On the Markov-Dyck shifts of vertex type

TL;DR

This paper explores the properties of Markov-Dyck shifts of vertex type derived from finite directed graphs, revealing their relationship with edge type shifts and providing formulas for their zeta functions.

Contribution

It establishes that Markov-Dyck shifts of vertex type are finite-to-one factors of edge type shifts with equal entropy, and derives a new expression for their zeta functions.

Findings

Vertex type shifts are finite-to-one factors of edge type shifts.

Both shift types have the same topological entropy.

A new formula for the zeta function of vertex type shifts is provided.

Abstract

For a given finite directed graph $G$, there are two types of Markov-Dyck shifts, the Markov-Dyck shift $D_G^V$ of vertex type and the Markov-Dyck shift $D_G^E$ of edge type. It is shown that, if $G$ does not have multi-edges, the former is a finite-to-one factor of the latter, and they have the same topological entropy. An expression for the zeta function of a Markov-Dyck shift of vertex type is given. It is different from that of the Markov-Dyck shift of edge type.