2017/10/10 by Dupont, Christophe, Rogue, Axel
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1710.03508
LLet f be a holomorphic endomorphism of \mathbb P^ 2 of degree d ≥ 2. We estimate the local directional dimensions of closed positive currents S with respect to ergodic dilating measures ν. We infer several applications. The first one shows that the currents S containing a measure of entropy h_ν> 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.