Poincaré-Treshchëv mechanism in integrable Hamiltonian systems under almost-periodic perturbations

In this paper, a quasi-periodic Poincaré theorem is established for almost periodic perturbed nearly integrable Hamiltonian systems

$ \begin{eqnarray*} H = N(y)+\epsilon P(x, y, \omega t, \epsilon), \end{eqnarray*} $

where $ y\in G, \, G\subset \mathbb{R}^d $ is a bounded closed region, $ x\in \mathbb{T}^d $ with $ d $ being the degree of freedom, $ \epsilon>0 $ is a small parameter, $ N(y) $ is a real analytic function and $ P(x, y, \omega t, \epsilon) $ is real analytic in $ (x, y, t) $ and almost periodically in $ t $ with the frequency $ \omega = (\cdots, \omega_\lambda, \cdots)\in \mathbb{R}^\mathbb{Z} $. By applying the quasi-linear iterative scheme to deal with small divisors, we show the persistence of the majority of infinite dimensional resonant invariant tori associated to non-degenerate relative equilibria on any internal resonant surface under the usual Kolmogorov non-degenerate condition. Furthermore, by introducing spatial structure for almost-periodic function and non-resonant condition for infinite dimensional frequency, the fibre space of general resonance's Poincaré theorem is generalized from finite dimension to infinite dimension.