Unital shift equivalence

TL;DR

The paper develops a unital variant of shift equivalence for finite nonnegative integer matrices and shows its characterization via isomorphism of unital dimension data, rather than a direct dynamical equivalence. It proves that unital shift equivalence implies continuous orbit equivalence for one-sided shifts, yielding isomorphisms of topological full groups and Leavitt path algebras and tying into Hazrat's graded classification conjecture. Through explicit examples, it clarifies that unital SE is distinct from existing one-sided conjugacy notions and discusses open questions about a dynamical description of unital SE. This work provides a one-sided analogue to flow-equivalence results and strengthens the interplay between symbolic dynamics and algebraic invariants.

Abstract

We introduce and study a unital version of shift equivalence for finite square matrices over the nonnegative integers. In contrast to the classical case, we show that unital shift equivalence does not coincide with one-sided eventual conjugacy. We also prove that unital shift equivalent matrices define one-sided shifts of finite type that are continuously orbit equivalent. Consequently, unitally shift equivalent matrices have isomorphic topological full groups and isomorphic Leavitt path algebras, the latter being related to Hazrat's graded classification conjecture in algebra.

Theorems & Definitions (26)

• Definition 3.1
• Proposition 3.2
• proof
• Remark 3.3
• Lemma 3.4
• proof
• Example 3.5
• Remark 3.6
• Lemma 3.7
• proof