arXiv:1408.5193v1 [math.DS] 22 Aug 2014

EXISTENCE OF NONCONTRACTIBLE PERIODIC ORBITS OF HAMILTONIAN SYSTEM SEPARATING TWO LAGRANGIAN TORI ON T ∗ Tn WITH APPLICATION TO NON CONVEX HAMILTONIAN SYSTEMS

arXiv:1408.5193v1 [math.DS] 22 Aug 2014

JINXIN XUE UNIVERSITY OF CHICAGO

Abstract. In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on T ∗ Tn separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in [P] in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on T ∗ T2 . This solves a problem of Arnold in [A].

1. Introduction In a recent paper [P], Polterovich proved the existence of invariant measure µ for Hamiltonian systems in the following setting. Consider a symplectic manifold (M, ω) and a pair of compact subsets X, Y ⊂ M with the following properties: (P1) Y cannot be displaced from X by any Hamiltonian diffeomorphism θ: θ(Y ) ∩ X 6= ∅ for every θ ∈ Ham(M, ω). (P2) There exists a path {φt }, t ∈ [0, 1], φ0 = id of symplectomorphisms so that φ1 displaces Y from X : φ1 (Y ) ∩ X = ∅. We put X 0 := φ1 (Y ) and a := Flux({φ1 }) = in [P] states as follows.

R1 0

[iφ˙ t ω] dt, then the main theorem

Theorem 1 (Theorem 1.1 of [P]). For every F ∈ C ∞ (M, R) with F |X ≤ 0,

F |X 0 ≥ 1,

the Hamiltonian flow ψt of F possesses an invariant probability measure µ with (1.1)

ha, ρ(µ)i ≥ 1

where ρ(µ) is the rotation vector of µ. Date: August 25, 2014. Email: [email protected]. 1

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A similar result is obtained in [V] as Proposition 5.10 using a different approach. It is natural to ask if the invariant measures are supported on periodic orbits. In the same paper [P], the author asks the following question: Can one, under assumptions of Theorem 1, deduce existence of a closed orbit of the Hamiltonian flow so that the corresponding rotation vector satisfies inequality (1.1)? Finding periodic orbit is an important theme in symplectic dynamics. As remarked in [G], “there is a general principle in symplectic dynamics that a compactly supported function with sufficiently large variation must have fast nontrivial periodic orbits or even one-periodic orbits if the function is constant near its maximum” (recall for instance the Hofer–Zehnder capacity and etc). However, this principle is not correct in full generality. There is a famous counter-example of Zehnder (see Example 2.1 of [P]). Periodic orbit does not exist for the manifold (T4 = R4 /Z4 , ω = dp1 ∧ dq1 + γdp2 ∧ dq1 + dp2 ∧ dq2 ) where γ is irrational, F (p, q) = sin 2(πp1 ), and X = {p = 0}, X 0 = {p = (1/2, 0)} two Lagrangian tori. So we specialize to the case M = T ∗ Tn ,

ω0 =

n X

dqi ∧ dpi

i=1

and X being the zero section {p = 0}, X 0 another Lagrangian torus corresponding to {p = p∗ 6= 0}, and ask for the existence of noncontractible periodic orbits. However, there is an immediate counterexample given by the Hamiltonian (1.2)

F (p, q) =

hα, pi hα, p∗ i

where α ∈ Rn is completely irrational and hα, p∗ i 6= 0. Any composition σ ◦ F has no periodic orbits, where σ : R → R smooth. We choose σ(x) = 0 for x ≤ 0, σ = 1 for x ≥ 1 and monotone, then multiply σ ◦ F by a compactly supported function η(p) : Rn → R that decays sufficiently slowly outside a big ball. We can make the resulting Hamiltonian system η(p) · σ ◦ F (p, q) have no noncontractible periodic orbits of period 1. In this paper, we show the existence of noncontractible periodic orbits of Hamiltonian systems in the following setting. Consider the symplectic manifold (T ∗ Tn , ω0 ). Consider two Lagrangian tori of T ∗ Tn : X being the zero section {p = 0} and X 0 the section {p = p∗ }, where p∗ = (p∗1 , . . . , p∗n ) ∈ Rn \{0} with p∗i > 0, i = 1, . . . , n is a constant vector. We choose some large R  kp∗ k (where k·k is the Euclidean norm) and denote by RT ∗ Tn the open set RT ∗ Tn := {(p, q) ∈ T ∗ Tn | kpk < R} .

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(b) The dual cone C ∗

(a) The cone C

Figure 1. The cone C and the dual cone C ∗ For a nonempty closed set V in the complement of p∗ , we consider time-periodic Hamiltonians H(p, q, t) in the following set  ∞ Hc (RT ∗ Tn ; V, p∗ ) := H(p, q, t) ∈ Ccpt (RT ∗ Tn × T1 , R) | (1.3) H(p∗ , q, t) ≥ c, and H(p, q, t) = 0, for p ∈ V } , where we use cpt to mean compactly supported. In this paper, the set V is either Rn \ C in the following Theorem 2 or W in Theorem 3. Our first result is the following Theorem 2. Consider a cone C positively spanned by linearly independent vectors v1 , . . . , vn ∈ Rn , ( n ) X C = span+ {v1 , v2 , . . . , vn } := ci vi | ci > 0, ∀ i = 1, 2, . . . , n . i=1

Denote by A = (v1 , . . . , vn ) the matrix formed by vi , i = 1, . . . , n as column vectors. Then for any point p∗ lying in the interior of the cone C, for all H(p, q, t) ∈ Hc (RT ∗ Tn ; Rn \ C, p∗ ), any homology class α ∈ H1 (Tn , Z) \ {0} satisfying n
−1
−1
−1 o hp∗ , αi ≤ c, and α ∈ C ∗ := span+ AT e 1 , AT e2 , . . . , AT en , where ei , i = 1, . . . , n are the standard basis vectors of Rn , there exists a periodic orbit of H in the homology class α of period 1. A positively spanned cone cannot contain any lines, so the cone C in the assumption is necessary to rule out the counterexample (1.2). As the angle at the tip of the cone C becomes more obtuse, the set of homology classes admitting periodic orbits becomes smaller. See Figure 1 for the picture of the cone C and its dual cone C ∗ in the two dimensional case (we choose v1 = (1, 3), v2 = (3, 1)). Our Theorem 2 is sharp in view of the counterexample (1.2).

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We get the next theorem when we choose A to be identity in Theorem 2. For simplicity of notations, we introduce the following closed set W := {p ∈ Rn | pi ≤ 0,

for some i = 1, . . . , n}.

Theorem 3. For all H(p, q, t) ∈ Hc p∗ ∈ Rn \ W , and homology class α = (α1 , . . . , αn ) ∈ H1 (Tn , Z) \ {0} satisfying (RT ∗ Tn ; W, p∗ ),

(1.4)

hα, p∗ i ≤ c, and αi > 0,

∀ i = 1, . . . , n,

there exists a 1-periodic orbit of the Hamiltonian flow of H in the homology class α. The main body of the paper is devoted to proving Theorem 3. Our other theorems are derived from Theorem 3. A closely related result is the following Theorem B of [BPS]. To state the theorem, we first define the symplectic action as Z 1 (H(x(t), t) − λ(x(t))) ˙ dt, for x ∈ C ∞ (T1 , RT ∗ Tn ), (1.5) AH (x) = 0

where λ = p dq is the Liouville 1-form. Theorem 4 (Theorem B of [BPS]). For every compactly supported smooth ∞ (RT ∗ Tn × T1 , R) with R = 1 and every e ∈ Zn Hamiltonian function H ∈ Ccpt such that kek ≤ c := inf H(0, q, t), (q,t)∈Tn ×T1

the Hamiltonian system has a periodic solution x(t) in the homotopy class e with action AH (x) ≥ c. Remark 1. The cut-off R is only to guarantee compactness when constructing Floer theory. This is a reminiscent of the setting of Theorem 4. However, our result is not an immediate consequence of Theorem 4. Suppose we choose R large enough and consider Hamiltonians H whose oscillation (Hofer norm) is much smaller than R. If we rescale the fiber by p 7→ p/R, correspondingly we should rescale the Hamiltonian H 7→ H/R. The oscillation of the rescaled Hamiltonian is not large enough to produce noncontractible 1-periodic orbits applying Theorem 4. Our Theorem 3 gives plenty of 1-periodic orbits whose homology class satisfying (1.4). We will see in Lemma 3.1 that the periodic orbits that we find are not produced by the cut-off but by the oscillation on W and p∗ . We also get the following dense existence result. Theorem 5. Consider H(p, q) : T ∗ Tn → R an autonomous Hamiltonian in Hc (RT ∗ Tn ; W, p∗ ). Then for each nontrivial homology class α = (α1 , . . . , αn ) ∈ H1 (Tn , Z), there exists a dense subset Sα ⊂ (0, minq H(p∗ , q)) with the property that for each s ∈ Sα , the level set {H = s} contains a closed orbit (not necessarily period 1) in the class α.

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As an application of Theorem 3, we answer a question of Arnold in the following Theorem 6. For the problem, see Section 1.8 of [A], where Arnold asked p2 p2 for the existence of periodic orbits of the non convex system H = 1 − 2 + 2 2 V (q1 , q2 ), (p, q) ∈ T ∗ T2 in each homology class. This system appears naturally when one wants to prove Arnold diffusion for non convex type Hamiltonian systems and finding periodic orbits is the first thing one needs to do. Arnold remarked that “It seems that the contemporary technique of the calculus of variation in the large has no ready methods for this problem”, which seems still the case nowadays. The next result shows the strength of our theorem when applied to nonconvex type Hamiltonian systems. Theorem 6. Consider Hamiltonnian system of the form H(p, q) = p1 p2 + V (q1 , q2 ), (p, q) ∈ T ∗ T2 , V (q) ∈ C ∞ (T2 , R). Denote by M := max V (q) − min V (q) . For each homology class α ∈ H1 (T2 , Z)\ q∈T2 q∈T2 {0}, α1 > 0, α2 > 0, there exists a dense subset Sα of (M, ∞) such that for each s ∈ Sα , there exists a periodic orbit lying on the energy level {H = s} and with homology class α.

(1.6)

The idea is to notice that the function p1 p2 is positive in the interior of the first quadrant and is zero on the boundary. After proper scaling and translation of the Hamiltonian to handle the bounded perturbation V , then composing it with σ that we used in the paragraph of (1.2), we get a modified Hamiltonian to which Theorem 3 is applicable. We obtain a periodic orbit lying on the energy level of the modified Hamiltonian, which is also a periodic orbit of the Hamiltonian (1.6). See Section 5.3 for more details. Remark 2. • Our system (1.6) is equivalent to Arnold’s original one up to linear symplectic transformation (see Section 5.3). • If we want homology class α ∈ H1 (T2 , Z) \ {0}, with α1 < 0, α2 < 0, we make the coordinates change α 7→ −α,

(p, q) 7→ (−p, −q).

• If we want homology class α ∈ H1 (T2 , Z) \ {0}, with α1 < 0, α2 > 0, we make the coordinates change (α1 , α2 ) 7→ (−α1 , α2 ),

(p1 , p2 , q1 , q2 ) 7→ (p1 , −p2 , q1 , −q2 ),

H 7→ −H.

• If we want homology class α ∈ H1 (T2 , Z) \ {0}, with α1 > 0, α2 < 0, we make the coordinates change (α1 , α2 ) 7→ (α1 , −α2 ),

(p1 , p2 , q1 , q2 ) 7→ (−p1 , p2 , −q1 , q2 ),

H 7→ −H.

Then apply the above Theorem 6. It is interesting to notice that the inequalities (1.1) in [P, V] go in the opposite direction as ours (1.4) (In our case, if we rescale the energy oscillation from c to

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1, the corresponding time rescaling will take the rotation vector α to α/c). The invariant measure µ found in [V] verifies the equality (see Proposition 5.10 and Corollary 5.8 of [V]) (1.7)

A(µ) = α(p∗ ) − hp∗ , ρ(µ)i,

where α(p∗ ) is Mather’s α function, which can be considered as energy c, and A is the symplectic action (see [V] and our definition in (1.5)). On the other hand, in our proof, we always guarantee our periodic orbits satisfy the inequality A ≤ c − hp∗ , αi (see Section 4.2). In Mather theory for positive definite Lagrangian system [M], Equation (1.7) implies that µ is action minimizing. So we may think that the invariant measures found by [P, V] resemble the action minimizing measure of Mather. However, in our case, strict inequality may happen. Notice our action carries a negative sign compared to Mather’s action. This shows that our periodic orbits may not be action minimizing in Mather’s setting. It seems highly nontrivial that on the critical energy level when equality holds, invariant measure can be supported on periodic orbits. Let us now review the literature briefly. The existence of certain periodic orbits in Hamiltonian systems is part of the story of Weinstein conjecture. Please refer to [G] for a review. We focus on results mostly relevant to ours. In [HV], the authors prove the existence of periodic orbits for Hamiltonians separating neighbourhoods of two points on CPn using J-holomorphic curve techniques. Using the method of [HV], Gatien and Lalonde [GL] showed the existence of noncontractible periodic orbits for compactly supported Hamiltonians separating two Lagrangian tori on T ∗ K where K is the Klein bottle as well as the case when kp∗ k is sufficiently small for T ∗ Tn . In [L], Y.-J. Lee generalized the result of [GL] by introducing a Gromov-Witten type invariant. Notice T ∗ Tn is exactly a case when the invariant of [L] vanishes, so that we have counterexample (1.2) and the Gromov-Witten invariant approach does not work in out setting. On the other hand, there is a Floer theoretical approach developed in [BPS], the authors obtain several results including Theorem 4. Their results are further generalized by [W, SW] to general manifolds T ∗ M where M is closed. The method in our proof is to implement the machinery of [BPS]. We will show in the following sections that the method of [BPS] goes through. The paper is organized as follows. In Section 2, we set up the machinery of Floer homology. This part follows mainly from [BPS] with some variations following [W]. Our new contribution is Lemma 2.5. We define the filtered Floer homology group in Section 2.1 and the inverse and direct limits of the groups in Section 2.2 induced by the monotone homotopies of Hamiltonians. We introduce exhausting sequences in Section 2.3, which would reduce the computation of the Floer homology group for any Hamiltonian to that for an exhausting sequence. In Section 2.4, we introduce a BPS type capacity which is suitable to find periodic orbits and another homological relative capacity which is accessible to computation and bound the other capacity. Next, in Section 2.5, we introduce the Morse-Bott theory which would be used to compute the Floer homology group

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of the exhausting sequence. In Section 4, we prove Theorem 3. In this section, we construct a family of profile functions as an exhausting sequence and study their first and second order derivatives carefully. We use Morse-Bott theory to compute the Floer homology group for the profile functions. Finally in Section 5, we prove Theorem 2, 5 and 6. 2. Floer homology In this section, we set up the framework of [BPS]. Since we specialize to the manifold T ∗ Tn , we get some simplification in the presentation. 2.1. Floer theory and spectral invariants. 2.1.1. Symplectic actions. We consider the standard symplectic form ω0 = dqi and the Liouville 1-form λ =

n X

n X

dpi ∧

i=1

pi dqi such that dλ = ω0 .

i=1 ∞ (RT ∗ Tn × T1 , R). In this section, we define Floer homology for functions in Ccpt We denote by L Tn := C ∞ (T1 , RT ∗ Tn ) the space of free loops of RT ∗ Tn . For each x ∈ L Tn , we can associate to it a free homotopy class [x] ∈ π1 (Tn ) = H1 (Tn , Z). Given a homotopy class α ∈ π1 (Tn ) we write the loop space

Lα Tn := {x ∈ L Tn | [x] = α}. ∞ (RT ∗ Tn × T1 , R) determines a time-periodic Hamiltonian Each H(p, q, t) ∈ Ccpt vector field XH through the relation

i(XH )ω0 = −dH. The space of 1-periodic solutions of the Hamiltonian equation representing a class α is denoted by P(H, α) := {x ∈ Lα Tn | x˙ = XH (x(t))} . Elements of P(H, α) are the critical points of the symplectic action AH (x) (1.5) for x ∈ Lα Tn . 2.1.2. Action spectrum and periodic orbits. The action spectrum is defined as S(H; α) = AH (P(H; α)) = {AH (x) | x ∈ Lα Tn ,

x(t) ˙ = XH (x(t))}.

P [a,b) (H; α)

Consider −∞ ≤ a < b ≤ ∞ and denote by the set of 1-periodic solutions of the Hamiltonian system H representing the class α and with action lying in the interval [a, b): P [a,b) (H; α) := P b (H; α)\P a (H; α),

P a (H; α) := {x ∈ P(H; α) | AH (x) < a}.

∞ (RT ∗ Tn × T1 , R) under consideration satisfies We need to assume the H ∈ Ccpt the following nondegeneracy condition:

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(?) a, b ∈ / S(H; α) and every 1-periodic orbit x ∈ P(H; α) is nondegenerate in the sense that the derivative dφ1H (x(0)) of the time-1 map φ1H does not have 1 in its spectrum. This nondegeneracy condition can be achieved by perturbing H near each periodic orbit (see Section 2.1 of [W]). 2.1.3. Floer homology group. We next define the Floer homology group HF[a,b) with Z2 coefficient as the homology of the chain complex CF[a,b) (H; α) over Z2 which is generated by the 1−periodic orbits in P [a,b) (H; α), where we define M CF[a,b) (H; α) := CFb (H; α)/CFa (H; α), CFa (H; α) := Z2 x. x∈P a (H;α)

2.1.4. The boundary operator, energy. To define the boundary operator, we consider the perturbed Cauchy-Riemann equation ∂s u + J0 (∂t u − XH (u)) = 0.

(2.1)

For a smooth solution u(s, t) : R1 × T1 → T ∗ Tn of (2.1), we define its energy as Z Z E(u) := k∂s uk2 ds dt. T1

R

If u is a finite energy solution of (2.1), then the limits exist (2.2)

lim u(s, t) = x± (t),

s→±∞

lim ∂s u(s, t) = 0

s→±∞

and are uniform in t. Moreover, we have x± ∈ P(H; α) and the energy identity (2.3)

E(u) = AH (x− ) − AH (x+ ).

This energy identity, the exactness of ω0 imply the space of finite energy solutions of (2.1) is compact with respect to compact-open topology. Namely, only the splitting into a finite sequence of adjacent Floer connecting orbits can occur in the limit. 2.1.5. Compactness and nondegeneracy issues. Throughout  the paper, we fix the 0 −idn almost complex structure to be the standard one J0 = . In order idn 0 to guarantee the linearized operator of (2.1) to be surjective, we do not perturb the almost complex structure. Instead, we perturb the Hamiltonian H in an arbitrarily small neighborhood U of the image of u in (R − ε)T ∗ Tn × T1 where ε ∞ ((R − ε)T ∗ Tn × T1 , R). We can choose the is chosen to be so small that H ∈ Ccpt perturbation to vanish up to second order along the orbits x± (see Section 2.1 of [W] and Theorem 5.1 (ii) of [FHS]). Namely, there exists a small neighborhood ∞ (U, R) and a subset U U of zero in Ccpt reg of regular perturbations of Baire’s second category such that the linearized operator of (2.1) is surjective, for all u solving equation (2.1) for the Hamiltonian H + h with h ∈ Ureg , and satisfying (2.2) with x± ∈ P (a,b) (H + h; α).

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2.1.6. Moduli space and Floer homology. For every H + h with h ∈ Ureg and every pair of periodic orbits x± ∈ P(H; α) the space M (x− , x+ ; H, J0 ; α) of solutions of (2.1) with boundary conditions (2.2) is a smooth manifold whose dimension near a solution u of (2.1) and (2.2) is the difference of the Conley– Zehnder indices µCZ of x− and x+ (relative to u). We denote the subspace of solutions of index one by M 1 (x− , x+ ; H, J0 ; α). It follows from Section 2.1.5 that the quotient M 1 (x− , x+ ; H, J0 ; α)/R (modulo time shift) is a finite set for every pair of periodic orbits x± ∈ P(H; α) with Conley–Zehnder index difference being 1. The Floer boundary operator ∂ H on the chain complex CFb (H; α) is defined as X #(M 1 (x, y; H, J0 ; α)/R) y ∂ H x := y∈P b (H;α)

for every x ∈ P b (H; α) with µCZ (y) = µCZ (x) + 1. The energy identity (2.3) shows that CFa (H; α) is a subcomplex, i.e. it is invariant under ∂ H . We therefore  H get a boundary operator ∂ on the quotient complex CF[a,b) (H; α). We finally define the homology of the quotient complex as    ker ∂ H : CF[a,b) (H; α) → CF[a,b) (H; α)  . HF[a,b) (H, α) := im [∂ H ] : CF[a,b) (H; α) → CF[a,b) (H; α)

2.1.7. Homotopic invariance. The above homology group HF[a,b) (H, α) is defined for a fixed Hamiltonian. When we have a smooth homotopy of Hamiltonians Hs : s ∈ R with Hs = H0 when s ≤ 0 and Hs = H1 when s ≥ 1, we consider the following Cauchy-Riemann equation ∂s u + J0 ∂t u − ∇Hs (u, t) = 0.

(2.4)

The smooth solutions u : R × T1 → RT ∗ Tn of (2.4) is a connecting orbit between two periodic orbits with the same Conley–Zehnder index. Namely we have uniformly in t ∈ T1 the limits (2.5)

lim u(s, t) = z0 (t),

s→−∞

lim u(s, t) = z1 (t),

s→+∞

lim ∂s u(s, t) = 0

s→±∞

where zi (t) ∈ P(Hi , α), i = 0, 1 and µCZ (z0 ) = µCZ (z1 ). We have the energy identity Z ∞Z (2.6) E(u) = AH0 (z0 ) − AH1 (z1 ) − (∂s Hs ) (u(s, t), t) dt ds. −∞

T1

Similar to Section 2.1.5 we can find a second category subset of regular homotopies among all homotopies such that the linearized operator of (2.4) is surjective, for all elements u of the moduli spaces M (z0 , z1 ; Hs , J0 ; α) (see also Section 2.1 of [W]). Solution of (2.4) defines a Floer chain map from CF(H0 , α) to CF(H1 , α).

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2.1.8. Monotone homotopy. Next, we define  ∞ (2.7) H a,b (RT ∗ Tn ; α) := H ∈ Ccpt (RT ∗ Tn × T1 , R) | a, b ∈ / S(H, α) . Suppose there are two Hamiltonians H0 , H1 ∈ H a,b (RT ∗ Tn ; α) satisfy H0 (p, q, t) ≥ H1 (p, q, t) for all (p, q, t) ∈ RT ∗ Tn × T1 as well as being nondegenerate in the sense of (?). Then there exists a homotopy s 7→ Hs from H0 to H1 such that ∂s Hs ≤ 0. We call such a homotopy monotone. Every monotone homotopy s 7→ Hs induces a natural monotone homomorphism σH1 H0 : HF[a,b) (H0 ; α) → HF[a,b) (H1 ; α),

(2.8)

which is independent of the choice of the monotone homotopy of Hamiltonians used to define it. We have the composition rule σH2 H1 ◦ σH1 H0 = σH2 H0 ,

(2.9)

whenever H0 , H1 , H2 ∈ H a,b (RT ∗ Tn ; α) satisfy H0 ≥ H1 ≥ H2 , and σHH = id for every H ∈ H a,b (RT ∗ Tn ; α). To make the monotone homomorphism σH1 H0 an isomorphism, we need the following proposition. Proposition 2.1 (Proposition 4.5.1 of [BPS]). Let −∞ ≤ a < b ≤ ∞, α ∈ π1 (T ∗ Tn ) be a nontrivial homotopy class, and H0 , H1 ∈ H a,b (RT ∗ Tn ; α) be such that H0 ≥ H1 . Suppose that there exists a monotone homotopy {Hs }0≤s≤1 from H0 to H1 such that Hs ∈ H a,b (RT ∗ Tn ; α) for every s ∈ [0, 1]. Then σH1 H0 : HF[a,b) (H0 ; α) → HF[a,b) (H1 ; α) is an isomorphism. 2.2. Direct and inverse limits. ∞ (RT ∗ Tn × T1 , R). We introduce a partial order on 2.2.1. Partial order on Ccpt ∞ ∗ n Ccpt (RT T × T, R) by

H0  H1

⇔

H0 (p, q, t) ≥ H1 (p, q, t)

∀ (p, q, t) ∈ RT ∗ Tn × T1 .

The monotone homomorphisms σH1 H0 of Section 2.1.8 give rise to a partially ordered system (HF, σ) of Z2 -vector spaces over H a,b (RT ∗ Tn ; α) defined in (2.7). By definition, this means that HF assigns to each H ∈ H a,b (RT ∗ Tn ; α) the Z2 -vector space HF[a,b) (H; α), and σ assigns to all elements H0  H1 of H a,b (RT ∗ Tn ; α) the monotone homomorphism σH1 H0 subject to composition rule (2.9). 2.2.2. Inverse limit on H a,b (RT ∗ Tn ; W ; α). We restrict the partially ordered system (H a,b (RT ∗ Tn ; α), ) to a partially ordered system (H a,b (RT ∗ Tn ; W ; α),  ) where we define (2.10) n o \ ∞ H a,b (RT ∗ Tn ; W ; α) := H ∈ H a,b (RT ∗ Tn ; α) Ccpt ((Rn \ W ) × Tn × T1 , R) ,

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We abbreviate H a,b (RT ∗ Tn ; W ; α) as H a,b (W ). The partial order set (H a,b (W ), ) is downward directed: For all H1 , H2 ∈ H a,b (W ) there exists H0 ∈ H a,b (W ) such that H0  H1 and H0  H2 . The functor (HF, σ) is called an inverse system of Z2 -vector spaces over H a,b (W ). Its inverse limit (the absolute symplectic homology) is defined by [a,b) SH (RT ∗ Tn ; W ; α) := lim HF[a,b) (H; α) ←− ← − H∈H a,b (W )   Y HF[a,b) (H; α) := {aH }H∈H a,b (W ) ∈  a,b H∈H

For H ∈ H

a,b (W ),

(W )

  H0  H1 ⇒ σH H (aH ) = aH . 1 0 1 0 

let

[a,b) πH : ← SH (RT ∗ Tn ; W ; α) → HF[a,b) (H; α) − be the projection to the component corresponding to H. It holds πH1 = σH1 H0 ◦ πH0 , whenever H0  H1 .

(2.11)

2.2.3. Direct limit on Hca,b (W ). To define relative symplectic homology, fix c > 0 and consider the subset (2.12) n o Hca,b (RT ∗ Tn ; W, p∗ ; α) := H ∈ H a,b (RT ∗ Tn ; W ; α) | H(p∗ , q, t) > c . We abbreviate it as Hca,b (W, p∗ ). This set is upward directed: For all H0 , H1 ∈ Hca,b (W, p∗ ) there exists H2 ∈ Hca,b (W, p∗ ) such that H0  H2 and H1  H2 . The functor (HF, σ) is called a direct system of Z2 -vector spaces over Hca,b (W, p∗ ). Its direct limit is defined to be the quotient [a,b);c SH (RT ∗ Tn ; W, p∗ ; α) := −→

lim −→

HF[a,b) (H; α)

H∈Hca,b (W,p∗ )

n o. := (H, aH ) H ∈ Hca,b (W, p∗ ) , aH ∈ HF[a,b) (H; α) ∼, where (H0 , a0 ) ∼ (H1 , a1 ) iff there exists H2 ∈ Hca,b (W, p∗ ) such that H0  H2 , H1  H2 and σH2 H0 (a0 ) = σH2 H1 (a1 ). This is an equivalence relation, since Hca,b (W, p∗ ) is upward directed. The direct limit is a Z2 -vector space with the operations k[H0 , a0 ] := [H0 , ka0 ],

[H0 , a0 ] + [H1 , a1 ] := [H2 , σH2 H0 (a0 ) + σH2 H1 (a1 )],

for all k ∈ Z2 and H2 ∈ Hca,b (W, p∗ ) such that H0  H2 and H1  H2 . For H ∈ Hca,b (W, p∗ ) define the homomorphism (2.13)

[a,b);c ιH : HF[a,b) (H; α) → SH (RT ∗ Tn ; W, p∗ ; α), −→

It satisfies ιH0 = ιH1 ◦ σH1 H0 , whenever H0  H1 .

aH 7→ [H, aH ].

12

JINXIN XUE UNIVERSITY OF CHICAGO

2.3. Exhausting sequence. To compute direct and inverse limits we introduce the notion of exhausting sequences following [BPS]. Let (G, σ) be a partially ordered system of R-modules over (I, ) and denote Z± := {ν ∈ Z | ± ν > 0} . A sequence {iν }ν∈Z+ is called upward exhausting for (G, σ) iff the following holds • For every ν ∈ Z+ we have iν  iν+1 and σiν+1 iν : Giν → Giν+1 is an isomorphism. • For every i ∈ I there exists a ν ∈ Z+ such that i  iν . A sequence {iν }ν∈Z− is called downward exhausting for (G, σ) iff the following holds • For every ν ∈ Z− we have iν−1  iν and σiν iν−1 : Giν−1 → Giν is an isomorphism. • For every i ∈ I there exists a ν ∈ Z− such that iν  i. We use exhausting sequences to simplify computations of direct and inverse limits. Lemma 2.2 (Lemma 4.7.1 of [BPS]). Let (G, σ) be a partially ordered system of R-modules over (I, ). (1) If {iν }ν∈Z+ is an upward exhausting sequence for (G, σ) then (I, ) is upward directed and the homomorphism ιiν : Giν → lim G is an isomor−→ phism for every ν ∈ Z+ . (2) If {iν }ν∈Z− is a downward exhausting sequence for (G, σ) then (I, ) is downward directed and the homomorphism πiν : lim G → Giν is an ←− isomorphism for every ν ∈ Z− . 2.4. Capacities.

2.4.1. Symplectic homology. We cite the following proposition from [BPS] about the existence of a homomorphism between absolute and relative symplectic homologies which factors through Floer homology. Proposition 2.3 (Proposition 4.8.1 of [BPS]). Let α ∈ π1 (Tn ) be a nontrivial homotopy class and suppose that −∞ ≤ a < b ≤ ∞. Then, for every c ∈ R, there exists a unique homomorphism Tα[a,b);c : SH[a,b) (RT ∗ Tn ; W ; α) → SH[a,b);c (RT ∗ Tn ; W, p∗ ; α) ←− −→

PERIOIDIC ORBITS OF HAMILTONIANS SEPARATING LAGRANGIAN TORI

13

such that for any two Hamiltonian functions H0 , H1 ∈ Hca,b (RT ∗ Tn ; W, p∗ ; α) with H0 ≥ H1 the following diagram commutes: [a,b);c

Tα

SH[a,b) (RT ∗ Tn ; W ; α) ←− πH

0

[a,b)

HF

/ SH[a,b);c (RT ∗ Tn ; W, p∗ ; α) −→ O ιH



σH

(H0 ; α)

1 ,H0

/ HF

[a,b)

1

(H1 ; α)

Here πH0 : SH[a,b) (RT ∗ Tn ; W ; α) → HF[a,b) (H0 ; α) ←− and ιH1 : HF[a,b) (H1 ; α) → SH[a,b);c (RT ∗ Tn ; W, p∗ ; α) −→ are the canonical homomorphisms introduced in Section 2.2.2 and 2.2.3. In particular, since σHH = id for every H ∈ Hca,b (RT ∗ Tn ; W, p∗ ; α), we have [a,b);c

Tα

SH[a,b) (RT ∗ Tn ; W ; α) ←− πH

(

/ SH[a,b);c (RT ∗ Tn ; W, p∗ ; α) −→ 5 ιH

[a,b)

HF

(H; α)

2.4.2. The homological relative capacity. Following [BPS] we define two capacities. For every nontrivial homotopy class α ∈ π1 (Tn ) and every real number c > 0 we define the set n o Ac (RT ∗ Tn ; W, p∗ ; α) := a ∈ R The homomorphism Tα[a,∞);c does not vanish . The homological relative capacity of the triple (RT ∗ Tn ; W, p∗ ) is the function ∗ n b C(RT T ; W, p∗ ) : π1 (Tn ) × [−∞, ∞) → [0, ∞]

which assigns to the class α ∈ π1 (Tn ) and the following number for a ≥ −∞  ∗ n b (2.14) C(RT T ; W, p∗ ; α, a) := inf c > 0 sup Ac (RT ∗ Tn ; W, p∗ ; α) > a . Here we use the convention that inf ∅ = ∞ and sup ∅ = −∞. For a = −∞ we abbreviate  ∗ n ∗ n b b C(RT T ; W, p∗ ; α) := C(RT T ; W, p∗ ; α, −∞) = inf c > 0 Ac (RT ∗ Tn ; W, p∗ ; α) 6= ∅ .

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JINXIN XUE UNIVERSITY OF CHICAGO

2.4.3. A relative symplectic capacity. We define the BPS type relative symplectic capacity by (2.15)

C(RT ∗ Tn ; W, p∗ ; α, a) := inf {c > 0 | ∀H ∈ Hc (RT ∗ Tn ; W, p∗ ), ∃ x ∈ P(H; α) such that AH (x) ≥ a} .

We get the existence of periodic orbits provided we bound C(RT ∗ Tn ; W, p∗ ; α, a) ∗ Tn ; W, p∗ ; α, a) is computable and does bound b from above. The capacity C(RT ∗ n ∗ C(RT T , W, p ; α, a), as said by the next proposition. Proposition 2.4 (Proposition 4.9.1 of [BPS]). Let α ∈ π1 (Tn ) and a ∈ R. If ∗ Tn ; W, p∗ ; α, a) < ∞ then every Hamiltonian H ∈ H (RT ∗ Tn ; W, p∗ ) with b C(RT c ∗ Tn ; W, p∗ ; α, a) has a 1-periodic orbit in the homotopy class α with b c ≥ C(RT action AH (x) ≥ a. In particular, ∗ n b C(RT T ; W, p∗ ; α, a) ≥ C(RT ∗ Tn ; W, p∗ ; α, a).

The proof of this proposition is a word by word translation of that of Proposition 4.9.1 of [BPS]. We remark here that the function class Hca,b (W, p∗ ) (2.12) (with ∗ Tn ; W, p∗ ; α, a) (2.14) differs from the function b b = ∞) in the definition of C(RT ∗ n ∗ class Hc (RT T ; W, p ) (1.3) in the definition of C(RT ∗ Tn ; W, p∗ ; α, a) (2.15) by the strict inequality “> c”. Functions in Hc (RT ∗ Tn ; W, p∗ ) can be approximated by that in Hca,b (W, p∗ ). See the proof of Proposition 4.9.1 of [BPS] for the approximation argument. 2.5. Morse-Bott theory in Floer homology. We need to use Morse-Bott theory to compute Floer homology. We first give the definition of Morse-Bott manifolds (Section 5.2 of [BPS]). Definition 1. A subset P ⊂ P(H; α) is called a Morse-Bott manifold of periodic orbits if the set C0 := {x(0) | x ∈ P } is a compact submanifold of of a symplectic manifold M and Tx0 C0 = Ker(Dψ1 (x0 ) − id) for every x0 ∈ C0 , where ψ1 is the time-1 map of the Hamiltonian flow induced by the Hamiltonian H(p, q, t) ∈ ∞ (M, R). Ccpt ∗ n For a compactly supported Hamiltonian system H(p)  defined on (RT T , ω0 ) ∂H (p), p ∈ Rn × Tn is and depending only on variables in the fibers, the set ∂p   ∂H foliated into invariant tori labeled by frequencies (p), p ∈ Rn according ∂p to Liouville-Arnold theorem. If we consider a torus corresponding to frequency ∂H q˙ = (p0 ) ∈ Zn \ {0}. This is an invariant torus foliated by periodic orbits ∂p of period 1. We pick any point q(0) in the torus as initial condition to solve our Hamiltonian equation, the resulting periodic orbit lies completely on the torus. We have the following easy criteria to determine when such a torus is a Morse-Bott manifold.

PERIOIDIC ORBITS OF HAMILTONIANS SEPARATING LAGRANGIAN TORI

15

Lemma 2.5. For a Hamiltonian system H(p) defined on (T ∗ Tn , ω0 ) and depending only on variables in the fibers, the set   ∂H n (p0 ) ∈ Z \ {0} P = (p, q) p = p0 , q˙ = ∂p is a Morse-Bott manifold of periodic orbits for H(p) iff   2 ∂ H (p0 ) 6= 0, i, j = 1, 2, . . . n. det ∂pi ∂pj

Proof. The Hamiltonian equations are q˙ =

∂H (p), p˙ = 0. The linearized equa∂p

tion has the following form d dt



δq δp



 =

0 0

  ∂2H  δq  ∂p2 δp 0

where (δp, δq) ∈ T(p,q) (T ∗ Tn ). This equation can be integrated explicitly, whose fundamental solution at time 1 is     ∂2H ∂2H 0 0 Dψ1 = exp  ∂p2  = id2n +  ∂p2  . 0 0 0 0 According to Definition 1, we only need to check for (p0 , q0 ) ∈ P T(p0 ,q0 ) P = Ker(Dψ1 (p0 , q0 ) − id) On the one hand, the set P in consideration is an n-torus P = {(p, q) | p = p0 , q ∈ Tn }, whose tangent space at (p, q) is T(p,q) = {(δp, δq) | δp = 0, δq ∈ Rn }. One the other hand we have     ∂2H ∂2H 0 (p0 )  n  2 Ker(Dψ1 (p0 , q0 ) − id) = Ker = Ker 2 (p0 ), δq0 ∈ R . ∂p ∂p 0 0 These tell us that to guarantee P is a Morse-Bott manifold, we need and only ∂2H need to have that the matrix is nondegenerate at p0 .  ∂p2 Next, we cite the following theorem of Pozniak from [BPS] in order to compute Floer homology using Morse-Bott manifold. Theorem 7 (Theorem 5.2.2 of [BPS]). Let −∞ ≤ a < b ≤ ∞, α ∈ π1 (M ), and H ∈ H a,b (M ; α). Suppose that the set P := {x ∈ P(H; α) | a < AH (x) < b} is a connected Morse–Bott manifold of periodic orbits. Then HF[a,b) (H; α) ∼ = H∗ (P ; Z2 ).

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JINXIN XUE UNIVERSITY OF CHICAGO

3. Construction of the profile functions In this section, we prove Theorem 3. The idea is to construct a family of profile functions Hs (p), s ∈ R that is both upward and downward exhausting. We will show that for a satisfying 0 ≤ a ≤ c − hp∗ , αi, all the homology groups HF[a,∞) (Hs , α) are isomorphic to each other and nonvanishing as s varies. The main result of this section is Lemma 3.1. We have the following list of requirements for the family of profile functions ∞ (RT ∗ Tn , R). Hs ∈ Ccpt • On the C 0 level: Hs (p) is both upward and downward exhausting, i.e. for each cc (RT ∗ Tn ; W, p∗ ) := {H(p, q, t) ∈ H∈H

(3.1)

∞ Ccpt (RT ∗ Tn × T1 ∩ ((Rn \ W ) × Tn × T1 ), R) | H(p∗ , q, t) > c},

there exist s < s0 such that Hs0 < H < Hs . Functions in (1.3) can be cc defined here. approximated by functions in H ∂Hs (ps ) = α, the ho• On the C 1 level: there exists a unique ps such that ∂p mology class in Theorem 3, and the action of the corresponding periodic orbit is greater than a ∈ [0, c − hp∗ , αi]. ∂ 2 Hs (ps ) • On the C 2 level: det 6= 0, ∀s, so that Lemma 2.5 is satisfied. ∂p2

3.1. Profile functions. We first construct two families of profile functions for s ≥ 1 and s ≤ −1. Then we construct a homotopy from s = 1 to s = −1.

|x|2

3.1.1. A model function in one dimensional case. Consider the function e− 2δ where δ√is sufficiently small. The second order derivative vanishes at the √ point x = ± δ (the turning points) and the first order derivative at x = ± δ is √ 1 1 ∓ √ e− 2 . The value of the function at x = ± δ is e−1/2 ' 0.61. δ We define a C 1 function u as follows. We consider one copy of e− copy −e

−

|x|2 2δ

|x|2 2δ

and one

+ 1. After shifting horizontally the first function properly we use a 1 1 piece of straight line of slope √ e− 2 to join their turning points smoothly. The δ

PERIOIDIC ORBITS OF HAMILTONIANS SEPARATING LAGRANGIAN TORI

(a) The model function u

17

(b) The profile functions seen from a section: Fs (tp∗ ) as a function of t

Figure 2. Profile functions

explicit expression of this function is given as follows (see (A) of Figure 2).

(3.2)

  1,    |x−b|2   e− 2δ ,    −1/2 e u(x) = √ x + 1 − 2e−1/2 ,  δ 2    |x|  − 2δ  −e + 1,    0,

√ if x ≥ (4 − e1/2 ) δ := b, √ if x ∈ [b − δ, b], √ √ if x ∈ [ δ, b − δ], √ if x ∈ [0, δ], if x ≤ 0.

The function u is C ∞ everywhere except at the two turning points x = b − √ √ δ, δ where u000 is discontinuous as well as the two points x = 0, b where u00 is discontinuous. We smoothen u in a δ 3/2 neighborhoods of the four points to get a function in C ∞ (R, R), which is still denoted by u. The smoothing can be done as follows. We use a partition of unity to localize in a δ 3/2 neighbourhood of each 00 point. √ The second derivative u decreases to zero continuously in a neighborhood of √δ and decreases from zero to √ negative √ continuously in a neighborhood of b− δ. In neighbourhoods of b− δ and δ, we convolute u00 with a nonnegative compactly supported C ∞ approximating Dirac-δ function. At x = b, the second derivative u00 jumps from −δ −1 to zero. We join the two pieces smoothly such 1 that u00 (x) = 0 for x ≥ b − δ 3/2 and u00 (x) ≤ 0 for x ≤ b. Similar for x = 0. 3 We also multiply a scalar to u if necessary, such that it takes values from 0 to 1. The following are satisfied by the smoothed function u: 1 1 • u(x) = 0 for x ≤ δ 3/2 and u(x) = 1 for x ≥ b − δ 3/2 . 3 3 • u00 does not change sign in the above four neighbourhoods so that u0 is monotone there.

18

JINXIN XUE UNIVERSITY OF CHICAGO

Finally, we define p b 1/2 = (4 − e ) δs , ∀s ∈ R. 2 2|s| In the function u we replace the parameters δ by δs and b by bs to get a function called us (x). So we have us (x) = u−s (x) and u(x) = u0 (x). δ

, 2|s|

δs :=

bs :=

3.1.2. Profile functions when s ≥ 1. For s ≥ 1, our profile function is defined as a proper shift and rescaling of the products of the function u. We define s

(3.3)

Fs (p) = (c + 2 )

n Y i=1

 us

pi p∗i

 ,

s ≥ 1.

So this function is equal to 0 on W and c + 2s for pi ≥ p∗i bs for all i = 1, . . . , n. See the upper two curves in (B) of Figure 2. 3.1.3. Profile functions when s ≤ −1. We define for s ≤ −1, (3.4)   v   u n 2   −s + 2s X pi u −2 − 1  − 2−s + 2s , Fs (p) = max (c + 2−s )us bs − t F (p) . −s p∗   c + 2−s i i=1

When p is close to p∗ , Fs takes the former expression in (3.4). The value at p∗ is Fs (p∗ ) = c + 2s and the function quickly decays to −2−s + 2s . When p is close to W , Fs takes the latter expression in (3.4). Except the two cases, we have Fs = −2−s + 2s . See the lower two curves in (B) of Figure 2. 3.1.4. Homotopy from s = 1 to s = −1. From s = 1 to s = 0, we use a shift     b1 p∗ − (1 − s) · 3/4, s ∈ [0, 1]. Fs (p) = F1 p − (1 − s) 1 − 2 In the horizontal direction, the translation moves the point b1 p∗ to p∗ . In the vertical direction, the function moves down by 3/4. Now we see that F0 (p) = F−1 (p), where   v u n 2  [  3/2 uX p pi − 1 ≤ δ−1 , pi < p∗i , for all i = 1, . . . , n p∈W p δ−1 < t p∗   i i=1

2

2

because of e−x · e−y = e−x F−1 .

2 −y 2

, where the LHS is for F0 and the RHS is for

Next, from s = 0 to s = −1, we use a linear homotopy Fs (p) = −sF−1 (p) + (1 + s)F0 (p),

s ∈ [−1, 0].

PERIOIDIC ORBITS OF HAMILTONIANS SEPARATING LAGRANGIAN TORI

To summarize the above, we have defined a function ((B) of Figure (3.5)    n Y pi  s  , (c + 2 ) us    p∗i  i=1        b1    p − (1 − s) 1 − F p∗ − (1 − s) · 3/4, 1   2  Fs (p) = −sF−1 (p) + (1 + s)F0 (p),s (  2 !   P  p n i  − 2−s + 2s ,  max (c + 2−s )us bs −  i=1 ∗ − 1  p  i      −2−s + 2s   F−s (p) , c + 2−s

19

2) if s ≥ 1, if s ∈ [0, 1], if s ∈ [−1, 0],

if s ≤ −1.

Notice at the points s = −1, 0, 1, the homotopy Fs is not smooth. We smoothen Fs as a function of s in neighbourhoods of the three points by localizing to a ε neighborhood of each point using a partition of unity then convolute an approximatnig Dirac-δ function of s only, which is C ∞ and compactly supported. ∞ (RT ∗ Tn × 3.1.5. The cut-off. Finally, we need to cut off Fs properly to make it in Ccpt T1 , R). We define

(3.6)

ws (x) = us ((1 − bs ) − |x|),

which is zero for |x| ≥ 1 − bs and 1 for |x| ≤ 1 − 2bs . Our profile function is defined to be   1 kpk . (3.7) Hs (p) = Fs (p) · ws R   1 The function ws kpk is radially symmetric and its radial derivatives are all R nonpositive. cc (RT ∗ Tn ; W, p∗ ). It is easy to check that Hs is an exhausting sequence for Hamiltonians in H cc (RT ∗ Tn ; W, p∗ ), there exist s > s0 such that Hs0 < H < Hs . Namely for ∀H ∈ H 3.1.6. Location of Morse-Bott manifolds. In this section, we find the ps satisfying ∂H (ps ) = α, where α satisfies the assumption of Theorem 3. The heuristics ∂p are simple. In the function u, if we want to solve u0 (x) = a ≥ 0 where a is independent √ √ of δ. For small δ, we have two roots lying in the intervals (b − δ, b), (0, δ). We do √ not expect to find solutions in the linear part of u since 0 u is either 0 or O(1/ δ) there. Lemma 3.1. Consider α ∈ H1 (Tn , Z) \ {0} as in Theorem 3. For each s there ∂Hs is a unique solution p+ (p) = α satisfying the following as δ → 0: s of ∂p

20

JINXIN XUE UNIVERSITY OF CHICAGO

• For s ≥ 1 + ε, the point p+ s satisfies p+ s,i p∗i

  p ∈ bs − δs , bs ,

s −3s Hs (p+ , s ) = c + 2 − O(δ)2

• For s ≤ −1 − ε, the point p+ s satisfies v u n 2 p uX pi s 3s t − 1 < δs , p+ ≤ p∗i , ∀i, Hs (p+ s ) = c + 2 − O(δ)2 , s,i p∗ i=1

i

where O(δ) in the above two cases are positive and independent of s. For s ∈ [−1 − ε, 1 + ε], the solution p+ s stays arbitrarily close to the following cases without smoothing with respect to s. + ∗ • As s goes from 1 to 0, the solution p+ s moves to p−1 + p with constant speed. + + ∗ • For −1 ≤ s ≤ 0, we have p+ s = p−1 = p1 + p . If there is any other solution denoted by p− s , it must satisfy − Hs (p− s ) − hps , αi < 0.

Proof. We first forget about the cut-off ws and consider only Fs . Close to the end of the proof, we study the effect of ws . We also forget about the smoothing with ∂Hs = α for respect to s when working with p+ s , since once we have a solution of ∂p the function Hs without smoothing with respect to s, we get a solution of the smoothed one using implicit function theorem. The nondegeneracy condition ∂ 2 Hs + det (p ) 6= 0 is given by the next Lemma 3.2. ∂p2 s Step 1, existence and uniqueness of p+ s. Substep 1.1, the case s ≥ 1. pi In the proof, we define yi = ∗ , i = 1, . . . , n. We consider first s ≥ 1 and p i √ yi − bs ∈ (− δs , 0). We introduce a new function (3.8) ! n Y X |yi − bs |2 p fs (y) := Fs (p) = (c+2s ) us (yi ) = (c+2s ) exp − , yi ∈ (bs − δs , bs ). 2δs i=1

i

We also forget about the smoothing when defining u for a moment for the simplicity of notations and study it in the next paragraph. Consider of P level sets 2 = fs (y) = (c + 2s ) · C where C ∈ [e−n/2 , 1]. We get a√sphere |y − b | i s i 2δs (− ln C) for each C, whose radius ranges from 0 to nδs . Next consider (3.9)

yi − bs ∂fs = fs · (ln us (yi ))0 = − C(c + 2s ) ∂yi δs

PERIOIDIC ORBITS OF HAMILTONIANS SEPARATING LAGRANGIAN TORI

21

evaluated on each level set C(c + 2s ). When y moves on the sphere, the unit 1 (bs − y1 , . . . , bs − yn ) achieves any vector of the portion of vector P 2 i |yi − bs | Sn−1 lying in the first quadrant since yi < bs . Moreover the modulus s s

X |yi − bs |2

∂fs 2(− ln C) s

, = C(c + 2s )

∂y = C(c + 2 ) 2 δs δs i

p ranges from 0 to e−1/2 (c + 2s ) 1/δs and is monotone with respect to C when √ −2 ln C − 1 C ∈ [e−1/2 , 1] since we have (C − ln C)0 = √ . This shows that the 2 − ln C ∂fs image of the map covers the first quadrant part of a ball of radius e−1/2 (c + ∂p p p 2s ) 1/δs = e−1/2 (c + 2s )2s 1/δ centered at the origin. Moreover we have P ∂fs 2 that is one-to-one in the domain {y : i |yi − bs | < δs }, since fs (p) is ∂y monotone in the radial direction centered at the point bs (1, . . . , 1). Therefore, for α ∈ H1 (Tn , Z) in Theorem 3, if δ is small enough, we can always find a ∂Fs (p) : Rn → Rn for y in the region unique preimage p+ s of α under the map ∂p P ∂Fs 2 {y : (p) = α in the i |yi − bs | < δs }. We do not expect to find any root of ∂p

p P

s −n/2 (c + 2s ) n/δ  kαk domain {y : nδs < i |yi − bs |2 ≤ δs } since ∂f s ∂y ≥ e there. Substep 1.2, the smoothing. 3/2

In the definition of us , within a δs

neighbourhood of b smoothing s where the  2 b − x |x − b | s s takes effect, u0s goes monotonically from 0 to exp − which is δs 2δs √ √ bounded by δ s , so that the partial derivatives of fs are bounded by O( δ) when c + 2s is considered, which cannot be αi > 0, ∀i for δ small enough. The √ 3/2 smoothing in a δs neighbourhood of bs − δ s does not create new root either −1/2 since u0s ≥ const.δs , which is too large to be α/(c + 2s ). Substep 1.3, the cases s ≤ −1 and s ∈ [−1, 1]. To show the existence and uniqueness of the p+ s in the second bullet point in the lemma, we apply the same argument to the former expression in (3.4) after “max” when s ≤ −1 by considering the function obtained from fs with bs replaced by 1. The solution p+ s is again unique since we require αi > 0, which forces yi − 1 < 0 (see (3.9) with bs replaced by 1). The smoothing does not produce new roots for the same reason as the previous substep. Moreover, the latter function in (3.4) after “max” does not create any root since all of its partial derivatives is −2−s + 2s nonpositive due to the negative factor , which cannot be αi . c + 2−s

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JINXIN XUE UNIVERSITY OF CHICAGO

The homotopy from s = 1 to s = 0 does not change the derivative of Fs up to a translation of p. Moreover, the linear homotopy from s = 0 to s = −1 does not ∂Fs produce new solutions of (p) = α. ∂p Step 2, the value Fs (p+ s ). + Next, we evaluate Fs (p+ s ). It follows from (3.9) that at ps we have

bs − yi = δs αi p∗i /fs .

(3.10)

We plug this back to the expression of Fs (3.8) to get that as δ → 0 and s ≥ 1, s s −2 s −3s Fs (p+ . s ) = (c + 2 )(1 − O(δs )(c + 2 ) ) = (c + 2 ) − O(δ)2

For s ≤ −1 with bs replaced by 1 in (3.8), we have −s Fs (p+ − 2s = (c + 2−s )(1 − O(δs )(c + 2−s )−2 ) = (c + 2−s ) − O(δ)23s . s )+2

This completes the proof of the p+ s part statement. Step 3, the inequality satisfied by p− s. Substep 3.1, the case s ≥ 1. √ 3/2 When s ≥ 1, we consider possible roots p− / (bs − δs − δs , bs ) for s with yj ∈ some j. Again consider the function fs (3.8). We have the calculation u0 (yj ) ∂fs = s fs (y) = p∗j αj . ∂yj us (yj )

(3.11) We then get − Fs (p− s ) − hps , αi = fs (y) −

X

yi p∗i αi < fs (y) − yj p∗j αj ≤ fs (y)

i

us (yj ) − yj u0s (yj ) . us (yj )

R yj

Notice us (yj ) = 0 u0s (t) dt < yj u0s (yj ) since u0s is monotone and u00s ≥ 0 in the − domain of yj under consideration. This shows Fs (p− s ) − hps , αi < 0. Substep 3.2, the cases s ≤ −1 as well as s ∈ [−1, 1]. Next Possible roots p− / s must have yj − 1 ∈ √ we consider the case of s ≤ −1. √ ∗ (− δs , 0) for some j so that ky − 1 k ≥ δs where 1∗ := (1, 1, . . . , 1) ∈ Rn . We need to invoke the former expression of (3.4) after “max” (the other one is excluded in Substep 1.3), fs (y) = Fs (p) = (c + 2−s )us (bs − ky − 1∗ k) − 2−s + 2s . We denote by xs = bs − ky − 1∗ k and take derivative directly we get ∂fs (−yi + 1) = (c + 2−s )u0s (xs ) = p∗i αi , ∂yi ky − 1∗ k

∀i.

PERIOIDIC ORBITS OF HAMILTONIANS SEPARATING LAGRANGIAN TORI

This gives us

sX

∂fs

= (c + 2−s )u0s (xs ) =

(p∗i αi )2 = O(1),

∂y

23

as δ → 0.

i

√ 3/2 3/2 √ / [bs − δs , bs ], otherwise This shows that xs can only be in [δs , δs −δs ] if xs ∈

∂fs −1/2 1/2

∂y = O(δs ) or O(δs ). We next invoke the fourth one in (3.2) to get     p |xs |2 −s + 1 − 2−s + 2s , for xs ∈ [δs3/2 , δs − δs3/2 ]. fs (y) = (c + 2 ) − exp − 2δs Taking derivative directly we get   ∂fs |xs |2 xs (−y + 1∗ ) −s = (c + 2 ) exp − . ∂y 2δs δs ky − 1∗ k

∂fs

= O(1) Since the exponential term is bounded from below by from ∂y as δ → 0, we get xs = O(δs )/(c + 2−s ), ∀i. Plugging this back to fs we get for δ small and all s ≤ −1 that e−1/2 ,

−s −s Fs (p− + 2s = O(δ)23s − 2−s + 2s < −1 < 0. s ) ' O(δs )/(c + 2 ) − 2 n − − We always have p− s ∈ R \ W so that hps , αi > 0 hence Fs − hps , αi < 0.

The case s ∈ [−1, 1] is obtained by the above two cases s = ±1. The smoothing with respect to s can be made such that the smoothed Fs is sufficiently close to the nonsmoothed one in C 1 (RT ∗ Tn × T1 , R) norm, so that the deviation of the root p− s from the non smoothed case is also sufficiently small using implicit − function theorem. We get Fs (p− s ) − hps , αi < 0 for the smoothed Fs with s ∈ [−1 − ε, 1 + ε]. Step 4, the cut-off ws . Finally, let us show that the cut-off ws does not create any root of the equation ∂Hs = α. We only need to consider the region where p ∈ Rn \W and kpk/R ' 1. ∂p First consider when Fs (p) ≥ 0, hence s ≥ −1. We have ∂Hs ∂Fs p = ws + Fs ws0 . ∂p ∂p kpkR Since we have kpk/R ' 1, there must be at least one j such that pj > bs p∗j , so ∂Fs ∂Hs that u0s (pj /p∗j ) = 0 hence = 0. Therefore we have ≤ 0 for this j, since ∂pj ∂pj p all the entries of Fs ws0 are nonpositive. However, we require αi > 0. kpkR Next consider the case Fs ≤ 0. We have Fs − hp, αi < 0 since Fs < 0 and all the entries of p and α are positive. This completes the proof. 

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3.1.7. Hessian isnondegenerate. In this section, we show that the condition in  ∂2H + (p+ Lemma 2.5, det s ) 6= 0 is satisfied at the point ps in Lemma 3.1. We ∂p2 have the following lemma. Lemma 3.2. Consider α ∈ H1 (Tn , Z) satisfying the assumption of Theorem 3 ∂Hs and the point p+ (p) = α. Then we have s in Lemma 3.1 solving the equation ∂p ∂ 2 Hs + (p ) is negative definite ∀s. for δ small enough and fixed, the matrix ∂p2 s Proof. We know from Lemma 3.1 that p+ s is not close to the boundary of {kpk ≤ R} so we have Hs = Fs . We forget about the smoothing with respect to s for a moment. As in the proof of Lemma 3.1, we consider   the function (3.8). Once ∂ 2 fs 1 we get (y) we multiply the matrix diag ∗ to both the left and right ∂yi ∂yj pi ∂ 2 Fs + of it to get (p ), which does not change the signature. We have ∂pi ∂pj s  0 us (yi )u0s (yj )   , i 6= j,  2 ∂ fs us (yi )us (yj ) (y) = fs · u00 (yi )  ∂yi ∂yj  , i = j.  s us (yi ) We can rewrite

(3.12)

∂ 2 fs (y) as a matrix form ∂yi ∂yj

∂ 2 fs (y) = fs · (Λ + V ⊗ V ), where ∂yi ∂yj  00  us (yi ) (u0s (yi ))2 1 Λ = diag − = diag{(ln us (yi ))00 } = − id, us (yi ) (us (pi ))2 δs yi − bs u0 (yi ) = (ln us (yi ))0 = − . Vi = s us (yi ) δs

Using (3.10), we see that the matrix V ⊗ V is O(1)(c + 2s )−2 as δ → 0 where O(1) does not depend on s. As a result the Hess Fs is diagonally dominant and negative definite. The case s ≤ −1 follows the same line of argument. The case Fs with s ∈ [−1, 1] is only a translation of F1 , which does not change the Hessian. When the smoothing with respect to s is taken into account, we have the same calculation as (3.12) except that we need to convolute with an approximating Dirac-δ in the s variable. We make the smoothed Fs be sufficiently close to the nonsmoothed one in C 1 (RT ∗ Tn × T1 , R) norm so that the deviation of the root p+ s from the non smoothed case is  δ using implicit function theorem, which implies V ⊗ V = O(1) for |s| < 2 using the calculation of Vi in (3.12), so that HessFs is still diagonally dominant. 

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25

4. Proof of Theorem 3 In this section, we proof Theorem 3 using Lemma 3.1 and the machinery set up in Section 2. 4.1. Computation of the action. We obtain Morse-Bott manifolds corre± sponding to p± s denoted by Ps . These Morse-Bott manifolds are Lagrangian n tori T . Along each periodic orbit x ⊂ Ps± we evaluate the action Z 1 Hs − hp, qidt. ˙ AHs (x) = 0

For our profile function Hs , we have q˙ =

∂Hs = α and pi > 0 for all i. We have ∂p

the following four cases. • Case 1, the action of Ps+ when s ≥ 1 + ε. The value of the profile function Hs (p+ s ) is obtained in Lemma 3.1, + and ps → 0 as δ → 0 or s → ∞. The action is estimated as s AHs (Ps+ ) = c + 2s − O(δ)2−3s − hp+ s , αi → c + 2 ,

δ → 0.

Ps+

• Case 2, the action of when s ≤ −1 − ε. The value of the profile function Hs (p+ s ) is also obtained in Lemma ∗ as δ → 0 or s → ∞. The action is estimated as → p 3.1, and p+ s s ∗ AHs (Ps+ ) = c + 2s − O(δ)23s − hp+ s , αi → c + 2 − hp , αi,

δ → 0.

• Case 3, the action of Ps+ when −1 − ε ≤ s ≤ 1 + ε. Consider first Fs in (3.5) without smoothing with respect to s. As s goes from 1 to 0, the point p+ s moves from a neighbourhood of 0 to a neighbourhood of p∗ with linear speed, so we get the action is 1 1 − O(δ) − hp+ − (1 − s)hp∗ , αi, δ → 0. s , αi → c + 2 2 When s goes from 0 to −1, the linear homotopy does not influence a neighbourhood of p+ s , so the action is the same as s = 0 case. The smoothing of Fs with respect to s around the points ±1, 0 add only an error to the action that can be made as small as we wish using implicit function theorem. The O term in the above cases are positive. • Case 4, the action of Ps− . − Using the last statement in Lemma 3.1, we get Fs (p− s ) − hps , αi < 0, − − so we get for ps , the action satisfies AHs (Ps ) < 0.

AHs (Ps+ ) = c +

4.2. Proof of the main theorem. Proof of the main Theorem 3. There are 4 steps.

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Step 1. If 0 ≤ a ≤ c − hp∗ , αi, then SH[a,∞) (T ∗ Tn ; α) ∼ = H∗ (Tn ; Z2 ) for ∀s ∈ R. ←− Moreover, the homomorphism πs : SH[a,∞) (T ∗ Tn ; α) → HF[a,∞) (Hs ; α) ←− is an isomorphism whenever Hs (p∗ ) > c. We use our action calculation in Section 4.1. Notice that component-wisely ∗ p+ s,i < pi , ∀s ∈ R, ∀i we have ∗ hp+ s , αi < hp , αi. ∗ This means that when c − hp+ s , αi > c − hp , αi ≥ a, we have

(4.1)

AHs (Ps− ) < 0 ≤ a ≤ c − hp∗ , αi < AHs (Ps+ ),

∀s ∈ R

when a satisfies 0 ≤ a ≤ c − hp∗ , αi. Hence, by Theorem 7, HF[a,∞) (Hs ; α) ∼ = H∗ (Tn ; Z2 ) since the Morse-Bott manifold Ps+ is a torus, and by Proposition 2.1 the monotone homomorphism σFs1 Fs0 : HF[a,∞) (Hs0 ; α) → HF[a,∞) (Hs1 ; α) is an isomorphism. Hence Step 1 follows from Lemma 2.2 (ii). Step 2. If a > c − hp∗ , αi, then SH[a,∞);c (T ∗ Tn ; α) = 0 for s  −1. −→ For s  −1, using the Step 2 in Section 4.1, we get that both the sets Ps± have actions less than a for −s sufficiently large. Hence HF[a,∞) (Hs ; α) = 0 for −s sufficiently large. Hence Step 2 follows from Lemma 2.2 (i). Step 3. If 0 ≤ a ≤ c − hp∗ , αi, then SH[a,∞);c (T ∗ Tn ; α) ∼ = H∗ (Tn ; Z2 ). Moreover, −→ the homomorphism ιs : HF[a,∞) (Hs ; α) → SH[a,∞);c (T ∗ Tn ; α) −→ is an isomorphism for s  −1. We use the calculation in Case 1 of Section 4.1 again to obtain the same inequality (4.1) in Step 1. By Theorem 7, we have HF[a,∞) (Hs ; α) ∼ = H∗ (Ps+ ; Z2 ). By Proposition 2.1, the monotone homomorphism σHs1 Hs0 : HF[a,∞) (Hs0 ; α) → HF[a,∞) (Hs1 ; α) is an isomorphism. Step 3 follows now from Lemma 2.2 (i). Step 4. If 0 ≤ a ≤ c − hp∗ , αi then the homomorphism Tα[a,∞);c : SH[a,∞) (T ∗ Tn ; α) → SH[a,∞);c (T ∗ Tn ; α) ←− −→ is an isomorphism. According to its definition, Hs (p∗ ) > c for every s. Hence, by Step 1, πs is an isomorphism for every s ∈ R. Moreover, by Step 3, ιs is an isomorphism for [a,∞);c [a,∞);c every s. By Proposition 2.3, Tα = ιs ◦ πs for every s. Hence Tα is an isomorphism.

PERIOIDIC ORBITS OF HAMILTONIANS SEPARATING LAGRANGIAN TORI

27

∗ Tn ; W, p∗ ; α) in (2.14), we get that b According to the definition of C(RT ∗ n b C(RT T ; W, p∗ ; α) = a + hp∗ , αi.

According to Proposition 2.4, we get ∗ n b C(RT ∗ Tn ; W, p∗ ; α) ≤ C(RT T ; W, p∗ ; α) = a + hp∗ , αi < ∞. This implies that periodic orbits exist as claimed in Theorem 3. The proof is complete.  5. Proof of Theorem 2, 5 and 6 5.1. More general type of wedges. In this section, we prove Theorem 2. Proof of Theorem 2. Let us consider more general wedges than W . Suppose the cone C is positively spanned by vectors v1 , . . . , vn assumed in the statement. We put these vectors together as columns of a matrix A. This matrix A transforms the first quadrant to the given cone C. We suppose p = AP and correspondingly q = (AT )−1 Q. so that the transformation (P, Q) 7→ (p, q) is symplectic. We start with Hamiltonian systems H(p, q) ∈ Hc (RT ∗ Tn ; Rn \ C, p∗ ) where q ∗ lies in the interior of the cone C and W in (1.3) is replaced by Rn \ C. After the symplectic transformation above induced by A, we get a Hamiltonian system h(P, Q) = H(AP, (AT )−1 Q) in Hc (R0 T ∗ Tn ; W, p∗ ) for some large R0 . Consider homology class α ∈ H1 (Tn , Z) \ {0} for the (p, q) coordinates. So the homology class correspondingly for (P, Q) is AT α using q = (AT )−1 Q. If we have hP ∗ , AT αi ≤ c where P ∗ = (AT )−1 p∗ and all the components of AT α are positive, then Theorem 3 implies that there is a 1-periodic orbit of the system T α. Theorem 3 should be applied with the group h(P, Q) in the homology class A n H1 (T , Z) finitely generated by AT e1 , . . . , AT en over Z, where ei , i = 1, . . . , n, are the standard basis of Rn . Now we go back to the H(p, q) system. We have hp∗ , αi = hAP ∗ , αi = hP ∗ , AT αi ≤ c. We need to assume that all the components of AT α are positive in order to apply Theorem 3. This AT α ∈ span+ {e1 , . . . , en }. This is equivalent to saying  means T −1 that α ∈ span+ (A ) e1 , . . . , (AT )−1 en . This completes the proof.  5.2. Dense existence. In this section we prove Theorem 5. The argument follows that of Theorem 3.4.1 of [BPS]. Proof of Theorem 5. We show for each a, b satisfying minq H(p∗ , q) > b > a > 0, there exists s ∈ (a, b) such that the level set {H = s} carries a closed orbit in the class α. Define a smooth function σ : R → R with the following properties: • σ(r) = 0, for r ≤ 0,

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JINXIN XUE UNIVERSITY OF CHICAGO

• σ(r) = 1, for r ≥ 1, • σ 0 (r) > 0, for 0 < r < 1.  H −a in Hc (RT ∗ Tn ; W, p∗ ). Then we get a compactly supported function F := cσ b−a We apply Theorem 3 to get that F has a 1-periodic orbit x in the class α. The orbit x lies on a level set of {F = ρ} where ρ ∈ (0, c). Since σ takes the interval (a, b) injectively to (0, c), there exists s ∈ (a, b) such that {F = ρ} = {H = s}, hence x lies on the level set {H = s}.  

5.3. Arnold’s problem. In this section, we show Theorem 6. Proof of Theorem 6. Our Hamiltonian system (1.6) is related to Arnold’s original one through the following symplectic transformation           1 1 p1 1 −1 p1 q1 1 −1 q1 √ √ 7→ , 7→ . p2 p2 q2 q2 2 1 1 2 1 1 Suppose we want to find periodic orbits in an homology class α with αi > 0, ∀i = 1, 2, . . . , n. We can always find p∗ ∈ Rn \ W and a, b such that (5.1)

H(p∗ , q) = p∗1 p∗2 + V (q) ≥ p∗1 p∗2 − M > b > a > M.

We choose c = hp∗ , αi and define a Hamiltonian function using σ in Section 5.2    c · σ H(p, q) − a · w (kpk/R), p ∈ Rn \ W, 0 b−a F (p, q) =  0, p ∈ W. where w0 (kpk/R) is the cut-off function introduced in (3.6) with s = 0. We see easily F ∈ Hc (RT ∗ Tn ; W, p∗ ) using (5.1). We apply Theorem 3 to F to get that there exists a periodic orbit of F in the homology class α with period one. Let us assume for a moment that the periodic orbit is not created by w0 6= 1, namely kpk/R is not close to 1. We get a periodic orbit on the energy level {H = s} where s ∈ (a, b). Since b > a can be arbitrary numbers greater than hp∗ , αi. We also get dense existence. Namely, there exists a dense subset Sα of (M, p∗1 p∗2 − M ), such that for each s ∈ Sα , the energy level {H = s} contains a periodic orbit with homology class α. The argument can be done for any p∗ ∈ Rn \ W satisfying (5.1), so we get dense existence in the set of energy levels (M, ∞). Once p∗ is chosen, we need to choose R much larger than p∗ . Finally, we show that the periodic orbit is not created by w0 6= 1. We assume kpk/R ' 1. We only need to consider p ∈ Rn \W , since F (p, q) = 0 when p ∈ W .

PERIOIDIC ORBITS OF HAMILTONIANS SEPARATING LAGRANGIAN TORI

29

We have the Hamiltonian equations −c 0 ∂V ∂F = σ · w0 · , p˙ = − ∂q b−a ∂q   (5.2) ∂F p c ∂H 0 q˙ = + cσ · w0 · = σ · w0 · . ∂p b−a ∂p kpkR   H(p, q) − a = 1 Consider first p1 p2 > b + M , then H(p, q) > b so that σ b−a and σ 0 = 0. So we get p˙ = 0, and q˙ have nonpositive entries since p ∈ Rn \ W and w0 ≤ 0. In this case, the homology class of a periodic orbit of F cannot be α whose entries are positive. Notice once a periodic orbit enters the region {p1 p2 > b + M }, it always stays there because of p˙ = 0 and the periodicity. It remains to consider a periodic orbit with p1 p2 ≤ b + M during time 1. When kpk/R ' 1, since p˙ is bounded, we must have for all time either p1 ≤ 2(b + M )/R, p2 ≥ R/2,

or

p2 ≤ 2(b + M )/R, p1 ≥ R/2.

c ∂H σ 0 · w0 in front of = (p2 , p1 )T b−a ∂p is bounded, and the second term has nonpositive entries. For large enough R, either p1 or p2 is close to zero. However, since we assume α ∈ H1 (Tn , Z), α1 > 0, α2 > 0, a 1-periodic orbit of F with p1 p2 ≤ b + M and kpk/R ' 1 cannot have homology class α for R large enough. This completes the proof. 

In the q˙ equation of (5.2), the factor

Acknowledgment I would like to thank Prof. L. Polterovich for introducing me to the problem, valuable suggestions and constant encouragements. I would also thank Prof. A. Wilkinson for her interests in the work and her patience to read the manuscript for motivating us to think about the dense existence and discover Theorem 6. My thanks also go to my former coadvisor V. Kaloshin from whom I learnt the problem of Arnold. References [A]

Arnold, V. I. “Mathematical problems in classical physics”. Trends and perspectives in applied mathematics, Appl. Math. Sci., vol. 100, Springer, New York, (1994), pp. 1-20. [BPS] Biran, Paul, Leonid Polterovich, and Dietmar Salamon. “Propagation in Hamiltonian dynamics and relative symplectic homology.” Duke Mathematical Journal 119.1 (2003): 65-118. [FHS] Floer, Andreas, Helmut Hofer, and Dietmar Salamon. “Transversality in elliptic Morse theory for the symplectic action.” Duke Mathematical Journal 80.1 (1995): 251-292. [G] Ginzburg, Viktor L. “The Weinstein conjecture and theorems of nearby and almost existence.” The breadth of symplectic and Poisson geometry. Birkhuser Boston, 2005. 139-172. [GL] Gatien, Daniel, and Franois Lalonde. “Holomorphic cylinders with Lagrangian boundaries and Hamiltonian dynamics.” Duke Mathematical Journal 102.3 (2000): 485-512.

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[HV] [L] [M] [P] [V] [SW]

[W]

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Hofer, Helmut, and C. Viterbo. “The Weinstein conjecture in the presence of holomorphic spheres.” Communications on pure and applied mathematics 45.5 (1992): 583-622. Lee, Yi-Jen. “Non-contractible periodic orbits, Gromov invariants, and Floer-theoretic torsions.” arXiv preprint math/0308185 (2003). Mather, John N. “Action minimizing invariant measures for positive definite Lagrangian systems.” Mathematische Zeitschrift 207.1 (1991): 169-207. Polterovich, Leonid. “symplectic intersections and invariant measures”, Annales math´ematiques du Qu´ebec (2014) Vichery, Nicolas. “Spectral invariants towards a Non-convex Aubry-Mather theory”, arXiv preprint math/1403.2058(2014) Saloma˜ o, Pedro A.S., Weber, Joa. “An almost existence theorem for non-contractible periodic orbits in cotangent bundles”, S˜ ao Paulo Journal of Mathematical Sciences, 6, no. 2, (2012), pp. 385-394. Weber, Joa. “Noncontractible periodic orbits in cotangent bundles and Floer homology.” Duke Mathematical Journal 133.3 (2006): 527-568.