On certain labelled directed graphs of symbolic dynamics

TL;DR

This paper introduces a new class of coded symbolic dynamical systems called hyposynchronizing subshifts, explores their properties, and extends the family of Dyck shifts with explicit constructions and invariants.

Contribution

It defines hyposynchronization and restricted complexity for subshifts, and constructs a family of hyposynchronizing subshifts extending Dyck shifts with invariants including the Artin-Mazur zeta function.

Findings

Restricted complexity is not invariant under topological conjugacy.

Constructed explicit examples of hyposynchronizing subshifts.

Characterized these subshifts using invariants like the Artin-Mazur zeta function.

Abstract

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.