An infinite-dimensional Kolmogorov theorem and the construction of almost periodic breathers

In this paper, we present two infinite-dimensional Kolmogorov theorems based on non-resonant frequencies of Bourgain’s Diophantine type or even weaker conditions. To be more precise, under a Legendre-type nondegeneracy condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with a universally prescribed frequency independent of any spectral asymptotics. As an application, we prove that for a class of perturbed networks with weakly coupled oscillators described by $$\frac{{\mathrm{d}}^2{x}_n}{\mathrm{d}{t}^2}+V^{\mathrm{\prime }}\left(x_n\right)={\epsilon }_n{W}^{\mathrm{\prime }}\left(x_{n+1}-x_n\right)-{\epsilon }_{n-1}{W}^{\mathrm{\prime }}\left(x_n-x_{n-1}\right),n\in \mathbb{Z},$$ or even for more general perturbed networks, frequency-preserving almost periodic breathers do persist, provided that the local potential $V$ and the coupling potential $W$ satisfy certain assumptions. In particular, this yields the first frequency-preserving result for the Aubry–MacKay conjecture [MA94,Aub95].