Infinite-Dimensional Kolmogorov Theorems

• Infinite-dimensional Kolmogorov theorems extend classical KAM theory by proving the persistence of invariant tori in function spaces with infinitely many degrees of freedom.
• They employ weighted analytic norms and iterative solutions of homological equations to manage small-divisor problems and secure superexponential convergence.
• These theorems enable frequency preservation in almost periodic breathers for perturbed lattice systems, with broad applications to Hamiltonian PDEs and extended dynamical structures.

Infinite-dimensional Kolmogorov theorems generalize classical results from finite-dimensional dynamical systems and probability theory to infinite-dimensional contexts arising in partial differential equations, stochastic analysis, and Hamiltonian systems. These theorems address core questions about the persistence of invariant structures (such as KAM tori), the quantitative approximation of functions, and the solvability and stability of evolution equations in function spaces with infinitely many degrees of freedom.

1. Kolmogorov Theorems in Infinite-dimensional Hamiltonian Systems

Classical Kolmogorov–Arnold–Moser (KAM) theory establishes persistence of invariant tori in nearly integrable, finite-dimensional Hamiltonian systems with a nondegeneracy and strong nonresonance (Diophantine) condition on the frequencies. Infinite-dimensional extensions, as developed in recent work (Tong et al., 19 Aug 2025, Wu et al., 2018), adapt this framework to phase spaces indexed by a countable lattice (e.g., $\mathbb{Z}$), which are typical, for instance, of lattice dynamical systems and Hamiltonian PDEs.

In (Tong et al., 19 Aug 2025), two main infinite-dimensional Kolmogorov theorems are presented, parameterized by:

• Analytic Hamiltonians $H(x, y)$ defined in a weighted norm on an infinite product of thickened tori.
• A nondegeneracy condition imposing that the second derivative in the action variables $y$ is close to a Hermitian target operator $\mathcal{Q}(x,y)$ which is invertible in the analytic norm.
• A frequency vector $\omega \in \mathbb{R}^{\mathbb{Z}}$ satisfying an infinite-dimensional Diophantine condition of Bourgain’s type: for each nonzero $k \in \mathbb{Z}^{* \infty}$ (finitely supported), $$|\langle k, \omega \rangle| \geq \gamma \prod_{j \in \mathbb{Z}} (1 + |k_j|)^{-\mu}$$ for fixed $\gamma, \mu > 0$.

The main conclusion is the existence of an analytic, symplectic coordinate transformation which conjugates the perturbed Hamiltonian to a normal form where the invariant torus $\{ y = 0 \}$ persists and carries the prescribed frequency $\omega$. A key innovation is decoupling the persistence mechanism from spectral asymptotics, relying only on universal (measure-theoretic) nonresonance. The approach can be extended to even weaker control-function-based nonresonance (allowing for substantial generalization of Diophantine conditions).

This result contrasts with earlier infinite-dimensional KAM approaches (Wu et al., 2018), which typically required rapid decay (short-range) of interactions and made essential use of finite-dimensional truncations. The (Tong et al., 19 Aug 2025) theorems directly construct the infinite-dimensional conjugacy and norm control using weighted analytic norms and iterative solutions to infinite-dimensional homological equations, with explicit exponential bounds on loss of analyticity and perturbation decrement.

2. Construction of Frequency-Preserving Almost Periodic Breathers

As an application, frequency-preserving infinite-dimensional Kolmogorov theorems are used to construct almost periodic breathers in infinite networks of weakly coupled oscillators, completing longstanding conjectures in nonlinear lattice dynamics.

Consider the perturbed network,

$$\frac{{\rm d}^2}{{\rm d} t^2} x_n + V'(x_n) = \varepsilon_n \left[ W'(x_{n+1} - x_n) - \varepsilon_{n-1} W'(x_n - x_{n-1}) \right], \quad n \in \mathbb{Z}$$

with analytic local potential $V$ and coupling potential $W$, and small (possibly spatially decaying) coupling coefficients $\varepsilon_n$.

The method proceeds by transforming each local Hamiltonian $1$-degree-of-freedom oscillator into action-angle coordinates where the local dynamics are integrable and characterized by energy levels (actions) and phase (angles). The nondegeneracy condition becomes a "twist" condition on $V$ ensuring $H_0''(\rho) > 0$ in the energy-action relationship, while analyticity of $W$ ensures the smallness of perturbation when the $\varepsilon_n$ decay rapidly.

By verifying all assumptions of the infinite-dimensional KAM theorem (analyticity, nondegeneracy, Diophantine frequency, and small perturbative norm), a full-dimensional invariant torus with prescribed frequency vector $\omega$ persists. The torus corresponds to a time–almost periodic, spatially exponentially localized solution—a frequency-preserving almost periodic breather.

Notably, this is the first rigorous proof of the frequency-preserving variant of the Aubry–MacKay conjecture, which previously lacked control over the drift of embedded frequencies under perturbations.

3. Mathematical Formulation and Key Estimates

The analytic norm for the function spaces is specified by

$$\|u\|_{\sigma} = \sum_{k \in \mathbb{Z}^{* \infty}} |\hat{u}(k)| e^{\sigma |k|_{\eta}},$$

with $|k|_{\eta}$ a weighted $\ell^1$-norm to control analytic regularity loss and convergence rates in the frequency lattice.

Persistence of the invariant torus is established under smallness conditions like

$$\|\mathscr{H}(x,0) - \int_{\mathbb{T}^\infty} \mathscr{H}(\xi,0) d\xi\|_\sigma \leq e^{-K},$$

$$\|\mathscr{H}_y(x,0) - \omega\|_\sigma \leq e^{-K\sigma^{-2/\eta}},$$

$$\|\mathscr{H}_{yy}(x,y) - \mathcal{Q}(x,y)\|_{(\sigma,\sigma)} \leq \sigma^{-1} e^{-K},$$

for large $K$ and appropriate analytic width $\sigma$ shrinking along the iteration.

Homological equations of the form

$\omega \cdot \partial_x f = g$

are solved with loss of analyticity controlled by the Diophantine or control-function bounds, allowing the iterative machinery to proceed without spectral gap restrictions.

In the KAM iteration for the conjugating transformation, error terms satisfy exponential decay

$$\epsilon_{\nu} \leq \exp(-2^\nu K \sigma^{-2/\eta}),$$

ensuring rapid convergence and persistence of the embedded torus for exponentially small perturbations.

4. Generalizations and Implications

• The analytic approach is flexible enough to admit Hamiltonians that are not integrable at zeroth order, greatly expanding the class of systems where persistence may be proved.
• Nonresonant frequencies are selected from "typical" sets in the sense of prevalence in $\mathbb{R}^{\mathbb{Z}}$, reflecting genericity of the KAM tori.
• The methodology is robust to generalizations of the small-divisor conditions, only requiring estimates governed by a control function, beyond the algebraic decay of Diophantine sequences (see Definitions FULLCONTROL and weakdio in (Tong et al., 19 Aug 2025)).
• The result explicitly shows that frequency selection can be achieved independently of spectral asymptotics, as neither the frequency vector nor the weight depends on the lattice site for measure-theoretic genericity.

5. Comparison with Previous Approaches

Classically, infinite-dimensional KAM results relied on finite-dimensional approximations and rapid decay of the perturbation (short-range interactions) to control small divisors (Wu et al., 2018). The present approach (Tong et al., 19 Aug 2025) offers the first direct, full-dimensional, frequency-preserving results for analytic Hamiltonians, removing the need for truncation or external modulation, with uniform control over the norm and explicit construction of the conjugacy.

6. Applications Beyond Coupled Oscillator Networks

While the immediate application is to perturbed oscillator chains and lattice dynamical systems (including FPU arrays, nonlinear Klein–Gordon chains, and discrete nonlinear Schrödinger lattices), the overall framework is directly applicable to Hamiltonian PDEs on infinite domains, continuum limits, and spatially localized/coherent structures in noncompact systems that admit similar analytic structure and nondegeneracy.

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In summary, infinite-dimensional Kolmogorov theorems provide a rigorous mechanism for the persistence of invariant tori with prescribed frequency vectors in analytic infinite-dimensional Hamiltonian systems, under nondegeneracy and broad nonresonance conditions. The technical machinery developed ensures control on small-divisor problems and superexponential convergence, enabling the explicit construction of frequency-preserving almost periodic breathers in nonlinear oscillator networks and establishing a new paradigm for the analysis of high-dimensional or spatially extended dynamical systems (Tong et al., 19 Aug 2025, Wu et al., 2018).