Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariant sets through resonant normal form for infinite dimensional holomorphic vector fields

Jessica Elisa Massetti, Michela Procesi, Laurent Stolovitch

TL;DR

This work develops an infinite-dimensional normal-form/linearization theory for analytic vector fields on sequence spaces with a fixed point at 0. By combining momentum preservation, a majorant-norm Banach framework, and a KAM-style Newton iteration under a Diophantine modulo $\Delta_\lambda$ condition, the authors construct a near-identity change of coordinates that conjugates the field to a resonant normal form and produces an analytic invariant manifold $\Sigma$ through the origin on which the flow is linear. The main result guarantees the existence of analytic invariant submanifolds through the fixed point, with the restricted dynamics analytically conjugate to the linear part ${\mathtt{D}}(\lambda)$ on $\Sigma$, under bounded-resonance and arithmetic hypotheses. The approach provides a dimension-uniform, convergent scheme applicable to infinite-dimensional Hamiltonian dynamics and discretized PDEs, enabling stable/unstable/center manifolds and almost-periodic behavior on invariant sets.

Abstract

In this paper, we study infinite dimensional holomorphic vector fields on sequence spaces, having a fixed point at $0$. Under suitable hypotheses we prove the existence of analytic invariant submanifolds passing through the fixed point. The restricted dynamics is analytically conjugate to the linear one under some Diophantine-like condition.

Invariant sets through resonant normal form for infinite dimensional holomorphic vector fields

TL;DR

This work develops an infinite-dimensional normal-form/linearization theory for analytic vector fields on sequence spaces with a fixed point at 0. By combining momentum preservation, a majorant-norm Banach framework, and a KAM-style Newton iteration under a Diophantine modulo condition, the authors construct a near-identity change of coordinates that conjugates the field to a resonant normal form and produces an analytic invariant manifold through the origin on which the flow is linear. The main result guarantees the existence of analytic invariant submanifolds through the fixed point, with the restricted dynamics analytically conjugate to the linear part on , under bounded-resonance and arithmetic hypotheses. The approach provides a dimension-uniform, convergent scheme applicable to infinite-dimensional Hamiltonian dynamics and discretized PDEs, enabling stable/unstable/center manifolds and almost-periodic behavior on invariant sets.

Abstract

In this paper, we study infinite dimensional holomorphic vector fields on sequence spaces, having a fixed point at . Under suitable hypotheses we prove the existence of analytic invariant submanifolds passing through the fixed point. The restricted dynamics is analytically conjugate to the linear one under some Diophantine-like condition.

Paper Structure

This paper contains 16 sections, 17 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Analytic vector fields, constants of motions and resonances
  3. Functional setting
  4. The Lie derivative operator
  5. Diagonal vector fields and commuting flows
  6. Structure of resonant monomials
  7. Assumptions and Diophantine conditions
  8. Vector fields tangent to $\Sigma$
  9. Main Result and examples
  10. Examples and Applications
  11. Proof of the main result
  12. Homological equation and Technical Lemmata
  13. KAM algorithm
  14. Properties of regular vector fields and proof of Homological equation
  15. Proof of Lemma \ref{['monotone']}
  16. ...and 1 more sections

Key Result

Lemma 2.3

${\mathcal{H}}_{s,r}$ and ${\mathcal{V}}_{s,r}$ are scales of Banach spaces w.r.t. $s$, namely more precisely The norms are also compatible with the degree namely for all $f\in {\mathcal{H}}_{s,r}^{{\mathtt{d}}}$, resp $X\in {\mathcal{V}}_{s,r}^{{\mathtt{d}}}$

Theorems & Definitions (48)

  • Definition 2.1: Holomorphic functions
  • Definition 2.2: Admissible vector fields
  • Lemma 2.3: Inclusion of spaces
  • proof
  • Definition 2.4: Projections
  • Definition 2.5: Degree projections
  • Proposition 2.6
  • proof
  • Lemma 2.7: Flow
  • Remark 2.8
  • ...and 38 more