Convergence of the composed KAM transformations, with derivative and frequency-Lipschitz bounds

Domain inclusions. On $D_{j+1}$ the step moves actions by at most $C{r}_j{E}_j$, angles by at most $C{\sigma }_j{E}_j$ and parameters by at most $B_j$. Take the step smallness so that $C{E}_j\le 1\mathrm{/}8$. Then $r_{j+1}\mathrm{/}2+C{r}_j{E}_j\le r_j\mathrm{/}16+r_j\mathrm{/}8<r_j\mathrm{/}2$. The angular loss $s_j-s_{j+1}$ is at least $5{\sigma }_j$, so $s_{j+1}-8{\sigma }_{j+1}+C{\sigma }_j{E}_j\le s_j-9{\sigma }_j+{\sigma }_j\mathrm{/}8<s_j-8{\sigma }_j$. Moreover $h_{j+2}+B_j\le h_{j+1}\mathrm{/}4+h_{j+1}\mathrm{/}(16n)<h_{j+1}$. These inequalities also put the input to ${\varphi }_j$ in $O_{h_j\mathrm{/}4}$ and its image in $O_{h_j\mathrm{/}2}$. Thus $T_j(D_{j+1})\subseteq D_j$, and induction defines the extended composition on $D_j$ with image in $D_0$. The exact sequence formula for $s_j$ gives $s_j-8{\sigma }_j\ge s_0-24{\sigma }_0\ge s_0\mathrm{/}4$, and in fact for the sequence with loss $8{\sigma }_j$ it gives $s_j-8{\sigma }_j\ge s_0-16{\sigma }_0\ge s_0\mathrm{/}2$. Hence the common larger angular strip $S_t$, $t:=s_0\mathrm{/}4+2{\sigma }_0$, lies in every $D_j$ at $I=0$. Equivariance and fixed-parameter canonicity follow by composition; real parameters remain real by the stated real one-step construction.

Phase Jacobians in a fixed weight. Because the phase Jacobians are upper triangular in $(I,\vartheta )$, conjugating their error matrices from $W_j$ to $W_0$ leaves diagonal blocks unchanged, multiplies the upper right block by $(r_j\mathrm{/}{\sigma }_j)\mathrm{/}(r_0\mathrm{/}{\sigma }_0)\le 1$, and leaves the lower left block zero. Thus $\mathrm{\parallel }{W}_0(D_z{T}_j-I)W_0^{-1}\mathrm{\parallel }\le C{E}_j$. The parameter block satisfies $\mathrm{\parallel }D{\varphi }_j\mathrm{\parallel }\le 1+(16n\mathrm{/}7)B_j\mathrm{/}{h}_j$. Their products are bounded by $\exp (C\sum E_j)$ and $\exp ((16n\mathrm{/}7)\sum B_j\mathrm{/}{h}_j)$ respectively. Here $\sum E_j\le 2E_0$ and $\sum B_j\mathrm{/}{h}_j\le 2B_0\mathrm{/}{h}_0$, since $B_j\mathrm{/}{h}_j=(B_0\mathrm{/}{h}_0)(E_j\mathrm{/}{E}_0)2^{\mu j}\le (B_0\mathrm{/}{h}_0)4^{-j}$. Decrease the initial smallness so the phase product is at most $2$; the parameter product is already at most $\exp (2\mathrm{/}(7\cdot 4^{\mu }))<2$, since $B_0\mathrm{/}{h}_0\le (16n{4}^{\mu }{)}^{-1}$. The chain rule therefore gives $\mathrm{\parallel }{W}_0{D}_z{\mathrm{\Psi }}_j{W}_0^{-1}\mathrm{\parallel }\le 2$ and $\mathrm{\parallel }D{\chi }_j\mathrm{\parallel }\le 2$.

The parameter block of the phase composition. The single-step phase-parameter block in the fixed weight is bounded by $C{E}_j{2}^{-j}\mathrm{/}{h}_j$, because $\mathrm{\parallel }{W}_0{W}_j^{-1}\mathrm{\parallel }\le 2^{-j}$ and $\mathrm{\parallel }D{\varphi }_j\mathrm{\parallel }\le 2$. Expanding the parameter block of the product of block triangular extended Jacobians yields one term for each step, with a phase product on its left and a parameter product on its right. Step 2.1 bounds these products by $2$, so $\mathrm{\parallel }{W}_0{\mathrm{\partial }}_{\nu }{\mathrm{\Psi }}_j\mathrm{\parallel }\le 4C{\sum }_{k<j}{E}_k{2}^{-k}\mathrm{/}{h}_k\le A{E}_0\mathrm{/}{h}_0\mathrm{.}$ The last inequality follows from $E_k\mathrm{/}{h}_k\le (E_0\mathrm{/}{h}_0)4^{-k}$. The same product expansion for $D{\chi }_j-I$ gives $\mathrm{\parallel }D{\chi }_j-I\mathrm{\parallel }\le A{B}_0\mathrm{/}{h}_0$. All bounds hold throughout the respective $D_j$; earlier factors are evaluated at the parameter images established in step 1.1.

Holomorphic increments. For $(0,\vartheta ,\nu )$ with $\vartheta \in S_t$, $\nu \in O_{h_{j+2}}(\mathrm{\Lambda })$, the straight segment from this point to $T_j(0,\vartheta ,\nu )$ lies in $D_j$. For the parameter segment this follows by choosing a label within $h_{j+2}$ of $\nu$: its entire segment stays within $h_{j+2}+B_j<h_{j+1}$ of that label. The action and angle segments stay inside their convex domains by step 1.1, with the same stronger margins on $S_t$. Integrating the derivatives of ${\mathrm{\Psi }}_j$ along this segment and using steps 2.1-3.1 gives $\mathrm{\mid }{W}_0{\mathrm{\Delta }}_j\mathrm{\Psi }\mathrm{\mid }\le 2C{E}_j{2}^{-j}+A(E_0\mathrm{/}{h}_0)B_j\le A{E}_j{2}^{-j}\mathrm{.}$ Indeed $(E_0\mathrm{/}{h}_0)B_j=(B_0\mathrm{/}{h}_0)E_j{2}^{-\mu j}$ and $B_0\mathrm{/}{h}_0$ is bounded by a dimensional small constant. This estimates both changes of phase and changes of the parameter fed into every earlier factor. Likewise $\mathrm{\mid }{\chi }_{j+1}-{\chi }_j\mathrm{\mid }\le 2B_j$ on this parameter tube. Cauchy's estimate with angular margin ${\sigma }_0$ gives the displayed first derivative bound on $S_{s_0\mathrm{/}4}$ and, for every multi-index $\beta$, the bound $\mathrm{\mid }{W}_0{\mathrm{\partial }}_{\vartheta }^{\beta }{\mathrm{\Delta }}_j\mathrm{\Psi }\mathrm{\mid }\le A\beta !{\sigma }_0^{-\mathrm{\mid }\beta \mathrm{\mid }}{E}_j{2}^{-j}$.

Frequency Lipschitz bounds without convexity of the label set. A holomorphic function $f$ bounded by $M$ on $O_a(\mathrm{\Lambda })$ has $\mathrm{\mid }f(\nu )-f({\nu }^{\mathrm{\prime }})\mathrm{\mid }\le 4n(M\mathrm{/}a)\mathrm{\mid }\nu -{\nu }^{\mathrm{\prime }}\mathrm{\mid }$ for labels $\nu ,{\nu }^{\mathrm{\prime }}$. If $\mathrm{\mid }\nu -{\nu }^{\mathrm{\prime }}\mathrm{\mid }<a\mathrm{/}2$, their segment stays within $a\mathrm{/}2$ of $\nu \in \mathrm{\Lambda }$, and Cauchy bounds on coordinate discs of radius $a\mathrm{/}2$ give the derivative bound $2nM\mathrm{/}a$ there. If their distance is at least $a\mathrm{/}2$, use $\mathrm{\mid }f(\nu )-f({\nu }^{\mathrm{\prime }})\mathrm{\mid }\le 2M$. Apply this argument to the weighted phase increment of step 4.1 with $a=h_{j+2}$. Since $E_j{2}^{-j}\mathrm{/}{h}_{j+2}\le 4^{2\mu }(E_0\mathrm{/}{h}_0)4^{-j}{2}^{-j},$ it proves clause 3 with a dimensional $C_L$. Apply it also to ${\chi }_{j+1}-{\chi }_j$, bounded by $2B_j$ on the same tube. The bound $B_j\mathrm{/}{h}_{j+2}\le 4^{2\mu }(B_0\mathrm{/}{h}_0)4^{-j}$ makes its Lipschitz series summable. This argument uses no segment contained in $\mathrm{\Lambda }$ and therefore applies to disconnected Diophantine label sets.

Limits and sizes. The bounds of step 4.1 are summable for every fixed angular multi-index, uniformly on $\mathcal{L}$. Their series therefore define ${\mathrm{\Psi }}_{\mathrm{\infty }}$, with uniform convergence of all angular derivatives. For each label, holomorphic convergence on $S_t$ and Cauchy differentiation prove holomorphy and identify these derivative limits with derivatives of ${\mathrm{\Psi }}_{\mathrm{\infty }}$. Summing the component bounds gives clause 4 after enlarging $A$; decreasing the initial smallness gives $nA{E}_0\le 1\mathrm{/}2$. The identity phase map has zero parameter derivative, so summing step 5.1 gives clause 3 for the limit. The parameter increments have summable sizes $2B_j$ and summable Lipschitz constants, yielding ${\chi }_{\mathrm{\infty }}$, its displacement bound and its asserted Lipschitz bound relative to the identity.

Exact invariance data. By the stated one-step identity, $H_j\circ T_j=H_{j+1}$. Induction on the extended compositions gives $H_0({\mathrm{\Psi }}_j(z,\nu ),{\chi }_j(\nu ))=H_j(z,\nu )=e_j(\nu )+\nu \cdot I+P_j(z,\nu )$ on $D_j$. The perturbation bound holds there because $D_j\subseteq D_{r_j,s_j,h_j}(\mathrm{\Lambda })$. The parameter coordinate is essential: for a parameter-dependent $H_0$ the Hamiltonian transported by the phase map at stage $j$ is $H_0(\cdot ,{\chi }_j(\nu ))$. This proves clause 5.