A Complete Invariant for Shift Equivalence for Boolean Matrices and Finite Relations

TL;DR

This work addresses the problem of classifying shift equivalence for finite Boolean relations by introducing a complete invariant. The authors develop a canonical form theory and show that a finite relation is characterized up to shift equivalence by the triple $(R_\ge, p, [\xi])$, where $R_\ge$ is the partial order on recurrent components, $p$ is the period, and $[\xi]$ is the cohomology class of a cocycle $\xi: R_\ge \to \mathcal{L}_p$. They provide a constructive reduction to canonical form and prove that two canonical-form relations are isomorphic exactly when they are shift equivalent, with the invariant robust under coboundaries and reducibility addressed via irreducible quotients. The results have implications for dynamical classifications of subshifts, Conley index constructions, and computational analyses of dynamical approximations. The framework leverages Szymczak category theory and a cocycle-based encoding of inter-component dynamics, yielding a precise and implementable invariant for shift equivalence in this discrete setting.

Abstract

We give a complete invariant for shift equivalence for Boolean matrices (equivalently finite relations), in terms of the period, the induced partial order on recurrent components, and the cohomology class of the relation on those components.

Theorems & Definitions (39)

• Theorem
• Definition 2.1
• Theorem 2.2: KR2Proposition 3.5
• Definition 2.3
• Theorem 2.4: MMWTheorems 12, 13
• Definition 2.5
• Proposition 2.6
• proof
• Lemma 2.7
• proof