The existence of invariant curves of a kind of almost periodic twist mappings

The paper proves an invariant-curve theorem for almost periodic twist maps via a KAM step that subtracts the ξ-mean [g](η) to solve the cohomological equations, obtains an explicit small-divisor estimate with an exponential-in-1/δ^2 loss (Lemma 3.1), and completes an iteration to produce an invariant graph with internal dynamics ξ1=ξ+α (Theorem 2.18) . By contrast, the model’s Newton–KAM sketch assumes a polynomial tame bound ‖Lα−1‖≲γ−1σ−Q and uses a simple composition estimate in AP spaces; these are not justified in the infinite-dimensional almost-periodic setting, where the paper carefully documents exponentially large losses and sensitivity of the weighted norms under composition (Remark 2.16; Appendix 6.4) . The model’s constant counterterm β and its elimination via intersection property are standard, but the paper handles the drift by removing [g](η) internally and does not require a global β, while still relying on the intersection property in the overall framework .