Persistence of Invariant Tori in Infinite-Dimensional Hamiltonian Systems

In this paper, we consider the persistence of invariant tori in infinite-dimensional Hamiltonian systems H=⟨ω,I⟩+P(θ,I,ω),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H=\langle \omega ,I \rangle +P(\theta ,I,\omega ), \end{aligned}$$\end{document}where θ∈TΛ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in \mathbb {T}^\Lambda $$\end{document}, I∈RΛ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I\in \mathbb {R}^\Lambda $$\end{document}, the frequency ω=(⋯,ωλ,⋯)λ∈Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega }=(\cdots ,{\omega }_\lambda ,\cdots )_{\lambda \in \Lambda }$$\end{document} is regarded as parameters varying freely over some subset ℓ∞(Λ,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^\infty (\Lambda ,\mathbb {R})$$\end{document} of the parameter space RΛ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^\Lambda $$\end{document}, ω=(⋯,ωλ,⋯)λ∈Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega }=(\cdots ,{\omega }_\lambda ,\cdots )_{\lambda \in \Lambda }$$\end{document} is a bilateral infinite sequence of rationally independent frequency, in other words, any finite segments of ω=(⋯,ωλ,⋯)λ∈Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega }=(\cdots ,{\omega }_\lambda ,\cdots )_{\lambda \in \Lambda }$$\end{document} are rationally independent.