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Dimension of ergodic measures and currents on $\mathbb{C}\mathbb{P}(2)$

Part of: Smooth dynamical systems: general theory Holomorphic mappings and correspondences Pluripotential theory Complex dynamical systems

Published online by Cambridge University Press:  04 January 2019

CHRISTOPHE DUPONT  and
AXEL ROGUE
CHRISTOPHE DUPONT
Affiliation:
Univ Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, France email [email protected], [email protected]
AXEL ROGUE
Affiliation:
Univ Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, France email [email protected], [email protected]
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Abstract

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.

Keywords

dimension theoryLyapunov exponentsnormal forms

MSC classification

Primary: 37C45: Dimension theory of dynamical systems 37F10: Polynomials; rational maps; entire and meromorphic functions 32H50: Iteration problems
Secondary: 32U40: Currents
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 8 , August 2020 , pp. 2131 - 2155
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Copyright
© Cambridge University Press, 2019

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