The parameter KAM limit: invariant quasi-periodic motion with frequency $\nu$ and Lagrangian images

Finite-step residual. Fix $j\ge 0$ and $\nu \in {\mathrm{\Omega }}_{\alpha ,\tau }$. By [F1], on the real points of $D_j$ one has $H({\mathrm{\Psi }}_j(I,\vartheta ,\nu );{\chi }_j(\nu ))=e_j(\nu )+\nu \cdot I+P_j(I,\vartheta ,\nu )$ with ${\mathrm{\Psi }}_j$ canonical in the phase variables for each fixed target $\nu$. Define $H_j^{\ast }(z):=H(z;{\chi }_j(\nu ))$; since ${\chi }_j$ does not depend on $z$, the ordinary phase transport identity applies, so the transport identity of [F2] applied to ${\mathrm{\Psi }}_j$ gives $D{\mathrm{\Psi }}_j{X}_{H_j^{\ast }\circ {\mathrm{\Psi }}_j}=X_H({\mathrm{\Psi }}_j;{\chi }_j(\nu ))$; by [F2] and linearity of the gradient, $X_{H_j^{\ast }\circ {\mathrm{\Psi }}_j}=X_{e_j+\nu \cdot I}+X_{P_j}=(0,\nu )+X_{P_j}$; hence $X_H({\mathrm{\Psi }}_j;{\chi }_j(\nu ))-D{\mathrm{\Psi }}_j(0,\nu )=D{\mathrm{\Psi }}_j{X}_{P_j}$ on the real points of $D_j$.

Weighted phase Jacobian product bound. For $x_j=(z,\nu )\in D_j$, recursively put $x_i=(z_i,{\lambda }_i):=T_i(x_{i+1})$ for $i=j-1,\dots ,0$, with $T_i$ the extended maps of [F1]'s composition supplier; thus $x_i\in D_i$ and ${\lambda }_i={\varphi }_i({\lambda }_{i+1})$. Parameters are independent of the input phase, so $D_z{\mathrm{\Psi }}_j=M_0\cdots M_{j-1}$, where $M_i:=D_z{\tilde{\mathrm{\Phi }}}_i(z_{i+1},{\lambda }_i)$. The supplier's domain inclusions put these evaluations in [F5]'s one-step domain. Since $W_i{W}_{i+1}^{-1}=\mathrm{diag}({\eta }_i{I}_n,\frac{1}{2}{I}_n)$, $\mathrm{\parallel }{W}_i{M}_i{W}_{i+1}^{-1}\mathrm{\parallel }\le (1+C{E}_i)\mathrm{/}2\le 1$. Insert adjacent weights in the product to obtain $\mathrm{\parallel }{W}_0{D}_z{\mathrm{\Psi }}_j{W}_j^{-1}\mathrm{\parallel }\le 1$, including the identity product at $j=0$.

The transported residual tends to zero. By [F4] and step 1.2, on the real points of $D_j$, $\mid X_H({\mathrm{\Psi }}_j;{\chi }_j(\nu ))-D{\mathrm{\Psi }}_j(0,\nu ){\mid }_{(W_0)}=\mid D{\mathrm{\Psi }}_j{X}_{P_j}{\mid }_{(W_0)}\le \mathrm{\parallel }{W}_{0}D{\mathrm{\Psi }}_j{W}_j^{-1}\mathrm{\parallel }\mid W_j{X}_{P_j}\mid \le \frac{2{\epsilon }_j}{r_j{\sigma }_j}=2\alpha E_j{\sigma }_j^{\mu -1},$ and by [F1] $E_j\le 4^{-j}{E}_0$ while ${\sigma }_j={\sigma }_0{2}^{-j}$, so $2\alpha E_j{\sigma }_j^{\mu -1}\le 2\alpha E_0{\sigma }_0^{\mu -1}{4}^{-j}{2}^{-(\mu -1)j}\to 0$ as $j\to \mathrm{\infty }$.

Passing to the limit. By [F3] the maps ${\mathrm{\Psi }}_j$ and their first $\vartheta$-derivatives converge uniformly on $\mathcal{L}$ to those of ${\mathrm{\Psi }}_{\mathrm{\infty }}$, and ${\chi }_j\to {\chi }_{\mathrm{\infty }}$ uniformly with the phase-parameter images in the original domain by [F1]. Since $X_H$ is continuous there, $X_H({\mathrm{\Psi }}_j;{\chi }_j(\nu ))\to X_H({\mathrm{\Psi }}_{\mathrm{\infty }};{\chi }_{\mathrm{\infty }}(\nu ))$ and $D{\mathrm{\Psi }}_j(0,\nu )={\sum }_k{\nu }_k{\mathrm{\partial }}_{{\vartheta }_k}{\mathrm{\Psi }}_j\to {\sum }_k{\nu }_k{\mathrm{\partial }}_{{\vartheta }_k}{\mathrm{\Psi }}_{\mathrm{\infty }}$ for each fixed label and real angle, while $X_H({\mathrm{\Psi }}_j;{\chi }_j(\nu ))-D{\mathrm{\Psi }}_j(0,\nu )=D{\mathrm{\Psi }}_j{X}_{P_j}\to 0$ in the $W_0$-norm by step 2.1; therefore $X_H({\mathrm{\Psi }}_{\mathrm{\infty }}(0,\vartheta ,\nu );{\chi }_{\mathrm{\infty }}(\nu ))={\sum }_k{\nu }_k{\mathrm{\partial }}_{{\vartheta }_k}{\mathrm{\Psi }}_{\mathrm{\infty }}(0,\vartheta ,\nu )$ at every real $\vartheta$, which is the real-slice part of clause 1.

Extension to the complex strip. Both sides of the identity of step 3.1 are holomorphic in $\vartheta$ on $S_{s_0\mathrm{/}4}$ for fixed $\nu$: the right side by [F3], and the left side because $H$ is holomorphic on the open domain and ${\mathrm{\Psi }}_{\mathrm{\infty }}$ maps $\{0\}\times S_{s_0\mathrm{/}4}\times {\mathrm{\Omega }}_{\alpha ,\tau }$ into its interior (by [F1]'s size bounds and $C{E}_0\le 1\mathrm{/}16$, $\mathrm{\mid }b\mathrm{\mid }\le 2C{r}_0{E}_0\le r_0\mathrm{/}8$ and $\mathrm{\mid }w\mathrm{\mid }\le 2C{\sigma }_0{E}_0\le {\sigma }_0\mathrm{/}8$); their difference $f$ vanishes on ${\mathbb{R}}^n$, so at every real ${\vartheta }_0$ all mixed complex derivatives of $f$ vanish (induction on the order: ${\mathrm{\partial }}_{{\vartheta }_i}f$ is holomorphic and vanishes on the real slice, since the difference quotient is taken along real directions), the power series of $f$ at ${\vartheta }_0$ has all coefficients zero, and $f=0$ on a polydisc around ${\vartheta }_0$; [F6] applied to the connected open set $S_{s_0\mathrm{/}4}$ gives $f\equiv 0$ there. This proves the extension assertion of clause 1.

Quasi-periodic motion. By [F3] the map $\vartheta \mapsto {\mathrm{\Psi }}_{\mathrm{\infty }}(0,\vartheta ,\nu )$ is holomorphic, hence $C^1$ on the real slice, and the chain rule gives $\frac{d}{ds}{\mathrm{\Psi }}_{\mathrm{\infty }}(0,{\vartheta }_0+s\nu ,\nu )={\sum }_j{\nu }_j{\mathrm{\partial }}_{{\vartheta }_j}{\mathrm{\Psi }}_{\mathrm{\infty }}(0,{\vartheta }_0+s\nu ,\nu )$, which equals $X_H({\mathrm{\Psi }}_{\mathrm{\infty }}(0,{\vartheta }_0+s\nu ,\nu );{\chi }_{\mathrm{\infty }}(\nu ))$ by the real-slice identity of step 3.1; this is clause 2.

Lagrangian images. The unperturbed parametrisation ${\iota }_0(\vartheta ):=(0,\vartheta )$ satisfies ${\iota }_0^{\ast }\omega =0$ because each term $d{\vartheta }_i\wedge d{I}_i$ pulls back to zero along $I\equiv 0$; each ${\mathrm{\Psi }}_j$ is canonical on real points by [F1], so $({\mathrm{\Psi }}_j\circ {\iota }_0{)}^{\ast }\omega ={\iota }_0^{\ast }{\mathrm{\Psi }}_j^{\ast }\omega =0$; the maps ${\mathrm{\Psi }}_j\circ {\iota }_0$ and their first $\vartheta$-derivatives converge uniformly to ${\mathrm{\Psi }}_{\mathrm{\infty }}\circ {\iota }_0$ and its derivatives on the real slice by [F3], so the coefficient functions of the pullback two-forms converge and the limit pullback ${\mathrm{\Psi }}_{\mathrm{\infty }}(\cdot ,\nu {)}^{\ast }\omega$ vanishes; this is clause 3.