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

Abstract

LLet $f$ be a holomorphic endomorphism of ${\mathbb{P}}^2$ of degree $d\ge 2$. 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 Thélin-Vigny. The second application asserts that the Dujardin's semi-extremal endomorphisms are close to suspensions of one-dimensional Lattès maps. Finally, we obtain an upper bound for the dimension of the equilibrium measure, towards the formula conjectured by Binder-DeMarco.