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

TL;DR

This paper investigates the local directional dimensions of positive currents in complex projective space under holomorphic endomorphisms, providing new bounds and characterizations related to entropy and specific dynamical systems.

Contribution

It introduces estimates for directional dimensions of currents with respect to ergodic measures, addressing questions about entropy, characterizing certain endomorphisms, and bounding equilibrium measure dimensions.

Findings

Currents with entropy > log d have directional dimension > 2

Dujardin's semi-extremal endomorphisms are near suspensions of Lattès maps

Upper bounds for the dimension of equilibrium measures

Abstract

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.