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.

中文翻译：

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关于Green的半极限同态的Green电流的规律性。 \ begin {document} $ \ mathbb {P} ^ 2 $ \ end {document}

我们研究了$电流\ mathbb {P} ^ 2 $的半极端内同态的Green电流的规律。在适当的假设下，我们表明，相对于一维Lebesgue测度，稳定切片的逐点下Radon-Nikodym导数在平衡测度的几乎每个点上都有界。这为沿全同态圆盘的格林电流提供了微弱的度量规则性。