A dynamical characterization of diagonal preserving $*$-isomorphisms of graph $C^*$-algebras

Abstract: We characterize when there exists a diagonal preserving $$-isomorphism between two graph $C^$-algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of "orbit equivalence" between the boundary path spaces of the directed graphs $E$ and $F$ and show that this is a necessary and sufficient condition for the existence of a diagonal preserving $$-isomorphism between the graph $C^$-algebras $C^*(E)$ and $C^*(F)$.