Intuitively this should happen for a large class of dynamical systems, but I don't know the right necessary and sufficient conditions.
A class of examples satisfying this is given by polyhedral billiards, where you assign a symbol to each face and correspond orbits to sequences in the obvious manner. It is a result of G Galperin, T Krüger and S Troubetzkoy in their paper "Local instability of orbits in polygonal and polyhedral billiards" (1995), that if a trajectory has a periodic sequence then it must be a periodic trajectory and that if there are two different trajectories with the same symbolic sequences then the symbolic sequences aren't are periodic.
