A generic approach via relative singularity and controllability: Frequency-preserving with arbitrarily weak regularity in parameterized Hamiltonian systems
Zhicheng Tong, Yong Li
TL;DR
The article addresses the longstanding problem of preserving frequency in parameterized Hamiltonian systems under arbitrarily weak regularity. It introduces Internal Condition (H1), Relative Singularity (H2) and Controllability (H3) and proves a frequency-preserving KAM theorem (Theorem ['FPKAMT1']) for a perturbed Hamiltonian of the form $H(y,x,\xi,\varepsilon)=\langle\omega(\xi),y\rangle+\varepsilon P(y,x,\xi)$, yielding a nearby parameter $\xi^*$ with frequency $\Upsilon=\omega(\xi_0)$ and a continuous family $\xi^*(\hat{\xi}_0)$ under perturbations. The framework accommodates explicit systems with arbitrarily weak regularity, including Hölder and even discontinuous parameter dependence, and extends to partial and infinite-dimensional KAM settings with no spectral asymptotics. The paper also provides comprehensive counterexamples showing the indispensability of the proposed conditions, and demonstrates the robustness of the approach via parameter translations and a quasi-linear KAM iteration. Collectively, these results establish a generic, broadly applicable route to frequency preservation in parameterized Hamiltonian dynamics, with implications for both finite- and infinite-dimensional systems. The methods offer explicit convergence rates and flexibility through extra parameters, enabling partial preservation and infinite-dimensional extensions under minimal regularity assumptions.
Abstract
In this paper, we introduce a novel and generic approach to prove the persistence of frequency-preserving invariant tori in parameterized Hamiltonian systems, addressing irregular continuity with respect to parameters. Unlike traditional methods that strongly rely on domain extraction techniques or uniform weak convexity of the frequency mapping, we propose the concepts of relative singularity and controllability for the first time. These concepts enable us to deal with a wide range of explicit parameterized Hamiltonian systems with arbitrarily weak regularity, thereby overcoming a previously insurmountable challenge. We also construct several counterexamples to highlight the indispensability of our new conditions in the sense of frequency-preserving. Furthermore, we demonstrate the broad applicability of our results to various cases with explicit arbitrarily weak regularity, including the partial frequency-preserving case and the infinite-dimensional case without any spectral asymptotics. Overall, our approach, based on the concepts of relative singularity and controllability, illustrates its genericity in the frequency-preserving KAM theory.