Analytic KAM data: angular tori, action-angle neighbourhoods, Fourier cutoffs and Diophantine frequency labels

Throughout, fix an integer $n\ge 1$. The space ${\mathbb{R}}^n$ is read through Complex $m$-space and its real coordinate dictionary and ${\mathbb{C}}^n$ carries the polydisc geometry of Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$. We relabel the suppliers' coordinates indexed by $0,\dots ,m-1$ as $z_i:=z(i-1)$ for $1\le i\le m$; every coordinate tuple and derivative below uses this relabelling. All norms below are supremum norms: for $z\in {\mathbb{C}}^m$,

$\mathrm{\mid }z{\mathrm{\mid }}_{\mathrm{\infty }}:={\max }_{1\le i\le m}\mathrm{\mid }{z}_i\mathrm{\mid },\mathrm{\mid }z{\mathrm{\mid }}_1:={\sum }_{i=1}^m\mathrm{\mid }{z}_i\mathrm{\mid }\mathrm{.}$

Angular torus. Let $\mathbb{Z}$ be the integers (The integers as equivalence classes of pairs of naturals). Write $k=(k_1,\dots ,k_n)\in {\mathbb{Z}}^n$, $\mathrm{\mid }k{\mathrm{\mid }}_1=\mathrm{\mid }{k}_1\mathrm{\mid }+\cdots +\mathrm{\mid }{k}_n\mathrm{\mid }$, and let $(2\pi \mathbb{Z}{)}^n:=\{2\pi k:k\in {\mathbb{Z}}^n\}\subseteq {\mathbb{R}}^n$. On ${\mathbb{R}}^n$ define the relation $\vartheta \sim {\vartheta }^{\mathrm{\prime }}$ by $\vartheta -{\vartheta }^{\mathrm{\prime }}\in (2\pi \mathbb{Z}{)}^n$. This is an equivalence relation, and the angular torus is the quotient set

${\mathbb{T}}^n:=(\mathbb{R}\mathrm{/}2\pi \mathbb{Z}{)}^n:={\mathbb{R}}^n\mathrm{/}\text{\hspace{0.17em}⁣}\sim$

with the quotient topology (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection, The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space), and canonical projection $\pi :{\mathbb{R}}^n\to {\mathbb{T}}^n$, which is continuous (Continuity of a map of topological spaces at a point and globally). A map $f:{\mathbb{R}}^n\to X$ is $2\pi$-periodic if $f(\vartheta +2\pi k)=f(\vartheta )$ for all $\vartheta \in {\mathbb{R}}^n$, $k\in {\mathbb{Z}}^n$; such an $f$ factors uniquely through $\pi$, and a map on ${\mathbb{T}}^n$ is, by definition, a $2\pi$-periodic map on ${\mathbb{R}}^n$ read through this factorisation. A function on ${\mathbb{T}}^n$ is continuous, resp. real analytic, when its $2\pi$-periodic representative has that property. Here real analytic has the componentwise meaning of Real analytic germs in several variables: near every real point each real component is represented by an absolutely convergent power series in the real coordinates, and the same coefficients define a holomorphic function on a complex polydisc whose restriction is the given real component. For a vector-valued map this condition applies to each of its finitely many scalar components.

Action-angle neighbourhoods. For $0<r$ and $0<s\le \mathrm{\infty }$ put

$B_r:=\{I\in {\mathbb{C}}^n:\mathrm{\mid }I{\mathrm{\mid }}_{\mathrm{\infty }}<r\},S_s:=\{\vartheta \in {\mathbb{C}}^n:\mathrm{\mid }\mathrm{I}\mathrm{m}\vartheta {\mathrm{\mid }}_{\mathrm{\infty }}<s\},$

and, for a set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ (always measurably or topologically qualified in the uses below) and $h\ge 0$,

$O_h(\mathrm{\Omega }):=\{\nu \in {\mathbb{C}}^n:{\mathrm{dist}}_{\mathrm{\infty }}(\nu ,\mathrm{\Omega })<h\},{\mathrm{dist}}_{\mathrm{\infty }}(\nu ,\mathrm{\Omega }):={\inf }_{\omega \in \mathrm{\Omega }}\mathrm{\mid }\nu -\omega {\mathrm{\mid }}_{\mathrm{\infty }}\mathrm{.}$

For $\mathrm{\Omega }=\mathrm{\varnothing }$ the infimum is taken in the extended nonnegative reals with $\inf \mathrm{\varnothing }=+\mathrm{\infty }$, so $O_h(\mathrm{\varnothing })=\mathrm{\varnothing }$ for every finite $h\ge 0$. Supremum norms of functions on an empty domain are $0$ (the supremum in $[0,+\mathrm{\infty }]$). Thus an empty Diophantine label set gives an empty parameter family; results quantified over its labels are vacuous, and positive-measure assertions require their separately stated hypotheses.

The action-angle neighbourhood of width $(r,s,h)$ over $\mathrm{\Omega }$ is the open product

$D_{r,s,h}(\mathrm{\Omega }):=B_r\times S_s\times O_h(\mathrm{\Omega })\subseteq {\mathbb{C}}^{2n}\times {\mathbb{C}}^n,$

and $D_{r,s}(\mathrm{\Omega }):=B_r\times S_s$. In the KAM construction the parameter set is a frequency label set ${\mathrm{\Omega }}_{\alpha }\subseteq {\mathbb{R}}^n$ (defined below) and the relevant parameter neighbourhoods are $O_h({\mathrm{\Omega }}_{\alpha })$; the scale of a step is measured by the three widths $r,s,h$ and an angular margin $\sigma >0$.

Holomorphic data and norms. A function $g:D_{r,s,h}(\mathrm{\Omega })\to \mathbb{C}$ is holomorphic on $D_{r,s,h}(\mathrm{\Omega })$ when it is continuous and separately holomorphic in each of the $2n+n$ complex variables (Separately holomorphic functions, Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions); equivalently, by A continuous separately holomorphic function is the sum of an absolutely convergent power series with Cauchy-integral coefficients on every smaller polydisc, it is given near every point by an absolutely convergent power series in the $(I,\vartheta ,\nu )$ variables. The sup-norms are

$\mathrm{\mid }g{\mathrm{\mid }}_{r,s}:={\sup }_{B_r\times S_s}\mathrm{\mid }g\mathrm{\mid },\mathrm{\mid }g{\mathrm{\mid }}_{r,s,h}:=\mathrm{\mid }g{\mathrm{\mid }}_{r,s,h}(\mathrm{\Omega }):={\sup }_{D_{r,s,h}(\mathrm{\Omega })}\mathrm{\mid }g\mathrm{\mid }\in [0,+\mathrm{\infty }]\mathrm{.}$

The first norm applies to a function of $(I,\vartheta )$ alone; the second applies to a function of $(I,\vartheta ,\nu )$ on its declared product domain. They are distinct notations: $O_0(\mathrm{\Omega })=\mathrm{\varnothing }$, so the three-width norm at $h=0$ is $0$, not the phase-only norm.

A phase function $g:B_r\times S_s\to \mathbb{C}$ is $2\pi$-periodic in the angles if $g(I,\vartheta +2\pi k)=g(I,\vartheta )$ for every $I\in B_r$, $\vartheta \in S_s$ and $k\in {\mathbb{Z}}^n$. For a parameter-dependent function on $D_{r,s,h}(\mathrm{\Omega })$ this means $g(I,\vartheta +2\pi k,\nu )=g(I,\vartheta ,\nu )$ for every point $(I,\vartheta ,\nu )$ of that domain and every $k\in {\mathbb{Z}}^n$. Real angular translations preserve $S_s$, so these equalities are defined on the entire declared domain.

Where no parameter occurs we write $g\in {\mathcal{H}}_{r,s}$ if $g$ is holomorphic on $B_r\times S_s$, $2\pi$-periodic in the angles and $\mathrm{\mid }g{\mathrm{\mid }}_{r,s}<+\mathrm{\infty }$; and $g\in {\mathcal{H}}_{r,s,h}$ if $g$ is holomorphic on $D_{r,s,h}(\mathrm{\Omega })$, $2\pi$-periodic in the angles and $\mathrm{\mid }g{\mathrm{\mid }}_{r,s,h}<+\mathrm{\infty }$. Functions of $\vartheta$ alone use the same periodicity condition with $I$ omitted. Thus at fixed non-angle variables the restriction to real angles factors through ${\mathbb{T}}^n$; the strip norms are suprema over complex angular representatives and the integrals below are integrals of the descended real-angle functions.

Fourier coefficients, mean and cutoff. Let $g\in {\mathcal{H}}_{r,s}$. For $k\in {\mathbb{Z}}^n$ and $I\in B_r$ the Fourier coefficient is the iterated Riemann integral

$g_k(I):=\frac{1}{(2\pi {)}^n}{\int }_0^{2\pi }\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\cdots \text{\hspace{0.17em}⁣}{\int }_0^{2\pi }g(I,\vartheta )e^{-ik\cdot \vartheta }d{\vartheta }_n\cdots d{\vartheta }_1,$

as in Sections, lower and upper section integrals, and iterated Riemann integrals on product rectangles and Jordan sets; by Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections it equals the multiple Riemann integral over $[0,2\pi {]}^n$ and is independent of the order of integration. The angular mean of $g$ is $[g]:=g_0$, so $[g](I)=\frac{1}{(2\pi {)}^n}{\int }_{[0,2\pi {]}^n}g(I,\vartheta )d\vartheta$; $g$ is zero-mean if $[g]=0$ for every $I\in B_r$. For a real number $K\ge 0$ the Fourier cutoff $T_{K}g$ is the trigonometric polynomial

$T_{K}g(I,\vartheta ):={\sum }_{k\in {\mathbb{Z}}^n,\mathrm{\mid }k{\mathrm{\mid }}_1\le K}{g}_k(I)e^{ik\cdot \vartheta }\mathrm{.}$

The sum is finite because $\mathrm{\mid }k{\mathrm{\mid }}_1\le K$ bounds each $\mathrm{\mid }{k}_i\mathrm{\mid }$ by $K$. For a function $g$ of $(I,\vartheta ,\nu )$ the same notations apply with the $\nu$-variable carried along; e.g. $[g]$, $g_k$ and $T_{K}g$ are then functions of $(I,\nu )$.

Frequency label sets. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, $\alpha >0$ and $\tau >n-1$. A frequency $\nu \in {\mathbb{R}}^n$ is $(\alpha ,\tau )$-Diophantine if

$\mathrm{\mid }k\cdot \nu \mathrm{\mid }\ge \alpha \mathrm{\mid }k{\mathrm{\mid }}_1^{-\tau }\text{for every }k\in {\mathbb{Z}}^n\setminus \{0\},$

with $k\cdot \nu ={\sum }_i{k}_i{\nu }_i$. The label set is

${\mathrm{\Omega }}_{\alpha ,\tau }:=\mathrm{\Omega }\cap \{\nu \in {\mathbb{R}}^n:\nu \text{ is }(\alpha ,\tau )\text{-Diophantine}\}\mathrm{.}$

${\mathrm{\Omega }}_{\alpha ,\tau }$ is closed in $\mathrm{\Omega }$ and contained in the global $(\alpha ,\tau )$-Diophantine set; it need not have interior points. Write $\widehat{\mathrm{\Omega }}:=\mathrm{\Omega }$ when $\tau ,\alpha$ are fixed and no confusion arises. The parameter neighbourhoods used below are $O_h({\mathrm{\Omega }}_{\alpha ,\tau })$.

Hamiltonians, canonical form and Poisson bracket. A Hamiltonian datum on $D_{r,s,h}(\mathrm{\Omega })$ is a holomorphic $H:D_{r,s,h}(\mathrm{\Omega })\to \mathbb{C}$ which is $2\pi$-periodic in the angles and real-valued on real points; in particular we consider families $H(I,\vartheta ;\nu )=e(\nu )+\nu \cdot I+P(I,\vartheta ,\nu )$ where $e$ is holomorphic on $O_h(\mathrm{\Omega })$ and real on real points, and $P\in {\mathcal{H}}_{r,s,h}$ is real on real points. On the real phase slice pair the coordinates as $q=\vartheta$, $p=I$ in the phase-domain notation of A phase domain, its canonical one-form and its standard symplectic two-form. On the complex phase domain $B_r\times S_s$, identify phase tangent vectors in angle-first order with $v=(v_{\vartheta },v_I)\in {\mathbb{C}}^{2n}$, and define the standard complex two-form to be the constant alternating complex bilinear form

$\omega ={\sum }_{i=1}^{n}d{\vartheta }_i\wedge d{I}_i,\omega (v,w):={\sum }_{i=1}^n((v_{\vartheta }{)}_i(w_I{)}_i-(v_I{)}_i(w_{\vartheta }{)}_i)\mathrm{.}$

Here $d{\vartheta }_i(v)=(v_{\vartheta }{)}_i$, $d{I}_i(v)=(v_I{)}_i$, and the wedge denotes the alternating product displayed above. No complex conjugation occurs. Fix any parameter $\lambda \in O_h(\mathrm{\Omega })$, real or complex. All derivatives of $H(\cdot ;\lambda )$ below are complex derivatives in the phase variables alone. The power-series coefficient bounds of A continuous separately holomorphic function is the sum of an absolutely convergent power series with Cauchy-integral coefficients on every smaller polydisc meet the hypotheses of An absolutely convergent multi-indexed power series is holomorphic and differentiates termwise, so these derivatives exist, are holomorphic, and give the complex linear phase differential

$d_{\mathrm{p}\mathrm{h}}H(v):=H_{\vartheta }\cdot v_{\vartheta }+H_I\cdot v_I\mathrm{.}$

The Hamiltonian vector field is the holomorphic phase map

$X_H=(H_I,-H_{\vartheta })\text{in }(\vartheta ,I)\text{ order},{\theta }^{\mathrm{\prime }}=H_I,I^{\mathrm{\prime }}=-H_{\vartheta }\text{ (Hamilton’s equations)},$

These are the component equations for a $C^1$ curve $\gamma :J\to S_s\times B_r$ on a nonempty open real interval $J$, in angle-first order, with ${\gamma }^{\mathrm{\prime }}(t)=X_H(\gamma (t);\lambda )$ for every $t\in J$; such a curve is called an integral curve at fixed parameter $\lambda$. It is characterised by ${\iota }_{X_H}\omega =d_{\mathrm{p}\mathrm{h}}H$, meaning $\omega (X_H,v)=d_{\mathrm{p}\mathrm{h}}H(v)$ for every phase vector $v$: substitution gives $H_I\cdot v_I+H_{\vartheta }\cdot v_{\vartheta }$, and comparison of the coefficients of $v_I,v_{\vartheta }$ determines $X_H$ uniquely. For holomorphic phase functions $f,g$ on the same domain, with parameters held fixed, define their Poisson bracket by

$\{f,g\}:=\omega (X_f,X_g)=f_{\vartheta }\cdot g_I-f_I\cdot g_{\vartheta }={\mathrm{\nabla }}_{\mathrm{p}\mathrm{h}}{f}^{\mathsf{T}}J{\mathrm{\nabla }}_{\mathrm{p}\mathrm{h}}g,$

where ${\mathrm{\nabla }}_{\mathrm{p}\mathrm{h}}f=(f_{\vartheta },f_I)$ and $J=\left(\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right)$ in angle-first order. The equalities follow by substitution into the displayed bilinear form. For real $\lambda$, the real-valued restriction of a Hamiltonian datum to $(B_r\cap {\mathbb{R}}^n)\times {\mathbb{R}}^n$ is smooth by the same power-series suppliers, hence satisfies the real $C^2$ hypotheses of The Hamiltonian field of a $C^2$ function on a phase domain is $J\mathrm{\nabla }H$, and the Poisson bracket is ${\sum }_i({\mathrm{\partial }}_{q^i}f{\mathrm{\partial }}_{p_i}g-{\mathrm{\partial }}_{p_i}f{\mathrm{\partial }}_{q^i}g)$. That supplier identifies the restricted field with its real Hamiltonian field. Holomorphic $f,g$ that are real on real points likewise have smooth real restrictions, to which that supplier's bracket formula applies. More generally, for $C^1$ functions $H,f,g:U\to \mathbb{R}$ on an open real phase domain $U$ we use the same coordinate formulas with real partial derivatives: they define continuous fields and brackets, and the same coefficient comparison gives ${\iota }_{X_H}\omega =dH$ without invoking the $C^2$ supplier.

In particular $\{f,e+\nu \cdot I\}=\nu \cdot f_{\vartheta }$ for every holomorphic phase function $f$, and for every real $C^1$ phase function $f$ when $e,\nu$ and the phase variables are real.

Angular parametrisations and Lagrangian images. Fix a Hamiltonian datum $H$. For parameter-dependent data, fix a value $\lambda \in O_h(\mathrm{\Omega })$ and use $H$ below for the phase function $(I,\vartheta )\mapsto H(I,\vartheta ;\lambda )$; its vector field differentiates only the phase variables. An angular parametrisation is a map $\psi =({\psi }_I,{\psi }_{\vartheta })$ with ${\psi }_I:{\mathbb{R}}^n\to {\mathbb{C}}^n$ $2\pi$-periodic and ${\psi }_{\vartheta }:{\mathbb{R}}^n\to {\mathbb{C}}^n$ satisfying ${\psi }_{\vartheta }(\vartheta +2\pi k)={\psi }_{\vartheta }(\vartheta )+2\pi k$ for all $k\in {\mathbb{Z}}^n$. Define the complex angular quotient ${\mathbb{T}}_{\mathbb{C}}^n:={\mathbb{C}}^n\mathrm{/}(2\pi \mathbb{Z}{)}^n$ by the relation $z\sim z^{\mathrm{\prime }}$ if $z-z^{\mathrm{\prime }}\in (2\pi \mathbb{Z}{)}^n$, with its quotient topology and projection ${\pi }_{\mathbb{C}}:{\mathbb{C}}^n\to {\mathbb{T}}_{\mathbb{C}}^n$. Then $\vartheta \mapsto ({\psi }_I(\vartheta ),{\pi }_{\mathbb{C}}({\psi }_{\vartheta }(\vartheta )))$ is periodic and induces a map $\widehat{\psi }:{\mathbb{T}}^n\to {\mathbb{C}}^n\times {\mathbb{T}}_{\mathbb{C}}^n$. When both components are real-valued this instead induces $\widehat{\psi }:{\mathbb{T}}^n\to {\mathbb{R}}^n\times {\mathbb{T}}^n$ using $\pi$ on the angular component. The parametrisation is holomorphic on the strip $S_s$ when both components extend holomorphically to $S_s$, and real analytic when in addition $\psi ({\mathbb{R}}^n)\subseteq {\mathbb{R}}^{2n}$ and the components are real analytic in the real variables. In the action-first order of $\psi$, the same field is $X_H=(-H_{\vartheta },H_I)$. It is faithful if $\psi (\vartheta )=\psi ({\vartheta }^{\mathrm{\prime }})$ implies $\vartheta -{\vartheta }^{\mathrm{\prime }}\in (2\pi \mathbb{Z}{)}^n$, and an immersion if it is $C^1$ and $D\psi (\vartheta )$ has rank $n$ for every $\vartheta \in {\mathbb{R}}^n$; a tuple that is both is an embedded torus parametrisation. Its image is Lagrangian (with respect to $\omega$) if $\psi$ is $C^1$ and ${\psi }^{\ast }\omega ={\sum }_{i=1}^{n}d({\psi }_{\vartheta }{)}_i\wedge d({\psi }_I{)}_i=0$ as a two-form on the parameter space of the representative, and invariant with frequency $\nu$ if $\psi$ is $C^1$, lies in the domain of $X_H$, and

$X_H(\psi (\vartheta ))={\sum }_{j=1}^n{\nu }_j{\mathrm{\partial }}_{{\vartheta }_j}\psi (\vartheta )\text{for all real }\vartheta \mathrm{.}$

• Conventions. The torus is the $2\pi$-periodic one, so $e^{ik\cdot \vartheta }$ is $2\pi$-periodic; the real restrictions of the symplectic form, Hamiltonian field and bracket are exactly the $\vartheta ,I$ coordinate model of A phase domain, its canonical one-form and its standard symplectic two-form and The Hamiltonian field of a $C^2$ function on a phase domain is $J\mathrm{\nabla }H$, and the Poisson bracket is ${\sum }_i({\mathrm{\partial }}_{q^i}f{\mathrm{\partial }}_{p_i}g-{\mathrm{\partial }}_{p_i}f{\mathrm{\partial }}_{q^i}g)$ for real $C^2$ data. The complex bilinear extension is defined and checked above. The dimension hypothesis $n\ge 2$ used by the KAM theorems below is imposed there, not here.
• Axiom of Countable Choice. The measure-theoretic frequency-set estimates in this batch use the Axiom of Countable Choice; analytic suppliers are used with their exact stated hypotheses, and no extra choice is required for the determined Newton iteration. The assumption is carried by the positive-measure theorem and its consumers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).
• Scope. Nothing here is a statement about a physical system: $I,\vartheta ,\nu$ are complex or real variables, and the "Hamiltonian" is a function whose Hamilton equations are an ODE system. In particular "frequency" means the vector $\nu$ in the invariance identity above, not a measured frequency.
• Sup versus $r,s,h$. The norms $\mathrm{\mid }\cdot {\mathrm{\mid }}_{r,s,h}$ are finite only on bounded functions; every function considered below is bounded on its declared neighbourhood by hypothesis, and every estimate is an inequality between these suprema. The quadratic Taylor step and the Fourier estimates below use the margins between $r,s,h$ and the smaller widths they produce.