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A flower theorem in $\mathbb{C}^n$

Kémo Morvan

TL;DR

This work extends the Leau-Fatou flower picture to holomorphic germs $f:(\\mathbb{C}^n,0)\\to(\\mathbb{C}^n,0)$ tangent to the identity that fix coordinate hyperplanes. By normalizing to $\\langle a,M\\rangle = -1$ and exploiting the one-dimensional reduction along $\\mathbf{x}^m$, the authors construct $d = \\gcd(M)$ forward and backward parabolic domains on which forward or backward orbits converge to $0$, and provide explicit Fatou coordinates that conjugate $f$ (or $f^{-1}$) to a translation in suitable coordinates. The central result is a higher-dimensional flower theorem: these domains, together with the fixed set $\\{\\mathbf{x}^M=0\\}$, form a neighborhood of the origin and yield a full foliation by parabolic petals with holomorphic conjugacies to translations; a complementary divergence result shows nonexistence of such petals when a certain real-part condition fails. Overall, the paper generalizes the classical one-dimensional dynamics to several complex variables, employing invariant functions, approximate Fatou coordinates, and patching techniques to obtain exact Fatou coordinates on invariant domains.

Abstract

We prove an analog of the flower theorem for non-degenerate reduced tangent to the identity germs that fix the coordinate hyperspaces in any dimension.

A flower theorem in $\mathbb{C}^n$

TL;DR

This work extends the Leau-Fatou flower picture to holomorphic germs tangent to the identity that fix coordinate hyperplanes. By normalizing to and exploiting the one-dimensional reduction along , the authors construct forward and backward parabolic domains on which forward or backward orbits converge to , and provide explicit Fatou coordinates that conjugate (or ) to a translation in suitable coordinates. The central result is a higher-dimensional flower theorem: these domains, together with the fixed set , form a neighborhood of the origin and yield a full foliation by parabolic petals with holomorphic conjugacies to translations; a complementary divergence result shows nonexistence of such petals when a certain real-part condition fails. Overall, the paper generalizes the classical one-dimensional dynamics to several complex variables, employing invariant functions, approximate Fatou coordinates, and patching techniques to obtain exact Fatou coordinates on invariant domains.

Abstract

We prove an analog of the flower theorem for non-degenerate reduced tangent to the identity germs that fix the coordinate hyperspaces in any dimension.
Paper Structure (7 sections, 13 theorems, 96 equations, 3 figures)

This paper contains 7 sections, 13 theorems, 96 equations, 3 figures.

Table of Contents

  1. Preliminaries
  2. Existence of parabolic domains
  3. Invariant functions
  4. Approximate Fatou coordinates
  5. Existence of Fatou coordinates
  6. The flower theorem
  7. Proof of theorem \ref{['thm:div']}

Key Result

Theorem A

Let f be a biholomorphism of form eq:f and such that for all $i \in \{1, \ldots, n\}$, we have $\mathop{\mathrm{Re}}\nolimits \left(\frac{a_i}{\langle a,M \rangle} \right) > 0$. Then in any neighbourhood of the origin there exists $d$ pairwise disjoint connected open sets $\Omega^{+}_0, \ldots, \Ome

Figures (3)

  • Figure 1: The sets $C$ and $\tilde{C}$
  • Figure 2: The sets $\tilde{S}_{-1}$ (in red) and $\tilde{S}_{1}$ (in green) for p=3.
  • Figure :

Theorems & Definitions (37)

  • Remark 1
  • Theorem A
  • Theorem B
  • Definition 1.1
  • Definition 1.2: see figure \ref{['fig:S']}
  • Theorem 1.3: Leau-Fatou flower theorem
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • ...and 27 more