3.1 Frequency-preserving KAM

Before starting, let us make some preparations. Following remark 2.4, there exists a modulus of continuity $\varpi _{1}(x)=x$ such that $f$ and $g$ are automatically $\varpi _{1}$ continuous about $r$. Besides, the following assumptions are crucial to our KAM theorems.

1. (A1) Let $p\in \mathbb {R}^{n}$ be given in advance and denote by $E^{\circ }$ the interior of $E$. Assume that

   (3.1)\begin{equation} \deg\left(\omega({\cdot}),E^{{\circ}},p\right)\neq 0. \end{equation}

2. (A2) Assume that $\omega (r_{*})=p$ with some $r_{*}\in E^{\circ }$ by (3.1), and

   \[ |\langle k,\omega(r_{*})\rangle -k_{0}|\geq \frac{\gamma}{|k|^{\tau}},\quad \forall\ k\in \mathbb{Z}^{n}\backslash \{0\},\ k_{0}\in\mathbb{Z}, \ |k_{0}|\leq M_{0}|k|, \]
   where $\gamma >0$, $\tau >n-1$ is fixed, and $M_{0}$ is assumed to be the upper bound of $|\omega |$ on $E$.

3. (A3) Assume that $B(r_{*},\,\delta )\subset E^{\circ }$ with $\delta >0$ is a neighbourhood of $r_{*}$. There exists a modulus of continuity $\varpi _{2}$ such that

   \[ |\omega(r')-\omega(r'')|\geq \varpi_{2}(|r'-r''|),\quad r',r''\in B(r_{*},\delta), \quad 0<|r'-r''|\leq 1. \]

Via these assumptions, we are now in a position to present the following KAM theorem for twist mapping with intersection property, which is the first frequency-preserving result on Moser's theorem to the best of our knowledge.

Theorem 3.1 Consider mapping (1.1) with intersection property. Assume that the perturbations are real analytic about $(\theta,\,r)$, and the frequency $\omega$ is continuous about $r$. Moreover, ${\rm (A1)}$–${\rm (A3)}$ hold. Then there exists a sufficiently small $\varepsilon _{0}$, a transformation $\mathscr {W}$ when $0<\varepsilon <\varepsilon _{0}$. The transformation $\mathscr {W}$ is a conjugation from $\mathscr {F}$ to $\hat {\mathscr {F}}$, and $\hat {\mathscr {F}}(\theta,\,r)=(\theta +\omega (r_{*}),\,r-\tilde {r})$ is the integrable rotation on $\mathbb {T}^{n}\times E$ with frequency $\omega (r_{*} )=p$, where $\tilde {r}$ is the translation about the action $r$ resulting from the transformation $\mathscr {W}$, and the constant $\tilde {r} \rightarrow 0$ as $\varepsilon \rightarrow 0$. That is, the following holds$:$

\[ \mathscr{W}\circ\hat{\mathscr{F}}=\mathscr{F}\circ\mathscr{W} . \]

When $n=1$, consider the area-preserving mapping of form (1.1), which obviously satisfies the intersection property. Actually, if a closed curve does not intersect its image under the mapping, the areas enclosed by these two curves will be different. Correspondingly, we obtain Moser's invariant curve theorem with frequency-preserving as stated in the following corollary.

Corollary 3.2 Consider mapping (1.1) for $n=1$. Assume that the perturbations are real analytic about $(\theta,\,r)$, and the frequency $\omega$ is continuous and strictly monotonic concerning $r$. Moreover, $({\rm A2})$ and $({\rm A3})$ hold. Then there exists a sufficiently small $\varepsilon _{0}$, a transformation $\mathscr {W}$ when $0<\varepsilon <\varepsilon _{0}$. The transformation $\mathscr {W}$ is a conjugation from $\mathscr {F}$ to $\hat {\mathscr {F}}$, and $\hat {\mathscr {F}}(\theta,\,r)=(\theta +\omega (r_{*}),\,r-\tilde {r})$ is the integrable rotation on $\mathbb {T}\times E$ with frequency $\omega (r_{*})=p$ for $p\in \omega (E^\circ )^\circ$ fixed, where $\tilde {r}$ is the translation about $r$ resulting from the transformation $\mathscr {W}$, and the constant $\tilde {r}\rightarrow 0$ as $\varepsilon \rightarrow 0$. That is, the following holds$:$

\[ \mathscr{W}\circ\hat{\mathscr{F}}=\mathscr{F}\circ\mathscr{W}. \]

Besides concentrating on twist mappings with action-angular variables, Herman [Reference Herman12, Reference Herman13] first considered the smooth mappings that contain only angular variables. It inspires us to investigate the perturbed mappings on $\mathbb {T}^n$ as well. We therefore consider the following mapping $\mathscr {F}:\mathbb {T}^{n}\times \Lambda \rightarrow \mathbb {T}^{n}$ defined by

(3.2)\begin{equation} \theta^{1}=\theta+\omega(\xi)+\varepsilon f(\theta,\xi,\varepsilon), \end{equation}

where $\theta \in \mathbb {T}^{n}=\mathbb {R}^{n}/ 2\pi \mathbb {Z}^{n}$, $\xi \in \Lambda \subset \mathbb {R}^{n}$ is a parameter, $\Lambda$ is a connected closed bounded domain with interior points, and $\varepsilon$ is a sufficiently small scalar. Assume that the perturbation $f(\theta,\,\xi,\,\varepsilon )$ is real analytic about $\theta$, continuous about the parameter $\xi$ and the frequency $\omega (\xi )$ is continuous about $\xi$. We will prove that mapping (3.2) has an invariant torus with the frequency unchanged during the iteration process. Moreover, the assumptions ${\rm (B1)}$–${\rm (B3)}$ corresponding to ${\rm (A1)}$–${\rm (A3)}$ are respectively

1. (B1) Let $q\in \mathbb {R}^{n}$ be given in advance and denote by $\Lambda ^{\circ }$ the interior of the parameter set $\Lambda$. Assume that

   (3.3)\begin{equation} \deg\left(\omega({\cdot}),\Lambda^{{\circ}},q\right)\neq 0. \end{equation}

2. (B2) Assume that $\omega (\xi _{*})=q$ with some $\xi _*\in \Lambda ^{\circ }$ by (3.3), and

   \[ | \langle k,\omega(\xi_{*})\rangle-k_{0}|\geq \frac{\gamma}{| k|^{\tau}}, \quad \forall\ k\in \mathbb{Z}^{n}\backslash\{0\},\ k_{0}\in\mathbb{Z}, \ |k_{0}|\leq M_{1}| k|, \]
   where $\tau >n-1$, $\gamma >0$ and $M_{1}$ is assumed to be the upper bound of $|\omega |$ on $\Lambda$.

3. (B3) Assume that $B(\xi _{*},\,\delta )\subset \Lambda ^{\circ }$ with $\delta >0$ is the neighbourhood of $\xi _{*}$. There exists a modulus of continuity $\varpi _{2}$ with $\varpi _{1}\lesssim \varpi _{2}$ such that

   \[ | \omega(\xi')-\omega(\xi'')|\geq \varpi_{2}(| \xi'-\xi''|),\quad \xi',\xi''\in B(\xi_{*},\delta),\quad 0<|\xi'-\xi''|\leq1. \]

Similar to theorem 3.1, we have the following theorem on $\mathbb {T}^n$, where the parameter-dependence for the perturbations is shown to be only continuous. This result is new, and unexpected, thanks to the parameter translation technique introduced in [Reference Du, Li and Zhang9, Reference Tong, Du and Li36] as we forego.

Theorem 3.3 Consider mapping (3.2). Assume that the perturbation $f(\theta,\,\xi,\,\varepsilon )$ is real analytic about $\theta$ on $D(h)$, $\varpi _{1}$ continuous about $\xi$ on $\Lambda$ and $\omega$ is continuous about $\xi$ on $\Lambda$. Moreover, ${\rm (B1)}$–${\rm (B3)}$ hold. Then there exists a sufficiently small $\varepsilon _{0}$, a transformation $\mathscr {U}$ when $0<\varepsilon <\varepsilon _{0}$. The transformation $\mathscr {U}$ is a conjugation from $\mathscr {F}$ to $\hat {\mathscr {F}}$, and $\hat {\mathscr {F}}(\theta )=\theta +\omega (\xi _{*})$ is the integrable rotation on $\mathbb {T}^{n}\times \Lambda$ with frequency $\omega (\xi _{*})=q$. That is, the following holds$:$

(3.4)\begin{equation} \mathscr{U}\circ\hat{\mathscr{F}}=\mathscr{F}\circ\mathscr{U}. \end{equation}

The main difference between theorems 3.1 and 3.3 is that the analyticity of the perturbation $f$ about $r$ can be used in theorem 3.1 to ensure that $f$ is at least Lipschitz continuous about $r$, that is, there exists a modulus of continuity $\varpi _{1}(x)=x$. In fact, it prohibits us from extending $r$ to complex strips in the KAM scheme if $f$ is assumed to be only continuous about $r$. However, for theorem 3.3 we consider the case where there is no action variable $r$ but only parameter $\xi$. In this situation, the perturbation $f$ being continuous about $\xi$ is enough, and we will employ condition $({\rm B3})$ directly in the proof of frequency-preserving. Explicitly, the parameter-dependence for $f$ could be very weak, such as the arbitrary $\alpha$-Hölder continuity $\varpi _{\rm H}^{\alpha }(x)=x^{\alpha }$ with any $0<\alpha <1$, then $\varpi _{2}$ in (B3) being the Logarithmic Lipschitz type $\varpi _{\rm {LL}}(x)\sim (-\log x)^{-1}$ as $x\to 0^+$ allows for theorem 3.3 due to remark 2.4. Actually, in view of theorem 2.5, we could deal with the case which admits extremely weak regularity, at least nowhere differentiable. In order to show the wide applicability of theorem 3.3, we directly give the following corollary.

3.2 Further discussions

Here we make some further discussions, including how to touch the parameter without dimension limitation problem under our approach, as well as the importance of the weak convexity in preserving prescribed frequency.

4.1 Description of the $0$-th KAM step

For sufficiently large integer $m$, let $\rho$ be a constant with $0<\rho <1$, and assume $\eta >0$ such that $(1+\rho )^{\eta }>2$. Define

\[ \gamma=\varepsilon^{\frac{1}{4(n+m+2)}}. \]

The parameters in the $0$-th KAM step are defined by

\[ h_{0}=h,\quad s_{0}= s,\quad \gamma_{0}= \gamma,\quad \mu_{0}= \varepsilon^{\frac{1}{8\eta(m+1)}}, \]

\begin{align*} D(h_{0})& =\{\theta\in \mathbb{C}^{n}: {\rm Re}\ \theta\in\mathbb{T}^{n}, \ |{\rm Im}\ \theta|\leq h_{0}\},\\ G(s_{0})& =\{r\in \mathbb{C}^{n}:{\rm Re}\ r\in E, \ |{\rm Im}\ r|\leq s_{0}\}, \end{align*}

where $0< s_{0},\,h_{0},\,\gamma _{0},\,\mu _{0}\leq 1$, and denote $\mathcal {D}_{0}:=\mathcal {D}(h_{0},\,s_{0})=D(h_{0})\times G(s_{0})$ for simplicity.

The mapping at $0$-th KAM step is

\[ \mathscr{F}_{0}: \begin{cases} \theta^{1}_{0}=\theta_{0}+\omega_{0}(r_{0})+f_{0}(\theta_{0},r_{0},\varepsilon),\\ r^{1}_{0}=r_{0}+g_{0}(\theta_{0},r_{0},\varepsilon), \end{cases} \]

where $\omega _{0}(r_{0})=\omega (r_{*})=p$, $f_{0}(\theta _{0},\,r_{0},\,\varepsilon )=\varepsilon f(\theta _{0},\,r_{0},\,\varepsilon )$, and $g_{0}(\theta _{0},\,r_{0},\,\varepsilon )=\varepsilon g(\theta _{0}, r_{0},\,\varepsilon )$. The following lemma states the estimates on perturbations $f_{0}$ and $g_{0}$.

Lemma 4.1 Assume $\varepsilon _{0}$ is sufficiently small so that

\[ \varepsilon^{\frac{3}{4}}(||f||_{\mathcal{D}_{0}}+||g||_{\mathcal{D}_{0}})\leq s^{m}_{0}\varepsilon^{\frac{1}{8\eta(m+1)}}, \]

for $0<\varepsilon <\varepsilon _{0}$. Then

\[ ||f_{0}||_{\mathcal{D}_{0}}+ ||g_{0}||_{\mathcal{D}_{0}}\leq \gamma^{n+m+2}_{0}s^{m}_{0}\mu_{0}. \]

Proof. Following $\gamma _{0}^{n+m+2}=\varepsilon ^{\frac {1}{4}}$ and $\mu _{0}=\varepsilon ^{\frac {1}{8\eta (m+1)}}$, one has

\begin{align*} \gamma^{n+m+2}_{0}s^{m}_{0}\mu_{0}& = s^{m}_{0}\varepsilon^{\frac{1}{4}}\varepsilon^{\frac{1}{8\eta(m+1)}}\\ & \geq s^{m}_{0}\varepsilon^{\frac{1}{4}}\varepsilon^{\frac{1}{8\eta(m+1)}}s^{{-}m}_{0}\varepsilon^{\frac{3}{4}}\varepsilon^{-\frac{1}{8\eta(m+1)}}(||f||_{\mathcal{D}_{0}}+||g||_{\mathcal{D}_{0}})\\ & =\varepsilon(||f||_{\mathcal{D}_{0}}+||g||_{\mathcal{D}_{0}})\\ & =||f_{0}||_{\mathcal{D}_{0}}+ ||g_{0}||_{\mathcal{D}_{0}}. \end{align*}

4.2 Description of the $\nu$-th KAM step

We now define the parameters appear in $\nu$-th KAM step as:

\[ h_{\nu}=\frac{h_{\nu-1}}{2}+\frac{h_{0}}{4},\quad s_{\nu}=\frac{s_{\nu-1}}{2},\quad \mu_{\nu}=\mu^{1+\rho}_{\nu-1},\quad \mathcal{D}_{\nu}=\mathcal{D}(h_{\nu},s_{\nu}). \]

After $\nu$ KAM steps, the mapping becomes

\[ \mathscr{F}_{\nu}: \begin{cases} \theta^{1}_{\nu}=\theta_{\nu}+\omega_{0}(r_{0})+f_{\nu}(\theta_{\nu},r_{\nu},\varepsilon),\\ r^{1}_{\nu}=r_{\nu}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+g_{\nu}(\theta_{\nu},r_{\nu},\varepsilon), \end{cases} \]

where $\sum \limits ^{\nu }_{i=0}r^{*}_{i}$ is the translation about $r_{0}$. Moreover,

(4.1)\begin{equation} ||f_{\nu}||_{\mathcal{D}_{\nu}}+ ||g_{\nu}||_{\mathcal{D}_{\nu}}\leq \gamma^{n+m+2}_{0}s^{m}_{\nu}\mu_{\nu}. \end{equation}

Define

\begin{align*} h_{\nu+1}& =\frac{h_{\nu}}{2}+\frac{h_{0}}{4},\\ s_{\nu+1}& =\frac{s_{\nu}}{2},\\ \mu_{\nu+1}& =\mu^{1+\rho}_{\nu},\\ K_{\nu+1}& =([\log\frac{1}{\mu_{\nu}}]+1)^{3\eta},\\ \mathscr{D}_{i}& =\mathcal{D}(h_{\nu+1}+\frac{i-1}{4}(h_{\nu}-h_{\nu+1}),is_{\nu+1}),\quad i=1,2,3,4,\\ \mathcal{D}_{\nu+1}& =\mathcal{D}(h_{\nu+1},s_{\nu+1}),\\ \hat{\mathcal{D}}_{\nu+1}& =\mathcal{D}(h_{\nu+2}+\frac{3}{4}(h_{\nu+1}-h_{\nu+2}),s_{\nu+2}),\\ \Gamma(h_{\nu}-h_{\nu+1})& =\sum_{0<|k|\leq K_{\nu+1}}|k|^{\tau}e^{-|k|\frac{h_{\nu}-h_{\nu+1}}{4}}\leq \frac{4^{\tau}\tau!}{(h_{\nu}-h_{\nu+1})^{\tau}}. \end{align*}

For simplicity of notation, we also denote

\begin{align*} & \mathscr{D}_{3}:=\mathscr{D}_{*}\times\mathscr{G}_{*}:=D(h_{\nu+1}+\frac{1}{2}(h_{\nu}-h_{\nu+1}))\times G(3s_{\nu+1}),\\ & \mathscr{D}_{4}:=\mathscr{D}_{**}\times\mathscr{G}_{**}:=D(h_{\nu+1}+\frac{3}{4}(h_{\nu}-h_{\nu+1}))\times G(4s_{\nu+1}).\\ \end{align*}

4.2.1 Truncation.

The Fourier series expansion of $f_{\nu }(\theta _{\nu +1},\,r_{\nu +1},\,\varepsilon )$ is

(4.2)\begin{equation} f_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)=\sum_{k\in\mathbb{Z}^{n}}f_{k,\nu}(r_{\nu+1})\,{\rm e}^{{\rm i}\langle k,\theta_{\nu+1}\rangle}, \end{equation}

where $f_{k,\nu }(r_{\nu +1})=\int _{\mathbb {T}^n}f_{\nu }(\theta _{\nu +1},\,r_{\nu +1},\,\varepsilon )\,{\rm e}^{-{\rm i}\langle k,\,\theta _{\nu +1}\rangle }\,{\rm d}\theta _{\nu +1}$ is the Fourier coefficient of $f_{\nu }$. The truncation and remainder of $f_{\nu }(\theta _{\nu +1},\,r_{\nu +1},\,\varepsilon )$ are respectively

\begin{align*} \mathcal{T}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)& =\sum_{0<|k|\leq K_{\nu+1}}f_{k,\nu}\,{\rm e}^{{\rm i}\langle k,\theta_{\nu+1}\rangle},\\ \mathcal{R}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)& =\sum_{|k|> K_{\nu+1}}f_{k,\nu}\,{\rm e}^{{\rm i}\langle k,\theta_{\nu+1}\rangle}. \end{align*}

Thus, $f_{\nu }$ has an equivalent expression of the form

\[ f_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)=f_{0,\nu}(r_{\nu+1})+\mathcal{T}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)+\mathcal{R}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon). \]

Furthermore, we have the following estimate.

Lemma 4.2 If

\[ {({\rm H1})}\quad\int^{+\infty}_{K_{\nu+1}}l^{n}\,{\rm e}^{{-}l\frac{h_{\nu}-h_{\nu+1}}{4}}\,{\rm d}l\leq \mu_{\nu}, \]

then there exists a constant $c_{1}$ such that

\[ || \mathcal{R}_{K_{\nu+1}}f_{\nu}||_{\mathscr{D}_{3}}\leq c_{1}\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu^{2}_{\nu}. \]

Proof. Since the Fourier coefficients decay exponentially, one has

\begin{align*} | f_{k,\nu}|_{\mathscr{G}_{*}} & \leq | f_{\nu}| _{\mathscr{D}_{4}}\,{\rm e}^{-| k| (h_{\nu+1}+\frac{3}{4}(h_{\nu}-h_{\nu+1}))}\\ & \leq \gamma^{n+m+2}_{0}s^{m}_{\nu}\mu_{\nu}\,{\rm e}^{-| k| (h_{\nu+1}+\frac{3}{4}(h_{\nu}-h_{\nu+1}))}, \end{align*}

then

\begin{align*} | \mathcal{R}_{K_{\nu+1}}f_{\nu}|_{\mathscr{D}_{3}}& \leq\sum_{| k|>K_{\nu+1}}| f_{k,\nu}|_{\mathscr{G}_{*}}\,{\rm e}^{| k|(h_{\nu+1}+\frac{1}{2}(h_{\nu}-h_{\nu+1}))}\\ & \leq\sum_{| k|> K_{\nu+1}}| f_{\nu}| _{\mathscr{D}_{4}}\,{\rm e}^{-| k| \frac{h_{\nu}-h_{\nu+1}}{4}}\\ & \leq\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu_{\nu}\sum_{| k|>K_{\nu+1}}\,{\rm e}^{-| k|\frac{h_{\nu}-h_{\nu+1}}{4}}\\ & \leq\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu_{\nu}\int^{+\infty}_{K_{\nu+1}}l^{n}\,{\rm e}^{{-}l\frac{h_{\nu}-h_{\nu+1}}{4}}\,{\rm d}l\\ & \leq\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu^{2}_{\nu}. \end{align*}

Moreover,

\begin{align*} & [\mathcal{R}_{K_{\nu+1}}f_{\nu}]_{\varpi_{1}}\\ & \quad=\sup_{\theta_{\nu+1}\in \mathscr{D}_{*} }\sup_{\substack{ r'_{\nu+1},r''_{\nu+1}\in \mathscr{G}_{*}\\ r'_{\nu+1}\neq r''_{\nu+1}}}\frac{| \mathcal{R}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},r'_{\nu+1},\varepsilon)-\mathcal{R}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},r''_{\nu+1},\varepsilon)|}{\varpi_{1}(| r'_{\nu+1}-r''_{\nu+1}|)}\\ & \quad=\sup_{\theta_{\nu+1}\in \mathscr{D}_{*} }\sup_{\substack{ r'_{\nu+1},r''_{\nu+1}\in \mathscr{G}_{*}\\ r'_{\nu+1}\neq r''_{\nu+1}}}\\ & \qquad\times\frac{\left|\sum\limits_{| k|>K_{\nu+1}} f_{k,\nu}(r'_{\nu+1})\,{\rm e}^{{\rm i}\langle k,\theta_{\nu+1}\rangle}-\sum\limits_{| k|>K_{\nu+1}}f_{k,\nu}(r''_{\nu+1})\,{\rm e}^{{\rm i}\langle k,\theta_{\nu+1}\rangle}\right|}{\varpi_{1}(| r'_{\nu+1}-r''_{\nu+1}|)}\\ & \quad\leq\sup_{\theta_{\nu+1}\in \mathscr{D}_{*} }\sup_{\substack{ r'_{\nu+1},r''_{\nu+1}\in \mathscr{G}_{*}\\ r'_{\nu+1}\neq r''_{\nu+1}}}\frac{\sum\limits_{| k|>K_{\nu+1}}| f_{k,\nu}(r'_{\nu+1})-f_{k,\nu}(r''_{\nu+1})|\,{\rm e}^{|k|(h_{\nu+1}+\frac{1}{2}(h_{\nu}-h_{\nu+1}))}}{\varpi_{1}(| r'_{\nu+1}-r''_{\nu+1}|)}\\ & \quad\leq\sup_{\theta_{\nu+1}\in \mathscr{D}_{**} }\sup_{\substack{ r'_{\nu+1},r''_{\nu+1}\in \mathscr{G}_{**}\\ r'_{\nu+1}\neq r''_{\nu+1}}} \\ & \qquad\times\frac{| f_{\nu}(\theta_{\nu+1},r'_{\nu+1},\varepsilon)-f_{\nu}(\theta_{\nu+1},r''_{\nu+1},\varepsilon)| \sum\limits_{|k|>K_{\nu+1}}\,{\rm e}^{-|k|\frac{h_{\nu}-h_{\nu+1}}{4}}}{\varpi_{1}(|r'_{\nu+1}-r''_{\nu+1}|)}\\ & \quad\leq[f_{\nu}]_{\varpi_{1}}\sum_{| k|>K_{\nu+1}}\,{\rm e}^{-|k|\frac{h_{\nu}-h_{\nu+1}}{4}}\\ & \quad\leq[f_{\nu}]_{\varpi_{1}}\sum_{| k|>K_{\nu+1}}k^{n}\,{\rm e}^{-|k|\frac{h_{\nu}-h_{\nu+1}}{4}}\\ & \quad\leq[f_{\nu}]_{\varpi_{1}}\int^{+\infty}_{K_{\nu+1}}l^{n}\,{\rm e}^{{-}l\frac{h_{\nu}-h_{\nu+1}}{4}}\,{\rm d}l\\ & \quad\leq\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu^{2}_{\nu}, \end{align*}

i.e.,

\[ ||\mathcal{R}_{K_{\nu+1}}f_{\nu}||_{\mathscr{D}_{3}}=|\mathcal{R}_{K_{\nu+1}}f_{\nu}|_{\mathscr{D}_{3}}+[\mathcal{R}_{K_{\nu+1}}f_{\nu}]_{\varpi_{1}}\leq c_{1}\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu^{2}_{\nu}. \]

Similarly, we get

\[ ||\mathcal{R}_{K_{\nu+1}}g_{\nu}||_{\mathscr{D}_{3}}\leq c_{1}\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu^{2}_{\nu}. \]

4.2.2 Transformation

For $\mathscr {F}_{\nu }$ on $\mathcal {D}_{\nu +1}$, introduce a transformation $\mathscr {U}_{\nu +1}:={\rm id}+(U_{\nu +1},\,V_{\nu +1})$ that satisfies

(4.3)\begin{equation} \mathscr{U}_{\nu+1}\circ\bar{\mathscr{F}}_{\nu+1}=\mathscr{F}_{\nu}\circ\mathscr{U}_{\nu+1}, \end{equation}

where ${\rm id}$ denotes the identity mapping. Since

\[ \mathscr{U}_{\nu+1}: \begin{cases} \theta^{1}_{\nu}=\theta^{1}_{\nu+1}+U_{\nu+1}(\theta^{1}_{\nu+1},r^{1}_{\nu+1}),\\ r^{1}_{\nu}=r^{1}_{\nu+1}+V_{\nu+1}(\theta^{1}_{\nu+1},r^{1}_{\nu+1}), \end{cases} \]

and

\[ \bar{\mathscr{F}}_{\nu+1}: \begin{cases} \theta^{1}_{\nu+1}=\theta_{\nu+1}+\omega_{0}(r_{0})+\bar{f}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon),\\ r^{1}_{\nu+1}=r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+\bar{g}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon), \end{cases} \]

with $r^{*}_{0}=0$, from the left side of (4.3), we can derive that

\begin{align*} \theta^{1}_{\nu}& =\theta^{1}_{\nu+1}+U_{\nu+1}(\theta^{1}_{\nu+1},r^{1}_{\nu+1})\\ & = \theta_{\nu+1}+\omega_{0}(r_{0})+\bar{f}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)\\ & \quad+ U_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0})+\bar{f}_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+\bar{g}_{\nu+1}\right),\\ r^{1}_{\nu}& =r^{1}_{\nu+1}+V_{\nu+1}(\theta^{1}_{\nu+1},r^{1}_{\nu+1}) \\ & = r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+\bar{g}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)\\ & \quad+ V_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0})+\bar{f}_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+\bar{g}_{\nu+1}\right). \end{align*}

Also, one has

\[ \mathscr{F}_{\nu}:\begin{cases} \theta^{1}_{\nu}=\theta_{\nu}+\omega_{0}(r_{0})+f_{\nu}(\theta_{\nu},r_{\nu},\varepsilon),\\ r^{1}_{\nu}=r_{\nu}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+g_{\nu}(\theta_{\nu},r_{\nu},\varepsilon), \end{cases} \]

and

\[ \mathscr{U}_{\nu+1}: \begin{cases} \theta_{\nu}=\theta_{\nu+1}+U_{\nu+1}\left(\theta_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right),\\ r_{\nu}=r_{\nu+1}+V_{\nu+1}\left(\theta_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right). \end{cases} \]

By the right side of (4.3), we obtain

\begin{align*} \theta^{1}_{\nu}& =\theta_{\nu}+\omega_{0}(r_{0})+f_{\nu}(\theta_{\nu},r_{\nu},\varepsilon)\\ & = \theta_{\nu+1}+U_{\nu+1}\left(\theta_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)+\omega_{0}(r_{0})\\ & \quad+ f_{\nu}(\theta_{\nu+1}+U_{\nu+1},r_{\nu+1}+V_{\nu+1},\varepsilon),\\ r^{1}_{\nu}& =r_{\nu}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+g_{\nu}(\theta_{\nu},r_{\nu},\varepsilon)\\ & = r_{\nu+1}+V_{\nu+1}\left(\theta_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)-\sum\limits^{\nu}_{i=0}r^{*}_{i}\\ & \quad+ g_{\nu}(\theta_{\nu+1}+U_{\nu+1},r_{\nu+1}+V_{\nu+1},\varepsilon). \end{align*}

Therefore, (4.3) implies that

(4.4)\begin{align} & \omega_{0}(r_{0})+\bar{f}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)+U_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0})+\bar{f}_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+\bar{g}_{\nu+1}\right)\nonumber\\ & = U_{\nu+1}\left(\theta_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)+\omega_{0}(r_{0})+f_{\nu}(\theta_{\nu+1}+U_{\nu+1},r_{\nu+1}+V_{\nu+1},\varepsilon), \end{align}

(4.5)\begin{align} & \bar{g}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)+V_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0})+\bar{f}_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+\bar{g}_{\nu+1}\right)\nonumber\\ & = V_{\nu+1}\left(\theta_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)+g_{\nu}(\theta_{\nu+1}+U_{\nu+1},r_{\nu+1}+V_{\nu+1},\varepsilon). \end{align}

Following the iteration process before, the $\omega _{0}(r_{0})$ on the right side of (4.4) is actually

\[ \omega_{0}(r_{0})=\omega_{0}(r_{\nu})+\sum^{\nu-1}_{i=0}f_{0,i}(r_{\nu}). \]

Since the frequency $\omega _{0}$ is continuous about $r$, there exists a modulus of continuity $\varpi _{*}$ such that

(4.6)\begin{equation} \frac{|\omega_{0}(r_{\nu})-\omega_{0}(r_{\nu+1})|}{\varpi_{*}(|r_{\nu}-r_{\nu+1}|)} \leq [\omega_0]_{\varpi_{*}} <{+}\infty. \end{equation}

The perturbation $f$ is $\varpi _{1}$ continuous about $r$ with $\varpi _{1}\lesssim \varpi _{*}$ due to $\varpi _{1}(x)=x$, one has

(4.7)\begin{equation} \frac{|f_{0,i}(r_{\nu})-f_{0,i}(r_{\nu+1})|}{\varpi_{1}(|r_{\nu}-r_{\nu+1}|)}\leq [f_{0,i}]_{\varpi_{1}}<{+}\infty,\quad 0\leq i\leq \nu-1 . \end{equation}

Consequently, from (4.6) and (4.7), we may deduce that

\[ \omega_{0}(r_{\nu})=\omega_{0}(r_{\nu+1})+\mathcal{O}(\varpi_{*}(|V_{\nu+1}|)), \]

and

\[ f_{0,i}(r_{\nu})=f_{0,i}(r_{\nu+1})+\mathcal{O}(\varpi_{1}(|V_{\nu+1}|)). \]

It allows the actions $r_{\nu }$ and $r_{\nu +1}$ in (4.4) to be unified and the KAM iteration process to continue. Therefore, (4.4) and (4.5) are equal to the following

\begin{align*} & U_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0})+\bar{f}_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+\bar{g}_{\nu+1}\right)\\& \qquad -U_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0}),r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)\nonumber\\ & \qquad+ U_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0}),r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)-U_{\nu+1}\left(\theta_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)\nonumber\\ & \qquad+ \omega_{0}(r_{0})+\bar{f}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)\nonumber\\ & \quad= \omega_{0}(r_{\nu+1})+\sum^{\nu-1}_{i=0}f_{0,i}(r_{\nu+1})+\mathcal{O}(\varpi_{*}(|V_{\nu+1}|))+\mathcal{O}(\varpi_{1}(|V_{\nu+1}|))\nonumber\\ & \qquad+ f_{\nu}(\theta_{\nu+1}+U_{\nu+1},r_{\nu+1}+V_{\nu+1},\varepsilon)-f_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)\nonumber\\ & \qquad+f_{0,\nu}(r_{\nu+1})+\mathcal{T}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)\\ & \qquad+ \mathcal{R}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon),\\ & V_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0})+\bar{f}_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+\bar{g}_{\nu+1}\right)\nonumber\\ & \qquad-V_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0}),r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right) \nonumber\\ & \qquad+ V_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0}),r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)\nonumber\\ & \qquad-V_{\nu+1}\left(\theta_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)+\bar{g}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)\nonumber\\ & \quad = g_{\nu}(\theta_{\nu+1}+U_{\nu+1},r_{\nu+1}+V_{\nu+1},\varepsilon)-g_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)+g_{0,\nu}(r_{\nu+1})\\ & \qquad+\mathcal{T}_{K_{\nu+1}}g_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)+ \mathcal{R}_{K_{\nu+1}}g_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon). \end{align*}

The transformation $\mathscr {U}_{\nu +1}={\rm id}+(U_{\nu +1},\,V_{\nu +1})$ needs to satisfy the homological equations

(4.8)\begin{equation} \begin{cases} U_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0}),r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)-U_{\nu+1}\left(\theta_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)\\\quad =\mathcal{T}_{K_{\nu+1}}f_{\nu}\left(\theta_{\nu+1},r_{\nu+1},\varepsilon\right),\\ V_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0}),r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)-V_{\nu+1}\left(\theta_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)\\\quad =\mathcal{T}_{K_{\nu+1}}g_{\nu}\left(\theta_{\nu+1},r_{\nu+1},\varepsilon\right). \end{cases} \end{equation}

The new perturbations are respectively

(4.9)\begin{align} \bar{f}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)& =f_{\nu}(\theta_{\nu+1}+U_{\nu+1},r_{\nu+1}+V_{\nu+1},\varepsilon)-f_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)\nonumber\\ & \quad+\mathcal{R}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)+\mathcal{O}(\varpi_{*}(|V_{\nu+1}|))+\mathcal{O}(\varpi_{1}(|V_{\nu+1}|))\nonumber\\ & \quad+U_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0}),r_{\nu+1}-\sum^{\nu}_{i=0}r^{*}_{i}\right)\nonumber\\ & \quad-U_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0})+\bar{f}_{\nu+1},r_{\nu+1}-\sum^{\nu}_{i=0}r^{*}_{i}+\bar{g}_{\nu+1}\right), \end{align}

(4.10)\begin{align} \bar{g}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)& =g_{\nu}(\theta_{\nu+1}+U_{\nu+1},r_{\nu+1}+V_{\nu+1},\varepsilon)-g_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)\nonumber\\ & \quad+g_{0,\nu}(r_{\nu+1})+\mathcal{R}_{K_{\nu+1}}g_{\nu}(\theta_{\nu+1},r_{\nu+1},\varepsilon)\nonumber\\ & \quad+V_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0}),r_{\nu+1}-\sum^{\nu}_{i=0}r^{*}_{i}\right)\nonumber\\ & \quad-V_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0})+\bar{f}_{\nu+1},r_{\nu+1}-\sum^{\nu}_{i=0}r^{*}_{i}+\bar{g}_{\nu+1}\right). \end{align}

The homological equations (4.8) are uniquely solvable on $\mathcal {D}_{\nu +1}$. Let us start by considering the first equation in (4.8). Formally, denote $U_{\nu +1}(\theta _{\nu +1},\,r_{\nu +1}-\sum ^{\nu }_{i=0}r^{*}_{i})$ as

\[ U_{\nu+1}\left(\theta_{\nu+1},r_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}\right)=\sum_{0<|k|\leq K_{\nu+1}}U_{k,\nu+1}\,{\rm e}^{{\rm i}\langle k,\theta_{\nu+1}\rangle }, \]

taking it into the first equation in (4.8), one has

\begin{align*} & \sum_{0<|k|\leq K_{\nu+1}}U_{k,\nu+1}\,{\rm e}^{{\rm i}\langle k,\theta_{\nu+1}+\omega_{0}(r_{0})\rangle}-\sum_{0<|k|\leq K_{\nu+1}}U_{k,\nu+1}\,{\rm e}^{{\rm i}\langle k,\theta_{\nu+1}\rangle}\\& \quad=\sum_{0<|k|\leq K_{\nu+1}}f_{k,\nu}\,{\rm e}^{{\rm i}\langle k,\theta_{\nu+1}\rangle}. \end{align*}

By comparing the coefficients above, we have

(4.11)\begin{equation} U_{k,\nu+1}({\rm e}^{{\rm i}\langle k,\omega_{0}(r_{0})\rangle}-1)=f_{k,\nu}. \end{equation}

The details of estimating (4.11) can be seen in the following lemma.

Lemma 4.3 Equation (4.11) has a unique solution $U_{k,\nu +1}$ on $G(s_{\nu +1})$ satisfying the following estimate

\[ ||U_{k,\nu+1}||_{\mathscr{G}_{*}}\leq c_{2}||f_{\nu}||_{\mathscr{D}_{4}}\gamma^{{-}1}_{0}|k|^{\tau}\,{\rm e}^{-|k|(h_{\nu+1}+\frac{3}{4}(h_{\nu}-h_{\nu+1}))}. \]

Proof. We notice that the coefficients $f_{k,\nu }$ decay exponentially, i.e.,

\[ ||f_{k,\nu}||_{\mathscr{G}_{**}}\leq ||f_{\nu}||_{\mathscr{D}_{4}}\,{\rm e}^{-|k|(h_{\nu+1}+\frac{3}{4}(h_{\nu}-h_{\nu+1}))}. \]

It is well known that $|\sin \phi |\geq \frac {2}{\pi }|\phi |$ when $-\frac {\pi }{2}\leq \phi \leq \frac {\pi }{2}$. There exists a $k_{0}\in \mathbb {Z}$ satisfying $|\frac {\langle k,\,\omega _{0}(r_{0})\rangle -k_{0}}{2}|\leq \frac {\pi }{2}$ such that

\begin{align*} ||e^{{\rm i}\langle k,\omega_{0}(r_{0})\rangle}-1||^{2}_{\mathcal{G}_{**}}& =||\cos \langle k,\omega_{0}(r_{0})\rangle+{\rm i}\sin\langle k,\omega_{0}(r_{0})\rangle-1||^{2}_{\mathcal{G}_{**}}\\ & = ||\cos \langle k,\omega_{0}(r_{0})\rangle-1||^{2}_{\mathcal{G}_{**}}+||\sin\langle k,\omega_{0}(r_{0})\rangle||^{2}_{\mathcal{G}_{**}}\\ & \geq 4||\sin\frac{\langle k,\omega_{0}(r_{0})\rangle}{2}||^{2}_{\mathcal{G}_{**}}\\ & = 4||\sin\frac{\langle k,\omega_{0}(r_{0})\rangle-k_{0}}{2}||^{2}_{\mathcal{G}_{**}}. \end{align*}

As a consequence,

\begin{align*} ||e^{{\rm i}\langle k,\omega_{0}(r_{0})\rangle}-1||_{\mathscr{G}_{**}}& \geq2||\sin\frac{\langle k,\omega_{0}(r_{0})\rangle-k_{0}}{2}||_{\mathscr{G}_{**}}\\ & \geq\frac{4}{\pi}||\frac{\langle k,\omega_{0}(r_{0})\rangle-k_{0}}{2}||_{\mathscr{G}_{**}}\\ & \geq c||\langle k,\omega_{0}(r_{0})\rangle-k_{0}||_{\mathscr{G}_{**}}\\ & \geq\frac{c\gamma_{0}}{|k|^{\tau}}. \end{align*}

Therefore,

(4.12)\begin{align} ||U_{k,\nu+1}||_{\mathscr{G}_{*}}& \leq \frac{||f_{k,\nu}||_{\mathscr{G}_{**}}}{||{\rm e}^{{\rm i}\langle k,\omega_{0}(r_{0})\rangle}-1||_{\mathscr{G}_{**}}}\nonumber\\ & \leq c_{2}||f_{\nu}||_{\mathscr{D}_{4}}\gamma^{{-}1}_{0}|k|^{\tau}\,{\rm e}^{-|k|(h_{\nu+1}+\frac{3}{4}(h_{\nu}-h_{\nu+1}))}. \end{align}

In the same way, we get

\[ V_{k,\nu+1}({\rm e}^{{\rm i}\langle k,\omega_{0}(r_{0})\rangle}-1)=g_{k,\nu}. \]

Thus,

(4.13)\begin{align} ||V_{k,\nu+1}||_{\mathscr{G}_{*}}& \leq\frac{||g_{k,\nu}||_{\mathscr{G}_{**}}}{||{\rm e}^{{\rm i}\langle k,\omega_{0}(r_{0})\rangle}-1||_{\mathscr{G}_{**}}}\nonumber\\ & \leq c_{2}||g_{\nu}||_{\mathscr{D}_{4}}\gamma^{{-}1}_{0}|k|^{\tau}\,{\rm e}^{-|k|(h_{\nu+1}+\frac{3}{4}(h_{\nu}-h_{\nu+1}))}. \end{align}

4.2.3 Translation

In the usual iteration process, one has to find out a decreasing series of domains that the Diophantine condition fails. To avoid this, we will construct a translation to keep the frequency unchanged in this section. Consider the translation

\[ \mathscr{V}_{\nu+1}:\theta_{\nu+1}\rightarrow \theta_{\nu+1},\quad r_{\nu+1}\rightarrow r_{\nu+1}+r^{*}_{\nu+1}:=\hat{r}_{\nu+1}, \]

where $\hat {r}_{\nu +1}\in B_{c\mu _{\nu }}(\hat {r}_{\nu })$. The action has a shift under the translation $\mathscr {V}_{\nu +1}$, but the angular variable is unchanged. To make the frequency-preserving, it requests that

\[ \omega_{0}(\hat{r}_{\nu+1})+\sum^{\nu}_{i=0}f_{0,i}(\hat{r}_{\nu+1})-\omega_{0}(r_{0})+\omega_{0}(r_{0})=\omega_{0}(r_{0}), \]

i.e.,

\[ \omega_{0}(\hat{r}_{\nu+1})+\sum^{\nu}_{i=0}f_{0,i}(\hat{r}_{\nu+1})=\omega_{0}(r_{0}). \]

We will demonstrate the equation in the following subsection. After the translation $\mathscr {V}_{\nu +1}$, the mapping $\bar {\mathscr {F}}_{\nu +1}$ becomes $\mathscr {F}_{\nu +1}=\bar {\mathscr {F}}_{\nu +1}\circ \mathscr {V}_{\nu +1}$, i.e.,

\[ \mathscr{F}_{\nu+1}: \begin{cases} \theta^{1}_{\nu+1}=\theta_{\nu+1}+\omega_{0}(r_{0})+f_{\nu+1}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon),\\ \hat{r}^{1}_{\nu+1}=\hat{r}_{\nu+1}-\sum\limits^{\nu}_{i=0}r^{*}_{i}+g_{\nu+1}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon). \end{cases} \]

The corresponding new perturbations are

(4.14)\begin{align} f_{\nu+1}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon)& =f_{\nu}(\theta_{\nu+1}+U_{\nu+1},\hat{r}_{\nu+1}+V_{\nu+1},\varepsilon)-f_{\nu}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon)\nonumber\\ & \quad+\mathcal{R}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon)+\mathcal{O}(\varpi_{*}(|V_{\nu+1}|))+\mathcal{O}(\varpi_{1}(|V_{\nu+1}|)) \nonumber\\ & \quad+U_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0}),\hat{r}_{\nu+1}-\sum^{\nu}_{i=0}r^{*}_{i}\right)\nonumber\\ & \quad-U_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0})+f_{\nu+1},\hat{r}_{\nu+1}-\sum^{\nu}_{i=0}r^{*}_{i}+g_{\nu+1}\right), \end{align}

and

(4.15)\begin{align} g_{\nu+1}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon)& =g_{\nu}(\theta_{\nu+1}+U_{\nu+1},\hat{r}_{\nu+1}+V_{\nu+1},\varepsilon)-g_{\nu}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon)\nonumber\\ & \quad+g_{0,\nu}(\hat{r}_{\nu+1})+\mathcal{R}_{K_{\nu+1}}g_{\nu}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon)\nonumber\\ & \quad+V_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0}),\hat{r}_{\nu+1}-\sum^{\nu}_{i=0}r^{*}_{i}\right)\nonumber\\ & \quad-V_{\nu+1}\left(\theta_{\nu+1}+\omega_{0}(r_{0})+f_{\nu+1},\hat{r}_{\nu+1}-\sum^{\nu}_{i=0}r^{*}_{i}+g_{\nu+1}\right). \end{align}

4.2.4 Frequency-preserving

In this section, we will show that the frequency is unchanged during the iteration process under the conditions ${\rm (A1)}$ and ${\rm (A3)}$. The topological degree condition $({\rm A1})$ states that we can find a $\hat {r}_{\nu +1}$ such that the frequency remains preserved. Besides, the weak convexity condition $({\rm A3})$ ensures that $\{\hat {r}_{\nu }\}$ is a Cauchy sequence. The following lemma is crucial to our consideration.

Lemma 4.4 Assume that

\[ {({\rm H2})}\quad\left\|\sum^{\nu}_{i=0}f_{0,i}\right\|_{G(s_{\nu+1})}\leq c\mu^{\frac{1}{2}}_{0}. \]

Then there exists a $\hat {r}_{\nu +1}\in B_{c\mu _{\nu }}(\hat {r}_{\nu })$ such that

(4.16)\begin{equation} \omega_{0}(\hat{r}_{\nu+1})+\sum^{\nu}_{i=0}f_{0,i}(\hat{r}_{\nu+1})=\omega_{0}(r_{0}). \end{equation}

Proof. The proof is an induction on $\nu$. Obviously, $\omega _{0}(r_{0})=\omega _{0}(r_{0})$ when $\nu =0$. Now assume that for some $\nu \geq 1$, one has

(4.17)\begin{equation} \omega_{0}(\hat{r}_{j})+\sum^{j-1}_{i=0}f_{0,i}(\hat{r}_{j})=\omega_{0}(r_{0}), \quad \hat{r}_{j}\in B_{c\mu_{j-1}}(\hat{r}_{j-1})\subset B(r_{0},\delta),\quad 1\leq j\leq \nu. \end{equation}

We need to find out a $\hat {r}_{\nu +1}$ in the neighbourhood of $\hat {r}_{\nu }$ that satisfies

(4.18)\begin{equation} \omega_{0}(\hat{r}_{\nu+1})+\sum^{\nu}_{i=0}f_{0,i}(\hat{r}_{\nu+1})=\omega_{0}(r_{0}). \end{equation}

Since $\mu _{0}^{\frac {1}{2}}$ is sufficiently small and the condition ${\rm (A1)}$ holds, we have

(4.19)\begin{align} \deg\left(\omega_{0}({\cdot})+\sum^{\nu}_{i=0}f_{0,i}({\cdot}),B(r_{0},\delta),\omega_{0}(r_{0})\right)=\deg\left(\omega_{0}({\cdot}),B(r_{0},\delta),\omega_{0}(r_{0})\right)\neq 0, \end{align}

where $\omega _{0}(r_0)=p$, and $p \in \mathbb {R}^n$ is given in advance. This shows that there exists at least a $\hat {r}_{\nu +1}\in B(r_{0},\,\delta )$ with some $\delta >0$ such that (4.16) holds. Remark 2.4 tells us that there exists a modulus of continuity $\varpi _{1}(x)=x$ such that $f$ is $\varpi _{1}$ continuous about $r$, and following (4.1), one has

\[ [f_{0,i}]_{\varpi_{1}}\leq c\mu_{i},\quad 0\leq i\leq \nu, \]

i.e.,

\[ |f_{0,i}(\hat{r}_{\nu+1})-f_{0,i}(\hat{r}_{\nu})|\leq c\mu_{i}\varpi_{1}(|\hat{r}_{\nu+1}-\hat{r}_{\nu}|),\quad 0\leq i\leq \nu. \]

Following definitions 2.1 and 2.3, and together with $({\rm A3})$, one has

\[ \varlimsup_{x\rightarrow 0^+}\frac{x}{\varpi_{2}(x)}<{+}\infty, \]

this means $\varpi _{1}\lesssim \varpi _{2}$. Equations (4.17) and (4.18) imply that

\[ \omega_{0}(\hat{r}_{\nu+1})+\sum^{\nu}_{i=0}f_{0,i}(\hat{r}_{\nu+1})=\omega_{0}(\hat{r}_{\nu})+\sum^{\nu-1}_{i=0}f_{0,i}(\hat{r}_{\nu}). \]

Then

\begin{align*} |f_{0,\nu}(\hat{r}_{\nu+1})|& =|\omega_{0}(\hat{r}_{\nu})-\omega_{0}(\hat{r}_{\nu+1})+\sum\limits^{\nu-1}_{i=0}(f_{0,i}(\hat{r}_{\nu})-f_{0,i}(\hat{r}_{\nu+1}))|\nonumber\\ & \geq|\omega_{0}(\hat{r}_{\nu})-\omega_{0}(\hat{r}_{\nu+1})|-\sum\limits^{\nu-1}_{i=0}|f_{0,i}(\hat{r}_{\nu})-f_{0,i}(\hat{r}_{\nu+1})|\nonumber\\ & \geq\varpi_{2}(|\hat{r}_{\nu}-\hat{r}_{\nu+1}|)-c\left(\sum\limits^{\nu-1}_{i=0}\mu_{i}\right)\varpi_{1}(|\hat{r}_{\nu}-\hat{r}_{\nu+1}|)\nonumber\\ & \geq\frac{\varpi_{2}(|\hat{r}_{\nu}-\hat{r}_{\nu+1}|)}{2}. \end{align*}

The last inequality holds since $\varepsilon$ is sufficiently small such that $c(\sum^{\nu -1}_{i=0}\mu _{i})\leq \frac {1}{2}$, and $\varpi _1 \lesssim \varpi _2$. Therefore,

(4.20)\begin{equation} |\hat{r}_{\nu}-\hat{r}_{\nu+1}|\leq \varpi^{{-}1}_{2}(2|f_{0,\nu}(\hat{r}_{\nu+1})|)\leq \varpi^{{-}1}_{2}(2c\mu_{\nu})\leq c\varpi^{{-}1}_{1}(2c\mu_{\nu})\leq c\mu_{\nu}, \end{equation}

where the last inequality is due to definition 2.1, i.e., $\varlimsup \limits _{x\rightarrow 0^+}\frac {x}{\varpi _{1}(x)}<+\infty$. This implies that $\{\hat {r}_{\nu }\}$ is a Cauchy sequence and $\hat {r}_{\nu +1}\in B_{c\mu _{\nu }}(\hat {r}_{\nu })$.

4.2.5 Estimates on new transformations

According to (4.12) and (4.13), the estimates on transformations are given in the lemma below.

Lemma 4.5 Assume that there exists a constant $c_{3}$ such that

\[ {({\rm H3})}\quad c_{3}\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1})\leq \min \left\{s_{\nu+1},\frac{h_{\nu}-h_{\nu+1}}{4}\right\}. \]

Then the followings hold.

1. (a) $||\mathscr {U}_{\nu +1}-{\rm id}||_{\mathscr {D}_{3}}\leq c_{3}\gamma ^{n+m+1}_{0}s^{m}_{\nu }\mu _{\nu }\Gamma (h_{\nu }-h_{\nu +1})$.

2. (b) $\mathscr {U}_{\nu +1}:\mathcal {D}_{\nu +1}\rightarrow \mathcal {D}_{\nu }$.

3. (c) Setting $\mathscr {W}_{\nu +1}:=\mathscr {U}_{\nu +1}\circ \mathscr {V}_{\nu +1}$, one has $\mathscr {W}_{\nu +1}:\mathcal {D}_{\nu +1}\rightarrow \mathcal {D}_{\nu }$, and

   \[\ \ ||\mathscr{W}_{\nu+1}-{\rm id}||_{\mathcal{D}_{\nu+1}}\leq c_{3}\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1}). \]

Proof. $({\rm i})$ Since $U_{\nu +1}(\theta _{\nu +1},\,r_{\nu +1}-\sum^{\nu }_{i=0}r^{*}_{i})=\sum \limits _{0<|k|\leq K_{\nu +1}}U_{k,\nu +1}e^{{\rm i}\langle k,\,\theta _{\nu +1}\rangle }$, we get

(4.21)\begin{align} ||U_{\nu+1}||_{\mathscr{D}_{3}}& \leq ||U_{k,\nu+1}||_{\mathscr{G}_{*}}\sum_{0<|k|\leq K_{\nu+1}}\,{\rm e}^{|k|(h_{\nu+1}+\frac{1}{2}(h_{\nu}-h_{\nu+1}))}\nonumber\\ & \leq||f_{\nu}||_{\mathscr{D}_{4}}\gamma^{{-}1}_{0}\sum_{0<|k|\leq K_{\nu+1}}|k|^{\tau}\,{\rm e}^{-|k|\frac{h_{\nu}-h_{\nu+1}}{4}}\nonumber\\ & \leq c_{3}\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1}). \end{align}

Besides,

(4.22)\begin{align} ||V_{\nu+1}||_{\mathscr{D}_{3}}& \leq ||V_{k,\nu+1}||_{\mathscr{G}_{*}}\sum_{0<|k|\leq K_{\nu+1}}\,{\rm e}^{|k|(h_{\nu+1}+\frac{1}{2}(h_{\nu}-h_{\nu+1}))}\nonumber\\ & \leq||g_{\nu}||_{\mathscr{D}_{4}}\gamma^{{-}1}_{0}\sum_{0<|k|\leq K_{\nu+1}}|k|^{\tau}\,{\rm e}^{-|k|\frac{h_{\nu}-h_{\nu+1}}{4}}\nonumber\\ & \leq c_{3}\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1}). \end{align}

Thus, $({\rm i})$ is due to (4.1), (4.21) and (4.22).

$({\rm ii})$ By $(\theta _{\nu +1},\,r_{\nu +1})\in \mathscr {D}_{3}$, $({\rm H3})$ implies that

\begin{align*} |\theta_{\nu}-\theta_{\nu+1}|& =||U_{\nu+1}||_{\mathscr{D}_{3}}\\ & \leq c_{3}\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1})\\ & \leq\frac{h_{\nu}-h_{\nu+1}}{4},\\ |r_{\nu}-r_{\nu+1}|& =||V_{\nu+1}||_{\mathscr{D}_{3}}\\ & \leq c_{3}\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1})\\ & \leq s_{\nu+1}. \end{align*}

Thus, $\mathscr {U}_{\nu +1}:\mathcal {D}_{\nu +1}\subset \mathscr {D}_{3}\rightarrow \mathscr {D}_{4}\subset \mathcal {D}_{\nu }$.

$({\rm iii})$ now follows from $({\rm i})$ and $({\rm ii})$ immediately.

4.2.6 Estimates on the new perturbations

In what follows, we are able to show the estimates on the new perturbations.

Lemma 4.6 Assume that

\[ {({\rm H4}) } \quad\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1})\leq \varpi^{{-}1}_{*}(\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu^{2}_{\nu}). \]

Then there exists a constant $c_{4}$ such that

\begin{align*} & ||\bar{f}_{\nu+1}||_{\mathcal{D}_{\nu+1}}+||\bar{g}_{\nu+1}||_{\mathcal{D}_{\nu+1}}\\ & \quad \leq c_{4}\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu^{2}_{\nu}(\gamma^{n+m+1}_{0}s^{m}_{\nu}(h_{\nu+1}-h_{\nu+2})^{{-}1}\Gamma(h_{\nu}-h_{\nu+1})\\& \qquad+\gamma^{n+m+1}_{0}s^{m-1}_{\nu}\Gamma(h_{\nu}-h_{\nu+1})+1). \end{align*}

Moreover, if

\begin{align*} & {({\rm H5})}\quad 2^{m}c_{4}\mu^{1-\rho}_{\nu}(\gamma^{n+m+1}_{0}s^{m}_{\nu}(h_{\nu+1}-h_{\nu+2})^{{-}1}\Gamma(h_{\nu}-h_{\nu+1})\\& \qquad\qquad +\gamma^{n+m+1}_{0}s^{m-1}_{\nu}\Gamma(h_{\nu}-h_{\nu+1})+1)\leq 1, \end{align*}

then

\[ ||f_{\nu+1}||_{\mathcal{D}_{\nu+1}}+||g_{\nu+1}||_{\mathcal{D}_{\nu+1}}\leq \gamma^{n+m+2}_{0}s^{m}_{\nu+1}\mu_{\nu+1}. \]

Proof. Note that $\bar {f}_{\nu +1}$ and $\bar {g}_{\nu +1}$ are solved by the implicit function theorem from (4.9) and (4.10). Thus,

(4.23)\begin{align} ||\bar{f}_{\nu+1}||_{\mathcal{D}_{\nu+1}}& \leq c ||\partial_{\theta_{\nu+1}}f_{\nu}||_{\hat{\mathcal{D}}_{\nu+1}}||U_{\nu+1}||_{\mathcal{D}_{\nu+1}}+c||\partial_{r_{\nu+1}}f_{\nu}||_{\hat{\mathcal{D}}_{\nu+1}}||V_{\nu+1}||_{\mathcal{D}_{\nu+1}}\nonumber\\ & \quad+c||\mathcal{R}_{K_{\nu+1}}f_{\nu}||_{\mathcal{D}_{\nu+1}}+c\varpi_{*}(|V_{\nu+1}|), \end{align}

where $\varpi _{1}(|V_{\nu +1}|)\lesssim \varpi _{*}(|V_{\nu +1}|)$ is due to $\varpi _{1}\lesssim \varpi _{*}$.

The intersection property implies that there exists a $\theta ^{0}_{\nu +1}$ such that for each $r^{0}_{\nu +1}\in G(s_{\nu +1})$, one has

\[ \bar{g}_{\nu+1}(\theta^{0}_{\nu+1},r^{0}_{\nu+1},\varepsilon)=0, \]

i.e.,

\begin{align*} & \sup_{\theta_{\nu+1}\in D(h_{\nu+1})}||\bar{g}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)||\\ & \quad= \sup_{\theta_{\nu+1}\in D(h_{\nu+1})}||\bar{g}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)-\bar{g}_{\nu+1}(\theta^{0}_{\nu+1},r^{0}_{\nu+1},\varepsilon)||\\ & \quad= \mathop{{\rm osc}}_{\theta_{\nu+1}\in D(h_{\nu+1})}\bar{g}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)\\ & \quad= \mathop{{\rm osc}}_{\theta_{\nu+1}\in D(h_{\nu+1})}(\bar{g}_{\nu+1}(\theta_{\nu+1},r_{\nu+1},\varepsilon)-\bar{h}), \end{align*}

where $\bar {h}$ is a function of $r_{\nu +1}$, and $\mathop {{\rm osc}}\limits _{\theta _{\nu +1}}$ denotes the oscillation about ${\theta _{\nu +1}}$. Specially, taking $\bar {h}=g_{0,\nu }(r_{\nu +1})$, one has

\[ \frac{1}{2}||\bar{g}_{\nu+1}||_{\mathcal{D}_{\nu+1}}\leq ||\bar{g}_{\nu+1}-g_{0,\nu}||_{\mathcal{D}_{\nu+1}}. \]

Therefore,

(4.24)\begin{align} ||\bar{g}_{\nu+1}||_{\mathcal{D}_{\nu+1}}& \leq c||\bar{g}_{\nu+1}-g_{0,\nu}||_{\mathcal{D}_{\nu+1}}\nonumber\\ & \leq c||\partial_{\theta_{\nu+1}}g_{\nu}||_{\hat{\mathcal{D}}_{\nu+1}}||U_{\nu+1}||_{\mathcal{D}_{\nu+1}}+c||\partial_{r_{\nu+1}}g_{\nu}||_{\hat{\mathcal{D}}_{\nu+1}}||V_{\nu+1}||_{\mathcal{D}_{\nu+1}}\nonumber\\ & \quad+c||\mathcal{R}_{K_{\nu+1}}g_{\nu}||_{\mathcal{D}_{\nu+1}}. \end{align}

Following (4.23), (4.24), $({\rm H4})$ and estimates obtained earlier, we have

\begin{align*} & ||\bar{f}_{\nu+1}||_{\mathcal{D}_{\nu+1}}+||\bar{g}_{\nu+1}||_{\mathcal{D}_{\nu+1}}\\ & \quad\leq c||\partial_{\theta_{\nu+1}}f_{\nu}||_{\hat{\mathcal{D}}_{\nu+1}}||U_{\nu+1}||_{\mathcal{D}_{\nu+1}}+c||\partial_{\theta_{\nu+1}}g_{\nu}||_{\hat{\mathcal{D}}_{\nu+1}}||U_{\nu+1}||_{\mathcal{D}_{\nu+1}}\\ & \qquad+ c||\partial_{r_{\nu+1}}f_{\nu}||_{\hat{\mathcal{D}}_{\nu+1}}||V_{\nu+1}||_{\mathcal{D}_{\nu+1}}+c||\partial_{r_{\nu+1}}g_{\nu}||_{\hat{\mathcal{D}}_{\nu+1}}||V_{\nu+1}||_{\mathcal{D}_{\nu+1}}\\ & \qquad+ ||\mathcal{R}_{K_{\nu+1}}f_{\nu}||_{\mathcal{D}_{\nu+1}}+c||\mathcal{R}_{K_{\nu+1}}g_{\nu}||_{\mathcal{D}_{\nu+1}}+c\varpi_{*}(|V_{\nu+1}|)\\ & \quad\leq c\frac{\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu_{\nu}}{h_{\nu+1}-h_{\nu+2}}\cdot\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1})\\ & \qquad+ c\frac{\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu_{\nu}}{s_{\nu+1}-s_{\nu+2}}\cdot\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1})\\ & \qquad+ c\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu^{2}_{\nu}\\ & \quad\leq c_{4}\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu^{2}_{\nu}\cdot\gamma^{n+m+1}_{0}s^{m}_{\nu}(h_{\nu+1}-h_{\nu+2})^{{-}1}\Gamma(h_{\nu}-h_{\nu+1})\\ & \qquad+ c_{4}\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu^{2}_{\nu}\cdot\gamma^{n+m+1}_{0}s^{m-1}_{\nu}\Gamma(h_{\nu}-h_{\nu+1})\\ & \qquad+ c_{4}\gamma^{n+m+2}_{0}s^{m}_{\nu}\mu^{2}_{\nu}. \end{align*}

Finally, $({\rm H5})$ implies that

\[ ||f_{\nu+1}||_{\mathcal{D}_{\nu+1}}+||g_{\nu+1}||_{\mathcal{D}_{\nu+1}}\leq \gamma^{n+m+2}_{0}s^{m}_{\nu+1}\mu_{\nu+1}. \]

4.3 The preservation of intersection property

In the previous § 4.2.3, we have constructed a translation $\mathscr {V}_{\nu +1}$ such that the frequency $\omega _{0}(r_{0})$ unchanged. The translation $\mathscr {V}_{\nu +1}$ truns $\bar {\mathscr {F}}_{\nu +1}$ into $\mathscr {F}_{\nu +1}=\bar {\mathscr {F}}_{\nu +1}\circ \mathscr {V}_{\nu +1}$ but drops the intersection property. For this purpose, we construct the conjugation of $\bar {\mathscr {F}}_{\nu +1}$ such that it has the same properties as $\bar {\mathscr {F}}_{\nu +1}$. Denote by $\hat {\mathscr {F}}_{\nu +1}$ the conjugation of $\bar {\mathscr {F}}_{\nu +1}$, that is, $\hat {\mathscr {F}}_{\nu +1}=\mathscr {V}^{-1}_{\nu +1}\circ \bar {\mathscr {F}}_{\nu +1}\circ \mathscr {V}_{\nu +1}$, where

\[ \mathscr{V}^{{-}1}_{\nu+1}: \theta^{1}_{\nu+1}\rightarrow \theta^{1}_{\nu+1},\quad \hat{r}^{1}_{\nu+1}\rightarrow \hat{r}^{1}_{\nu+1}-r^{*}_{\nu+1}. \]

Therefore, $\hat {\mathscr {F}}_{\nu +1}$ has the form

\[ \hat{\mathscr{F}}_{\nu+1}: \begin{cases} \theta^{1}_{\nu+1}=\theta_{\nu+1}+\omega_{0}(r_{0})+f_{\nu+1}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon),\\ \hat{r}^{1}_{\nu+1}=\hat{r}_{\nu+1}-\sum\limits^{\nu+1}_{i=0}r^{*}_{i}+g_{\nu+1}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon). \end{cases} \]

It ensures that the mapping $\hat {\mathscr {F}}_{\nu +1}$ still has the intersection property. For ease of notation, let $\mathscr {F}_{\nu +1}:=\hat {\mathscr {F}}_{\nu +1}$ in $(\nu +1)$-th KAM step.

5.1 Proof of theorem 3.1

5.1.1 Iteration lemma

The iteration lemma guarantees the inductive construction of the transformations in all KAM steps. Let $s_{0}$, $h_{0}$, $\gamma _{0}$, $\mu _{0}$, $\mathscr {F}_{0}$, $\mathcal {D}_{0}$ be given in § 4.1, and set $K_{0}=0$, $r^{*}_{0}=0$, $0<\rho <1$ is a constant. We define the following sequences inductively for all $\nu =0,\,1,\,2,\,\cdots$.

\begin{align*} h_{\nu+1}& =\frac{h_{\nu}}{2}+\frac{h_{0}}{4},\\ s_{\nu+1}& =\frac{s_{\nu}}{2},\\ \mu_{\nu+1}& =\mu^{1+\rho}_{\nu},\\ K_{\nu+1}& =([\log\frac{1}{\mu_{\nu}}]+1)^{3\eta},\\ \mathcal{D}_{\nu+1}& =\mathcal{D}(h_{\nu+1},s_{\nu+1}),\\ \hat{\mathcal{D}}_{\nu+1}& =\mathcal{D}(h_{\nu+2}+\frac{3}{4}(h_{\nu+1}-h_{\nu+2}),s_{\nu+2}),\\ \Gamma(h_{\nu}-h_{\nu+1})& =\sum_{0<|k|\leq K_{\nu+1}}|k|^{\tau}\,{\rm e}^{-|k|\frac{h_{\nu}-h_{\nu+1}}{4}}\leq \frac{4^{\tau}\tau!}{(h_{\nu}-h_{\nu+1})^{\tau}}. \end{align*}

Lemma 5.1 Consider mapping (1.1) for $\nu =0,\,1,\,2,\,\cdots$. If $\varepsilon _{0}$ is sufficiently small such that $({\rm H1})-({\rm H5})$ hold, and

\[ ||f_{\nu}||_{\mathcal{D}_{\nu}}+||g_{\nu}||_{\mathcal{D}_{\nu}}\leq \gamma^{n+m+2}_{0}s^{m}_{\nu}\mu_{\nu}, \]

then the iteration process described above is valid, and the following properties hold.

1. (a) There exists a real analytic transformation $\mathscr {W}_{\nu +1}:=\mathscr {U}_{\nu +1}\circ \mathscr {V}_{\nu +1}$ that satisfies $\mathscr {W}_{\nu +1}\circ \hat {\mathscr {F}}_{\nu +1}=\hat {\mathscr {F}}_{\nu }\circ \mathscr {W}_{\nu +1}$, where

   \[ \hat{\mathscr{F}}_{\nu+1}: \begin{cases} \theta^{1}_{\nu+1}=\theta_{\nu+1}+\omega_{0}(r_{0})+f_{\nu+1}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon),\\ \hat{r}^{1}_{\nu+1}=\hat{r}_{\nu+1}-\sum\limits^{\nu+1}_{i=0}r^{*}_{i}+g_{\nu+1}(\theta_{\nu+1},\hat{r}_{\nu+1},\varepsilon). \end{cases} \]

   Also, the transformation $\mathscr {W}_{\nu +1}$ has the estimate

   (5.1)\begin{equation} ||\mathscr{W}_{\nu+1}-{\rm id}||_{\mathcal{D}_{\nu+1}}\leq c_{3}\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1}). \end{equation}

2. (b) $\{\hat {r}_{\nu }\}$ is a Cauchy sequence and

   \[ |\hat{r}_{\nu+1}-\hat{r}_{\nu}|\leq c\mu_{\nu}. \]

3. (c) The estimate on new perturbations is

   \[ ||f_{\nu+1}||_{\mathcal{D}_{\nu+1}}+||g_{\nu+1}||_{\mathcal{D}_{\nu+1}}\leq \gamma^{n+m+2}_{0}s^{m}_{\nu+1}\mu_{\nu+1}. \]

Proof. The proof is an induction on $\nu$. It is easy to see that we can take sufficiently small $\varepsilon _{0}$ to ensure that $({\rm H1})-({\rm H5})$ hold. Since

\begin{align*} \hat{\mathscr{F}}_{\nu+1}& =\mathscr{V}^{{-}1}_{\nu+1}\circ\bar{\mathscr{F}}_{\nu+1}\circ\mathscr{V}_{\nu+1}\\ & =\mathscr{V}^{{-}1}_{\nu+1}\circ(\mathscr{U}^{{-}1}_{\nu+1}\circ\hat{\mathscr{F}}_{\nu}\circ\mathscr{U}_{\nu+1})\circ\mathscr{V}_{\nu+1}, \end{align*}

and $\mathscr {W}_{\nu +1}=\mathscr {U}_{\nu +1}\circ \mathscr {V}_{\nu +1}$, we have that $\mathscr {W}_{\nu +1}\circ \hat {\mathscr {F}}_{\nu +1}=\hat {\mathscr {F}}_{\nu }\circ \mathscr {W}_{\nu +1}$. For the estimate (5.1) in $({\rm i})$, see lemma 4.5. In addition, we notice that $({\rm ii})$ is due to (4.20), and $({\rm iii})$ follows from lemma 4.6.

5.1.2 Convergence

Observe that

(5.2)\begin{align} \hat{\mathscr{F}}_{\nu+1}& =\mathscr{W}^{{-}1}_{\nu+1}\circ\hat{\mathscr{F}}_{\nu}\circ\mathscr{W}_{\nu+1}\nonumber\\ & =\mathscr{W}^{{-}1}_{\nu+1}\circ\mathscr{W}^{{-}1}_{\nu}\circ\hat{\mathscr{F}}_{\nu-1}\circ\mathscr{W}_{\nu}\circ\mathscr{W}_{\nu+1}\nonumber\\ & =\cdots\nonumber\\ & =\mathscr{W}^{{-}1}_{\nu+1}\circ\cdots\circ\mathscr{W}^{{-}1}_{1}\circ\mathscr{F}_{0}\circ\mathscr{W}_{1}\circ\cdots\circ\mathscr{W}_{\nu+1}. \end{align}

Denote

\[ \mathscr{W}^{\nu+1} := \mathscr{W}_{1}\circ\mathscr{W}_{2}\circ\cdots\circ\mathscr{W}_{\nu+1}, \]

then (5.2) implies that

\[ \mathscr{W}^{\nu+1}\circ\hat{\mathscr{F}}_{\nu+1}=\mathscr{F}_{0}\circ\mathscr{W}^{\nu+1}. \]

The transformation $\mathscr {W}^{\nu +1}$ is convergent since

(5.3)\begin{align} & || \mathscr{W}^{\nu+1}-\mathscr{W}^{\nu}||_{\mathcal{D}_{\nu+1}}\nonumber\\& \quad=||\mathscr{W}_{1}\circ\cdots\circ\mathscr{W}_{\nu}\circ\mathscr{W}_{\nu+1}-\mathscr{W}_{1}\circ\cdots\circ\mathscr{W}_{\nu}||_{\mathcal{D}_{\nu+1}}\nonumber\\ & \quad\leq||\mathscr{W}^{\nu}||_{\mathcal{D}_{\nu}}||\mathscr{W}_{\nu+1}-{\rm id}||_{\mathcal{D}_{\nu+1}}\nonumber\\ & \quad\leq\prod\limits^{\nu}_{i=1}(1+c\gamma^{n+m+1}_{0}s^{m}_{i-1}\mu_{i-1}\Gamma(h_{i-1}-h_{i}))c\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1})\nonumber\\ & \quad \leq c\gamma^{n+m+1}_{0}s^{m}_{\nu}\mu_{\nu}\Gamma(h_{\nu}-h_{\nu+1}). \end{align}

Therefore, $\lim \limits _{\nu \rightarrow \infty }\mathscr {W}^{\nu }:=\mathscr {W}$, as well as $\mathscr {F}_{\infty }=\lim \limits _{\nu \rightarrow \infty }\hat {\mathscr {F}}_{\nu }$, we thus deduce that

\[ \mathscr{W}\circ\mathscr{F}_{\infty}=\mathscr{F}\circ\mathscr{W}. \]

It remains to consider the following convergence. By lemma 4.4, one has

(5.4)\begin{align} & \omega_{0}(\hat{r}_{1})+f_{0,0}(\hat{r}_{1})=\omega_{0}(r_{0}),\nonumber\\ & \omega_{0}(\hat{r}_{2})+f_{0,0}(\hat{r}_{2})+f_{0,1}(\hat{r}_{2})=\omega_{0}(r_{0}),\nonumber\\ & \qquad\qquad\qquad\vdots\nonumber\\ & \omega_{0}(\hat{r}_{\nu})+f_{0,0}(\hat{r}_{\nu})+\cdots+f_{0,\nu-1}(\hat{r}_{\nu})=\omega_{0}(r_{0}). \end{align}

Taking limits on both sides of (5.4), we obtain

\[ \omega_{0}(\hat{r}_{\infty})+\sum^{\infty}_{i=0}f_{0,i}(\hat{r}_{\infty})=\omega_{0}(r_{0})=\omega(r_{*}), \]

that is, for given $r_{*}\in E^{\circ }$, the mapping $\mathscr {F}_{\infty }$ on $\mathcal {D}_{\infty }$ becomes the integrable rotation

\[ \mathscr{F}_{\infty}: \begin{cases} \theta^{1}_{\infty}=\theta_{\infty}+\omega(r_{*}),\\ r^{1}_{\infty}=r_{\infty}-\tilde{r}, \end{cases} \]

where $\omega (r_{*})=p$, and $p$ is given in advance. Besides, $\tilde {r}=\sum \limits ^{\infty }_{i=0}r^{*}_{i} \to 0$ as $\varepsilon \to 0$. This completes the proof of the frequency-preserving KAM persistence in theorem 3.1.

5.2 Proof of corollary 3.2

This subsection is devoted to the proof of corollary 3.2. We will show that the assumption on $\omega (r)$ here contains the transversality condition $({\rm A1})$. In fact, the frequency mapping $\omega (r)$ is injective on $E^{\circ }$, and consequently, it is surjective from $E^{\circ }$ to $\omega ({E^{\circ }})$. Therefore, it is a homeomorphism and by Nagumo's theorem, we have that the Brouwer degree $\deg (\omega,\,E^{\circ },\,p)=\pm 1$ for some $p \in \omega (E^{\circ })^\circ$. Finally, by applying theorem 3.1 we directly obtain the desired frequency-preserving KAM persistence in corollary 3.2.

5.3 Proof of theorem 3.3

This section outlines the proof of theorem 3.3. We focus on describing the parts of the proof of theorem 3.3 that differ from those of theorem 3.1. We shall see that $\varpi _{1}\lesssim \varpi _{2}$ is used directly by assumption $({\rm A3})$ in the process of proving lemma 5.2. Moreover, there is no need to construct a conjugation of $\bar {\mathscr {F}}_{\nu +1}$ since the mapping we considered in theorem 3.3 does not have the intersection property. The parts we leave out in this section are similar to those described in § 4.

Consider the mapping $\mathscr {F}:\mathbb {T}^{n}\times \Lambda \rightarrow \mathbb {T}^{n}$ defined by

\[ \theta^{1}=\theta+\omega(\xi)+\varepsilon f(\theta,\xi,\varepsilon), \]

where $\xi \in \Lambda \subset \mathbb {R}^{n}$ is a parameter, $\Lambda$ is a connected closed bounded domain with interior points. For $\theta _{0}\in D(h_{0})$, and $\xi _{0}\in \Lambda _{0}:=\{\xi \in \Lambda : |\xi -\xi _{0}|<{\rm dist}\ (\xi _{0},\,\partial \Lambda )\}$, denote

\[ \theta^{1}_{0}=\theta_{0}+\omega(\xi_{0})+f_{0}(\theta_{0},\xi_{0},\varepsilon), \]

where $\omega (\xi _{0})=\omega (\xi _{*})=q$ for given $q\in \mathbb {R}^n$ in advance and $f_{0}(\theta _{0},\,\xi _{0},\,\varepsilon )=\varepsilon f(\theta _{0},\,\xi _{0},\,\varepsilon )$. The estimate on $||f_{0}||_{D(h_{0})}$ is

\[ ||f_{0}||_{D(h_{0})}\leq \gamma^{n+m+2}_{0}\mu_{0}, \]

if

\[ \varepsilon^{\frac{3}{4}}\varepsilon^{-\frac{1}{8\eta(m+1)}}||f||_{D(h_{0})}\leq 1. \]

Set

\[ \Lambda_{\nu}:=\{\xi:{\rm dist}\ (\xi,\partial\Lambda_{\nu-1}) <\mu_{\nu-1}\}, \quad \nu \in \mathbb{N}^+. \]

Suppose that after $\nu$ KAM steps, for $\theta _{\nu }\in D(h_{\nu })$ and $\xi _{\nu }\in \Lambda _{\nu }$, the mapping becomes

\[ \theta^{1}_{\nu}=\theta_{\nu}+\omega(\xi_{0})+f_{\nu}(\theta_{\nu},\xi_{\nu},\varepsilon), \]

and one has

\[ ||f_{\nu}||_{D(h_{\nu})}\leq \gamma^{n+m+2}_{0}\mu_{\nu}. \]

Introduce a transformation $\mathscr {U}_{\nu +1}:={\rm id}+U_{\nu +1}$ that satisfies $\mathscr {U}_{\nu +1}\circ \bar {\mathscr {F}}_{\nu +1}=\mathscr {F}_{\nu }\circ \mathscr {U}_{\nu +1}$. Then the conjugation $\bar {\mathscr {F}}_{\nu +1}$ of mapping $\mathscr {F}_{\nu }$ is

\[ \bar{\mathscr{F}}_{\nu+1}: \theta^{1}_{\nu+1}=\theta_{\nu+1}+\omega(\xi_{0})+\bar{f}_{\nu+1}(\theta_{\nu+1},\xi_{\nu},\varepsilon). \]

We obtain the homological equation

(5.5)\begin{equation} U_{\nu+1}(\theta_{\nu+1}+\omega(\xi_{0}))-U_{\nu+1}(\theta_{\nu+1})=\mathcal{T}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},\xi_{\nu},\varepsilon), \end{equation}

and the new perturbation

\begin{align*} \bar{f}_{\nu+1}(\theta_{\nu+1},\xi_{\nu},\varepsilon)& =f_{\nu}(\theta_{\nu+1}+U_{\nu+1},\xi_{\nu},\varepsilon)-f_{\nu}(\theta_{\nu+1},\xi_{\nu},\varepsilon)+\mathcal{R}_{K_{\nu+1}}f_{\nu}(\theta_{\nu+1},\xi_{\nu},\varepsilon)\\ & \quad+U_{\nu+1}(\theta_{\nu+1}+\omega(\xi_{0}),\xi_{\nu},\varepsilon)-U_{\nu+1}(\theta_{\nu+1}+\omega(\xi_{0})+\bar{f}_{\nu+1},\xi_{\nu},\varepsilon). \end{align*}

The homological equation (5.5) is uniquely solvable on $D(h_{\nu +1})$, and the new perturbation $\bar {f}_{\nu +1}$ can be solved by the implicit function theorem.

To keep the frequency unchanged, construct a translation

(5.6)\begin{equation} \mathscr{V}_{\nu+1}: \theta_{\nu+1}\rightarrow \theta_{\nu+1},\quad \tilde{\xi}_{\nu}\rightarrow \tilde{\xi}_{\nu}+\xi_{\nu+1}-\xi_{\nu}, \end{equation}

where $\xi _{\nu +1}$ is to be determined. This translation changes the parameter alone, and the mapping becomes $\mathscr {F}_{\nu +1}=\bar {\mathscr {F}}_{\nu +1}\circ \mathscr {V}_{\nu +1}$, that is,

\[ \mathscr{F}_{\nu+1}: \theta^{1}_{\nu+1}=\theta_{\nu+1}+\omega(\xi_{0})+f_{\nu+1}(\theta_{\nu+1},\xi_{\nu+1},\varepsilon), \]

where the frequency $\omega (\xi _{0})=\omega (\xi _{\nu +1})+\sum \limits ^{\nu }_{i=0}f_{0,i}(\xi _{\nu +1})$. The following lemma states that the frequency is preserved.

Lemma 5.2 Assume that

\[ ({\rm H6})\quad \left\|\sum^{\nu}_{i=0}f_{0,i}(\xi_{\nu})\right\|_{\Lambda_{\nu}}\leq c\mu^{\frac{1}{2}}_{0}. \]

Then there exists a $\xi _{\nu +1}\in B_{c\mu _{\nu }}(\xi _{\nu })\subset \Lambda _{0}$ such that

\[ \omega(\xi_{\nu+1})+\sum^{\nu}_{i=0}f_{0,i}(\xi_{\nu+1})=\omega(\xi_{0}). \]

Proof. The proof is an induction on $\nu \in \mathbb {N}$. When $\nu =0$, obviously, $\omega (\xi _{0})=\omega (\xi _{0})$. When $\nu \geq 1$, let

(5.7)\begin{equation} \omega(\xi_{j})+\sum^{\nu-1}_{i=0}f_{0,i}(\xi_{j})=\omega(\xi_{0}),\quad j=1,2,\cdots,\nu, \end{equation}

then

(5.8)\begin{equation} \omega(\xi_{\nu+1})+\sum^{\nu}_{i=0}f_{0,i}(\xi_{\nu+1})=\omega(\xi_{0}) \end{equation}

needs to be verified. Taking the assumptions $({\rm B1})$ and $({\rm B3})$, one has

(5.9)\begin{equation} \deg\left(\omega(\xi_{\nu+1})+\sum^{\nu}_{i=0}f_{0,i}(\xi_{\nu+1}),\Lambda_{0},\omega(\xi_{0})\right)=\deg\left(\omega(\xi_{\nu+1}),\Lambda_{0},\omega(\xi_{0})\right)\neq0. \end{equation}

This means that there exists at least one parameter $\xi _{\nu +1}\in \Lambda _{0}$ such that $\omega (\xi _{\nu +1})+\sum \limits ^{\nu }_{i=0}f_{0,i}(\xi _{\nu +1})=\omega (\xi _{0})$ holds. Next, let us verify that $\xi _{\nu +1}\in B_{c\mu _{\nu }}(\xi _{\nu })\subset \Lambda _{\nu }$. From (5.7) and (5.8), one has

\[ \omega(\xi_{\nu})+\sum^{\nu-1}_{i=0}f_{0,i}(\xi_{\nu})=\omega(\xi_{\nu+1})+\sum^{\nu}_{i=0}f_{0,i}(\xi_{\nu+1}), \]

i.e.,

\[ f_{0,\nu}(\xi_{\nu+1})=\omega(\xi_{\nu})-\omega(\xi_{\nu+1})+\sum^{\nu-1}_{i=0}(f_{0,i}(\xi_{\nu})-f_{i}(\xi_{\nu+1})). \]

Since $||f_{i}||_{D(h_{i+1})}\leq \gamma ^{n+m+2}_{0}\mu _{i}$ for each $i\geq 0$, one has $[f_{0,i}]_{\varpi _{1}}\leq c\mu _{i}$. Therefore,

\[ | f_{0,i}(\xi_{\nu})-f_{0,i}(\xi_{\nu+1})|\leq c\mu_{i}\varpi_{1}(| \xi_{\nu}-\xi_{\nu+1}|). \]

Following $({\rm A3})$, we have

\begin{align*} | f_{0,\nu}(\xi_{\nu+1})|& =|\omega(\xi_{\nu})-\omega(\xi_{\nu+1})+\sum^{\nu-1}_{i=0}(f_{0,i}(\xi_{\nu})-f_{0,i}(\xi_{\nu+1}))|\\ & \geq|\omega(\xi_{\nu})-\omega(\xi_{\nu+1})|-\sum^{\nu-1}_{i=0}|f_{0,i}(\xi_{\nu})-f_{0,i}(\xi_{\nu+1})|\\ & \geq\varpi_{2}(| \xi_{\nu}-\xi_{\nu+1}|)-c\varpi_{1}(|\xi_{\nu}-\xi_{\nu+1}|)\sum^{\nu-1}_{i=0}\mu_{i}\\ & \geq\frac{\varpi_{2}(|\xi_{\nu}-\xi_{\nu+1}|)}{2}. \end{align*}

The last inequality is due to $\varpi _{1}\lesssim \varpi _{2}$, and $\varepsilon$ is sufficiently small such that $c(\sum^{\nu -1}_{i=0}\mu _{i})\leq \frac {1}{2}$. Thus,

(5.10)\begin{equation} |\xi_{\nu}-\xi_{\nu+1}|\leq \varpi^{{-}1}_{2}(2|f_{0,\nu}(\xi_{\nu+1})|)\leq \varpi^{{-}1}_{2}(2c \mu_{\nu})\leq \varpi^{{-}1}_{1}(2c\mu_{\nu})\leq c\mu_{\nu}, \end{equation}

which is similar to (4.20). Moreover,

\[ | \xi_{\nu+1}-\xi_{0}|\leq\sum^{\nu}_{i=0}| \xi_{i+1}-\xi_{i}|\leq c\sum^{\nu}_{i=0}\mu_{i}\leq 2c\mu_{0}. \]

The above illustrates that $\{\xi _{\nu }\}$ is a Cauchy sequence and $\xi _{\nu +1}\in B_{c\mu _{\nu }}(\xi _{\nu })\subset \Lambda _{0}$.

We now summarize the standard convergence. Denote

\[ \mathscr{U}^{\nu+1}:=\mathscr{U}_{1}\circ\cdots\circ\mathscr{U}_{\nu+1}, \]

and

\[ \mathscr{W}^{\nu+1}:=\mathscr{W}_{1}\circ\cdots\circ\mathscr{W}_{\nu+1}:=(\mathscr{U}_{1}\circ\mathscr{V}_{1})\circ\cdots\circ(\mathscr{U}_{\nu+1}\circ\mathscr{V}_{\nu+1}). \]

Both of them are convergent, one therefore has $\mathscr {U}:=\lim \limits _{\nu \rightarrow \infty }\mathscr {U}^{\nu }$ and $\mathscr {W}:=\lim \limits _{\nu \rightarrow \infty }\mathscr {W}^{\nu }$. Moreover, we get

\[ \omega(\xi_{\infty})+\sum\limits^{\infty}_{i=0}f_{0,i}(\xi_{\infty})=\omega(\xi_{0})=\omega(\xi_{*})=q \]

for $\xi _{*}\in \Lambda ^{\circ }$ fixed. For (3.4), we know that the conjugation for dynamical system (3.2) only focuses on the angular variable $\theta \in \mathbb {T}^n$. As mentioned in (5.6), the angular variable is unchanged under the translation. Therefore, one has

\[ \mathscr{U}\circ\mathscr{F}_{\infty}=\mathscr{F}\circ\mathscr{U}, \]

here $\mathscr {F}_{\infty }$ denotes the limit of $\mathscr {F}_{\nu }$ on $\mathcal {D}_{\infty }$. More precisely, one obtains the integrable rotation

\[ \mathscr{F}_{\infty}:\theta^{1}_{\infty}=\theta_{\infty}+\omega(\xi_{*}) \]

with frequency $\omega (\xi _{*})=q$, where $q$ is given in advance. In other words, we prove the KAM persistence with prescribed frequency-preserving. This completes the proof of theorem 3.3.