Inverse images of a positive closed current for a holomorphic endomorphism of a compact K\"ahler manifold

In this paper, we prove that for a given surjective holomorphic endomorphism f of a compact K\"ahler manifold X and for some integer p with 1≤ p≤ k, there exists a proper invariant analytic subset E for f such that if S is smooth in a neighborhood of E, the sequence dₚ⁻ⁿ(fⁿ)^*(S-αS) converges to $0$ exponentially fast in the sense of currents where dₚ denotes the dynamical degree of order p and αS is a closed smooth form in the de Rham cohomology class of S.