Balanced strong shift equivalence, balanced in-splits, and eventual conjugacy

TL;DR

This paper introduces balanced strong shift equivalence and demonstrates its role in characterizing one-sided eventual conjugacy of finite graphs without sinks, extending classical shift equivalence concepts.

Contribution

It defines balanced strong shift equivalence and links it to eventual conjugacy, providing new characterizations involving out-splits and balanced in-splits.

Findings

Two graphs are one-sided eventually conjugate iff their adjacency matrices are balanced strong shift equivalent.

Eventual conjugacy can be achieved through sequences of out-splits and balanced in-splits.

The work relates eventual conjugacy to block maps on finite paths, extending classical shift theory.

Abstract

We introduce the notion of balanced strong shift equivalence between square nonnegative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits; the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.