  In this paper, we discuss the equidistribution phenomena for holomorphic
endomorphisms over $\mathbb{P}^k$ in the case of bidegree $(p,p)$ with $1<p<k$.
We prove that if $f:\mathbb{P}^k\to\mathbb{P}^k$ is a holomorphic endomorphism
of degree $d\geq 2$ and $T^p$ denotes the Green $(p,p)$-current associated with
$f$, then there exists a proper invariant analytic subset $E$ for $f$ such that
$d^{-pn}(f^n)^*(S)\to T^p$ exponentially fast in the current sense for every
positive closed $(p,p)$-current $S$ of mass 1 such that $S$ is smooth on $E$.
