Nondegenerate frequency reduction: translating the action variable by the inverse frequency map and the $\epsilon \le c{\alpha }^2$ smallness

Let $n\ge 2$, $\tau >n-1$, $\mu :=\tau +3n+2$, $\alpha >0$, $s_0>0$, $0<{\sigma }_0\le \min (1,s_0\mathrm{/}24)$, and let $c_1\in (0,1]$ be the one-step smallness constant (the constant $c$ of the estimates of One KAM step: the affine generator, its canonical time-one flow, and the quadratic remainder bound). Let the following data be given.

1. An open set $U\subseteq {\mathbb{C}}^n$, a compact set $K\subseteq U\cap {\mathbb{R}}^n$, a radius $0<r_0\le 1$ with $\{J\in {\mathbb{C}}^n:{\mathrm{dist}}_{\mathrm{\infty }}(J,K)<2r_0\}\subseteq U$, a nonempty set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ and a conjugation-invariant open complex neighbourhood ${\mathrm{\Omega }}^{\mathrm{\prime }}\subseteq {\mathbb{C}}^n$ containing $\mathrm{\Omega }$; and a holomorphic map $b:{\mathrm{\Omega }}^{\mathrm{\prime }}\to U$ with $Dh(b(\nu ))=\nu$ for every $\nu \in {\mathrm{\Omega }}^{\mathrm{\prime }}$, where $h:U\to \mathbb{C}$ is holomorphic and $b(\mathrm{\Omega })+(B_{r_0}\cap {\mathbb{R}}^n)\subseteq K$. (In the applications $b=(Dh{)}^{-1}$ on a regular frequency patch, supplied by the real analytic inverse function theorem; here it is part of the data.)
2. A constant $M_2\ge 1$ with $\mathrm{\mid }{I}^{\mathsf{T}}{D}^{2}h(J)I\mathrm{\mid }\le M_2\mathrm{\mid }I{\mathrm{\mid }}_{\mathrm{\infty }}^2$ for all $I\in {\mathbb{C}}^n$ and all $J\in \{J^{\mathrm{\prime }}:{\mathrm{dist}}_{\mathrm{\infty }}(J^{\mathrm{\prime }},K)<2r_0\}$.
3. A holomorphic function $F$ on $\{J:{\mathrm{dist}}_{\mathrm{\infty }}(J,K)<2r_0\}\times S_{s_0}$, $2\pi$-periodic in $\vartheta$, with $\mathrm{\mid }F\mathrm{\mid }\le \epsilon$.

Put $r:=\sqrt{\epsilon \mathrm{/}{M}_2}$ and assume $0<r\le r_0$. Then:

1. Taylor decomposition. For every $\nu \in \mathrm{\Omega }$ and $\mathrm{\mid }I{\mathrm{\mid }}_{\mathrm{\infty }}\le r$, $h(b(\nu )+I)=h(b(\nu ))+\nu \cdot I+P_h(I,\nu ),\mathrm{\mid }{P}_h(I,\nu )\mathrm{\mid }\le M_2\mathrm{\mid }I{\mathrm{\mid }}_{\mathrm{\infty }}^2\mathrm{.}$
2. Effective perturbation. The function $G(I,\vartheta ,\nu ):=h(b(\nu )+I)-h(b(\nu ))-\nu \cdot I+F(b(\nu )+I,\vartheta )$ is holomorphic on $D_{r,s_0,h_0^{\mathrm{\prime }}}(\mathrm{\Omega })$ whenever $O_{h_0^{\mathrm{\prime }}}(\mathrm{\Omega })\subseteq {\mathrm{\Omega }}^{\mathrm{\prime }}$ and $b(O_{h_0^{\mathrm{\prime }}}(\mathrm{\Omega }))+B_r\subseteq \{J:{\mathrm{dist}}_{\mathrm{\infty }}(J,K)<2r_0\}$, $2\pi$-periodic in $\vartheta$, and satisfies $\mathrm{\mid }G\mathrm{\mid }\le M_2{r}^2+\epsilon =2\epsilon \mathrm{.}$
3. ${\alpha }^2$-smallness. If $\epsilon \le \frac{c_1^2}{4}\cdot \frac{{\alpha }^2{\sigma }_0^{2\mu }}{M_2},$ then $\mathrm{\mid }G\mathrm{\mid }\le c_1\alpha r{\sigma }_0^{\mu }$; equivalently the relative smallness $E:=\mathrm{\mid }G\mathrm{\mid }\mathrm{/}(\alpha r{\sigma }_0^{\mu })$ satisfies $E\le c_1$.
4. Transfer to the physical Hamiltonian. Let $H(p,\vartheta ):=h(p)+F(p,\vartheta )$ and suppose that the parameter Hamiltonian $H_{\mathrm{p}\mathrm{a}\mathrm{r}}(I,\vartheta ,\nu ):=h(b(\nu ))+\nu \cdot I+G(I,\vartheta ,\nu )$ satisfies the hypotheses of The parameter KAM limit: invariant quasi-periodic motion with frequency $\nu$ and Lagrangian images on $D_{r,s_0,h_0^{\mathrm{\prime }}}({\mathrm{\Omega }}_{\alpha ,\tau })$ (same sequences, same one-step construction, with $P_0:=G$), with limit phase family ${\mathrm{\Phi }}_{\mathrm{\infty }}=({\mathrm{\Phi }}_I,{\mathrm{\Phi }}_{\vartheta })$ and limit original-parameter map ${\chi }_{\mathrm{\infty }}$. Then for every $\nu \in {\mathrm{\Omega }}_{\alpha ,\tau }$ the physical family ${\mathrm{\Psi }}_{\nu }(\vartheta ):=(b({\chi }_{\mathrm{\infty }}(\nu ))+{\mathrm{\Phi }}_I(0,\vartheta ,\nu ),{\mathrm{\Phi }}_{\vartheta }(0,\vartheta ,\nu ))$ satisfies $X_H({\mathrm{\Psi }}_{\nu }(\vartheta ))={\sum }_{j=1}^n{\nu }_j{\mathrm{\partial }}_{{\vartheta }_j}{\mathrm{\Psi }}_{\nu }(\vartheta ),{\mathrm{\Psi }}_{\nu }^{\ast }\omega =0,$ where $\omega ={\sum }_{j}d{\vartheta }_j\wedge d{I}_j$ and $X_H=(-H_{\vartheta },H_I)$ is taken in the action-angle coordinates $(p,\vartheta )$; that is, the image is an invariant Lagrangian torus of the physical Hamiltonian $H$ with frequency $\nu$.

• Where nondegeneracy enters. Only through the existence of the inverse frequency map $b$ and through the Hessian bound $M_2$; no convexity or Legendre-Fenchel duality is used, and $h$ need not be convex.
• The threshold. The reduction produces the classical condition $\epsilon \le c{\alpha }^2$ on a fixed analytic domain: ${\sigma }_0$ and $M_2$ are fixed once the domain and the integrable part are fixed, so the factor ${\alpha }^2{\sigma }_0^{2\mu }\mathrm{/}{M}_2$ is exactly a constant times ${\alpha }^2$. No statement is made about optimality of the constant.