Local regularity of super-potentials and equidistribution of positive closed currents on \({\mathbb {P}}^k\)
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 $E$. \par The notion of super-potentials of positive closed currents was introduced and developed by \textit{T.-C. Dinh} and \textit{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.]. \par The author also proves a similar equidistribution result in the case when $f$ is a regular polynomial automorphism of $\mathbb{C}^k$.
holomorphic endomorphisms of projective space
super-potentials of positive closed currents
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