A Categorical Interpretation of Continuous Orbit Equivalence for Partial Dynamical Systems

TL;DR

This paper provides a categorical framework for understanding continuous orbit equivalence in partial dynamical systems, linking it to algebraic isomorphisms and groupoid structures, with applications to C*-algebras.

Contribution

It introduces the concept of orbit morphisms and characterizes orbit equivalence through algebraic and groupoid isomorphisms, extending the theory of partial dynamical systems.

Findings

Orbit morphisms are isomorphisms iff continuous orbit equivalences exist.

Diagonal-preserving isomorphisms correspond to orbit equivalences under torsion-free abelian stabilisers.

Results apply to C*-algebras from Boolean dynamical systems.

Abstract

We define the orbit morphism of partial dynamical systems and prove that an orbit morphism being an isomorphism in the category of partial dynamical systems and orbit morphisms is equivalent to the existence of a continuous orbit equivalence between the given partial dynamical systems that preserves the essential stabilisers. We show that this is equivalent to the existence of a diagonal-preserving isomorphism between the corresponding crossed products when the essential stabilisers of partial actions are torsion-free and abelian. We also characterize when an \'etale groupoid is isomorphic to the transformation groupoid of some partial action. Additionally, we explore the implications in the context of semi-saturated orthogonal partial dynamical systems over free groups, establishing connections with Deaconu-Renault systems and the concept of eventual conjugacy. Finally, we apply our results to C*-algebras associated with generalized Boolean dynamical systems.