On certain labelled directed graphs of symbolic dynamics

We consider families of coded systems that contain the Dyck shifts and that are closed under topological conjugacy. We introduce a notion of hyposynchronization of subshifts. We introduce a notion of restricted complexity of hyposynchronizing subshifts. Restricted complexity of hyyposynchronizing subshift is not invariant under topological conjugacy. We construct explicitly a family of hyposynchronizing subshifts of restricted complexity that extends the family of Dyck shifts. The subshifts in this family are characterized by their restricted complexity in conjunction with a certain set of invariants of topological conjugacy that inclufdes the Artin-Mazur zeta function.