  In this paper, we introduce notions about local regularity of the
super-potential and prove equidistribution of positive closed $(p, p)$-currents
of $\mathbb{P}^k$ whose super-potentials are bounded near an invariant analytic
subset, for holomorphic endomorphisms of $\mathbb{P}^k$ and for regular
polynomial automorphisms of $\mathbb{C}^k$.
