A combinatorial approach to study subshifts associated with multigraphs

TL;DR

This paper introduces a combinatorial framework for analyzing subshifts related to multigraphs by defining generalized languages with forbidden and repeated words, enabling entropy calculation and spectral analysis of associated matrices.

Contribution

It extends classical subshift theory to multigraphs through generalized languages and provides formulas for entropy and spectral properties of the adjacency matrix.

Findings

Derived a combinatorial expression for the generating function of generalized language words.

Calculated the Perron root and eigenvectors of the adjacency matrix.

Obtained topological entropy and an alternative Parry measure for the subshift.

Abstract

A subshift of finite type over finitely many symbols can be described as a collection of all infinite walks on a digraph with at most a single edge from a vertex to another. The associated finite set $\F$ of forbidden words is a constraint which determines the language of the shift entirely. In this paper, in order to describe infinite walks on a multigraph, we introduce the notion of multiplicity of a word (finite walk) and define repeated words as those having multiplicity at least $2$. In general, for given collections $\F$ of forbidden words and $\R$ of repeated words with pre-assigned multiplicities, we define notion of a generalized language which is a multiset. We obtain a subshift associated with $\F$ and $\R$ such that its entropy is calculated using the generalized language. We also study the relationship between the language of this subshift and the generalized language. We then obtain a combinatorial expression for the generating function that enumerates the number of words of fixed length in this generalized language. This gives the Perron root and eigenvectors of the adjacency matrix with integer entries associated to the underlying multigraph. Using this, the topological entropy and an alternate definition of Parry measure for the associated edge shift are obtained. We also discuss some properties of Markov measures on this subshift.