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

Christophe Dupont (1) , Axel Rogue (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Lyapunov exponents; invariant measures; positive closed currents; dimension theory; normal forms

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Christophe Dupont, Axel Rogue. Directional dimensions of ergodic currents on $\mathbb C \mathbb P (2)$. Ergodic Theory and Dynamical Systems, 2020, 40 (8), pp.2131-2155. ⟨10.1017/etds.2018.137⟩. ⟨hal-01613791⟩