2.1. Definitions and assumptions

Our main results concern perturbations of the Hamiltonian function in equation (2.1) in the class of mechanical systems as

(2.3) $$ \begin{align} H_\varepsilon(x,p) = H_0(x,p) + \varepsilon U(x), \end{align} $$

where $\varepsilon \in \mathbb {R}$ and $U \in C^2(\mathbb {T}^2)$ denotes a perturbing potential, which is assumed to be a Morse function (or a constant) and have an absolutely convergent Fourier series:

$$ \begin{align*} U(x) = \sum_{k_1 \in \mathbb{Z}} U_{k_1}(x^2) e^{i 2 \pi k_1 x^1 }= \sum_{(k_1,k_2) \in \mathbb{Z}^2} U_{k_1,k_2} e^{i 2\pi (k_1 x^1 + k_2 x^2)}. \end{align*} $$

(Note that in two dimensions, $C^2$ -regularity is not sufficient for ensuring an absolutely convergent Fourier series, although in one dimension it is). In the following, we introduce several subsets of $\mathbb {Z}^2$ in such a way that their definitions immediately carry over in arbitrary dimension $d \in \mathbb {N}$ (see Remark 1.1). First, we define the spectrum of U, that is, the set of non-vanishing Fourier coefficients, as

(2.4) $$ \begin{align} \mathcal{S}_U := \{ \boldsymbol{k} = (k_1, k_2) \in \mathbb{Z}^2 : U_{\boldsymbol{k}} \neq 0 \}, \end{align} $$

while the non-singular spectrum is denoted by

(2.5) $$ \begin{align} \mathcal{S}_{U,0} := \{ \boldsymbol{k} \in \mathcal{S}_U : \text{ there exists } i \neq j \ \text{such that}\ k_i \cdot k_j \neq 0 \}. \end{align} $$

Moreover, we define the coprime set of the orthogonal complement of $\mathcal {S}_U$ as well as its non-singular subset via

(2.6) $$ \begin{align} \mathcal{B}(\mathcal{S}_U^\perp) := \{ \boldsymbol{b} \in \mathcal{S}_U^\perp : \boldsymbol{b} \ \mathrm{coprime} \} \quad \text{and} \quad \mathcal{B}_0(\mathcal{S}^\perp_U) := \bigg\{ \boldsymbol{b} \in \mathcal{B}(\mathcal{S}^\perp_U) : \prod_i b_i \neq 0 \bigg\}, \end{align} $$

respectively. Note that the orthogonal complement is taken within $\mathbb {Z}^2$ . For the proofs in §4 and the generalization in Appendix A, it is important to observe that for every $\boldsymbol {k} \in \mathcal {S}_{U,0}$ exists some $\boldsymbol {b} \in \mathcal {B}_0(\mathcal {S}_U^\perp )$ such that $\boldsymbol {b} \cdot \boldsymbol {k} = 0$ .

Our main results will be formulated under the following assumptions.

Assumptions on the preserved integrability of equation (2.3)

Let $H_0 \in C^2(T^*{\mathbb {T}}^2)$ denote the Hamiltonian function from equation (2.1) satisfying one of the Assumptions (A1)–(A3), and U a perturbing potential as in equation (2.3) such that the following statement concerning the perturbed Hamilton–Jacobi equation (HJE):

(2.9) $$ \begin{align} \alpha_{\varepsilon}({\boldsymbol{c}}) = H_\varepsilon(x, {\boldsymbol{c}} + \nabla_x u_{\varepsilon, {\boldsymbol{c}}}(x) ) \end{align} $$

as well as the preserved integrability of $H_\varepsilon $ is satisfied.

1. (P) There exists an energy $e>0$ such that for every $(n,m) \in \mathcal {B}_0(\mathcal {S}_U^\perp )$ (recall equation (2.6)) and $\mu _i \in [0,\tilde {\mu }_i]$ , $i \in \{1,2\}$ , there exists a sequence $(\varepsilon _k)_{k \in \mathbb {N}}$ with $\varepsilon _k \neq 0$ but $\varepsilon _k \to 0$ such that we have the following.

   1. (i) The resonant torus from Proposition 2.1, characterized by $\boldsymbol {c} \in H^1({\mathbb {T}}^2, \mathbb {R})$ with

      (2.10) $$ \begin{align} |c_i|> \sqrt{\mu_i}\, \mathfrak{c}(V_i) \end{align} $$
      in the isoenergy submanifold $T_e$ having rotation vector proportional to $(n,m)$ , is preserved under the sequence of deformations $(H_{\varepsilon _k})_{k \in \mathbb {N}}$ .

   2. (ii) For ${\boldsymbol {c}} \in H^1({\mathbb {T}}^2, \mathbb {R})$ satisfying equation (2.10), Mather’s $\alpha $ -function and a solution $u_{\varepsilon ,{\boldsymbol {c}}}:{\mathbb {T}}^2 \to \mathbb {R}$ of the HJE in equation (2.9) can be expanded to first order in $\varepsilon $ , that is,

      (2.11) $$ \begin{align} u_{\varepsilon, {\boldsymbol{c}}} = u_{{\boldsymbol{c}}}^{(0)} + \varepsilon u_{ {\boldsymbol{c}}}^{(1)} + \mathcal{O}_{\boldsymbol{c}}(\varepsilon^2)\quad \text{and} \quad \alpha_{\varepsilon} = \alpha^{(0)} + \varepsilon \alpha^{(1)} + \mathcal{O}(\varepsilon^2), \end{align} $$
      where $u_{{\boldsymbol {c}}}^{(0)} , u_{ {\boldsymbol {c}}}^{(1)} \in C^{1,1}({\mathbb {T}}^2)$ and $O_{\boldsymbol {c}}(\varepsilon ^2)$ is understood in $C^{1,1}$ -sense. (Having $C^1$ -regularity here would be sufficient for our proofs in §4. However, we chose $C^{1,1}$ -regularity for the formulation of Assumption (P) to be in agreement with the statement from Proposition 2.1(b). More precisely, $C^{1,1}$ -regularity is kind of a compromise between the true $C^3$ -regularity of $u_{\boldsymbol {c}}$ and the required $C^1$ -regularity of $u_{\varepsilon , {\boldsymbol {c}}}$ . In addition, $C^{1,1}$ is the optimal regularity for subsolutions of equation (2.9), which exist, even if the Hamiltonian $H_\varepsilon $ is not integrable (see [Reference Bernard12, Reference Fathi and Siconolfi46]).)

We comment on the validity of assuming Assumption (P) in Remark D.1 in Appendix D. Moreover, we shall also discuss an alternative to equation (2.11) in Remark D.3. Finally, one can easily see from the proofs given in §4 that the condition on a fixed isoenergy manifold $\{ H_\varepsilon = e\}$ can be relaxed to having preservation of invariant tori in isoenergy manifolds characterized by energies $e \ge e_0$ for some fixed $e_0> 0$ .

Note that the rational invariant tori are the most ‘fragile’ objects of an integrable system as the KAM theorem [Reference Arnold5, Reference Kolmogorov63, Reference Moser80] predicts that general (non-integrable) perturbations preserve only ‘sufficiently irrational’ (Diophantine) invariant tori.

3.1. Topological obstructions

The following theorem due to Kozlov [Reference Kozlov68, Reference Kozlov69] (see [Reference Bialy15] for a strengthened version of this result) categorizes two-dimensional compact manifolds regarding the possibility to endow them with an integrable metric (see Question (Q1)).

A result similar to Theorem 3.1 holds for polynomially integrable geodesic flows.

Recall that any two-dimensional compact manifold M can be represented either as the sphere with handles or the sphere with Möbius strips, in the orientable and non-orientable case, respectively. The Euler characteristic $\chi _M$ can be computed as

$$ \begin{align*} \chi_M = 2-2g \quad \text{respectively}\ \chi_M = 2-m, \end{align*} $$

where g is the number of handles (the genus) and m is the number of Möbius strips. To have integrability, the above theorem imposes the condition $\chi _M\ge 0$ on M and we thus know that the number of handles is at most $1$ and the number of Möbius strips is not greater than $2$ . Therefore, any real-analytic two-dimensional compact Riemannian manifold $(M,g)$ with real-analytic (or polynomial) additional integral is either the sphere $\mathbb {S}^2$ or the torus ${\mathbb {T}}^2$ (in the orientable case), or the projective plane $\mathbb {R}\mathbb {P}^2$ or the Klein bottle $\mathbb {K}^2$ (in the non-orientable case). In [Reference Bolsinov and Taimanov29], Bolsinov and Taimanov give a striking example of a real-analytic Riemannian manifold of dimension three, whose geodesic flow has the peculiar property, and that it is smoothly (but not analytically) integrable although it has positive topological entropy [Reference Adler, Konheim and McAndrew1]. The problem of proving (non-)existence of smoothly (but not analytically) integrable geodesic flows on compact surfaces of genus $g> 1$ is widely open (see [Reference Burns and Matveev31]).

In this work, we focus on integrable metrics on the torus ${\mathbb {T}}^2$ and refer to works by Bolsinov, Fomenko, Matveev, Kolokoltsov, and others [Reference Bolsinov, Fomenko and Matveev27, Reference Fomenko and Matveev48, Reference Kolokoltsov64, Reference Matveev78, Reference Nguyen, Polyakova and Selivanova81] for studies on integrable metrics on the sphere, the projective plane, and the Klein bottle. See [Reference Bolsinov, Matveev, Miranda and Tabachnikov24, Reference Burns and Matveev31] for recent surveys on open problems, and questions concerning geodesics and integrability of finite-dimensional systems in general.

3.2. Linearly and quadratically integrable metrics

The first non-trivial class of integrable metrics on the torus ${\mathbb {T}}^2$ is surfaces of revolution. Consider a two-dimensional surface $M \subset \mathbb {R}^3$ given by the equation $r = r(z)$ in standard cylindrical coordinates $(r, \varphi , z) \in (0,\infty ) \times [0,2\pi ) \times \mathbb {R}$ . As local coordinates on M, we take z and $\varphi $ . In the case where $r(z)$ is L-periodic and we identify $0$ and L, then M is diffeomorphic to the torus ${\mathbb {T}}^2$ and the Riemannian metric induced on M by the Euclidean metric on $\mathbb {R}^3$ has line element

(3.1) $$ \begin{align} d s^2 = (1+r'(z)^2) d z^2 + r(z)^2 d \varphi^2. \end{align} $$

Since the corresponding Hamiltonian function in equation (1.5) is independent of $\varphi $ , its associated momentum variable $p_\varphi $ is an additional first integral and thus the metric in equation (3.1) is integrable. Note that the additional first integral is linear in the momentum variables.

As discussed earlier, a Riemannian metric g on ${\mathbb {T}}^2$ is called a Liouville metric, whenever its line element can be written in the form in equation (1.1) in appropriate global coordinates $(x^1, x^2)$ , and where $f_1$ and $f_2$ are smooth positive periodic functions. The corresponding Hamiltonian function in equation (1.5) is given by

$$ \begin{align*} H(x^1,x^2,p_1,p_2) = \frac{p_1^2 + p_1^2}{2 (f_1(x^1)+f_2(x^2))} \end{align*} $$

and an additional first integral can easily be obtained as

$$ \begin{align*} F(x^1,x^2,p_1,p_2) = p_1^2 - f_1(x^1) H(x^1,x^2,p_1,p_2). \end{align*} $$

Therefore, clearly, also Liouville metrics are integrable. Note that the additional first integral F is quadratic in the momentum variables. It is not hard to see that a surface of revolution is just a particular case of a Liouville metric, where one can choose, e.g., $f_2 \equiv 0$ , by employing a simple change of variables.

The following proposition also provides the converse to the observation that surfaces of revolution and Liouville metrics admit additional first integrals which are linear and quadratic in the momenta, respectively. It collects several statements that have been proven in early works by Dini [Reference Dini43], Darboux [Reference Darboux37], and Birkhoff [Reference Birkhoff22], and were further developed by Babenko and Nekhoroshev [Reference Babenko and Nekhoroshev11], Kiyohara [Reference Kiyohara62], Kolokoltsov [Reference Kolokoltsov64], and others.

This classical result completely characterizes the integrable metrics g on ${\mathbb {T}}^2$ that admit an additional first integral that is linear or quadratic in the momentum variables. Similar results hold for Riemannian metrics on general two-dimensional manifolds [Reference Bolsinov, Fomenko and Matveev27, Reference Fomenko and Matveev48, Reference Kolokoltsov64, Reference Nguyen, Polyakova and Selivanova81].

3.3. Polynomially integrable metrics of higher degree

In the case of a sphere $\mathbb {S}^2$ , one can easily construct examples of metrics which admit an additional first integral that is cubic respectively quartic in the momentum variables. Using the Maupertuis principle, these can be obtained from the metrics constructed from Goryachev and Chaplygin [Reference Chaplygin33, Reference Goryachev54], and Kovaleskaya [Reference Kovalevskaya67] in the situation of the dynamics of a rigid body. Therefore, let $h>1$ be large enough (cf. equation (1.8)) and define the metrics $g_3$ and $g_4$ on $\mathbb {R}^3$ via their respective line elements

$$ \begin{align*} d s^2_3 = \frac{h-x^1}{4} \, \frac{(d x^1)^2 + (d x^2)^2 + 4 (d x^3)^2}{(x^1)^2 + (x^2)^2 + (x^3)^2/4}, \quad d s^2_4 = \frac{h-x^1}{2} \, \frac{(d x^1)^2 + (d x^2)^2 + 2 (d x^3)^2}{(x^1)^2 + (x^2)^2 + (x^3)^2/2}. \end{align*} $$

By restriction of $g_3$ and $g_4$ to the unit sphere $\mathbb {S}^2 \subset \mathbb {R}^3$ , the resulting metrics admit an additional first integral that is cubic respectively quartic in the momentum variables. It was shown by Bolsinov, Fomenko, and Kozlov [Reference Bolsinov and Fomenko25, Reference Bolsinov, Kozlov and Fomenko28] that these cannot be reduced to first integrals that are polynomially in the momentum variables of a lower degree, that is, they are not linearly or quadratically integrable. Since all attempts to construct such examples for the case of the torus have failed so far, the following folklore conjecture emerged.

In this general form, there is strong indication for the conjecture being false, as to be shown below (see Theorem 3.8). We will, however, provide existing results, which indicate that a certain weaker version of this conjecture, also formulated below, is indeed true.

It was proven by Korn and Lichtenstein [Reference Korn65, Reference Lichtenstein71] that on every point on a two-dimensional Riemannian manifold $(M,g)$ , there exist locally isothermal coordinates, that is, locally, the line element takes the form

(3.2) $$ \begin{align} d s^2 = \unicode{x3bb}(x^1,x^2) ((d x^1)^2 + (d x^2)^2), \end{align} $$

where $\unicode{x3bb} $ is some smooth positive function. In the case of a torus, it can be shown (by virtue of the uniformization theorem) that there exist global isothermal coordinates (not necessarily periodic), so the metric g is conformal equivalent to the Euclidean metric $g_{\mathrm {eucl}}$ . In particular, assuming that $(x^1,x^2)$ are just the angular coordinates on the torus ${\mathbb {T}}^2$ and in the special case of $\unicode{x3bb} $ being a trigonometric polynomial (this means that the spectrum $\mathcal {S}_\unicode{x3bb} $ defined in equation (2.4) is bounded), we have the following result due to Denisova and Kozlov.

Note that by Weierstrass’s theorem, any conformal factor $\unicode{x3bb} $ can be approximated as closely as required by a trigonometric polynomial. However, in the case of a general conformal factor $\unicode{x3bb} $ , there is the following theorem, again due to Denisova and Kozlov [Reference Denisova and Kozlov40].

In the following theorem, we collect several results from Bialy [Reference Bialy13], Denisova, Kozlov [Reference Denisova and Kozlov41] and Treshchev [Reference Denisova, Kozlov and Treshev42], Agapov and Aleksandrov [Reference Agapov and Aleksandrov2], and Mironov [Reference Mironov79].

Kozlov and Treshchev [Reference Kozlov and Treshev70] considered the problem from yet another point of view. They investigated the case of a mechanical Hamiltonian

$$ \begin{align*} H = \frac{1}{2} \sum_{ij} a_{ij} p_i p_j + V(x^1, \ldots , x^n), \end{align*} $$

where $A = (a_{ij})_{ij}$ is a positive definite matrix and V is a trigonometric polynomial of $(x^1, \ldots , x^n) \in {\mathbb {T}}^n$ . On the one hand, they show that there exist n polynomial first integrals if and only if the spectrum $\mathcal {S}_V$ of V is contained in $m \le n$ mutually orthogonal lines meeting at the origin. On the other hand, they showed that whenever there exist n polynomial integrals with independent forms of highest degree, then there exist n independent involutive polynomial first integrals of degree at most two. In the case where $a_{ij} = \delta _{i,j}$ (which can be achieved by diagonalization and scaling), Combot [Reference Combot34] improved the first result from the assumption of polynomial integrability to rational integrability, that is, the additional first integrals being rational functions of $p_i$ and $e^{i 2 \pi x^i}$ . More recently [Reference Heil, Moroianu and Semmelmann57, Reference Sharafutdinov83, Reference Sharafutdinov84], the problem was rephrased in the language of Killing tensor fields on ${\mathbb {T}}^2$ , where the order of an additional (polynomial) first integral is replaced by the rank of a Killing tensor field.

The results of Theorems 3.4, 3.5, and 3.6 support the validity of the following weaker version of the folklore conjecture formulated by Denisova and Kozlov [Reference Denisova and Kozlov39].

By Proposition 3.3, this means that polynomially integrable metrics on ${\mathbb {T}}^2$ are Liouville metrics. However, beside the partial results given above, a proof of this conjecture is still open. The numerous attempts on proving it used methods of complex analysis [Reference Babenko and Nekhoroshev11, Reference Birkhoff22] and the theory of partial differential equations (PDEs) [Reference Bialy and Mironov17, Reference Bialy and Mironov19]. More precisely, it is shown by Kolokoltsov [Reference Kolokoltsov64] that there exists an additional first integral quadratic in the momenta if and only if there exists a holomorphic function $R(z) = R_1(z) +i R_2(z)$ , with real valued $R_1$ and $R_2$ , and $z = x^1+i x^2$ , which solves

(3.3) $$ \begin{align} R_2 (\partial_{x^2}^2 \unicode{x3bb} - \partial_{x^1}^2 \unicode{x3bb}) + R_1 (\partial_{x^1} \partial_{x^2} \unicode{x3bb}) - 3 (\partial_{x^1} R_2)( \partial_{x^1} \unicode{x3bb}) +3 (\partial_{x^2} R_2) (\partial_{x^2} \unicode{x3bb}) + 2 (\partial_{x^2}^2 R_2) \unicode{x3bb} = 0, \end{align} $$

where $\unicode{x3bb} $ denotes the conformal factor from equation (3.2). Note that the second term in equation (3.3) disappears whenever $\unicode{x3bb} $ is the conformal factor of a Liouville metric. In this situation, the linear PDE in equation (3.3) always has a holomorphic solution $R = R_1+i R_2$ . The existence of first integrals of higher degree turns out to be equivalent to delicate questions about nonlinear PDEs of hydrodynamic type [Reference Bialy and Mironov17–Reference Bialy and Mironov19]. The PDE approach has also successfully been applied to generate new examples of integrable magnetic geodesic flows as analytic deformations of Liouville metrics on ${\mathbb {T}}^2$ without magnetic field (see [Reference Agapov, Bialy and Mironov3]). In fact, the examples from [Reference Agapov, Bialy and Mironov3] disprove the folklore conjecture when understood in the larger class of magnetic geodesic flows.

However, even for the original folklore conjecture stated above, there is a result due to Corsi and Kaloshin [Reference Corsi and Kaloshin35], which indicates it being false in the following (considerably weaker) sense.

Theorem 3.8. (Corsi and Kaloshin [Reference Corsi and Kaloshin35])

There exists a real-analytic mechanical Hamiltonian

$$ \begin{align*} H_\varepsilon(x^1,x^2,p_1,p_2) = \frac{p_1^2 + p_2^2}{2} + U(x^1,x^2;\varepsilon) \end{align*} $$

with a non-separable potential U and an analytic change of variables $\Phi $ such that $H_\varepsilon \circ \Phi = (p_1^2 + p_2^2)/2$ on the energy surface $\{ H_\varepsilon = 1/2 \}$ and $p \in \mathcal {P}$ , where $\mathcal {P}$ denotes a certain cone in the action space. (The function U is called non-separable whenever it cannot be written as a sum of two single-valued functions.)

If one assumes that the whole phase space $T^*{\mathbb {T}}^2$ is foliated by two-dimensional invariant Liouville tori (which is often called $C^0$ -integrability or complete integrability), then it follows from Hopf conjecture [Reference Burago and Ivanov30, Reference Hopf58] that the associated metric must be flat. This notion of integrability is thus too strong for a meaningful characterization of integrable metrics on ${\mathbb {T}}^2$ . (Similar results have been shown for geodesic flows of more general Finsler metrics on ${\mathbb {T}}^2$ preserving a sufficiently regular foliation of the phase space [Reference Gomes, Dias Carneiro and Ruggiero52, Reference Gomes and Ruggiero53].)

3.4. Analogy to integrable billiards

The fundamental Question (Q2) of characterizing integrable metrics on the torus ${\mathbb {T}}^2$ can be thought of as an analogue of identifying the class of integrable billiards [Reference Kaloshin and Sorrentino61]. For billiards, integrability is understood in a similar way as for the geodesic flow (see Definition 1.2). More precisely, integrability is characterized either through the existence of an integral of motion (near the boundary of the billiard table) for the so-called billiard ball map, or the existence of a foliation of the phase space (globally or near the boundary), consisting of invariant curves. The classical Birkhoff conjecture [Reference Birkhoff23, Reference Poritsky82] states that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. This corresponds to the folklore conjecture formulated above. Remarkably, while the Birkhoff conjecture is believed to be true, and there is strong evidence that this indeed the case [Reference Avila, De Simoi and Kaloshin10, Reference Bialy and Mironov20, Reference Glutsyuk49, Reference Kaloshin and Sorrentino60] (on the opposite side, Treshev constructed a non-elliptic billiard table which is formally integrable close to a two-periodic orbit [Reference Treshev88–Reference Treshev90]. This formal power series has recently been shown to be of Gervey class of order $\sigma> 9/4$ [Reference Wang and Zhang93]), the folklore conjecture in its general form was shown to be false by Theorem 3.8.

However, recall that if one assumes $C^0$ -integrability of a metric on ${\mathbb {T}}^2$ , the metric is actually flat [Reference Burago and Ivanov30, Reference Hopf58]. This corresponds to the following result from Bialy in the case of billiards.

Following a similar strategy leading to Theorem 3.9, Bialy and Mironov [Reference Bialy and Mironov21] proved the Birkhoff conjecture for centrally symmetric billiards, assuming only local $C^0$ -integrability, that is, the foliation of a suitable open proper subset of the phase space. In addition to this, the weakened version of the folklore conjecture (polynomial integrals can be reduced to integrals of degree at most two) corresponds to the so-called algebraic Birkhoff conjecture, which has recently been proven [Reference Bialy and Mironov20, Reference Glutsyuk49].

The main results of this paper in Theorems 2.2, 2.3, and 2.4 prove special cases of our conjecture that integrable deformations of Liouville metrics which preserve all (but finitely many) rational invariant tori are again Liouville metrics. This is related to the following conjecture in the case of billiards.

A first result in this direction was obtained by Delshams and Ramírez-Ros [Reference Delshams and Ramírez-Ros38]. More recently, Avila, De Simoi, and Kaloshin [Reference Avila, De Simoi and Kaloshin10] proved the conjecture for domains which are sufficiently close to a circle. The complete proof for domains sufficiently close to an ellipse of any eccentricity is given by Kaloshin and Sorrentino in [Reference Kaloshin and Sorrentino60]. Both works require the preservation of rational caustics (a curve $\Gamma $ is a caustic for the billiard in the domain $\Omega $ if every time a trajectory is tangent to it, then it remains tangent after every reflection according to the billiard ball map), which can be thought of as an analogue for the preservation of rational invariant tori as a fundamental assumption of our main results from §2. The result in [Reference Avila, De Simoi and Kaloshin10] was later extended by Huang, Kaloshin, and Sorrentino [Reference Huang, Kaloshin and Sorrentino59] to the case of local integrability close to the boundary and finally significantly improved by Koval [Reference Koval66].

Finally, as shown by Vedyushkina and Fomenko [Reference Vedyushkina (Fokicheva) and Fomenko92], linearly and quadratically integrable geodesic flows on orientable two-dimensional Riemannian manifolds are Liouville equivalent to topological billiards, glued from planar billiards bounded by concentric circles and arcs of confocal quadrics, respectively.

4.1. Proof of Theorem 2.2

The argument is divided into three steps.

Step (i). Fix an energy $e> 0$ . Since the Hamiltonian is already in action-angle coordinates, we simply change notation and write $(x^i,p_i) = (\theta ^i,I_i)$ for $i = 1,2$ as well as $\theta = (\theta ^1, \theta ^2)$ and $I = (I_1,I_2)$ , such that the perturbed Hamiltonian function $H_\varepsilon $ takes the form

$$ \begin{align*} H_\varepsilon(\theta, I) = \frac{I_1^2}{2} + \frac{I_2^2}{2} + \varepsilon U(\theta). \end{align*} $$

Step (ii). By Assumption (P), for any $(n,m) \in \mathcal {B}_0(\mathcal {S}_U^\perp )$ (recall equation (2.6)), we can find (in the isoenergy manifold $T_{e_\varepsilon }$ with energy $e = e_\varepsilon $ and $\varepsilon = \varepsilon _k $ for some $k \in \mathbb {N}$ ) a rational invariant invariant Liouville torus with rotation vector ${\boldsymbol {\omega }} = (\omega _1, \omega _2)$ which satisfies

(4.1) $$ \begin{align} \frac{\omega_1}{\omega_2} = \frac{n}{m} \in \mathbb{Q}. \end{align} $$

Moreover, we fix ${\boldsymbol {c}} \in H^1({\mathbb {T}}^2, \mathbb {R}) \cong \mathbb {R}^2$ to be given by ${\boldsymbol {c}} = (\omega _1, \omega _2)$ . We make this choice to cancel the average over a trajectory in equation (4.3) of the first term on the right-hand side of equation (4.2) (cf. also equations (4.6)–(4.8) below).

Using Assumption (P) again, we can expand the Hamilton–Jacobi equation in equation (2.9) as

$$ \begin{align*} \alpha_\varepsilon({\boldsymbol{c}}) &= H_\varepsilon(\theta, {\boldsymbol{c}} + \nabla u_{\varepsilon, {\boldsymbol{c}}}(\theta)) \\ &= \frac{\vert \partial_{\theta^1} u_{\varepsilon, {\boldsymbol{c}}}(\theta) + c_1\vert^2}{2} + \frac{\vert \partial_{\theta^2} u_{\varepsilon, {\boldsymbol{c}}}(\theta) + c_2\vert^2}{2} + \varepsilon U(\theta) \\ &= \frac{c_1^2}{2} + \frac{c_2^2}{2} + \langle {\boldsymbol{c}}, \nabla u_{\varepsilon, {\boldsymbol{c}}}(\theta) \rangle + \varepsilon U(\theta) + \frac{(\partial_{\theta^1} u_{\varepsilon, {\boldsymbol{c}}}(\theta) )^2}{2} + \frac{(\partial_{\theta^2} u_{\varepsilon, {\boldsymbol{c}}}(\theta))^2}{2}, \end{align*} $$

and it holds that

$$ \begin{align*} u_{\varepsilon, {\boldsymbol{c}}} = u_{{\boldsymbol{c}}}^{(0)} + \varepsilon u_{{\boldsymbol{c}}}^{(1)} + \mathcal{O}_{\boldsymbol{c}}(\varepsilon^2) \end{align*} $$

with $u_{{\boldsymbol {c}}}^{(0)} = u_{0,{\boldsymbol {c}}}$ . Since $H_0(\theta ,I)$ is integrable (and written in action-angle coordinates), one can choose $u_{0,{\boldsymbol {c}}} \equiv 0$ . By equation (D.6) in Proposition D.2 (see also [Reference Gomes50]), we have $\alpha ^{(1)}({\boldsymbol {c}}) = [U]_0$ , where

$$ \begin{align*} [U]_0 = \int_{{\mathbb{T}}^2} U(x^1, x^2)\,d x^1 \wedge dx^2. \end{align*} $$

Since the sequence $(\varepsilon _k)_{k \in \mathbb {N}}$ from Assumption (P) converges to zero, we compare coefficients and establish the first-order equation

(4.2)

Averaging equation (4.2) over the trajectory $\theta (t) = \theta _0 + {\boldsymbol {\omega }} t \in {\mathbb {T}}^2$ , with initial position $\theta _0 \in {\mathbb {T}}^2$ and where ${\boldsymbol {\omega }} = {\boldsymbol {c}}$ is chosen according to equation (4.1), such that the period $T_{{\boldsymbol {\omega }}}$ satisfies $T_{\boldsymbol {\omega }} \cdot {\boldsymbol {\omega }} = (n,m)$ , we get

(4.3) $$ \begin{align} [U]_0 = \frac{1}{T_{\boldsymbol{\omega}}} \int_{0}^{T_{\boldsymbol{\omega}}}\, \frac{d}{dt}u^{(1)}_{\varepsilon, {\boldsymbol{c}}}(\theta(t))\,dt + \frac{1}{T_{\boldsymbol{\omega}}} \int_{0}^{T_{\boldsymbol{\omega}}} U(\theta(t))\,dt. \end{align} $$

The first integral vanishes since $\theta (0) =\theta (T_{\boldsymbol {\omega }}) $ such that we are left with

(4.4) $$ \begin{align} \int_{0}^{1} (U(\theta_0^1 + n t,\theta_0^2 + m t) - [U]_0)\,dt = 0 \end{align} $$

for all $\theta _0 = (\theta _0^1, \theta _0^2) \in {\mathbb {T}}^2$ , which easily follows from equation (4.3) after a change of variables.

Before continuing with the third and final step, we have two important observations. First, by replacing $U \to U - [U]_0$ , we can assume without loss of generality that $[U]_0 = 0$ . Second, we define the separable part, $U_{\mathrm {sep}}$ , of U as

(4.5)

(recall the definition of the spectrum and the non-singular spectrum in equations (2.4) and (2.5)). Then, after a simple computation, we find that

$$ \begin{align*} \int_{0}^{1} U_{\mathrm{sep}}(\theta_0^1 + nt, \theta_0^2 + m t)\,dt = [U_{\mathrm{sep}}]_0 \quad \text{for all } (\theta_0^1, \theta_0^2) \in {\mathbb{T}}^2 \end{align*} $$

holds generally (that is, independent of the first-order relation in equation (4.2)) by means of equation (D.6) in Proposition D.2 (see also Remark D.1). We can thus split off the separable part and assume that $\mathcal {S}_U = \mathcal {S}_{U,0}$ in the following. Hence, the third step consists of showing that ${\mathcal {S}_U = \mathcal {S}_{U,0} = \emptyset }$ .

Step (iii). The goal of this final step is to establish the following lemma.

Since $(n,m) \in \mathcal {B}_0(\mathcal {S}_U^\perp )$ were arbitrary, this proves that

$$ \begin{align*} \mathcal{S}_U \subset(\{ 0 \} \times \mathbb{Z}) \cup (\mathbb{Z} \times \{ 0 \}), \end{align*} $$

or equivalently $\mathcal {S}_{U,0} = \emptyset $ and we have shown Theorem 2.2. It remains to prove Lemma 4.1.

Proof of Lemma 4.1

Starting from equation (4.4), we perform a Fourier decomposition to infer

$$ \begin{align*} \sum_{ k_1, k_2 \neq 0} \bigg[U_{k_1, k_2} \int_{0}^{1} e^{i 2 \pi k_1 n t} e^{i 2 \pi k_2 m t}\,d t\bigg] {e}^{{i} 2 \pi k_1 \theta_0^1} {e}^{{i} 2 \pi k_1 \theta_0^2} = 0 \quad \text{for all } (\theta_0^1, \theta_0^2) \in {\mathbb{T}}^2, \end{align*} $$

which implies that

$$ \begin{align*} U_{k_1, k_2} \cdot \delta_{k_1n+ k_2m, \, 0} = 0.\\[-34pt] \end{align*} $$

Applying Lemma 4.1 for every $(n,m) \in \mathcal {B}_0(\mathcal {S}_U^\perp )$ , we find that $\mathcal {S}_{U,0} = \emptyset $ , which finishes the proof of Theorem 2.2.

4.2. Proof of Theorem 2.3

For notational simplicity, we write $\mu \equiv \mu _2> 0$ and $V \equiv V_2 \in C^2({\mathbb {T}})$ .

Step (i). We fix an energy $e>0$ and consider the region of the phase space, where the subsystem in the second pair of coordinates is rotating, that is,

$$ \begin{align*} \frac{p_2^2}{2} - \mu V(x^2) = e^{(2)}>0 \end{align*} $$

and for ${p_1^2}/{2} = e^{(1)}> 0$ , we have $e = e^{(1)} + e^{(2)}$ . In a neighborhood of each of the two Liouville tori characterized by $H_0 = e$ and ${p_1^2}/{2} = e^{(1)}$ , we can find a change of variables $(x^2,p_2) = \Phi ^{(2)}_\mu (\theta ^2,I_2)$ (and we denote $(x^1, p_1) = (\theta ^1, I_1)$ ) such that the Hamiltonian function $H_0$ gets transformed in action-angle coordinates, that is,

$$ \begin{align*} H_0(\theta^1, I_1, \Phi^{(2)}_\mu(\theta^2,I_2)) = \frac{I_1^2}{2} + h^{(2)}_\mu(I_2) \end{align*} $$

for some smooth function $h^{(2)}_\mu $ agreeing with Mather’s $\alpha $ -function for the one-dimensional subsystem described by the Hamiltonian ${p_2^2}/{2} - \mu V(x^2)$ (see Appendix D). The change in the order of the four arguments of $H_0$ should not lead to confusion. Now, the perturbed Hamiltonian takes the form

$$ \begin{align*} H_\varepsilon(\theta^1, I_1, \Phi^{(2)}_\mu(\theta^2,I_2)) = \frac{I_1^2}{2} + h^{(2)}_\mu(I_2) + \varepsilon U(\theta^1, x^2(\theta^2,I_2,\mu)), \end{align*} $$

where we write $x^2_\mu (\theta ^2,I_2)$ for the first component of $\Phi ^{(2)}_\mu (\theta ^2,I_2)$ .

Step (ii). Assume without loss of generality that the $2$ -degree ${\deg }^{(2)}_U$ of U is at least $1$ (recall equation (2.7)), as otherwise, we had $U(x) = U_1(x^1)$ and Theorem 2.3 was proven. Then, for any $(n,m) \in \mathcal {B}_0(\mathcal {S}_U^\perp )$ , in particular with $| n| \le {\deg }_U^{(2)}$ , we can find (in the isoenergy manifold $T_{e_\varepsilon }$ with energy $ e= e_\varepsilon $ and $\varepsilon = \varepsilon _k$ for some $k \in \mathbb {N}$ ) a rational invariant Liouville torus with rotation vector $\boldsymbol {\omega } = (\omega _1, \omega _2)$ , which satisfies

(4.6) $$ \begin{align} \frac{\omega_1}{\omega_2} = \frac{n}{m} \in \mathbb{Q} \quad \text{and} \quad \boldsymbol{\omega} = (c_1, \nabla h^{(2)}_\mu(c_2)) \end{align} $$

for some $\boldsymbol {c} \in H^1({\mathbb {T}}^2, \mathbb {R}) \cong \mathbb {R}^2$ with $c_1 = \omega _1$ (as around equation (4.1)) and $\vert c_2 \vert> \gamma + \sqrt {\mu } \mathfrak {c}(V)$ for some $\gamma = \gamma (e, {\deg }^{(2)}_U)> 0$ , which we fix now. This new parameter $\gamma $ quantifies a safe distance (depending on the total energy $e>0$ and the degree of the trigonometric polynomial) to the region, opposite to where (i) the change of variables $\Phi ^{(2)}_\mu $ has bounded derivative (cf. equation (4.7)) and (ii) the function $h_\mu ^{(2)}$ is bounded from below (cf. equation (4.12)). In §4.3, we will have two such parameters, $\gamma _1, \gamma _2$ , for both coordinates directions which get transformed by some $\Phi $ .

By Assumption (P), we have

$$ \begin{align*} u_{\varepsilon, {\boldsymbol{c}}} = u_{{\boldsymbol{c}}}^{(0)} + \varepsilon u_{{\boldsymbol{c}}}^{(1)} + \mathcal{O}_{\boldsymbol{c}}(\varepsilon^2) \end{align*} $$

with $u_{{\boldsymbol {c}}}^{(0)} = u_{0,{\boldsymbol {c}}}$ and since $H_0(\theta ,I)$ is integrable (and written in action-angle coordinates), one can choose $u_{0,{\boldsymbol {c}}} \equiv 0$ . Therefore, by Assumption (P) again, we expand the Hamilton Jacobi equation in equation (2.9) as

(4.7) $$ \begin{align} \alpha_{\varepsilon}({\boldsymbol{c}}) &= H_\varepsilon(\theta, {\boldsymbol{c}} + \nabla u_{\varepsilon, {\boldsymbol{c}}}(\theta))\nonumber \\ & \nonumber= \frac{\vert \partial_{\theta^1} u_{\varepsilon, {\boldsymbol{c}}}(\theta) + c_1\vert^2}{2} + h_\mu^{(2)}(\partial_{\theta^2} u_{\varepsilon, {\boldsymbol{c}}}(\theta) + c_2)+ \varepsilon U(\theta^1,x^2_\mu(\theta^2, \partial_{\theta^2} u_{\varepsilon, {\boldsymbol{c}}}(\theta) + c_2)) \\ & \nonumber= \frac{c_1^2}{2} + h_\mu^{(2)}( c_2) + \varepsilon\langle (c_1, \nabla h_\mu^{(2)}( c_2)), \nabla u_{\varepsilon, {\boldsymbol{c}}}^{(1)}(\theta) \rangle + \varepsilon U(\theta^1, x^2_\mu(\theta^2, c_2)) \\ & \quad + \mathcal{O}( \Vert (\nabla^2 h_\mu^{(2)})\vert_{\{ \vert c_2 \vert> \gamma + \sqrt{\mu} \mathfrak{c}(V) \}} \Vert_{C^0} \varepsilon^2) + \mathcal{O} ( \Vert (\partial_{I_2} \Phi_\mu^{(2)})\vert_{\{ \vert c_2 \vert > \gamma + \sqrt{\mu} \mathfrak{c}(V) \}} \Vert_{C^0} \varepsilon^2). \end{align} $$

Since $|c_2|> \gamma + \sqrt {\mu } \mathfrak {c}(V)$ , both error terms are of the order $\mathcal {O}_\gamma (\varepsilon ^2)$ .

Analogously to the proof of Theorem 2.2, we thus obtain the first-order equation

(4.8) $$ \begin{align} [U]_0 = \langle (c_1, \nabla h_\mu^{(2)}( c_2)), \nabla u_{\varepsilon, {\boldsymbol{c}}}^{(1)}(\theta) \rangle + U(\theta^1, x^2_\mu(\theta^2, c_2)), \end{align} $$

where the constant $\alpha ^{(1)} \equiv [U]_0$ is given in equation (D.6) in Proposition D.2 (see also [Reference Gomes50]). Just as in the proof of Theorem 2.2, after averaging equation (4.8) over the trajectory $\theta (t) = \theta _0 + {\boldsymbol {\omega }} t \in {\mathbb {T}}^2$ , with initial position $\theta _0 \in {\mathbb {T}}^2$ and where ${\boldsymbol {\omega }}$ is chosen according to equation (4.6), such that the period $T_{{\boldsymbol {\omega }}}$ satisfies $T_{\boldsymbol {\omega }} \cdot {\boldsymbol {\omega }} = (n,m)$ , we find

(4.9) $$ \begin{align} \int_{0}^{1} (U(\theta_0^1 +n t, x^2_\mu(\theta_0^2 + m t,c_2)) - [U]_0)\,{d} t = 0 \end{align} $$

for all $\theta _0 = (\theta _0^1, \theta _0^2) \in {\mathbb {T}}^2$ .

Finally, analogously to §4.1, we may assume without loss of generality $[U]_0 = 0$ and observe that

$$ \begin{align*} \int_{0}^{1} U_{\mathrm{sep}}(\theta_0^1 + nt,x^2_\mu(\theta_0^2 + m t,c_2))\,dt = [U_{\mathrm{sep}}]_0 \quad \text{for all } (\theta_0^1, \theta_0^2) \in {\mathbb{T}}^2 \end{align*} $$

holds generally (that is, independent of the first-order relation in equation (4.8)) by a simple calculation based on equation (D.6) in Proposition D.2 (see also Remark D.1). We can thus split off the separable part $U_{\mathrm {sep}}$ of U defined in equation (4.5) and assume that $\mathcal {S}_U = \mathcal {S}_{U,0}$ in the following. Hence, the third step consists of showing that $\mathcal {S}_U = \mathcal {S}_{U,0} = \emptyset $ .

Step (iii). We begin this final step with performing a Fourier decomposition in equation (4.9) such that we obtain

$$ \begin{align*} \sum_{ k_1\neq 0}&\bigg[ \sum_{0 \neq |k_2| \le \deg_U^{(2)}} U_{k_1, k_2} \int_{0}^{1} {e}^{{i} 2 \pi k_1 n t} {e}^{{i} 2 \pi k_2 x_\mu^2(\theta_0^2 + mt, c_2)}\,d t \bigg] {e}^{{i} 2 \pi k_1 \theta_0^1} = 0\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \text{for all } (\theta_0^1, \theta_0^2) \in {\mathbb{T}}^2, \end{align*} $$

which implies that $[\cdots ] = 0$ for every and $\theta _0^2 \in {\mathbb {T}}$ .

After having eliminated $\theta _0^1 \in {\mathbb {T}}$ , we now fix some and consider the family of functions $(f^{(k_1, \mu )}_{k_2})_{0 \neq |k_2| \le \deg _U^{(2)}}$ in the Hilbert space $L^2({\mathbb {T}})$ , where

(4.10) $$ \begin{align} f_{k_2}^{(k_1, \mu)}: {\mathbb{T}} \to \mathbb{C}, \quad \theta_0^2 \mapsto \sum_{\substack{(n,m) \in \mathcal{B}_0(\mathcal{S}^\perp_U) \\ \exists 0 \neq |\tilde{k}_2| \le \deg_U^{(2)} : k_1n+\tilde{k}_2m = 0}}\! \int_{0}^{1} e^{i 2 \pi k_1 n t} e^{i 2 \pi k_2 x_\mu^2(\theta_0^2 + mt, c_2)}\,dt. \end{align} $$

Note that the sum in equation (4.10) is finite by Assumption (A2) (more precisely, it ranges over at most $2 \cdot \deg _U^{(2)}$ elements from $\mathcal {B}_0(\mathcal {S}_U^\perp )$ ) and we suppressed the dependence of $|c_2|> \gamma + \sqrt {\mu } \mathfrak {c}(V)$ on $(n,m) \in \mathcal {B}_0(\mathcal {S}_U^\perp )$ from the notation (recall equation (4.6)).

In this way, the problem of proving Theorem 2.3, that is, justifying $\mathcal {S}_{U,0} = \emptyset $ , reduces to a question about linear independence for the family of functions in equation (4.10) in the Hilbert space $L^2({\mathbb {T}})$ . Recall that the family $(f^{(k_1, \mu )}_{k_2})_{0 \neq |k_2| \le \deg _U^{(2)}}$ being linearly independent is equivalent to the Gram matrix

(4.11) $$ \begin{align} G^{(k_1, \mu)} = (G_{k_2, k^{\prime}_2}^{(k_1, \mu)})_{0 \neq |k_2|, |k^{\prime}_2| \le \deg_U^{(2)}} \quad \text{with } G_{k_2, k^{\prime}_2}^{(k_1, \mu)}:= \langle f_{k_2}^{(k_1, \mu)}, f_{k^{\prime}_2}^{(k_1, \mu)} \rangle_{L^2({\mathbb{T}})} \end{align} $$

being of full rank, where $\langle g,h\rangle _{L^2({\mathbb {T}})}$ denotes the standard inner product of $g,h \in L^2({\mathbb {T}})$ .

Proof. Using the version of Lemma B.1 for the inverse function, we find that

(4.12) $$ \begin{align} \Vert e^{i 2 \pi k_2 x^2_{\mu}( \, \cdot ,c_2)} - e^{i 2 \pi k_2 \, \cdot} \Vert_{C^0} = \mathcal{O}\bigg(\deg^{(2)}_U\frac{\mu \Vert V \Vert_{C^0}}{h_{\mu}(\gamma + \sqrt{\mu} \mathfrak{c}(V))}\bigg) =: \mathcal{O}(\mu) \end{align} $$

uniformly in $|k_2| \le \deg _U^{(2)}$ and $(n,m) \in \mathcal {B}_0(\mathcal {S}_U^\perp )$ .

With a slight abuse of notation for the error term, the elements $G^{(k_1, \mu )}_{k_2, k^{\prime }_2}$ of the Gram matrix can thus be computed as

$$ \begin{align*} &\int_{0}^{1}\,d \theta_0^2 \bigg( \bigg[\sum_{(n,m)} \int_{0}^{1}\,dt \, e^{-i 2 \pi k_1 n t} ( e^{-i 2 \pi k_2 m t} + \mathcal{O}(\mu)) \bigg] e^{-i 2 \pi k_2 \theta_0^2} \bigg. \\ &\quad\bigg. \times \ e^{i 2 \pi k^{\prime}_2 \theta_0^2} \bigg[\sum_{(n',m')} \int_{0}^{1}\,d t'e^{i 2 \pi k_1 n' t'} ( e^{i 2 \pi k^{\prime}_2 m' t'} + \mathcal{O}(\mu) ) \bigg]\bigg), \end{align*} $$

where the summations over $(n,m)$ and $(n',m')$ are understood as in equation (4.10). Using that for every $(k_1, k_2) \in \mathcal {S}_{U,0}$ , there exist exactly two elements from $\mathcal {B}_0(\mathcal {S}_U^\perp )$ (differing by a sign), we can evaluate both brackets $[\cdots ]$ being equal to $2 + \mathcal {O}(\deg _U^{(2)} \mu )$ .

From this, we conclude that

$$ \begin{align*} G^{(k_1, \mu)}_{k_2, k^{\prime}_2}\!=\!\int_{0}^{1}\!\!\!\,d \theta_0^2 [2 + \mathcal{O}(\deg_U^{(2)} \mu)] e^{i 2 \pi (k^{\prime}_2-k_2) \theta_0^2} [2 + \mathcal{O}(\deg_U^{(2)} \mu)] = 4 \delta_{k_2, k^{\prime}_2} + \mathcal{O}(\deg_U^{(2)}\! \mu). \end{align*} $$

Therefore, going back to equation (4.12), we infer the existence of $\tilde {\mu } = \tilde {\mu }(\mathcal {C}_2, \deg _U^{(2)}, e)> 0$ such that for all $\mu \in [0, \tilde {\mu }]$ , the Gram matrix $G^{(k_1, \mu )}$ from equation (4.11) is of full rank.

Since was arbitrary and Lemma 4.2 is independent of $k_1$ , this concludes the proof of Theorem 2.3(a).

For part (b), we note that $e^{i 2 \pi k_2 x_\mu ^2(\theta _0^2 + mt, c_2)} $ from equation (4.10) depends analytically on $\mu $ (see Appendix C). Therefore, the function $ \mu \mapsto G^{(k_1, \mu )}$ mapping to the Gram matrix in equation (4.11), for every fixed , is also analytic. (Using joint continuity of $(u,\mu ) \mapsto e^{i 2 \pi k_2 x_\mu ^2(u, c_2)}$ , it is an elementary exercise to show that the integrals over t and $\theta _0^2$ do not disturb the analyticity in $\mu $ .) This in turn implies that $\det (G^{(k_1, \mu )})$ is analytic in $\mu $ and thus, since $\det (G^{(k_1, \mu )}) \neq 0$ for $\mu \in (0,\tilde {\mu })$ (see Lemma 4.2), we find that the zero set

$$ \begin{align*} \mathcal{E}_0^{(k_1)} := \{ \mu \in (0,\infty) \mid \det(G^{(k_1, \mu)}) = 0 \} \subset (\tilde{\mu}, \infty) \end{align*} $$

of $\mu \mapsto \det (G^{(k_1, \mu )})$ is at most countable (finite in every compact subset), that is, in particular, a set of zero measure. Finally, setting

we constructed the exceptional null set, for which the conclusion $\mathcal {S}_{U,0} = \emptyset $ is not valid.

This finishes the proof of Theorem 2.3(b).

4.3. Proof of Theorem 2.4

As above, the argument is divided into three steps.

Step (i). We fix an energy $e>0$ and consider the region of the phase space, where both one-dimensional subsystems are rotating, that is,

$$ \begin{align*} \frac{p_1^2}{2} - \mu_1 V_1(x^1) = e^{(1)}>0\quad \text{and} \quad \frac{p_2^2}{2} - \mu_2 V_2(x^2) = e^{(2)}>0, \end{align*} $$

such that we have $e = e^{(1)} + e^{(2)}$ . In a neighborhood of each of the two Liouville tori characterized by $H_0 = e$ and $({p_1^2}/{2}) - \mu _1 V_1(x^1)= e^{(1)}$ , we can find two changes of variables $(x^1,p_1) = \Phi ^{(1)}_{\mu _1}(\theta ^1,I_1)$ and $(x^2,p_2) = \Phi ^{(2)}_{\mu _2}(\theta ^2,I_2)$ such that the Hamiltonian function $H_0$ gets transformed in action-angle coordinates, that is,

$$ \begin{align*} H_0(\Phi^{(1)}_{\mu_1}(\theta^1,I_1), \Phi^{(2)}_{\mu_2}(\theta^2,I_2)) = h^{(1)}_{\mu_1}(I_1) + h^{(2)}_{\mu_2}(I_2) \end{align*} $$

for some smooth functions $h^{(1)}_{\mu _1}$ and $h^{(2)}_{\mu _2}$ , which agree with Mather’s $\alpha $ -functions for the one-dimensional subsystem described by the Hamiltonians $({p_1^2}/{2}) - \mu _1 V(x^1)$ , respectively $({p_2^2}/{2}) - \mu _2 V(x^2)$ (see Appendix D). As in the proof of Theorem 2.3, the change in the order of the four arguments of $H_0$ should not lead to confusion.

Now, the perturbed Hamiltonian takes the form

$$ \begin{align*} H_\varepsilon(\Phi^{(1)}_{\mu_1}(\theta^1,I_1), \Phi^{(2)}_{\mu_2}(\theta^2,I_2)) = h^{(1)}_{\mu_1}(I_1)+ h^{(2)}_{\mu_2}(I_2) + \varepsilon U(x^1_{\mu_1}(\theta^1,I_1), x^2_{\mu_2}(\theta^2,I_2)), \end{align*} $$

where we write $x^i_{\mu _i}(\theta ^i,I_i)$ for the first component of $\Phi ^{(i)}_{\mu _i}(\theta ^i,I_i)$ , $i \in \{1,2\}$ .

Step (ii). Analogously to the proof of Theorem 2.3, we assume without loss of generality that the $1$ - and $2$ -degree ${\deg }^{(1)}_U$ and ${\deg }^{(2)}_U$ of U are at least $1$ (recall equation (2.8)), as otherwise, we had $U(x) = U_2(x^2)$ or $U(x) = U_1(x^1)$ and Theorem 2.4 was proven. Then, for any $(n,m) \in \mathcal {B}_0(\mathcal {S}_U^\perp )$ , in particular with $|m| \le {\deg }_U^{(1)}$ and $|n| \le {\deg }_U^{(2)}$ , we can find (in the isoenergy manifold $T_{e_\varepsilon }$ with energy $ e= e_\varepsilon $ and $\varepsilon = \varepsilon _k$ for some $k \in \mathbb {N}$ ) a rational invariant Liouville torus with rotation vector $\boldsymbol {\omega } = (\omega _1, \omega _2)$ which satisfies

(4.13) $$ \begin{align} \frac{\omega_1}{\omega_2} = \frac{n}{m} \in \mathbb{Q} \quad \text{and} \quad \boldsymbol{\omega} = (\nabla h^{(1)}_{\mu_1}(c_1), \nabla h^{(2)}_{\mu_2}(c_2)) \end{align} $$

for some $\boldsymbol {c} \in H^1({\mathbb {T}}^2, \mathbb {R}) \cong \mathbb {R}^2$ with $\vert c_1 \vert> \gamma _1 + \sqrt {\mu _1} \mathfrak {c}(V_1)$ and $\vert c_2 \vert> \gamma _2 + \sqrt {\mu _2} \mathfrak {c}(V_2)$ for some $\gamma _1 = \gamma _1(e, {\deg }^{(1)}_U)> 0$ , respectively $\gamma _2 = \gamma _2(e, {\deg }^{(2)}_U)> 0$ , which we fix now (see the paragraph below equation (4.6) for a discussion of the $\gamma $ parameters).

By Assumption (P), we have

$$ \begin{align*} u_{\varepsilon, {\boldsymbol{c}}} = u_{{\boldsymbol{c}}}^{(0)} + \varepsilon u_{{\boldsymbol{c}}}^{(1)} + O_{\boldsymbol{c}}(\varepsilon^2) \end{align*} $$

with $u_{{\boldsymbol {c}}}^{(0)} = u_{0,{\boldsymbol {c}}}$ and since $H_0(\theta ,I)$ is integrable (and written in action-angle coordinates), one can choose $u_{0,{\boldsymbol {c}}} \equiv 0$ . Therefore, by Assumption (P) again, we expand the Hamilton Jacobi in equation (2.9) as

$$ \begin{align*} \alpha_{\varepsilon}({\boldsymbol{c}}) &= H_\varepsilon(\theta, {\boldsymbol{c}} + \nabla u_{\varepsilon, {\boldsymbol{c}}}(\theta) ) \\ &= h_{\mu_1}^{(1)}(\partial_{\theta^1} u_{\varepsilon, {\boldsymbol{c}}}(\theta) + c_1) + h_{\mu_2}^{(2)}(\partial_{\theta^2} u_{\varepsilon, {\boldsymbol{c}}}(\theta) + c_2) \\[1mm] & \quad + \varepsilon U(x^1_{\mu_1}(\theta^1, \partial_{\theta^1} u_{\varepsilon, {\boldsymbol{c}}}(\theta) + c_1),x^2_{\mu_2}(\theta^2, \partial_{\theta^2} u_{\varepsilon, {\boldsymbol{c}}}(\theta) + c_2)) \\[1mm] & =\! \sum_{i=1}^{2}h_{\mu_i}^{(i)}( c_i)\! +\! \varepsilon \langle (\nabla h_{\mu_1}^{(1)}( c_1), \nabla h_{\mu_2}^{(2)}( c_2)), \nabla u_{\varepsilon, {\boldsymbol{c}}}^{(1)}(\theta) \rangle + \varepsilon U(x^1_{\mu_1}(\theta^1, c_1), x^2_{\mu_2}(\theta^2, c_2)) \\ & \quad + \mathcal{O}\bigg( \sum_{i=1}^{2}(\Vert (\nabla^2 h_{\mu_i}^{(i)})\vert_{\{ \vert c_i \vert> \gamma_i + \sqrt{\mu_i} \mathfrak{c}(V_i) \}} \Vert_{C^0} + \Vert (\partial_{I_i} \Phi_{\mu_i}^{(i)})\vert_{\{ \vert c_i \vert > \gamma_i + \sqrt{\mu_i} \mathfrak{c}(V_i) \}} \Vert_{C^0} ) \varepsilon^2\bigg). \end{align*} $$

Since $|c_i|> \gamma _i + \sqrt {\mu _i} \mathfrak {c}(V_i)$ , the error term is of order $\mathcal {O}_{\gamma _i}(\varepsilon ^2)$ .

Analogously to the proofs of Theorems 2.2 and 2.3, we thus obtain the first-order equation

(4.14) $$ \begin{align} [U]_0 = \langle (\nabla h_{\mu_1}^{(1)}( c_1), \nabla h_{\mu_2}^{(2)}( c_2)), \nabla u_{\varepsilon, {\boldsymbol{c}}}^{(1)}(\theta) \rangle + U(x^1_{\mu_1}(\theta^1, c_1), x^2_{\mu_2}(\theta^2, c_2)), \end{align} $$

where the constant $\alpha ^{(1)} \equiv [U]_0$ is again given by equation (D.6) in Proposition D.2 (see also [Reference Gomes50]). Just as in the proof of Theorems 2.2 and 2.3, after averaging equation (4.14) over the trajectory $\theta (t) = \theta _0 + {\boldsymbol {\omega }} t \in {\mathbb {T}}^2$ , with initial position $\theta _0 \in {\mathbb {T}}^2$ and where ${\boldsymbol {\omega }}$ is chosen according to equation (4.13) such that the period $T_{{\boldsymbol {\omega }}}$ satisfies $T_{\boldsymbol {\omega }} \cdot {\boldsymbol {\omega }} = (n,m)$ ,

(4.15) $$ \begin{align} \int_{0}^{1} (U(x^1_{\mu_1}(\theta_0^1 +n t, c_1), x^2_{\mu_2}(\theta_0^2 + m t,c_2)) - [U]_0)\,{d} t = 0 \end{align} $$

for all $\theta _0 = (\theta _0^1, \theta _0^2) \in {\mathbb {T}}^2$ .

Finally, analogously to §§4.1 and 4.2, we may assume without loss of generality $[U]_0 = 0$ and observe that

$$ \begin{align*} \int_{0}^{1} U_{\mathrm{sep}}(x^1_{\mu_1}(\theta_0^1 +n t, c_1),\ x^2_{\mu_2}(\theta_0^2 + m t,c_2))\,dt = [U_{\mathrm{sep}}]_0 \quad \text{for all } (\theta_0^1, \theta_0^2) \in {\mathbb{T}}^2 \end{align*} $$

holds generally (that is, independent of the first-order relation in equation (4.14)) by a simple calculation based on equation (D.6) in Proposition D.2 (see also Remark D.1). We can thus split off the separable part $U_{\mathrm {sep}}$ of U defined in equation (4.5) and assume that $\mathcal {S}_U = \mathcal {S}_{U,0}$ in the following. Hence, the third step consists of showing that $\mathcal {S}_U = \mathcal {S}_{U,0} = \emptyset $ .

Step (iii). We begin this final step with performing a Fourier decomposition in equation (4.15), such that we obtain

$$ \begin{align*} \sum_{\substack{ 0 \neq |k_1| \le \deg_U^{(1)} \\ 0 \neq |k_2| \le \deg_U^{(2)}}} U_{k_1, k_2}\! \int_{0}^{1} e^{i 2 \pi k_1 x_{\mu_1}^1(\theta_0^1 + nt, c_1)} e^{i 2 \pi k_2 x_{\mu_2}^2(\theta_0^2 + mt, c_2)}\,dt = 0 \quad \text{for all } (\theta_0^1, \theta_0^2) \in {\mathbb{T}}^2. \end{align*} $$

Analogously to the proof of Theorem 2.3, we now consider the family of functions

$$ \begin{align*} (f^{(\mu_1, \mu_2)}_{k_1, k_2})_{0 \neq |k_1| \le \deg_U^{(1)}, 0 \neq |k_2| \le \deg_U^{(2)}} \end{align*} $$

in the Hilbert space $L^2({\mathbb {T}}^2)$ , where

(4.16) $$ \begin{align} f_{k_1, k_2}^{(\mu_1, \mu_2)}: {\mathbb{T}}^2 \to \mathbb{C}, \quad (\theta_0^1, \theta_0^2) \mapsto \! \sum_{\substack{(n,m) \in \mathcal{B}_0(\mathcal{S}^\perp_U) }}\! \int_{0}^{1} e^{i 2 \pi k_1 x_{\mu_1}^1(\theta_0^1 + nt, c_1)} e^{i 2 \pi k_2 x_\mu^2(\theta_0^2 + mt, c_2)}\,dt. \end{align} $$

Note that the sum in equation (4.16) is finite by Assumption (A3) (more precisely, it ranges over the at most $(2 \deg _U^{(1)}) \cdot (2 \deg _U^{(2)}) $ elements from $\mathcal {B}_0(\mathcal {S}_U^\perp )$ ) and we suppressed the dependence of $|c_i|> \gamma _i + \sqrt {\mu _i} \mathfrak {c}(V_i)$ on $(n,m) \in \mathcal {B}_0(\mathcal {S}_U^\perp )$ from the notation (recall equation (4.13)).

In this way, the problem of proving Theorem 2.4, that is, justifying $\mathcal {S}_{U,0} = \emptyset $ , reduces to a question about linear independence for the family of functions in equation (4.16) in the Hilbert space $L^2({\mathbb {T}}^2)$ . Recall that the family $(f^{( \mu )}_{k_1, k_2})_{(k_1, k_2)}$ being linearly independent is equivalent to the Gram matrix $G^{( \mu )} $ with entries

(4.17) $$ \begin{align} G_{(k_1, k_2), (k^{\prime}_1, k^{\prime}_2)}^{(\mu_1, \mu_2)}:= \langle f_{k_1, k_2}^{( \mu_1, \mu_2)}, f_{k^{\prime}_1, k^{\prime}_2}^{(\mu_1, \mu_2)} \rangle_{L^2({\mathbb{T}}^2)} \quad \text{for } 0 \neq |k_i|, |k^{\prime}_i| \le \deg_U^{(i)}, \ i \in \{1,2\}, \end{align} $$

being of full rank, where $\langle g,h\rangle _{L^2({\mathbb {T}}^2)}$ denotes the standard inner product of $g,h \in L^2({\mathbb {T}}^2)$ .

Proof. Using the version of Lemma B.1 for the inverse function, we find that

(4.18) $$ \begin{align} \Vert e^{i 2 \pi k_i x^i_{\mu_i}( \, \cdot ,c_i)} - e^{i 2 \pi k_i \, \cdot} \Vert_{C^0} = \mathcal{O}\bigg(\deg^{(i)}_U\frac{\mu_i \Vert V_i \Vert_{C^0}}{h_{\mu_i}^{(i)}(\gamma_i + \sqrt{\mu_i} \mathfrak{c}(V_i))}\bigg) =: \mathcal{O}(\mu_i) \end{align} $$

uniformly in $|k_i| \le \deg _U^{(i)}$ and $(n,m) \in \mathcal {B}_0(\mathcal {S}_U^\perp )$ .

Similarly to Lemma 4.2, with a slight abuse of notation for the error term, the elements $ G_{(k_1, k_2), (k^{\prime }_1, k^{\prime }_32)}^{(\mu _1, \mu _2)}$ of the Gram matrix can thus be computed as

$$ \begin{align*} &\int_{0}^{1}\,d \theta_0^1 \int_{0}^{1}\,d \theta_0^2 \bigg( \bigg[\sum_{(n,m)} \int_{0}^{1}\,dt ( e^{-i 2 \pi k_1 n t} + \mathcal{O}(\mu_1)) ( e^{-i 2 \pi k_2 m t} + \mathcal{O}(\mu_2) ) \bigg] e^{-i 2 \pi k_1 \theta_0^1}e^{-i 2 \pi k_2 \theta_0^2} \\ &\quad \times \ e^{i 2 \pi k^{\prime}_1 \theta_0^1}e^{i 2 \pi k^{\prime}_2 \theta_0^2} \bigg[\sum_{(n',m')} \int_{0}^{1}\,d t' ( e^{i 2 \pi k^{\prime}_1 n' t'} + \mathcal{O}(\mu_1)) ( e^{i 2 \pi k^{\prime}_2 m' t'} + \mathcal{O}(\mu_2) ) \bigg]\bigg), \end{align*} $$

where the summations over $(n,m)$ and $(n',m')$ are understood as in equation (4.16). Using that for every $(k_1, k_2) \in \mathcal {S}_{U,0}$ , there exist exactly two elements from $\mathcal {B}_0(\mathcal {S}_U^\perp )$ (differing by a sign), we can evaluate both brackets $[\cdots ]$ being given by

$$ \begin{align*} 2 + \mathcal{O}(\deg_U^{(1)} \deg_U^{(2)}\mu_1)+ \mathcal{O} (\deg_U^{(1)}\deg_U^{(2)} \mu_2) =: 2 + \mathcal{O}( \deg_U^{(1)} \deg_U^{(2)} (\mu_1 + \mu_2 )), \end{align*} $$

after absorption of the second-order error in the first-order ones.

From this, we conclude that

$$ \begin{align*} G_{(k_1, k_2), (k^{\prime}_1, k^{\prime}_2)}^{(\mu_1, \mu_2)} &= \int_{0}^{1}\,d \theta_0^1 \int_{0}^{1}\,d \theta_0^2 ( [ 2 + \mathcal{O}(\deg_U^{(1)} \deg_U^{(2)} (\mu_1 + \mu_2 ))] e^{i 2 \pi (k^{\prime}_1-k_1)\theta_0^1} \\ &\quad \times \, e^{i 2 \pi (k^{\prime}_2-k_2) \theta_0^2} [ 2 + \mathcal{O}( \deg_U^{(1)} \deg_U^{(2)} (\mu_1 + \mu_2 ) ) ] ) \\ &=4 \delta_{k_1, k^{\prime}_1} \delta_{k_2, k^{\prime}_2}+\mathcal{O} ( \deg_U^{(1)} \deg_U^{(2)} (\mu_1 + \mu_2 )). \end{align*} $$

Therefore, going back to equation (4.18), we infer the existence of $\tilde {\mu }_i = \tilde {\mu }(\mathcal {C}_i,\deg _U^{(1)}, \deg _U^{(2)}, e)> 0$ , $i \in \{1,2\}$ , such that for all $\mu _i \in [0, \tilde {\mu }_i]$ , the Gram matrix $G^{(\mu _1, \mu _2)}$ from equation (4.17) is of full rank.

This finishes the proof of Theorem 2.4(a). For part (b), similarly to the proof of Theorem 2.3(b), we observe that for every fixed $\mu _1 \in [0,\tilde {\mu }_1]$ , the function $\mu _2 \mapsto \det (G^{(\mu _1, \mu _2)})$ is analytic. Since $\det (G^{(\mu _1, \mu _2)}) \neq 0$ for $\mu _2 \in (0,\tilde {\mu }_2)$ (see Lemma 4.3), we find that the zero set

$$ \begin{align*} \mathcal{E}_0^{(\mu_1)} := \{ \mu_2 \in (0,\infty) \mid \det(G^{(\mu_1, \mu_2)}) = 0 \} \subset (\tilde{\mu}_2, \infty) \end{align*} $$

of $\mu _2 \mapsto \det (G^{(\mu _1, \mu _2)})$ is at most countable (finite in every compact subset), that is, in particular, a (one-dimensional) set of zero measure.

Finally, for part (c), we note that, similarly to the proof of Theorem 2.3(b) and by means of Hartogs’s theorem on separate analyticity [Reference Hartogs56] (a separately analytic function is jointly analytic), the function $(\mu _1, \mu _2) \mapsto \det (G^{(\mu _1, \mu _2)})$ is (jointly) analytic. Since $\det (G^{(\mu _1, \mu _2)}) \neq 0$ for $(\mu _1, \mu _2) \in (0,\tilde {\mu }_1) \times (0,\tilde {\mu }_2)$ (see Lemma 4.3), we find that the zero set

$$ \begin{align*} \mathcal{E}_0 := \{ (\mu_1, \mu_2) \in (0,\infty)\times (0,\infty) \mid \det(G^{(\mu_1, \mu_2)}) = 0 \} \subset (\tilde{\mu}_1, \infty) \times (\tilde{\mu}_2, \infty) \end{align*} $$

of $(\mu _1, \mu _2) \mapsto \det (G^{(\mu _1, \mu _2)})$ is a (two-dimensional) set of zero measure.

This concludes the proof of Theorem 2.4(c).