On Conjugacy Invariants of $D_{\infty}$-Topological Markov Chains

TL;DR

This paper explores invariants of $D_{ abla}$-topological Markov chains, introducing new equivalence relations and examining their connections to conjugacy, zeta functions, and matrix representations.

Contribution

It introduces $D_{ abla}$-strong shift and shift equivalences for flip pairs, linking these to conjugacy and zeta functions in $D_{ abla}$-topological Markov chains.

Findings

$D_{ abla}$-strong shift equivalence relates to conjugacy.

$D_{ abla}$-shift equivalence relates to zeta functions.

New invariants help classify $D_{ abla}$-topological Markov chains.

Abstract

A $D_{\infty}$-topological Markov chain can be represented by a pair of zero-one square matrices, which is called a flip pair. We introduce the concepts of $D_{\infty}$-strong shift equivalence and $D_{\infty}$-shift equivalence, which are equivalence relations between flip pairs. We investigate the relationships between the existence of a $D_{\infty}$-conjugacy, the existence of a $D_{\infty}$-strong shift equivalence, the existence of a $D_{\infty}$-shift equivalence and the coincidence of the Lind zeta functions.