  LLet $f$ 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.
