Equidistribution towards the Green current for holomorphic maps

Dinh, Tien-Cuong; Sibony, Nessim

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 2, pp. 307-336

Let $f$ be a non-invertible holomorphic endomorphism of a projective space and $f^n$ its iterate of order $n$. We prove that the pull-back by $f^n$ of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to $f$ when $n$ tends to infinity. We also give an analogous result for the pull-back of positive closed $(1,1)$-currents and a similar result for regular polynomial automorphisms of ${ℂ}^k$.

Soient $f$ un endomorphisme holomorphe non-inversible d’un espace projectif et $f^n$ son itéré d’ordre $n$. Nous prouvons que l’image réciproque par $f^n$ d’une hypersurface générique (au sens de Zariski), proprement normalisée, converge vers le courant de Green associé à $f$ quand $n$ tend vers l’infini. Nous donnons également un résultat analogue pour les images réciproques des $(1,1)$-courants positifs fermés et un résultat similaire pour les automorphismes polynomiaux réguliers de ${ℂ}^k$.

Classification : 37F10, 32H50, 32U05
Keywords: Green current, exceptional set, plurisubharmonic function, Lelong number, regular automorphism
Mots-clés : courants de Green, ensemble exceptionnel, fonction plurisousharmonique, nombre de Lelong, automorphisme régulier

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