Ricci iteration, twisted and coupled K\"ahler-Einstein metrics.

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 Ding type functional, and show the equivalence to the 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.