Infinite Dimensional Holomorphic Vector Fields

• Infinite dimensional holomorphic vector fields are complex-analytic derivations on Banach and Hilbert manifolds that enable normal form reduction and invariant submanifold analysis.
• They employ resonant and nonresonant decompositions under Diophantine conditions to control small divisors and manage infinitely many resonances.
• These fields yield rigorous constructions of invariant manifolds, impacting the study of stability in infinite-dimensional dynamical systems and analytic PDEs.

Infinite dimensional holomorphic vector fields are holomorphic (complex-analytic) derivations on infinite-dimensional analytic manifolds, most concretely sequence or function spaces equipped with Banach or Hilbert topologies. These objects play a central role in complex geometry, the theory of dynamical systems in infinite dimensions (e.g., analytic PDEs), and infinite-dimensional Lie theory. Their paper encompasses algebraic, analytic, and geometric methods, with particular focus on the structure of Lie algebras of such vector fields, invariant sets, and reducibility and stability phenomena under analytic flows.

1. Functional-Analytic Setting for Infinite Dimensional Holomorphic Vector Fields

The most developed setting is that of analytic vector fields on Banach sequence spaces. A standard example, as in (Massetti et al., 6 Nov 2025), is the class of weighted $\ell_2$-spaces indexed by an infinite set:

• Index set $I = \mathbb{Z} \setminus \{0\} \times \{+,-\}$.
• For $s\geq0$, define the weighted Hilbert space:

$$\mathfrak{g}_s := \left\{ x = (x_k)_{k \in I} : |x|_s^2 = \sum_{k\in I} |k|^2 e^{2 s \sqrt{|k|}} |x_k|^2 < \infty \right\}$$

This sets the stage for analytic functions $f(x)$ and analytic vector fields $V(x) = (V_k(x))_{k\in I}$ expressible via convergent power series in $B_r(\mathfrak{g}_s)$, the open ball of radius $r$.

A holomorphic function is momentum-preserving if its power series coefficients vanish unless the total "momentum" $\sum_k k q_k = 0$. For vector fields, $V^{(k)}_q$ vanishes unless $\sum_h h q_h - k = 0$. The structure of these spaces is critical for ensuring well-defined algebraic operations, convergence of normal form transformations, and the existence of flows.

2. Resonant Normal Form Theory and the Structure of Resonances

Resonant normal form analysis for infinite dimensional holomorphic vector fields proceeds by decomposing the dynamics near a fixed point (typically at $0$) into linear and resonant/non-resonant components.

• Diagonal (Linear) Part: Given a frequency vector $\lambda = (\lambda_k)_{k\in I}$ with $\lambda_k \neq 0$, the linear part is

$$D_\lambda := \sum_{k\in I} \lambda_k x_k \partial_{x_k}$$

with Lie derivative $L_\lambda$. Monomials $x^q \partial_{x_k}$ are called resonant if $\lambda \cdot q - \lambda_k = 0$.

• Resonance Modules:

$$\Lambda_\lambda := \{ Q \in \mathbb{N}^{(I)} : Q \cdot \lambda = 0 \}$$

$$\Delta_\lambda := \bigcup_{k\in I} \{ P \in \mathbb{N}^{(I)} : P \cdot \lambda = 0,\, P+e_k \in \mathbb{N}^{(I)} \}$$

The resonant vector fields commuting with $D_\lambda$ are generated by monomials indexed by $\Lambda_\lambda$ and $\Delta_\lambda$.

• Diophantine-multiresonant Condition: To control infinite resonance phenomena, one imposes a Diophantine-type condition on the "frequency" $\lambda$ modulo $\Delta_\lambda$, ensuring that non-resonant monomials are separated from resonance by lower bounds on $|\lambda \cdot p|$ of the form:

$$|\lambda \cdot p| \geq \gamma \prod_{i \in I} (1 + p_i^2 i^2)^{-\tau}$$

for $p \notin \Delta_\lambda$ of finite support.

Normal form reduction seeks analytic, near-identity conjugacies bringing the field into a "simplest" form comprising only its resonant and higher-order terms. This is essential to identify invariant submanifolds and elucidate the geometric and dynamical structure.

3. Invariant Analytic Submanifolds: Main Existence Theorem and Formal Structures

The central result [(Massetti et al., 6 Nov 2025), Thm. 4.1] establishes the existence of analytic invariant submanifolds for this class of vector fields:

Let $\lambda \in \mathbb{R}^I$ satisfy:

• (i) Nondegeneracy: $\lambda_k \neq 0$,
• (ii) $(\gamma,1)$–Diophantine modulo $\Delta_\lambda$,
• (iii) $\Lambda_\lambda$ and $\Delta_\lambda$ generators have uniformly bounded support-size.

Given a momentum-preserving analytic vector field

$W = D_\lambda + Z + X + N$

where $Z$ is diagonal-resonant (of order at least 2), $X$ is nonresonant of order at least $*$, and $N$ higher-order nonresonant, there exists $r'>0$, $s'>s$, and an analytic, near-identity diffeomorphism $\Phi$ such that the pullback vector field

$$\Phi_* W = D_\lambda + Z + N_\infty, \qquad N_\infty \in (2),\quad \text{order} \geq *$$

where the (0) and (1) nonresonant parts have been removed.

The resonant manifold

$$\Sigma := \{ x \in B_{R}(\mathfrak{g}_s) : x^Q = 0 \ \forall Q \in \Lambda_\lambda \}$$

gets mapped under $\Phi$ to an analytic invariant submanifold, and restricted there, the conjugacy reduces $W$ to its linear part $D_\lambda$.

4. Analytic Conjugacy and Proof Techniques

The analytic conjugacy on $\Sigma$ is achieved by decomposing the Lie algebra of vector fields into summands $(0),(1),(2)$ according to the number of resonant factors, and iteratively solving the homological equation for the nonresonant components.

This is implemented via a Newton–Kolmogorov–Arnold–Moser (KAM) scheme using:

• Majorant norm estimates for control over analytic vector fields.
• Lie-derivative and commutator estimates (e.g., $$|L_X f|_{s,r}\leq (1+r/\rho)\|X\|_{s,r+\rho}|f|_{s,r+\rho}$$).
• Invertibility results for $L_\lambda$ restricted to the nonresonant sector, making essential use of the Diophantine condition for lower bounds on small denominators.
• Nilpotency arguments: For sufficiently high-order non-diagonal resonant terms, the action of $\mathrm{ad}_Z$ becomes nilpotent (specifically, squares to zero), enabling two-step inversion to kill lower-order terms.

Each step is realized via analytic time-1 flows $\Phi_i = \exp(L_{F_i})$, producing convergence through controlled shrinkage of radii and majorant norms.

5. Key Estimates and Structural Lemmas

The proof strategy rests on several detailed analytic estimates:

• Lie-derivative control: Ensures the closure of analytic vector fields under commutators, crucial for the KAM step convergence.
• Time-1 flow control: Uniform bounds on the deviation from the identity under exponentiation of small vector fields.
• Homological equation solubility: Explicit lower bounds on non-resonant denominators, bounding the growth of conjugating transformations.
• Nilpotency of $\mathrm{ad}_Z$: Ensures the iterative scheme for removing non-resonant terms stabilizes after finitely many steps for given truncation order.

These collectively ensure that the constructed invariant manifolds are genuinely analytic (not merely formal or $C^\infty$) and that the conjugacy is valid in a full neighborhood of the fixed point.

6. Applications, Implications, and Broader Context

This analysis demonstrates that, under explicit arithmetic (Diophantine) and support assumptions, infinite dimensional analytic dynamics exhibits a form of normal form theory analogous to classical finite-dimensional Poincaré–Dulac theory, but with nontrivial control over infinitely many resonances and small divisors. This has significant implications:

• Stability and Reducibility: The existence of analytic invariant submanifolds on which the flow reduces to the linear part guarantees regions of dynamically trivial behavior despite the possibility of pervasive resonance.
• Geometric Structure: The precise identification of $\Sigma$ as the vanishing locus of resonance-invariant monomials provides concrete geometric models for the phase space.
• Extension to PDEs and Infinite Dimensional Hamiltonian Systems: Since many evolution PDEs admit expansions in similar sequence spaces and can be cast as infinite-dimensional holomorphic vector fields, these results directly inform invariant set constructions for analytic nonlinear PDEs.

Furthermore, the use of a single-step, direct normal-form conjugacy—rather than stepwise Birkhoff–normal–form iteration—marks a conceptual advance in analytically rigorous infinite-dimensional KAM and normal-form theory.

A plausible implication is that similar techniques could yield further invariant structures for broader classes of infinite-dimensional flows, given suitable arithmetic separation of resonances.

7. Relation to Other Infinite Dimensional Holomorphic Vector Field Results

Key connections to other frameworks:

• The structure of infinite dimensional holomorphic vector fields (in the setting of $\mathbb{C}^n$, $n\to\infty$) and their Lie algebras—see (Andrist, 2018)—involves the dense generation of infinite-dimensional automorphism groups and Lie algebras by finitely many complete flows, a phenomenon relying on the flexibility of holomorphic functions and the Andersen–Lempert theory.
• The dichotomy for Bergman or $L^2$-holomorphic vector fields in domains of the Riemann sphere (Szőke, 2020) reveals that the $L^2$-space of holomorphic vector fields (sections of $T\mathbb{P}^1$) is either finite-dimensional (and equal to the full global space) or infinite-dimensional, reflecting the rigidity/flexibility duality of holomorphic sections in infinite dimensions.

The combination of normal form constructions, explicit resonance module calculations, and arithmetic separation thus defines the analytic, algebraic, and geometric landscape for infinite dimensional holomorphic vector fields and their invariant sets.