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Invariant foliations for endomorphims of $\mathbb{P}^2$ with a pluripotentialist product structure

Virgile Tapiero

Abstract

Let $f$ be a holomorphic endomorphism of $\mathbb{P}^2$, let $T$ be its Green current and $μ=T\wedge T$ be its equilibrium measure. We prove that if $μ$ has a local product structure with respect to $T$ then (an iterate of) $f$ preserves a local foliation $\mathcal{F}$ on a neighborhood of $\mathrm{Supp}(T )\backslash\mathcal{E}$,where $\mathcal{E}$ denotes the exceptional set of f . If the local foliation $\mathcal{F}$ extends through $\mathcal{E}$,then it extends to $\mathbb{P}^2$ and is an invariant pencil of lines.

Invariant foliations for endomorphims of $\mathbb{P}^2$ with a pluripotentialist product structure

Abstract

Let be a holomorphic endomorphism of , let be its Green current and be its equilibrium measure. We prove that if has a local product structure with respect to then (an iterate of) preserves a local foliation on a neighborhood of ,where denotes the exceptional set of f . If the local foliation extends through ,then it extends to and is an invariant pencil of lines.
Paper Structure (20 sections, 35 theorems, 76 equations)

This paper contains 20 sections, 35 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Holomorphic foliations and invariance
  3. Holomorphic foliations on TEXT
  4. Pull-back of foliations and invariance
  5. Poincaré-Dulac coordinates
  6. Lattès mappings on TEXT and multipliers on TEXT
  7. Endomorphisms of TEXT preserving a pencil of lines
  8. Poincaré maps for repelling fixed points
  9. The Patching Theorem
  10. Hausdorff dimension of measures
  11. Proof of the Patching Theorem
  12. Construction of a foliation near TEXT
  13. Pull back by the Poincaré map
  14. Proof of Theorem \ref{['thm:Main']}
  15. Proof of Theorem \ref{["thm:foliatedJ'"]}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $f$ be a holomorphic map of $\mathbb{P}^2$ of degree $d\geq2$, $T$ be its Green current, $\mu=T\wedge T$ be its equilibrium measure and $\mathcal{E}$ be its exceptional set. Assume that $\mathrm{Supp}(\mu)\cap\mathcal{E}=\emptyset$. Assume moreover that Then there exists an open neighborhood $\mathcal{V}$ of $\mathrm{Supp}(T)\backslash\mathcal{E}$, and there exists a holomorphic foliation $\m

Theorems & Definitions (42)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Definition 2.2
  • Theorem 2.3: Favre-Pereira FavPer09
  • Corollary 2.4
  • Proposition 2.5
  • Theorem 3.1: Berteloot-Loeb
  • Proposition 3.2: Berteloot-Dupont-Molino bdm07
  • Proposition 3.3
  • ...and 32 more