Equivariant rigidity of localized Bost-Connes C*-dynamical systems: the fixed-point algebra is the Cartan subalgebra

For localized Bost-Connes C*-dynamical systems the fixed-point algebra of the gauge action is identified with the canonical Cartan subalgebra, and equivariant isomorphisms of the systems are shown to be governed by this identification: an equivariant rigidity theorem reducing dynamical isomorphism to the underlying groupoid data.