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

LLet $f$ be a holomorphic endomorphism of ${\mathbb{P}}^2$ of degree $d \geq 2$\mathrm{.}Weestimatethelocaldirectionaldimensionsofclosedpositivecurrents$S$withrespecttoergodicdilatingmeasures$\nu$. We infer several applications. The first one shows that the currents $S$ containing a measure of entropy $h\mathrm{\_}\nu >\log d$ 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.