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

In this paper, we present two infinite-dimensional Kolmogorov theorems based on nonresonant frequencies of Bourgain's Diophantine type or even weaker conditions. To be more precise, under a nondegenerate condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with the universally prescribed frequency that is independent of any spectral asymptotics. As an application, we prove that for a class of perturbed networks with weakly coupled oscillators described by \[{{{{ d}^2}{x_n}}}{{{ d}{t^2}}} + V'( {x_n} ) = ε_n {W'( {xn + 1 - {x_n}} ) - εₙ₋₁W'( {{x_n} - {xn - 1}} )} ,\;\;n ∈ Z,\] or even for more general perturbed networks, almost periodic breathers with frequency-preserving do exist, provided that the local potential V and the coupling potential W satisfy certain assumptions. In particular, this yields the first frequency-preserving version of the Aubry-MacKay conjecture [8,21].