  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 $1\le p\le
k$, there exists a proper invariant analytic subset $E$ for $f$ such that if
$S$ is smooth in a neighborhood of $E$, the sequence
$d_p^{-n}(f^n)^*(S-\alpha_S)$ converges to $0$ exponentially fast in the sense
of currents where $d_p$ denotes the dynamical degree of order $p$ and
$\alpha_S$ is a closed smooth form in the de Rham cohomology class of $S$.
