Shift equivalence implies flow equivalence for shifts of finite type

TL;DR

This paper proves that shift equivalence over the nonnegative integers ($SE_{\\mathbb{Z}_+}$) implies flow equivalence (FE) for shifts of finite type (SFTs), and that for SFTs, eventual conjugacy coincides with $SE_{\\mathbb{Z}_+}$, making eventual conjugacy imply FE in this setting. The core method introduces a concrete polynomial shift equivalence (PSE) equation derived from a given SE, and develops a partitioned-matrix framework to handle reducible SFTs, culminating in a stabilized $SL_{\\mathcal{P}}(\\mathbb{Z})$-equivalence of $I - A$ and $I - B$. Leveraging Franks’ FE classification and related results (e.g., BPos2002), the paper shows that SE$_{\\mathbb{Z}_+}$ yields FE for general SFTs, including the necessary positivity on cycle components. The work also discusses the limits of these implications by presenting an example of eventually conjugate systems that are not flow equivalent, clarifying the landscape of dynamical classifications for SFTs and their algebraic invariants.

Abstract

Shifts of finite type defined from shift equivalent matrices must be flow equivalent.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Theorem 3.1
• proof
• Remark 3.3
• Remark 3.4
• Proposition 4.3
• proof