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Variance of vector fields -- Definition and properties

Jacky Cresson, Jordy Palafox

TL;DR

The paper addresses the variance of vector fields within the mould formalism and its role in analyzing the nilpotent parts of resonant vector fields. It provides a self-contained development of the variance concept, complete proofs of its fundamental formulas, and defines the associated derivations, notably showing $\nabla = \sum_{a\in A} Var_a$. It applies these results to derive and solve the Nilpotent mould equation $Var_c(Nil^{\bullet}) = I_c^{\bullet} \times Nil^{\bullet} - Nil^{\bullet} \times I_c^{\bullet}$ and computes $Nil^{\bullet}$ explicitly up to length four. The work clarifies algebraic mechanisms for handling small denominators and sets the stage for further analysis of analyticity and arborification of moulds in the correction problem.

Abstract

We give a self contained presentation of the notion of variance of a vector field introduced by Jean Ecalle and Bruno Vallet in \cite{ev} following a previous work of Jean Ecalle and Dana Schlomiuk in \cite{es}. We give complete proofs and definitions of various results stated in these articles. Following J. Ecalle and D. Schlomiuk, We illustrate the interest of the variance by giving a complete proof of the formulas for the mould defining the nilpotent part of a resonant vector field.

Variance of vector fields -- Definition and properties

TL;DR

The paper addresses the variance of vector fields within the mould formalism and its role in analyzing the nilpotent parts of resonant vector fields. It provides a self-contained development of the variance concept, complete proofs of its fundamental formulas, and defines the associated derivations, notably showing . It applies these results to derive and solve the Nilpotent mould equation and computes explicitly up to length four. The work clarifies algebraic mechanisms for handling small denominators and sets the stage for further analysis of analyticity and arborification of moulds in the correction problem.

Abstract

We give a self contained presentation of the notion of variance of a vector field introduced by Jean Ecalle and Bruno Vallet in \cite{ev} following a previous work of Jean Ecalle and Dana Schlomiuk in \cite{es}. We give complete proofs and definitions of various results stated in these articles. Following J. Ecalle and D. Schlomiuk, We illustrate the interest of the variance by giving a complete proof of the formulas for the mould defining the nilpotent part of a resonant vector field.
Paper Structure (9 sections, 7 theorems, 123 equations)

This paper contains 9 sections, 7 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Vector fields and infinitesimal automorphisms
  3. Universal mould and action of a mould
  4. Variance of a universal mould
  5. Derivation associated to the variance
  6. Variance and the Nilpotent part of resonant vector fields
  7. The Nilpotent part of a resonant vector field
  8. Explicit computation of the mould $Nil^{\bullet}$
  9. Conclusion and perspectives

Key Result

Theorem 1

Let $X$ be in prepared form and $A(X)$ the associated alphabet. We denote by $B_c$ a homogeneous vector field of degree at least $2$. Let $A_c =A(X)\cup \{c\}$. We have with where for $i=1,\dots ,l(\mathbf{n} )$ we have where the two operators $conf_i$ and $conb_i$ are respectively the forward (resp. backward) contraction of position $i$ defined by with Moreover, if $c\not\in A(X)$ and $\mat

Theorems & Definitions (17)

  • Definition 1: Action of a mould on vector fields
  • Definition 2: Variance of a vector field
  • Theorem 1: Variance of a universal mould
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 7 more