A generic approach via relative singularity and controllability: Frequency-preserving with arbitrarily weak regularity in parameterized Hamiltonian systems

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.