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Holonomy and equivalence of analytic foliations

Francisco Chaves

Abstract

The main goal of this paper is the analytic classification of the germs of singular foliations generated, up to an analytic change of coordinates, by the germs of vector fields of form the $x\partial_x+\sum_{i=1}^{n}a_i(x,\mathbf{z})\partial_{z_i}$, where $a_i(x,\mathbf{z})$ is a germ of analytic function with $a_i(x,0)=0$. We focus on the connection with the conjugation of the holonomies related to them. We prove, under some hypothesis, that these germs of singular foliations are analytically classified once their local holonomy along a given separatrix are analytically conjugated.

Holonomy and equivalence of analytic foliations

Abstract

The main goal of this paper is the analytic classification of the germs of singular foliations generated, up to an analytic change of coordinates, by the germs of vector fields of form the , where is a germ of analytic function with . We focus on the connection with the conjugation of the holonomies related to them. We prove, under some hypothesis, that these germs of singular foliations are analytically classified once their local holonomy along a given separatrix are analytically conjugated.

Paper Structure

This paper contains 7 sections, 21 theorems, 56 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Transversely Formally Objects.
  3. The Ring $\mathcal{O}_{r,R}[[\mathbf{z}]]$.
  4. Automorphisms and Derivations.
  5. The Lie Brackets and the Exponential map.
  6. Normal Form of $x$-normalized $D_R$-transversely Formal Vector Fields.
  7. The Crossing Type Foliations

Key Result

Proposition 1

Let $X$ and $Y$ be two germ vector fields, verifying $(1), (2), (3)$ and $(4)$. Denote by $h_X$ and $h_Y$ the holonomies of $X$ and $Y$ relatively to the separatrices of $X$ and $Y$ tangent to the eigenspace associated with the first eigenvalue, respectively. Then, if $h_X$ and $h_Y$ are analyticall

Figures (5)

  • Figure 1: Transversality of $H$ and $\Gamma$ at origin
  • Figure 2: No transverse negative resonance
  • Figure 3: Diagram of inclusions
  • Figure 4: Composed path
  • Figure 5: Siegel domain

Theorems & Definitions (52)

  • Proposition : reis2006equivalence, Theorem 1
  • Corollary 1
  • proof
  • Corollary 2
  • Theorem 3
  • Definition 1
  • Definition 2
  • Lemma 4
  • Claim 1
  • proof
  • ...and 42 more