Transverse foliations in the rotating Kepler problem

We construct finite energy foliations and transverse foliations of neighbourhoods of the circular orbits in the rotating Kepler problem for all negative energies. This paper would be a first step towards our ultimate goal that is to recover and refine McGehee's results on homoclinics and to establish a theoretical foundation to the numerical demonstration of the existence of a homoclinic-heteroclinic chain in the planar circular restricted three-body problem, using pseudoholomorphic curves.


Introduction
The rotating Kepler problem is the Kepler problem in rotating coordinates, obtained from the planar circular restricted three-body problem (PCR3BP) by setting the mass of one of the primaries to zero.Its Hamiltonian H : T * (R 2 \ {0}) → R is given by where q = (q 1 , q 2 ) and p = (p 1 , p 2 ).It admits two integrals of motion called the Kepler energy and the angular momentum, respectively.Throughout the paper, we tacitly assume that the Kepler energy is negative, so that its trajectories projected to the q-plane are ellipses.We call such trajectories Kepler ellipses.
The Hamiltonian H admits a unique critical value c = − 3 2 .If c < − 3  2 , then the energy level H −1 (c) consists of two connected components, denoted by Σ b c and Σ u c .
circular orbits degenerate into a circle of critical points.In the case c > − 3 2 , there is a unique circular orbit γ u retro on H −1 (c) that is retrograde.Fix c < − 3 (3) Pick any embedded disc D described in (b) above.Since the rotating Kepler problem is completely integrable, the solid torus Σ u direct is foliated by Liouville tori.This Liouville foliation induces a foliation of the disc D into concentric circles whose centre corresponds to the direct circular orbit γ u direct .The same holds for Σ u retro with γ u direct being replaced by γ u retro .The motivation of this paper comes from results on the PCR3BP by McGehee [23] and by Koon, Lo, Marsden and Ross [20].We briefly state below these results for readers' convenience.
The PCR3BP studies the motion of an infinitesimal body under the gravitational influence of two massive bodies moving along circular orbits around their center of mass.The infinitesimal body, denoted by C, is assumed to move in the plane spanned by the orbits of the two massive bodies, denoted by S and J.We rescale the total mass of the system to one, so that the mass of S equals 1 − µ and the mass of J equals µ for some µ ∈ [0, 1].If the center of mass is located at the origin, then the Hamiltonian of the PCR3BP in a rotating frame is given by where S = (−µ, 0) and J = (1 − µ, 0) denote the positions of S and J, respectively.Note that in the case µ = 0, this becomes the Hamiltonian of the rotating Kepler problem, see (1.1).We assume µ < 1  2 , so that S is heavier than J.The Hamiltonian H µ admits five equilibria L j , j = 1, 2, 3, 4, 5, ordered by action in the following way: These values tend to − 3  2 , the critical value of the rotating Kepler problem, as µ → 0. We are interested in energies slightly above the first two critical values H µ (L 1 ) and H µ (L 2 ).The corresponding Hill's regions are illustrated in Figure 1 L 2 are of saddle-center type, a theorem of Lyapunoff [21] shows that for energy slightly above H µ (L j ), the energy level carries a unique hyperbolic periodic orbit near L j , called the Lyapunoff orbit, j = 1, 2.
In his dissertation [23], McGehee established the presence of invariant tori on the energy level, as in Figure 1, for µ > 0 small enough, in other words, for J sufficiently light.By making use of these tori, he was able to find a homoclinic orbit to the Lyapunoff orbit associated with L 1 or L 2 .See Figure 2.
Using McGehee's result, Koon, Lo, Marsden and Ross provided a numerical demonstration of the existence of a homoclinic-heteroclinic chain in the Sun-Jupiter system [20]: let the mass µ of J be that of Jupiter and the energy be that of comet Oterma, so that it is slightly above H µ (L 2 ).For these values, the authors showed that the energy level carries the Lyapunoff orbits near L 1 and L 2 and a heteroclinic orbit between the two Lyapunoff orbits.These three periodic orbits consist of a so-called homoclinic-heteroclinic chain, which might be used to design spacecraft orbits exploring the interior and exterior regions as illustrated in Figure 2. A homoclinic-heteroclinic chain in the PCR3BP.The inner and outer black curves denote homoclinic orbits to the Lyapunoff orbits (red curves) near L 1 and L 2 , respectively.The blue curve indicates a heteroclinic orbit.
Our ultimate goal is to recover and refine McGehee's results and to establish a theoretical foundation to the numerical demonstration by Koon-Lo-Marsden-Ross, using pseudoholomorphic curves.As compactness is crucial in pseudoholomorphic curve theory, in order to achieve this goal, one first has to find an efficient way to deal with non-compact energy levels.This paper would provide a way to construct finite energy foliations of regions in the non-compact energy level in the case µ = 0, i.e. in the rotating Kepler problem, so it would be a first step towards our final goal.
Outline of the paper.We start by recollecting relevant facts about Reeb dynamics and pseudoholomorphic curves in symplectisations in Section 2. Then in Section 3 we review results on the rotating Kepler problem that will be needed in our argument later on.In Section 4 we recall the definition and some properties of Poincaré's coordinates, a modification of the well-known Delaunay coordinates that are not valid for the circular orbits.Constructions of finite energy folations and transverse foliations are provided in Sections 5 and 6.We present in Appendix A an alternative way to obtain a transverse foliation whose pages are annuli.

Basic notions in contact geometry
2.1.Periodic orbits.Let Σ = K −1 (0) ⊂ R 4 be a regular energy level of a Hamiltonian K : R 4 → R. We endow R 4 with coordinates (x 1 , x 2 , y 1 , y 2 ).A periodic orbit (of K) on Σ will be denoted by a pair P = (w, T ), where w : R → Σ solves the differential equation ẇ = X K • w, where X K denotes the Hamiltonian vector field of K, defined as ω 0 (X K , •) = −dK, and T > 0 is a period.Here, ω 0 = dy 1 ∧dx 1 +dy 2 ∧dx 2 denotes the standard symplectic form on R 4 .By abuse of notation, we may also denote by P the trace w(R) ⊂ Σ.If T is minimal, then P is called simple.Throughout the paper, when we consider a periodic orbit, then we tacitly assume that it is simple.If two periodic orbits P 1 = (w 1 , T 1 ) and P 2 = (w 2 , T 2 ) satisfy w 1 (R) = w 2 (R) and T 1 = T 2 , then they are identified, and we write P 1 = P 2 .
Suppose that K is invariant under an anti-symplectic involution ρ : R 4 → R 4 , meaning that ρ is an involution satisfying ρ * ω 0 = −ω 0 .The Hamiltonian vector field satisfies ρ * X K = −X K , so that K indicates the flow of X K , called the Hamiltonian flow of K.This shows that if P = (w, T ) is a periodic orbit, then P ρ = (w ρ , T ), defined as is a periodic orbit as well.In the case that w(R) = w ρ (R), we write P = P ρ and call it a symmetric periodic orbit (with respect to ρ).Note that every symmetric periodic orbit intersects the fixed point set Fix(ρ), which is assumed to be nonempty, precisely at two points.
Assume that Σ is compact and of contact type, i.e. there is a Liouville vector field Y that is transverse to Σ.The one-form λ = ω 0 (Y, •) restricts to a contact form on Σ, still denoted by λ.Denote by R = R λ the Reeb vector field of λ.Note that the Hamiltonian vector field X K and the Reeb vector field R are related by implying that their flows coincide up to reparametrisation.The Reeb period of a periodic orbit P is defined as the period of P with respect to the Reeb flow.
If the Liouville vector field Y is invariant under ρ, i.e. it satisfies ρ * Y = Y, then ρ becomes exact, meaning that it satisfies ρ * λ = −λ.In this case, ρ restricts to an anti-contact involution on the contact manifold (Σ, λ), again denoted by ρ.The triple (Σ, λ, ρ) is said to be a real contact manifold.Note that the Reeb vector field R and the Reeb flow Therefore, given a periodic orbit P = (w, T ), the Reeb periods of P and P ρ coincide.

Conley-Zehnder index.
As before, let Σ = K −1 (0) ⊂ R 4 be a compact energy level of K : R 4 → R, equipped with a transverse Liouville vector field Y.The corresponding contact form is denoted again by λ.
For every z = (x 1 , x 2 , y 1 , y 2 ) ∈ Σ the tangent space T z Σ is spanned by the orthogonal vectors X 1 , X 2 , X 3 , defined as where the 4 × 4 matrices A j , j = 1, 2, 3, are given by Note that X 3 = A 3 ∇K = X K is parallel to the Reeb vector field R. The projection π : T Σ → ξ = ker λ along R restricted to the tangent plane distribution span{X 1 , X 2 } induces an isomorphism from span{X 1 , X 2 } to the contact structure ξ.In particular, the latter is spanned by the vector fields Let T denote the unitary trivialisation of T Σ/RX 3 , induced by X 1 and X 2 .Set In the trivialisation T the transverse linearised flow of the Hamiltonian vector field X K along a periodic trajectory P = (w, T ) on Σ is described by solutions Let ϑ(t) be any continuous argument of a non-vanishing solution α(t), i.e. α 1 (t)+ iα 2 (t) ∈ R + e iϑ (t) .Define the rotation interval of P as which depends only on the initial condition α(0) ̸ = 0.It is a compact connected interval with length strictly less than π.The periodic orbit P is non-degenerate if and only if ∂I ∩ 2πZ = ∅.Given ε > 0 small enough, we set I ε := I − ε.Then the Conley-Zehnder index of P is defined as Note that in this definition the Conley-Zehnder index is lower semi-continuous with respect to the C 0 -topology.Suppose that the Hamiltonian K is invariant under an anti-symplectic involution ρ : R 4 → R 4 , and the Liouville vector field Y satisfies ρ * Y = Y, so that ρ restricts to an anti-contact involution of Σ, denoted again by ρ, as in the previous section.Since ρ * X j = X j , j = 1, 2, 3, we find by the definition of the Conley-Zehnder index that µ CZ (P ρ ) = µ CZ (P ).

2.3.
Pseudoholomorphic curves in symplectisations.Let (Σ, λ) be a closed contact three-manifold.We denote by J (λ) the set of dλ-compatible almost complex structures on ξ = ker λ.We extend each J ∈ J (λ) to a d(e r λ)-compatible almost complex structure J on T (R × Σ) that is given by where r denotes the coordinate on R. Note that J is R-invariant.
Given a closed Riemann surface (S, j) and a finite set Γ ⊂ S, we consider a smooth map ũ = (a, u) : S \ Γ → R × Σ, satisfying dũ • j = J • dũ and a finite energy condition where the supremum is taken over all monotone increasing smooth functions ϕ : R → [0, 1].In this paper we only consider the case S = S 2 and #Γ ∈ {1, 2}.If #Γ = 1 or #Γ = 2, then such a map is called a finite energy plane or a finite energy cylinder, respectively.
Points in Γ are called punctures of ũ = (a, u).If a is bounded in a small neighbourhood of a puncture z 0 ∈ Γ, then z 0 is called removable.In this case, ũ can be smoothly extended over z 0 .See [9].Otherwise, either a(z) → +∞ or a(z) → −∞ as z → z 0 .Then the puncture z 0 is called positive or negative, respectively.In the following we assume that every puncture is not removable, so that the set Γ is decomposed into Γ = Γ + ⊔ Γ − , where Γ ± consist only of positive/negative punctures, respectively.We assign ε ∈ {±1} to each puncture z 0 ∈ Γ ± according to its sign.
The following statement is taken from [9] and [11].
Then every sequence s n → +∞ admits a subsequence s n k and a τ -periodic orbit x of the Reeb vector field such that u(s The periodic orbit x in the theorem above is called an asymptotic limit of ũ at the puncture z 0 ∈ Γ.If the puncture z 0 is positive or negative, the corresponding asymptotic limit is also called positive or negative, respectively.A finite energy curve is called non-degenerate if its every asymptotic limit is non-degenerate.The theorem above shows that a non-degenerate finite energy plane admits a unique asymptotic limit. We now assume that (Σ, λ) is equipped with an anti-contact involution ρ.It defines the exact anti-symplectic involution ρ := Id R ×ρ on (R×Σ, d(e r λ)), meaning that ρ * (e r λ) = −e r λ.An almost complex structure J ∈ J (λ) is said to be ρ-antiinvariant if (2.6) Denote by J ρ (λ) ⊂ J (λ) the set of dλ-compatible and ρ-anti-invariant almost complex structures on ξ.If J ∈ J ρ (λ), then the associated d(e r λ)-compatible almost complex structure J as in (2.5) is ρ-anti-invariant.It follows that if ũ = (a, u) is a finite energy J-holomorphic curve, then ũρ := ρ • ũ • I, where I(z) = z, is a finite energy J-holomorphic curve with Hofer energy E(ũ ρ ) = E(ũ).Moreover, if x is a positive or negative asymptotic limit of ũ at z 0 ∈ Γ, then x ρ is a positive or negative asymptotic limit of ũρ at z0 ∈ I(Γ), respectively.If ũ = ũρ , then it is said to be invariant (with respect to ρ).Note that every asymptotic limit of invariant finite energy planes and cylinders has to be a symmetric periodic orbit.For more information on the behaviour of pseudoholomorphic curves under the symmetry, we refer the reader to [7,19].
2.4.Transverse foliations.Let ψ t be a smooth flow on an oriented closed threemanifold Σ.A transverse foliation F of Σ adapted to ψ t is formed by the following data: (1) the singular set P which consists of finitely many simple periodic orbits, called binding orbits.(2) a smooth foliation of Σ \ ∪ P ∈P P by properly embedded surfaces transverse to the flow.Each leaf F ∈ F has an orientation induced by the flow and the orientation of Σ.The closure F of F is a compact embedded surface in Σ whose boundary is a subset of P. Each end z of F is called a puncture.Every puncture z is associated with a binding orbit P z ∈ P, called the asymptotic limit of F at z.The asymptotic limit P z at z has two orientations, one induced by F and the other induced by the flow.If the two orientations coincide with each other, then the puncture z is said to be positive.Otherwise it is negative.
In the case where Σ has a non-empty boundary that is tangent to the flow, then we extend the definition of a transverse foliation in such a way that every binding orbit is required to be contained in the interior of Σ and each regular leaf is transverse to the boundary of Σ.
2.5.Finite energy foliations.Let λ be a contact form on the tight three-sphere (S 3 , ξ 0 ) and J ∈ J (λ).Definition 2.2.A (stable) finite energy foliation for (S 3 , λ, J) is a two-dimensional smooth foliation F of R × S 3 satisfying the following properties: (1) Every leaf F ∈ F is the image of an embedded finite energy J-holomorphic curve ũ F = (a F , u F ) having only one positive puncture.The number of negative punctures is any finite number.The energies of such finite energy surfaces are uniformly bounded.(2) Every asymptotic limit of F is unknotted and has Conley-Zehnder index 1, 2 or 3, where asymptotic limits of a leaf F is defined to be asymptotic limits of ũ F .(3) A map T : R × F → F, defines an R-action on F. If F is a fixed point of T , i.e.T (c, F ) = F , ∀c ∈ R, then its Fredholm index Ind( F ) := Ind(ũ F ) is equal to 0 and ũ F is a cylinder over a periodic orbit.Here, the Fredholm index Ind(ũ F ) represents the local dimension of the moduli space of unparametrised J-holomorphic curves near ũ F having the same asymptotic limits.In the case F is not a fixed point, then Ind( F ) ∈ {1, 2} and u F is embedded and transverse to the Reeb flow of λ.
Every non-degenerate contact form λ admits a finite energy foliation, as stated below.
The projection of a finite energy foliation, obtained in the theorem above, to S 3 provides a transverse foliation adapted to the Reeb flow of λ.
Suppose that (S 3 , λ) admits an anti-contact involution ρ and that J ∈ J ρ (λ).A finite energy foliation F is said to be symmetric (with respect to ρ) if ρ( F) = F.The projection of a symmetric finite energy foliation to S 3 produces a symmetric transverse foliation.

The rotating Kepler problem
Recall that the Hamiltonian of the rotating Kepler problem is given by Consider a smooth function It is strictly increasing for r < 1 and decreasing for r > 1 and attains a unique global maximum − 3 2 at r = 1.Therefore, for every c < − 3 2 the equation f (r) = c has precisely two roots r b c < 1 < r u c .It follows that for every c < − 3 2 the energy level Since {E, L} = 0, where the Kepler energy E and the angular momentum L are as in (1.2), the Hamiltonian flows of E and L commute: See, for instance, [8,Theorem 3.1.7].Therefore, given a trajectory γ(t) of H, i.e. a trajectory of the Hamiltonian vector field X H , there exists a Kepler ellipse α(t) such that γ(t) = exp(−it)α(t).
(3.1)Note that γ is not necessarily periodic.It is periodic if and only if the period of α and 2π are Q-commensurable.In other words, if we denote the period of α by T > 0, then γ is periodic if and only if kT = 2πl for some coprime positive integers k, l. (3.2) Let α(t) be a Kepler ellipse that projects to a circular orbit.Then γ(t), as in (3.1), is circular as well.This happens if and only if X E and X L are parallel along α(t).In order to study circular orbits, recall that the eccentricity e of a Kepler ellipse satisfies e 2 = 2EL 2 + 1.
(3.3) Since circular orbits have e = 0, we obtain 0 = 2E(H − E) 2 + 1. (3.4) Fix H = c.If c < − 3 2 , then equation (3.4) has three solutions respectively.Note that our coordinate system rotates in the counterclockwise direction.Therefore, γ b retro rotates in opposite direction to the coordinate system and γ b direct and γ u direct rotate in the same direction with the coordinate system.For this reason, γ b retro is called the retrograde circular orbit and γ b direct and γ u direct are called the direct circular orbits.
This unique circular orbit is retrograde.See Figure 3.   Suppose that α(t) is a non-circular Kepler ellipse and let γ(t) be the corresponding periodic trajectory of H, satisfying (3.1) and (3.2).Since L generates rotation, it implies that γ(t) lies in an S 1 -family of periodic trajectories of H satisfying the same condition.For this reason we call a periodic trajectory of H satisfying (3.2) a T k,l -type orbit and an S 1 -family of T k,l -type orbits a T k,l -torus.
The Kepler energy is constant along each T k,l -torus.Indeed, Kepler's third law implies and hence using (3.2) we obtain .
One can easily check that in the case c < − 3 2 all T k,l -type orbits satisfy k > l on Σ b c and k < l on Σ u c .For a fixed coprime k, l ∈ N, identity (3.3) determines a one-parameter family of T k,l -tori.Either the eccentricity e, the angular momentum L or the total energy H = c can be regarded as the parameter of such a family.We call this family the T k,l -torus family.See Figure 4.  Observe that along every T k,l -torus family, there is a unique T k,l -torus containing a collision orbit (and hence it consists only of collision orbits).Indeed, a T k,l -type orbit γ(t) is a collision orbit if and only if a Kepler ellipse α(t) is a collision orbit.This happens precisely at e = 1, and in view of (3.3) this implies L = 0, or equivalently H = E.A T k,l -type orbit is said to be direct or retrograde if it satisfies L > 0 or L < 0, respectively.
We summarize the discussion so far in the following.Recall that we consider orbits with E < 0.
and the angular momentum L satisfies When projected to Σ u c , we have In the case − 3 2 < c < 0, we have

Poincaré's coordinates
We briefly recall the definition of Poincaré's coordinates.For more details we refer the reader to [1, Chapter 9] and [24, Section 8.9].
Consider the following open subset of called the direct elliptical domain (recall that we have L > 0 along direct orbits, see Section 3), and define smooth maps α, β : E → R/2πZ and a : E → R as follows.
Choose (q, p) ∈ E and let γ denote the Kepler ellipse on which the point (q, p) lies.Then α = α(q, p) is defined as the argument of the position q, and β = β(q, p) is defined to be the argument of the perihelion, i.e. the point, lying on the projection of γ to the q-plane, that is nearest to the origin.Finally, a = a(q, p) is defined to be the semi-major axis of the Kepler ellipse γ.Note that a = − 1 2E , where E < 0 indicates the Kepler energy of γ.See Figure 5. .Some quantities associated with a Kepler ellipse of Kepler energy E < 0. The quantity a is the semi-major axis that is equal to − 1 2E .The quantity α is given by the argument of the present position, and β is given by the argument of the perihelion.
called the Kepler map.This implies that the domain E is a thickened two-torus.Theorem 4.1 (Lagrange, 1808).The Kepler map K satisfies Suggested by this result, Delaunay (1860) considered a diffeomorphism where However, since the argument of the perihelion is undefined for a circular orbit, the Delaunay map cannot be used to study the dynamics in a neighbourhood of a circular orbit.
Remark 4.2.The classical notations of the Delaunay variables are (g, l, G, L), not (l, k, L, K).However, we stick to our choice since we would like to denote the angular momentum by L.
To remedy this problem, Poincaré introduced new coordinates as follows.Define It turned out that Π is a symplectomorphism onto its image P = Π(D).We then define the Poincaré mapping which is a symplectomorphism as well.Namely, it satisfies We shall now add the circular orbits.Recall that for the circular orbits we have e = 0, α = β and L = K.Let If e(q 1 , q 2 , p 1 , p 2 ) = 0, then we set P(q 1 , q 2 , p 1 , p 2 ) = (0, λ, 0, Λ) by defining λ = α and Λ = L = K.This extends the Poincaré mapping to a symplectomorphism P : E → P. We conclude that in the study of direct orbits, i.e. orbits with L > 0, we can work with the transformed Hamiltonian P * H : P → R.

Below the critical energy
Throughout the section we fix c < − 3 2 and E 0 ∈ (E u direct , 0).We shall prove the assertion of Theorem 1.1 for the set diffeomorphic to a solid torus with core being the direct circular orbit γ u direct .Note that it contains only direct orbits and no collision orbits.Indeed, every periodic orbit on Σ u direct has L > 0. See Lemma 3.1.
Remark 5.1.The neighbourhood Σ u direct of γ u direct could be arbitrarily large: just take E 0 ∈ (E u direct , 0) to be sufficiently close to 0.
We shall work in Poincaré's coordinates.For simplicity we write (x 1 , x 2 , y 1 , y 2 ) = (η, λ, ξ, Λ).Note that Then the Hamiltonian (1.1) is given in Poincaré's coordinates by We set so that ) is fixed and we consider the subset Σ u direct ⊂ Σ u c , we have where Therefore, along every negative gradient flow line the x 2 -component is constant and the y 2 -component is strictly increasing.See Figure 6.Pick any x 2 ∈ R/2πZ and let h x2 be the corresponding negative gradient flow line of H 2 .Denote by Π : Σ u direct → R 2 the projection (x 1 , x 2 , y 1 , y 2 ) → (x 2 , y 2 ).Then the preimage Π −1 (h x2 ) is an embedded disc in Σ u direct that is transverse to the Hamiltonian flow.The point Π −1 (x 2 , Λ 1 ) is the intersection of the disc Π −1 (h x2 ) and the direct circular orbit γ u direct .By varying x 2 ∈ R/2πZ, we obtain the desired disc foliation.
Below we shall embed Σ u direct into a weakly convex three-sphere and construct a symmetric finite energy foliation of the three-sphere.We then obtain a disc foliation as the intersection of the projection of the finite energy foliation with Σ u direct .
where (x 2 , y 2 ) ∈ R/4πZ×R + , and denote by Σ the connected component of H−1 (c) that contains a double cover Σu direct of Σ u direct .Note that H is invariant under the anti-symplectic involution We claim that for y 2 ≥ Λ 1 , the smooth function K has precisely two critical points (0, Λ 3 ) that is a saddle and (2π, Λ 3 ) that is a global minimum.In fact, we show that K y2 < 0, ∀y 2 < Λ 3 and K y2 > 0, ∀y 2 > Λ 3 . (5.6) We assume y 2 ∈ [Λ 2 + ε 0 , Λ 2 + 2ε 0 ] and find that The first term in the right-hand side is non-positive due to the choice of the constant B. See (5.3).The second term is non-positive since 1 < Λ 2 , and the last term is also non-positive because of the choice of ε 0 .Note that at least one of the three terms is non-zero.This implies that K admits a critical point only if Notice that it satisfies K(x 2 , Λ max ) = c and that We now introduce Given ε 1 > 0, choose any non-decreasing smooth function g : R → [0, 1] satisfying g(x 2 ) = 0 for x 2 < −2ε 1 and g(x 2 ) = 1 for x 2 > −ε 1 and define We repeat a similar business for x 2 ≥ 4π in a symmetric way (note that the function K is symmetric in x 2 with respect to x 2 = 2π).Namely, we interpolate between the smooth functions K and using the non-increasing function g(x 2 ) := g(4π − x 2 ).This produces a smooth function H2 (x 2 , y 2 ), which is still symmetric in x 2 with respect to x 2 = 2π.Note that we have broken the periodicity in x 2 .See Figure 7.
Proof.By the definition and symmetry of H2 , it suffices to show that there are no critical points of H2 in Interpolations for y 2 (left) and x 2 (right) We assume −2ε 1 < x 2 < −ε 1 and find that If y 2 ≥ Λ 1 , then we obtain in view of (5.6) that and hence ( H2 ) y2 vanishes only if y 2 = Λ 3 .We then compute Note that ( H2 ) y2 may vanish for y 2 < 1.For y 2 > 0, a global minimum of the function is positive, and hence we have ( H2 ) x2 < 0, provided y 2 < 1.
For later use, we prove the following assertion.
Fix −2ε 1 < x 2 < −ε 1 and consider the smooth function There is a = a(x 2 ) ∈ (0, 1) such that the function Q is strictly increasing for y 2 ∈ (0, a) and is strictly decreasing for and hence we find This together with compactness of Σ implies that for any (x 1 , x 2 , y 1 , y 2 ) ∈ Σ with x 2 ∈ (−2ε 1 , −ε 1 ) the component y 2 is bigger than 1.The rest of the lemma is straightforward.This finishes the proof.□ 5.3.Construction of a contact form.In this section we construct a ρ-invariant Liouville vector field Y that is transverse to Σ. Then the restriction of the oneform ω 0 (Y, •) to Σ defines a contact form on Σ, denoted by λ.The anti-symplectic involution ρ : R 4 → R 4 restricts to an anti-contact involution on Σ, still denoted by ρ.Consequently, we obtain a real contact manifold (Σ, λ, ρ).Recall from Section 2.3 that ρ provides the exact anti-symplectic involution ρ = Id R × ρ on (R × Σ, d(e r λ)).
We first claim that the radial vector field with respect to p c1 is transverse to Σ along {x 2 < 2π}.By the definition of H, it suffices to consider the region {−2ε 1 < x 2 < 2π}.We find that The first two terms in the right-hand side are non-negative.The third term is also non-negative because of Lemma 5.3 and (5.6).For the last term, we find that By the choice of the constant B (see (5.3)), it is negative.This proves the claim.We now consider the Liouville vector field In order to interpolate between the two Liouville vector fields Y 0 and Y 1 , we define a smooth function Fix ε 2 > 0 small enough and pick any non-decreasing smooth function h : R → [0, 1] satisfying h(x 2 ) = 0 for x 2 < 2π − 2ε 2 and h(x 2 ) = 1 for ) where X hℓ indicates the Hamiltonian vector field associated with h(x 2 )ℓ(x 2 , y 2 ).
We claim that the Liouville vector field Ỹ is transverse to Σ along {x 2 < 4π}.In view of the argument above, it suffices to consider the region {2π − 2ε 2 < x 2 < 2π − ε 2 }.We find that The sum of the first two terms in the right-hand side above is positive.For the last term, we find This is non-negative because of (5.6).This proves the claim.We now repeat the same business as above in the region {x 2 > 2π}.More precisely, we consider the radial vector field with respect to p c2 and interpolate between Ỹ and Y 2 using the non-increasing function h(x 2 ) := h(4π − x 2 ).This provides the Liouville vector field Y that is transverse to Σ.By construction, it is invariant under ρ.
• γ 2 is constant; the red and blue curves in Figure 8.
In the last case, i.e. both γ 1 and γ 2 are non-constant, the periodic orbit γ lies in an S 1 -family of periodic orbits.
In this trivialisation the linearised Hamiltonian flow along P i restricted to (T Σ/RX K )| Pi is described by a solution to the ODE αi If i = 2, then we have x 2 = 2π, and hence This implies that the rotation interval (see (2.4)) of any non-trivial solution to the ODE above contains 0 as an interior point.Since the winding number of the projection of X 1 (and hence also of X 2 ) to the (x 2 , y 2 )-plane are equal to 1, it follows that the rotation interval of any non-trivial solution to ODE (2.3) contains 2π as an interior point.We conclude from the definition of the Conley-Zehnder index that P 2 is non-degenerate and satisfies Consider P 3 , so that x 2 = 0. Then a continuous argument of any non-vanishing solution α to the ODE above is of the form for some constant ϑ 0 ∈ R. Since the Hamiltonian period of P 3 equals 2π, the rotation interval of α is contained in (−2π, 0).Since the basis {X 1 , X 2 } has the negative orientation with respect to the basis {V 1 , V 2 }, and the winding number of the projection of X 1 to the (x 2 , y 2 )-plane equals 1, as in the previous case, this implies that the rotation interval of any non-trivial solution to ODE (2.3) is contained in (2π, 4π) from which we obtain that P 3 is non-degenerate and satisfies µ CZ (P 3 ) = 3.
For j = 1, 2, w j (t) is a closed curve which is oriented in the clockwise direction, and hence its winding number with respect to the standard basis is given by where the equality holds if and only if w j (t) is a simple closed curve.
We abbreviate by which are the projections of X 1 and X 2 along P to the (x 1 , y 1 )-plane.Since P does not correspond to the critical points of H2 , these vectors are non-vanishing and linearly independent.Hence, we may compute the rotation interval associated with the transverse linearised flow of X K along P using the trivialisation of the (x 1 , y 1 )-plane induced by Y 1 and Y 2 .We find the winding numbers of Y 1 and Y 2 with respect to the standard basis (5.10) Suppose that P corresponds to a black curve in Figure 8, so that ẇ1 (t) is nonvanishing.Since ẇ1 (t) is preserved by the linearised flow projected to the (x 1 , y 1 )plane, there is a solution α(t) to ODE (2.3) whose projection to the (x 1 , y 1 )-plane equals ẇ1 (t).We denote such a projection by ᾱ(t) and then compute its winding number with respect to the basis where we have used the fact that the basis B has negative orientation with respect to the standard basis.This implies that the rotation interval of P contains 4π, from which we conclude that µ CZ (P ) ≥ 3.
We now assume that P corresponds to the green curve in Figure 8, so that w 1 (t) ≡ (0, 0).In this case, the linearised Hamiltonian flow along P restricted to the (x 1 , y 1 )-plane is described by a solution to the ODE As in the case of P 3 , this implies that the rotation interval of β is contained in (−∞, 0), and hence we find using (5.10) that the rotation interval of any non-trivial solution to ODE (2.3) is contained in (2π, +∞).Consequently, we have µ CZ (P ) ≥ 3.This finishes the proof.□ 5.6.Construction of a finite energy foliation.Let X 1 , X 2 be the two vector fields that span the contact structure ξ = ker λ, see (2.2).We define the dλcompatible almost complex structure J : ξ → ξ by (5.11) Note that it is ρ-anti-invariant.See (2.6).
We only consider the case that we have x 2 ≡ 0, x 2 ≡ 2π, x 2 ≡ 4π or y 2 ≡ Λ 3 with 0 ≤ x 2 (s) ≤ 4π, ∀s.Recall that in this case we have H2 = K, where the latter is as in (5.4).Since u(s, t) ∈ Σ, this implies that Following the argument given in [6, Chapter 5] we shall show that the corresponding ũ is a finite energy plane asymptotic to P 2 , P 3 or P ′ 3 or a finite energy cylinder asymptotic to P 3 or P ′ 3 at its positive puncture and to P 2 at its negative puncture.
We first assume the case where x 2 (s) ∈ [0, 2π], ∀s, so that the Liouville vector field is given by Y = Y 0 + X hℓ , see (5.8), and hence the contact form λ equals We compute This together with the last identity of (5.12) implies that a is independent of t, so we may choose from which we see that a → ±∞ as s → ±∞.
The first identity of (5.12) implies that πu s and πu t are linearly independent, so that there exist A, B, C and D such that By definition of J, we have A = D and B = −C, and hence (5.16) Let P : R 4 → R 2 be the projection (x 1 , x 2 , y 1 , y 2 ) → (x 2 , y 2 ) and denote A direct computation using (5.14) shows that where By means of (5.16) we obtain and hence (5.17) We compute that Then solving (5.17) provides . (5.18) Suppose first that x 2 ≡ 0. Using (5.14) and (5.18) we find which may be seen as a differential equation of the type ẏ2 (s) = P (y 2 (s)). (5.20) Here, It follows from (5.6) that the function P satisfies the following properties: ( By construction such a solution yields a solution u(s, t) = (r(s) sin 2πt, 0, r(s) cos 2πt, y 2 (s)), (s, t) ∈ R × S 1 (5.22) to the first equation in (5.12).Here, the radius r(s) is determined by relation (5.14) with x 2 (s) = 0 and satisfies where r 3 is as in (5.9).This together with a(s, t), defined in (5.15), produces a J-holomorphic curve ũ = (a, u) : R × S 1 → R × Σ.Note that this solution depends on the choice of the initial condition y 2 (0) ∈ (Λ 1 , Λ 3 ) ∪ (Λ 3 , Λ max ).For any N > 0 and for any ϕ : R → [0, 1] with ϕ ′ ≥ 0, we have and hence Section 1].After removing it, we obtain an embedded finite energy J-holomorphic plane ũ = (a, u) : C → R×Σ asymptotic to P 3 at its positive puncture s = +∞.
In the case which may be seen as a differential equation satisfying the same properties as above.See (5.20) and (5.21).Note that the radius r(s), determined by (5.14) with x 2 (s) = 2π, satisfies lim where r 2 is as in (5.9).Then arguing in a similar manner we find an embedded finite energy J-holomorphic plane ũ, depending on the initial condition y 2 (0) ∈ (Λ 1 , Λ 3 )∪(Λ 3 , Λ max ), asymptotic to P 2 at its positive puncture s = +∞ and having energy equal to τ 2 = πr 2 2 .Note that it is invariant, i.e. ũ = ũρ .See Section 2.3.Finally, assume that which may be seen as a differential equation of the type Here, from which we see that any solution x 2 = x 2 (s) to ODE (5.24) with initial condition x 2 (0) ∈ (0, 2π) is strictly decreasing and satisfies In view of the construction such a solution gives rise to a solution u(s, t) = (r(s) sin 2πt, x 2 (s), r(s) cos 2πt, Λ 3 ), (s, t) ∈ R × S 1 (5.25) to the first equation in (5.12).The radius r(s) is determined by (5.14) with y 2 (s) = Λ 3 and satisfies lim where r 2 and r 3 are as in (5.9).This together with a(s, t), defined in (5.15), produces a J-holomorphic curve ũ = (a, u) : R × S 1 → R × Σ, depending on the choice of the initial condition x 2 (0) ∈ (0, 2π).The Hofer energy is given by τ 3 and the mass of ũ at −∞ is equal to τ 2 = πr 2 2 , implying that −∞ is non-removable.Thus, we obtain an embedded finite energy J-holomorphic cylinder ṽ = (b, v) : R×S 1 → R×Σ asymptotic to P 3 at its positive puncture s = +∞ and to P 2 at its negative puncture s = −∞.
We now consider the case x 2 ≡ 4π or y 2 ≡ Λ 3 with 2π ≤ x 2 (s) ≤ 4π, ∀s.Instead of the direct construction as above, we shall use the symmetry induced by the anti-contact involution ρ.
An application of the previous theorem to the finite energy plane ũ3,1 (recall that P 3 is non-degenerate and satisfies µ CZ (P 3 ) = 3, see Proposition 5.4) yields a maximal one-parameter family of embedded finite energy J-holomorphic planes ũτ = (a τ , u τ ) : C → R × Σ, τ ∈ (τ − , τ + ), all asymptotic to P 3 .This family is non-compact.Indeed, otherwise there is τ 1 ̸ = τ 2 with u τ1 (C) ∩ u τ2 (C) ̸ = ∅.Then it follows from [10,Theorem 1.4] that u τ1 (C) = u τ2 (C), and hence we find an open book decomposition of Σ with binding P 3 and disc-like pages.By the usual argument, every page is transverse to the Reeb vector field, implying that every periodic orbit other than P 3 is linked to P 3 .This contradicts to the presence of P 2 and P ′ 3 .We now take a look at the breaking of the family {ũ τ } as τ → τ ± .For simplicity we write τ − = 0 and τ + = 1.We assume that τ strictly increases in the direction of the Reeb vector field.
Pick a sequence τ n → 0 + as n → +∞ and write ũn := ũτn , n ∈ N. By a standard argument (see [6,14,16]), we find that ũn converges to a holomorphic building with height 2. The top consists of an embedded finite energy J-holomorphic cylinder ṽ = (b, v) : C\{0} → R×Σ asymptotic to P 3 at its positive puncture +∞ and to an index 2 orbit Q at at its negative puncture 0. The bottom consists of an embedded finite energy J-holomorphic plane ũ = (a, u) : C → R × Σ asymptotic to Q.Moreover, given a neighbourhood U ⊂ Σ of v(C \ {0}) ∪ Q ∪ u(C), we have u n (C) ⊂ U for n sufficiently large.See [13] and also [6,Proposition 9.5].Since P 2 is a unique index 2 orbit, see Proposition 5.4, we have Q = P 2 .The uniqueness property of finite energy planes asymptotic to P 2 and finite energy cylinders asymptotic to P 3 at +∞ and to P 2 at 0 (see [6,), based on Siefring's intersection theory [27], implies that ṽ = ṽ1 or ṽ = ṽ2 and ũ = ũ2,1 or ũ = ũ2,2 , up to reparametrisation and R-translation.
We now show that the planes u and u ′ are disjoint.We follow the argument given in the proof of [4,Proposition 3.17].Assume by contradiction that they have a non-empty intersection.Then Theorem 2.4 in [27] shows that u(C) = u ′ (C).Pick any point q ∈ Σ \ (P 3 ∪ v(C \ {0}) ∪ P 2 ∪ u(C)) and take a small neighbourhood U of P 3 ∪v(C\{0})∪P 2 ∪u(C), not containing q. Then there exists τ 0 and τ 1 contained in (0, 1), close enough to 0 and 1, respectively, such that u τ0 (C), u τ1 (C) ⊂ U.By Jordan-Brouwer separation theorem, the piecewise smooth embedded two-sphere S = u τ0 (C)∪P 3 ∪u τ1 (C) divides Σ into two components C 1 and C 2 having boundary S. It follows from the choice of the point q that one of the two components, say C 1 , contains q in its interior and the other contains We claim that the set A = τ ∈(0,1) u τ (C) ∩ int(C 1 ) is non-empty, open and closed in int(C 1 ).In view of Theorem 5.5 it is non-empty and open.In order to achieve the closedness of A, take a sequence w n ∈ A. Since w n is contained the closed subset C 1 , a limit of the sequence w n is contained in C 1 .On the other hand, since w n is also contained in the union τ ∈(0,1) u τ (C), the compactness property of the family {u τ } τ ∈(0,1) described above implies that its limit is contained either in τ ∈(0,1) u τ (C), in P 3 , or in v(C \ {0}) ∪ P 2 ∪ u(C).The last option does not occur because v(C \ {0}) ∪ P 2 ∪ u(C) is contained in the interior of C 2 .Therefore, a limit of the sequence w n is contained in τ ∈(0,1) u τ (C) ∪ P 3 ∩ C 1 , from which we see that the set τ ∈(0,1) u τ (C) ∪ P 3 ∩ C 1 is closed in Σ.This proves the claim.
We have constructed a stable finite energy foliation F of R × Σ, which is symmetric with respect to the exact anti-symplectic involution ρ and whose leaves are embedded finite energy planes and cylinders.Denote by F the projection of F to Σ, which is symmetric with respect to the anti-contact involution ρ.Since finite energy planes in F are asymptotic to P 2 or P 3 , their Σ-projections are transverse to the Reeb vector field [15].See also [8,Section 13.6].It is straightforward to see that the Σ-parts v 1 , v 2 of the cylinders ṽ1 , ṽ2 are transverse to the Reeb vector field as well.Therefore, F determines a transverse foliation of Σ whose binding orbits are P 2 , P 3 and P ′ 3 and whose regular leaves are embedded planes and cylinders.See Figure 9.
Remark 5.7.A transverse foliation satisfying the properties above is called a weakly convex foliation.For more details, see [5].
The projection of the finite energy foliation above defines a transverse foliation F of Σ, whose binding orbits are P 2 and P 3 and whose regular leaves are embedded planes and cylinders.
Denote by T the compact subset of the energy level Σ, which projects into the (x 2 , y 2 )-plane (modulo 4π in x 2 ) as the set

Note that the double cover Σu
direct of the set Σ u direct under consideration is contained in T and that T is diffeomorphic to a solid torus and has boundary The interior of T is filled with the images of planes u 2,1 and u τ = u 1,τ , τ ∈ (0, 1).The core of this solid torus is the second iterate of the direct circular orbit γ u direct .Let F ∈ F be a plane asymptotic to P 3 that is not rigid and contained in T 1 .For j = 1, 2, denote Π j (x 1 , x 2 , y 1 , y 2 ) = (x j , y j ), (x 1 , x 2 , y 1 , y 2 ) ∈ Σ.Then Π 1 (F ) is a closed disc of radius r 3 centred at the origin and Π 2 (F ) is an arc connecting (x 2 , Λ 1 ) for some x 2 ∈ (0, 4π) and (0, Λ (5.26) See (5.1).Note that the right-hand side above equals zero if and only if y 2 = Λ 1 .
This implies that d is independent of t, and hence we may write Note that d ϑ (s, t) → +∞ as s → ±∞.
We have constructed a finite energy J-holomorphic cylinder wϑ = (d ϑ , w ϑ ) : R × S 1 → R × Σ that is asymptotic to Q 1 and Q 2 at ∓∞, respectively.Note that both asymptotic limits are positive.Its Hofer energy equals the sum of the Reeb periods of Q 1 and Q 2 .
Since the above argument is independent of the choice of ϑ ∈ R/2πZ, by varying ϑ we obtain a finite energy foliation of R × Σ, which projects the open book decomposition of Σ with annulus-like pages, decribed above.Its intersection with the set Σ u direct yields the desired annulus foliation.Namely, the solid torus Σ u direct is foliated into annuli transverse to the Hamiltonian flow restricted to Σ u direct .The outer boundary of each annulus lies on the boundary torus ∂Σ u direct .The inner boundary coincides with the direct circular orbit γ u direct .
Figure 14.A slice of the annulus foliation.The dot and the dotted curves correspond to the direct circular orbit and annuli, respectively.The blue curve points in the direction of the Hamiltonian flow.

1 Figure 2 .
Figure 2. A homoclinic-heteroclinic chain in the PCR3BP.The inner and outer black curves denote homoclinic orbits to the Lyapunoff orbits (red curves) near L 1 and L 2 , respectively.The blue curve indicates a heteroclinic orbit.

implying that H − 1
(c) carries precisely three circular orbits: γ b retro with E = E b retro and γ b direct with E = E b direct , both lying on Σ b c , and γ u direct with E = E u direct , lying on Σ u c .Their angular momenta are

Figure 4 .
Figure 4. Some T k,l -tori with k = 9.The dashed line indicates the tori consisting only of collision orbits.

Figure 5
Figure 5.Some quantities associated with a Kepler ellipse of Kepler energy E < 0. The quantity a is the semi-major axis that is equal to − 1 2E .The quantity α is given by the argument of the present position, and β is given by the argument of the perihelion.

Figure 6 .
Figure 6.Negative gradient flow lines of H 2 .Dashed lines indicate Hamiltonian trajectories.

1 Figure 9 .
Figure 9.The projection of the transverse foliation F to the (x 2 , y 2 )-plane.The black curves indicate the families of planes.

2 Figure 10 .
Figure 10.An unpleasant scenario that is not the case.