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Green currents are central objects in holomorphic dynamics. They appear naturally as limit objects for the iteration of submanifolds, and they allow for a precise measure-theoretical study of the dynamics, especially beyond dimension one. In this talk, I will survey their construction and their main properties, including a suitable notion of regularity. In the second part of the talk, I will focus on the setting of holomorphic correspondences of Kähler manifolds.
This talk is partially based on a joint work with Muhan Luo.