Invariant foliations for endomorphims of ${\mathbb{P}}^2$ with a pluripotentialist product structure

Abstract

Let $f$ be a holomorphic endomorphism of ${\mathbb{P}}^2$, let $T$ be its Green current and $\mu =T\wedge T$ be its equilibrium measure. We prove that if $\mu$ has a local product structure with respect to $T$ then (an iterate of) $f$ preserves a local foliation $\mathcal{F}$ on a neighborhood of $\mathrm{Supp}(T)\backslash \mathcal{E}$,where $\mathcal{E}$ denotes the exceptional set of f . If the local foliation $\mathcal{F}$ extends through $\mathcal{E}$,then it extends to ${\mathbb{P}}^2$ and is an invariant pencil of lines.