On the regularity of the Green current for semi-extremal endomorphisms of $  \mathbb{P}^2 $

Christophe Dupont; Axel Rogue

doi:10.3934/dcds.2020163

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2020, Volume 40, Issue 12: 6767-6781. Doi: 10.3934/dcds.2020163

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On the regularity of the Green current for semi-extremal endomorphisms of $ \mathbb{P}^2 $

Christophe Dupont and
Axel Rogue

Université de Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Received: May 31, 2019

Revised: December 31, 2019

Early access: March 2020

Published: August 2020

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Abstract

We study the regularity of the Green current for semi-extremal endomorphisms of $ \mathbb{P}^2 $. Under suitable assumptions, we show that the pointwise lower Radon-Nikodym derivative of stable slices with respect to the one dimensional Lebesgue measure is bounded at almost every point for the equilibrium measure. This provides a weak amount of metric regularity for the Green current along holomorphic discs.

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Mathematics Subject Classification: 37F10, 32U40, 28A15.

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