the
evolution acts as a permutation on initial local number Fock states. We show
that at certain values of hopping and interactions, determined by a set of
Diophantine equations, such evolution can be realized. When only a subset of
the Diophantine equations is satisfied the Hilbert space can be fragmented into
frozen states, states obeying cellular automata like evolution and subspaces
where evolution mixes Fock states and is associated with eigenstates exhibiting
high entanglement entropy and level repulsion.
