  In this paper, we introduce the "coupled Ricci iteration", a dynamical system
related to the Ricci operator and twisted K\"ahler-Einstein metrics as an
approach to the study of coupled K\"ahler-Einstein (CKE) metrics. For negative
first Chern class, we prove the smooth convergence of the iteration. For
positive first Chern class, we also provide a notion of coercivity of the Ding
functional, and show its equivalence to existence of CKE metrics. As an
application, we prove the smooth convergence of the iteration on CKE Fano
manifolds assuming that the automorphism group is discrete.
