Let \(f:\mathbb{P}^k\to\mathbb{P}^k\) be a holomorphic map of algebraic degree \(d\geq2\). An important tool in the study of the dynamics of \(f\) is its dynamical Green current \(T\), defined by \(T=\lim_{n\to\infty}d^{-n}(f^n)^\star\omega\), where \(\omega\) is the Fubini-Study form on \(\mathbb{P}^k\). The author proves the following general equidistribution result towards \(T\): There exists an invariant analytic subset \(E\) for \(f\) such that if \(S\) is a positive closed current of bidegree \((p,p)\) and unit mass on \(\mathbb{P}^k\) with bounded super-potential near \(E\), then \(\lim_{n\to\infty}d^{-pn}(f^n)^\star S=T^p\) in the sense of currents. This generalizes an earlier result of the author [Trans. Am. Math. Soc. 368, No. 5, 3359–3388 (2016; Zbl 1418.37075)], where the assumption was that \(S\) is smooth near \(E\).
The notion of super-potentials of positive closed currents was introduced and developed by T.-C. Dinh and N. Sibony [Acta Math. 203, No. 1, 1–82 (2009; Zbl 1227.32024)], who gave many applications for it to complex dynamics in recent years. The notions of local continuity and local boundedness of super-potentials are introduced and studied by the author in the present paper, as main tool in proving the above theorem. In particular, it is shown that if a positive closed current \(S\) has bounded super-potential on an open set \(W\), then \(S\) has zero Lelong number at each point of \(W\). The author also shows that \(S\) has bounded, resp. continuous, super-potential near an analytic set \(E\) if and only if \(S\) is a PB, resp. PC, current near \(E\). The notions of PB/PC currents are due to Dinh and Sibony [loc. cit.].
The author also proves a similar equidistribution result in the case when \(f\) is a regular polynomial automorphism of \(\mathbb{C}^k\).