On slice measures of Green currents on CP(2)

Let $f$ be a holomorphic map of $\mathbb{C}{\mathbb{P}}^2$ of degree $d\ge 2$, let $T$ be its Green current and $\mu =T\wedge T$ be its equilibrium measure. We give a new proof of a theorem due to Dujardin asserting that $\mu \ll T\wedge {\omega }_{{\mathbb{P}}^2}$ implies ${\lambda }_2=\frac{1}{2}\log d$, where ${\lambda }_1\ge {\lambda }_2$ are the Lyapunov exponents of $\mu$. Then, assuming $\mu \ll T\wedge {\omega }_{{\mathbb{P}}^2}$, we study slice measures $\nu :=T\wedge d{d}^c\mathrm{\mid }W{\mathrm{\mid }}^2$, where $W$ is a holomorphic local submersion. We give sufficient conditions on the Radon-Nikodym derivative of $\mu$ with respect to the trace measure $T\wedge {\omega }_{{\mathbb{P}}^2}$ ensuring $\mu =\nu$. The involved submersion $W$ comes from normal coordinates for the inverse branches of the iterates of $f$.