Resonances and Phase Locking Phenomena for Foliation Preserving Torus Maps

The paper formulates the same a‑posteriori KAM problem for foliation-preserving torus maps via the functional F[h,λ]=0 (see (3.3)–(3.4)) and proves Theorem 3.1 with the Newton step, cohomological inversion under a Diophantine condition, and a quadratic estimate ∥e+∥_{ρ−δ} ≤ C δ^{-2τ} ∥e∥_ρ^2, yielding convergence and the tame bound |λ−λ0|, ∥h−h0∥_{ρ/2} ≤ C (4/ρ)^{2τ} ∥e0∥_ρ (Theorem 3.1). The key steps include the modified Newton equation, the preconditioning by (DH)^{-1} leading to a constant-coefficient cohomological equation, and the decomposition of the remainder into a term controlled by De and a second-order composition remainder; see the derivation and estimates in 3.1 and the iterative lemma and convergence in 3.2. The candidate solution implements the same Newton–KAM scheme with an “automatic reducibility” identity, the same cohomological inversion and analytic strip losses, and obtains the identical quadratic estimate and a‑posteriori bounds. The only discrepancy is a minor sign slip in Step 3 when rewriting B0=A0(1+u0)=A0∘Tω±r0 (the paper’s derivation corresponds to the plus sign), which does not affect the inequality used to bound |B0| away from zero. Overall, the approaches coincide in structure and detail, so both are correct and substantially the same proof. Citations: the functional set-up and theorem statement (3.3)–(3.4), Theorem 3.1 ; the Newton equation, preconditioning, and estimates for ΔV and Δλ (3.6)–(3.12) ; the quadratic error bound and iterative lemma Lemma 3.1 ; and the convergence scheme in 3.2 with the loss-of-analyticity schedule . The cohomological estimate used throughout is Lemma 1.1 .