On slice measures of Green currents on CP(2)
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929255522238464 |
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| author | Dupont, Christophe Tapiero, Virgile |
| author_facet | Dupont, Christophe Tapiero, Virgile |
| contents | Let $f$ be a holomorphic map of $\mathbb{C}\mathbb{P}^2$ of degree $d\geq 2$, let $T$ be its Green current and $μ=T\wedge T$ be its equilibrium measure. We give a new proof of a theorem due to Dujardin asserting that $μ\ll T\wedgeω_{\mathbb{P}^2}$ implies $λ_2=\frac{1}{2} \log\ d$, where $λ_1 \geq λ_2$ are the Lyapunov exponents of $μ$. Then, assuming $μ\ll T\wedgeω_{\mathbb{P}^2}$, we study slice measures $ν:=T\wedge dd^c|W|^2$, where $W$ is a holomorphic local submersion. We give sufficient conditions on the Radon-Nikodym derivative of $μ$ with respect to the trace measure $T\wedgeω_{\mathbb{P}^2}$ ensuring $μ=ν$. The involved submersion $W$ comes from normal coordinates for the inverse branches of the iterates of $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_16425 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On slice measures of Green currents on CP(2) Dupont, Christophe Tapiero, Virgile Complex Variables Dynamical Systems Let $f$ be a holomorphic map of $\mathbb{C}\mathbb{P}^2$ of degree $d\geq 2$, let $T$ be its Green current and $μ=T\wedge T$ be its equilibrium measure. We give a new proof of a theorem due to Dujardin asserting that $μ\ll T\wedgeω_{\mathbb{P}^2}$ implies $λ_2=\frac{1}{2} \log\ d$, where $λ_1 \geq λ_2$ are the Lyapunov exponents of $μ$. Then, assuming $μ\ll T\wedgeω_{\mathbb{P}^2}$, we study slice measures $ν:=T\wedge dd^c|W|^2$, where $W$ is a holomorphic local submersion. We give sufficient conditions on the Radon-Nikodym derivative of $μ$ with respect to the trace measure $T\wedgeω_{\mathbb{P}^2}$ ensuring $μ=ν$. The involved submersion $W$ comes from normal coordinates for the inverse branches of the iterates of $f$. |
| title | On slice measures of Green currents on CP(2) |
| topic | Complex Variables Dynamical Systems |
| url | https://arxiv.org/abs/2310.16425 |