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Dimension of ergodic measures and currents on 
$\mathbb{C}\mathbb{P}(2)$
Part of:
Pluripotential theory
Holomorphic mappings and correspondences
Complex dynamical systems
Smooth dynamical systems: general theory
Published online by Cambridge University Press: 04 January 2019
Abstract
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Let 
$f$ 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 
$\unicode[STIX]{x1D708}$. We infer several applications. The first one is an upper bound for the lower pointwise dimension of the equilibrium measure, towards a Binder–DeMarco’s formula for this dimension. The second one shows that every current 
$S$ containing a measure of entropy 
$h_{\unicode[STIX]{x1D708}}>\log d$ has a directional dimension 
${>}2$, which answers a question of de Thélin–Vigny in a directional way. The last one estimates the dimensions of the Green current of Dujardin’s semi-extremal endomorphisms.
MSC classification
Primary:
37C45: Dimension theory of dynamical systems
37F10: Polynomials; rational maps; entire and meromorphic functions
32H50: Iteration problems
Secondary:
32U40: Currents
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