Directional dimensions of ergodic currents on $\mathbb C \mathbb P (2)$

International audienceLLet $f$ be a holomorphic endomorphism of $\mathbb P^ 2$ of degree $d \geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $\nu$. We infer several applications. The first one shows that the currents $S$ containing a measure of entropy $h_\nu > \log d$ have a directional dimension $>2$, which answers a question by de Th\'elin-Vigny. The second application asserts that the Dujardin's semi-extremal endomorphisms are close to suspensions of one-dimensional Latt\`es maps. Finally, we obtain an upper bound for the dimension of the equilibrium measure, towards the formula conjectured by Binder-DeMarco