Brolin's theorem for curves in two complex dimensions

Favre, Charles; Jonsson, Mattias

Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1461-1501.

Pour toute application $f:{\mathbb{P}}^2\to {\mathbb{P}}^2$ de degré $d\ge 2$ nous donnons des conditions suffisantes sur un courant positif fermé $S$ de bidegré $(1,1)$, pour que la suite $d^{-n}{f}^{n*}S$ converge vers le courant de Green lorsque $n\to \infty$. Nous conjecturons aussi des conditions nécessaires pour ce problème de convergence.

Given a holomorphic mapping $f:{\mathbb{P}}^2\to {\mathbb{P}}^2$ of degree $d\ge 2$ we give sufficient conditions on a positive closed (1,1) current of $S$ of unit mass under which $d^{-n}{f}^{n*}S$ converges to the Green current as $n\to \infty$. We also conjecture necessary condition for the same convergence.

Classification : 37F10, 32U25
Keywords: holomorphic dynamics, currents, Lelong numbers, equidistribution, Kilseman numbers, volume estimates, asymptotic multiplicities
Mot clés : dynamique holomorphe, courants, nombre de Lelong, équidistribution, nombre de Kiselman, estimations de volume, multiplicités asymptotiques

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