Localized Bost-Connes systems III: invariants, entropy, and the measure class

This paper isolates which invariants of a localized Bost-Connes system depend on the inverse temperature and which do not. The measurable cohomology of the tail relation with circle coefficients is computed through its kernel, the beta-dependence is located exactly (Corollary 2.4), entropy invariants of the equilibrium states are evaluated, and the measure class of the KMS measure is identified as the invariant that survives when states, traces and weights do not.