Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles
Wenmin Gong
math.SG math.DG math.DS 53D40, 37J45, 70H12























Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles]Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles


Wenmin Gong]Wenmin Gong


School of Mathematical Sciences, Beijing Normal University,
    Beijing, 100875, China wmgong@bnu.edu.cn

[2010]53D40; 37J45; 70H12





The work is partially supported by  NSFC 11701313, the Fundamental Research Funds for the Central Universities 2018NTST18 and by China Postdoctoral Science Foundation grant 2017M620725.



 In this paper we establish the existence of periodic orbits belonging to any σ-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact 2-form σ admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form.

As a consequence, we also prove that, for any nontrivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran-Polterovich-Salamon capacity.




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    December 30, 2023
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§ INTRODUCTION


Let (M,g) be a closed connected Riemannian manifold with cotangent bundle π:T^*M→ M. Denote by ω_0=-dλ the canonical symplectic form on T^*M, where λ=pdq is the Liouville 1-form in  local coordinates (q,p) of T^*M. Let (M̃,g̃) be the universal cover of (M,g). For a closed 2-form σ∈Ω^2(M) we denote by σ̃ the lift of σ to M̃, and say that σ is weakly exact if σ̃ is exact. Let D_RT^*M denote the open disc cotangent bundle of finite radius R with respect to the metric g.  Denote by Ω_w^2(M) the set of all weakly exact 2-forms on M. For each σ∈Ω_w^2(M) we denote

    ω_σ:=ω_0+π^*σ

the twisted symplectic form, and call the symplectic manifold (T^*M,ω_σ) the twisted cotangent bundle and (D_RT^*M,ω_σ) the twisted disc bundle.


For a smooth function H:S^1× T^*M→ℝ we are interested in the existence of periodic solutions of the associated Hamiltonian system

    ẋ(t)=X_H,σ(t,x(t))  ∀ t∈ S^1,

where the Hamiltonian vector field X_H,σ on T^*M is determined by dH_t=ι(X_H,σ)ω_σ. The existence problem of contractible Hamiltonian periodic solutions on the twisted cotangent bundle has been studied in a number of papers; see, e.g., <cit.>. On the other hand, non-contractible periodic orbits of the Hamiltonian systems on the ordinary cotangent bundle have been previously investigated in several papers; see, e.g., <cit.>.




The aim of the present paper is to study certain existence results for
1-periodic orbits in given free homotopy classes of loops for compactly supported Hamiltonians on twisted cotangent bundles. The proof relies on the machinery of Floer homology for non-contractible periodic orbits.
In the course of the last two decades, this version of Floer homology has been studied and used among others
in 2003 by Biran, Polterovich and Salamon <cit.> on T^*M for
M=𝕋^n or M closed and
negatively curved, in 2006 by Weber <cit.> on T^*M for all closed Riemannian manifolds M, in 2013 by Gürel <cit.> on closed symplectic manifolds for simple non-contractible periodic orbits of Hamiltonian diffeomorphisms with arbitrarily large periods  and in 2016 by Ginzburg and Gürel <cit.> on closed symplectic manifolds for infinitely many non-contractible periodic orbits . For further references
concerning the existence of non-contractible orbits we refer to <cit.>.

The main difficulty in applying this technique to detect non-contractible periodic orbits of the Hamiltonian flow of H is to compute the filtered Floer homology HF_α^(a,b)(H). It is well known that the method adopted by
Biran, Polterovich and Salamon in <cit.> is applied successfully to many other situations <cit.>. The basic strategy is to examine the commutative diagram


    HF_α^(a,b)(H_0)
    [rr]^Ψ_H_1H_0[dr]_Ψ_HH_0       HF_α^(a,b)(H_1)
        HF_α^(a,b)(H)[ur]_Ψ_H_1H

where H_0≤ H≤ H_1 and H_0 and H_1 are time-independent Hamiltonians with HF_α^(a,b)(H_0) and HF_α^(a,b)(H_1)
being not hard to compute. However, in our cases, one can not directly use the method mentioned above which is based on Pozniak's Theorem <cit.> since the sets of critical points of the symplectic action functionals of the squeezing functions H_0 and H_1, in general, are not Morse-Bott in the sense of  <cit.>. Instead, we interpret the Floer homology in the twisted cotangent bundle as a “small perturbation” of the Floer homology in the ordinary cotangent bundle and show the isomorphism between them.  Of course we shall have to make suitable assumptions about the twisted symplectic forms.  For a nice introduction to the theories of deformations of Floer homology on a closed symplectic manifold under symplectic perturbations we refer to the papers by Ritter <cit.>, Zhang <cit.> and Bae and Frauenfelder <cit.>. It is worth to mention that the monotone homomorphisms (see Section <ref>) are preserved under the  isomorphisms which we obtain in this paper. This observation, together with the computation of Floer homology given by Weber <cit.>, helps us to circumvent the difficulty of computing the Floer homology on the twisted disc bundle directly. Our results can be seen as a twisted version of Weber's results <cit.> which, in particular, recover a result of Niche <cit.>.




 §.§ Main results





 A 1-form τ∈Ω^1(M̃) is said to be a primitive of σ with at most linear growth on the universal cover M̃ if dτ=σ̃ and, for any x∈M̃, there exists a constant ϱ_x>0 such that for all s≥ 0,


    sup_z∈ B_x(s)τ_z_g̃≤ϱ_x(s+1).


Here B_x(s) denotes the geodesic ball in M̃ of radius s>0 centered at x. Since M is compact, the above definition is independent of the choice of the metric g on M. Put



    u_σ,g,x(s)=inf_τ∈ℱ_σsup_z∈ B_x(s)τ_z_g̃  ∀ s>0,

where ℱ_σ={τ∈Ω^1(M̃)|dτ=σ̃}.
We call u_σ=u_σ,g,x:ℝ_+→ℝ_+ the cofilling function of σ∈Ω_w^2(M); see <cit.>. If g^' is another Riemannian metric on M and x^'∈M̃ is a different base point then it is easy to check that L^-1u_σ,g,x≤ u_σ,g^',x^'≤ L u_σ,g,x for some constant L>0.


We define the set 𝒫(M) as

    𝒫(M):={σ∈Ω_w^2(M)| there exists a constant ϵ>0 such that  u_σ(s)≤ϵ(s+1)}.

Notice that
 each σ∈𝒫(M), by definition, admits a primitive with at most linear growth on the universal cover M̃, and hence σ|_π_2(M)=0. So ω_σ is symplectically aspherical, which means that the symplectic form ω_σ vanishes on π_2(T^*M).


Let us list some manifolds on which there exists a weakly exact 2-form admitting a primitive with at most linear growth on the universal cover; see, e.g., Gromov <cit.>, Polterovich <cit.>, Sikorav <cit.> and references therein.


  
  * If a Riemannian manifold M is closed and of non-positive curvature then every closed 2-form σ on M admits a primitive with at most linear growth on M̃. Moreover, whenever M admits a metric of negative curvature, σ̃ admits a bounded primitive, that is, there exists a 1-form τ∈Ω^1 (M̃) such that dτ=σ̃ and
      
    sup_x∈M̃τ_x<∞.

      In particular, from (1) we have the following examples:
  
  * Assume that M is a closed oriented Riemannian surface with infinite fundamental group. Then the volume form on M admits a primitive with at most linear growth on M̃.
  
  * For the standard sympletic torus (𝕋^2n,∑_i=1^2ndx_i∧ dy_i) it holds that ∑_i=1^2ndx_i ∧ dy_i∈𝒫(𝕋^2n). In fact, for n-dimensional tori 𝕋^n any closed non-exact 2-form σ admits a primitive with at most linear growth on the universal cover ℝ^n but does not satisfy (<ref>); see <cit.>.







When (M,g) is a closed Riemannian manifold of negative curvature, the constant ϵ=ϵ(σ) in 𝒫(M) can be chosen to converge to zero as |σ|_g→0 Hereafter we denote |σ|_g def=sup_x∈ Mσ(x)_g. Indeed, there exists a universal constant ρ>0 such that every closed 2-form β∈Ω^2(M) with |β|_g≤ 1 satisfies u_β(s)≤ρ, ∀ s∈[0,∞); see Gromov <cit.>. Then for any closed 2-form σ∈Ω^2(M) the rescaling form σ̂=σ/|σ|_g satisfies |σ̂|_g=1, σ̂∈𝒫(M) and u_σ̂(s)≤ρ. This implies that u_σ(s)≤ϵ(σ):=ρ|σ|_g and ϵ(σ)→ 0 as |σ|_g→ 0.



Let S^1=ℝ/ℤ, and we denote the free loop space of M by 𝔏 M:=C^∞(S^1,M). Given a free homotopy class α∈[S^1,M] we define

    𝔏_α M:={γ∈𝔏 M|[γ]=α}.

We use the symbol Λ_α for the set  of lengths of all periodic geodesics in M with respect to g which represent α. It is a closed and nowhere dense subset of ℝ (see <cit.>), and hence there exists a periodic geodesic q such that

    l_α:=∫_S^1q̇(t)_gdt=infΛ_α.




Let σ be a two-form on M. A class α∈ [S^1,M] is said to be a σ-atoroidal class if any map f:S^1→𝔏_α M, thought of as a map f:𝕋^2→ M, satisfies ∫_𝕋^2f^*σ=0.


Note that if σ is weakly exact, then the class 0 of nullhomotopic loops is atoroidal, since both statements are equivalent to the statement that σ|_π_2(M)=0.

If M admits a metric of negative curvature, then every smooth map f:𝕋^2→ M induces the zero map
f^*:H^2_dR(M,ℝ)→ H_dR^2(𝕋^2,ℝ); see <cit.> or <cit.>. Therefore, for any closed 2-form σ∈Ω^2 (M), every free homotopy class α∈ [S^1,M] is a σ-atoroidal class. Another example is the 3-dimensional torus 𝕋^3={(x,y,z)|x,y,z∈ℝ/ℤ} with σ=dx∧ dy. It is easy to check that the homotopy class α=(0,0,n) for any integer n is σ-atoroidal, see also <cit.>.



Throughout this paper, for the sake of brevity, we always assume that σ∈𝒫(M) and α∈[S^1,M] is a σ-atoroidal class without any special statement. Given a homotopy class α of free loops in M, denote by 𝒫_α(H,σ) the set of periodic orbits of X_H,σ:

    𝒫_α(H,σ)={x∈ C^∞(S^1,T^*M)|ẋ(t)=X_H,σ(t,x(t))  ∀ t∈ S^1, [π(x)]=α}.

Our first main result establishes the following existence of non-contractible periodic orbits for compactly supported Hamiltonians on twisted cotangent bundles.


Let (M,g) be a closed connected Riemannian manifold, and let D_RT^*M be the open disc cotangent bundle of finite radius R with respect to the metric g. Assume that σ∈𝒫(M) and α∈[S^1,M] is any σ-atoroidal class. For every compactly supported Hamiltonian H∈ C^∞(S^1× D_RT^*M) with

    sup_S^1× MH< -Rl_α

there exists δ_0(H,g,σ,α)>0 such that if |δ|<δ_0(H,g,σ,α) then 𝒫_α(H,δσ)≠∅.


Moreover, if M is a closed and negatively curved manifold we have the following.

Assume that (M,g) is a closed Riemannian manifold of negative curvature, and that α∈[S^1,M] is any free homotopy class. Denote by σ a closed 2-form on M. Let H∈ C^∞(S^1× D_RT^*M) be a compactly supported function.
Then there exist some positive constants C=C(α) and δ_0=δ_0(H,g,α) such that if

    inf_S^1× MH>C
 then 𝒫_-α(H,σ)≠∅ whenever |σ|_g<δ_0.



In <cit.> the author used Pozniak's theorem to compute Floer homology of the squeezing functions K^± for a twisted cotangent bundle T^*M of a negatively curved manifold M, see <cit.>. He proved that the set 𝒫_ω(ρ) of 1-periodic orbits of Hamiltonian K^+ with respect to the twisted symplectic form ω is homeomorphic to S^1, see lines 20-25, page 627 in <cit.>. However, it seems very difficult for us to verify the condition that 𝒫_ω(ρ) is a Morse-Bott manifold, meaning that C_0={x(0):x∈𝒫_ω(ρ)} is a compact submanifold M and T_x_0C_0= Ker(dϕ_1(x_0)-Id) for every x_0∈ C_0, where ϕ_1 is the time one flow of K^+ with respect to ω (not to ω_0!). This condition is the key to make use of Pozniak's theorem. For this reason, we devise a different method to show Niche's result, which is a partial motivation of our work.  




Theorem <ref> (resp.Theorem <ref>) is a soft consequence of the invariance of Floer homology under symplectic deformations (see Theorem <ref> (resp. Theorem <ref>)).
As a result, to obtain periodic orbits one needs to bound the magnitude of the weak exact 2-form in terms of H. To get rid of the dependence on H, we introduce a class of Hamiltonian functions compactly supported in D_RT^*M, and show the finiteness of a symplectic capacity defined by it. To state this result, we need to put more restrictive assumptions on H, and the capacity defined here is slightly different from the Biran-Polterovich-Salamon (BPS) capacity (for the original definition see <cit.>). These additional constraints on the class of Hamiltonians to define the capacity are natural because we will apply the Theorem <ref> (resp. Theorem <ref>) to two fixed Hamiltonians sandwiching at the same time the whole class of functions to estimate the capacity, see Figure <ref>.



Let us denote 𝒫_α(H,σ;τ) the set of τ-periodic orbits of X_H,σ representing α∈[S^1,M] (identifying S^1  with ℝ/(τℤ)). We say a periodic orbit x∈𝒫_α(H,σ;τ) fast if 0<τ≤ 1; otherwise, we say it slow.
Let W be an open subset of T^*M containing M, and let U be an open subset of T^*M with compact closure U̅⊂ W.
Given a number A>0, denote by ℋ(W,U,A) the class of smooth functions H:W→ℝ such that

  
  (H0) H(x)≤ 0, for all x∈ W;
  
  (H1) H is compactly supported in U; and
  
  (H2) inf_WH>-A.

Let M⊂ V⊂ U, where V is an open subset of T^*M. For c∈(0,A], denote

    ℋ_c(W,U,V,A):={H∈ℋ(W,U,A) | sup_VH≤-c }.

Then the restricted BPS capacity ĉ_ BPS is defined as

    ĉ_ BPS(W,U,V,A;σ,α)=inf{c>0|∀ H∈ℋ_c(W,U,V,A), there is a fast periodic of H}.

Here we use the convention that inf∅=∞.

It is easy to check that if ĉ_ BPS(W,U,V,A;σ,α)<∞, then every H∈ℋ(W,U,A) satisfying sup_VH≤-ĉ_ BPS(W,U,V,A;σ,α) has at least a fast periodic orbit (with respect to the twisted symplectic form ω_σ=ω_0+π^*σ) whose projection to M represents α.


In what follows, let us denote U_R:=D_RT^*M,
let V be an open subset of U_R-ρ containing M with 0<ρ<R.



Let (M,g) be a closed connected Riemannian manifold. Denote V,A,ρ as above.  Assume that σ∈𝒫(M) and α∈[S^1,M] is any σ-atoroidal class. If A>Rl_α, then for any sufficiently small number ϵ>0 there exists a constant δ_0=δ_0(g,σ,α, V,A,ρ,ϵ)>0 such that for every δ∈(-δ_0,δ_0) it holds that

    ĉ_ BPS(U_R,U_R-ρ,V,A;δσ,α)≤ Rl_α+ϵ.







Let (M,g) be a closed Riemannian manifold of negative curvature, and let α∈[S^1,M] be a free homotopy class. Denote V,A,ρ as above, and denote by σ any closed 2-form on M. If A>Rl_α, then for any sufficiently small number ϵ>0 there exists a constant δ_0=δ_0(g,α, V,A,ρ,ϵ)>0 such that if |σ|_g<δ_0 then we have

    ĉ_ BPS(U_R,U_R-ρ,V,A;σ,α)≤ Rl_α+ϵ.



Fix ρ_0∈ (0,R-ρ) and take V=U_ρ_0. By the definition of ĉ_ BPS and a scaling argument like in the proof of Proposition <ref>, one can obtain 

    δ·ĉ_ BPS(U_R,U_R-ρ,U_ρ_0,A;σ,α)=ĉ_ BPS(U_δ R,U_δ(R-ρ),U_δρ_0,δ A;δσ,α)

for any positive number δ. Combining this with Theorem <ref> and Theorem <ref> we have the following two theorems. 



	Let (M,g) be a closed connected Riemannian manifold. Denote A,ρ_0, ρ as above.  Assume that σ∈𝒫(M) and α∈[S^1,M] is any σ-atoroidal class. If A>Rl_α, then for any sufficiently small number ϵ>0 there exists a constant δ_0=δ_0(g,σ,α, A,ρ,ρ_0,ϵ)>0 such that for any λ∈ (0,+∞) it holds that 
	
    ĉ_ BPS(U_λ R,U_λ(R-ρ),U_λρ_0,λ A;δ_0λσ,α)≤λ(Rl_α+ϵ).

	
	




	Let (M,g) be a closed Riemannian manifold of negative curvature, and let α∈[S^1,M] be a free homotopy class. Denote A,ρ,ρ_0 as above, and denote by σ any closed 2-form on M. If A>Rl_α, then for any sufficiently small number ϵ>0 there exists a constant δ_0=δ_0(g,α, A,ρ,ρ_0,ϵ)>0 such that 

    ĉ_ BPS(U_δ_0R|σ|_g,U_δ_0(R-ρ)|σ|_g,U_δ_0ρ_0|σ|_g,δ_0A|σ|_g;σ,α)≤δ_0(Rl_α+ϵ)|σ|_g.







In the definition of the restricted BPS capacity, one can also allow H to be time-dependent. In this case, we require that H_t,t∈[0,1] to be periodic in time, conditions (H0)-(H2) hold uniformly for t∈ S^1, and that sup_S^1× VH≤ -c in the definition of ℋ_c(W,U,V,A). Then one can show that Theorem <ref> – Theorem <ref> still hold.



Like the Hofer-Zehnder capacity, the finiteness of restricted BPS capacity implies almost existence of periodic orbits.
Consider a proper[1][1]A map is called proper if preimages of compact sets are compact. smooth function

    H:T^*M→ℝ

which is bounded from below. Then {H≤ s} is compact, and hence {H<s} is contained in D_rT^*M for some sufficiently large radius r=r(s). Suppose that M⊂V̅⊂{H< d}, where V is an open neighborhood of M in T^*M. Given ρ>0 and T>d, let A>(r(T+ρ)+ρ)l_α. Then
the finiteness of ĉ_ BPS(U_r(T+ρ)+ρ,U_r(T+ρ),V,A;δσ,α) implies the following.





Let (M,g) be a closed connected Riemannian manifold. Assume σ∈𝒫(M) and α∈[S^1,M] is a non-trivial σ-atoroidal class. Let H:T^*M→ℝ be a proper smooth function bounded from below, and denote d,V,A,T,ρ as above. Then there exists a constant δ_0=δ_0(g,σ,α, V, T, A, ρ)>0 such that
if δ∈(-δ_0,δ_0) then for almost all s∈[d,T] the level set {H=s} carries a periodic Hamiltonian orbit with respect to ω_δσ=ω_0+δπ^*σ whose projection to M represents α.



 Similarly, for manifolds of negative curvature the finiteness of  ĉ_ BPS(U_r(T+ρ)+ρ, U_r(T+ρ),V,A;σ,α) implies the following.


Let (M,g) be a closed Riemannian manifold of negative curvature, and let α∈[S^1,M] be any non-trivial free homotopy class. Let H:T^*M→ℝ be a proper smooth function bounded from below. Denote by σ any closed 2-form on M, and denote d,V,A,T,ρ as above. Then there exists a constant δ_0=δ_0(g,α, V, T, A,ρ)>0 such that
if |σ|<δ_0 then for almost every s∈[d,T] the level set {H=s} carries a periodic Hamiltonian orbit with respect to ω_σ=ω_0+π^*σ whose projection to M represents α.






 §.§ Applications


Recall that the Hamiltonian function H_g(q,p)=p^2_g/2 is said to be the standard kinetic energy on T^*M, the Hamiltonian flow of X_H_g,σ on T^*M, called a twisted geodesic flow, describes the motion of a charge on M in the magnetic field σ. For related results on the existence of periodic orbits of twisted geodesic flows, we refer to the papers by Ginzburg and Gürel <cit.> using the Floer theory for contractible periodic orbits, by Lu <cit.> applying pseudo symplectic capacities for magnetic fields given by symplectic forms and by Frauenfelder and Schlenk <cit.> using the spectral metric when the two-form σ is exact. Recently, Asselle and Benedetti <cit.> showed that for almost all energy levels above the maximum critical value of an autonomous Tonelli Hamiltonian  there exists a periodic magnetic geodesic based on variational methods. By the same method, Abbondandolo,  Macarini,  Mazzucchelli and  Paternain <cit.> proved the existence of infinitely many periodic orbits of exact magnetic flows on surfaces for almost every subcritical energy level. It is worth to mention that Ginzburg <cit.> also gave a counterexample and showed the nonexistence of closed trajectories of H_g; see <cit.> for a more comprehensive survey. Numerous results about periodic orbits of the Hamitonian flow in twisted cotangent bundles can be found in <cit.>. For H_g the function r(s) can be taken as √(2s). So by using Theorem <ref> we obtain the following.


Let (M,g) be a closed connected Riemannian manifold. Assume σ∈𝒫(M) and α∈[S^1,M] is a non-trivial σ-atoroidal class. Then for 0<d<T,
there exists a positive constant δ_0=δ_0(g,σ,α,d,T) such that if δ∈(-δ_0,δ_0), then for almost all s∈[d,T] the energy level {H_g=s} carries a periodic orbit of X_H_g,δσ whose projection to M represents α.


Similarly, Theorem <ref> implies the following.

Let (M,g) be a closed Riemannian manifold of negative curvature, and let α∈[S^1,M] be any non-trivial free homotopy class. Denote by σ any closed 2-form on M. Then for 0<d<T, there exists a positive constant δ_0=δ_0(g,α,d,T) such that if |σ|_g<δ_0, then for almost every s∈[d,T] the energy level {H_g=s} carries a periodic orbit of X_H_g,σ whose projection to M represents α.



The assertions of the above corollaries overlap with Merry's results <cit.>. Indeed, Corollary <ref> complements <cit.> in the case that magnetic field σ has no bounded primitive in the universal cover. On the other hand, <cit.> implies Corollary <ref>. Merry actually showed a stronger result whenever M admits a metric g of negative curvature:
for each d>0 and each nontrivial homotopy class α∈[S^1,M], there is a constant c=c(g,d)>0 such that for every s∈(d,∞) and every closed 2-form σ with |σ|_g<c the energy level {H_g=s} carries a periodic orbit of X_H_g,σ whose projection to M represents α.




When M is simply connected and σ is a non-exact closed 2-form on M,  our method in this paper does not apply to the twisted cotangent bundle (T^*M,ω_σ) since ω_σ is not weakly exact and hence the Floer action functional is not single-valued any more in this case. Fortunately,  very recently this weakly exact assumption is removed  in the context of a general symplectic cohomology for magnetic cotangent bundles, see <cit.>. In particular, the case M=S^2 is also considered in <cit.> by Benedetti and Ritter.






The paper is organized as follows. In Section <ref> we recall some background results and definitions of the filtered Floer homology on the twisted cotangent bundle. Section <ref> is mainly devoted to prove the invariance of Floer homology for symplectic deformations.
Before the proof,

the action spectrum properties (see Lemma <ref> and Lemma <ref>) are established in this section. The goal of Section <ref> is to compute the Floer homology in T^*M with its canonical symplectic form ω_0.
In Section <ref> we prove the main theorems and discuss the flows without non-contractible periodic orbits.





§ ACKNOWLEDGMENTS

I am deeply grateful to my Ph.D. advisor Guangcun Lu for introducing me to symplectic geometry and for valuable suggestions. My special thanks go to my colleague Xingpeng Dong for teaching me to draw beautiful graphs. I thank Rongrong Jin and Kun Shi for helpful discussions. I am also grateful to Viktor L. Ginzburg for useful comments and especially for his ideas in the proof of  Proposition <ref>. Finally, 
I am greatly indebted to the anonymous referee for the very carefully reading and helpful suggestions to improve the paper and for pointing out how to obtain Theorem <ref> and Theorem <ref>.







§ FLOER HOMOLOGY





 §.§ Preliminaries


The Riemannian metric g on M induces a metric  on TM  and a horizontal-vertical splitting of TT^*M, together with isomorphisms

    T_zT^*M=T^h_zT^*M⊕ T_z^vT^*M≅ T_qM⊕ T_q^*M≅ T_qM⊕ T_qM,   z=(q,p)∈ T^*M.

The above splitting gives rise to the almost complex structure J_g on T^*M represented by

    J_g=[  0 -I;  I  0 ].

Recall that an almost complex structure J on a symplectic manifold (W,ω) is  ω-compatible if the bilinear form ω(·,J·) defines a Riemannian metric on W.
It is easy to check that J_g is ω_0-compatible for every Riemmanian metric g on M.  Denote G_g(·,·):=ω_0(·,J_g·). When W is a manifold with a Riemanian metric G, various L^∞-norms |·|_G are defined by

    |v|_G:=sup_x∈ Wv(x)_G  ∀ v∈Γ(TW),   |θ|_g:=sup_x∈ Wθ(x)_g  ∀ θ∈Ω^k(W),


    |X|_G:=sup_x∈ Wsup{X(x)v_G|v∈ T_xW, v_G=1}  ∀ X∈Γ( End (TW)).

Let 𝒥(T^*M) be the set of one-periodic almost complex structures on T^*M with finite |·|_G_g-norm, and denote 
    𝒥(ω_σ):={J∈𝒥(T^*M):J is ω_σ-compatible on (T^*M,ω_σ)}.



Floer homology is the main tool utilized in this paper to prove the existence of periodic orbits of the Hamiltonian flow in twisted bundles. To define the Floer homology of compactly supported functions on a non-compact symplectically aspherical manifold we need to impose certain conditions on the manifold at infinity.

We say that a symplectic manifold (W,ω) without boundary is geometrically bounded if there exists an almost complex structure J and a complete Riemannian metric g on W such that

  
  * J is uniformly ω-tame, which means that for all tangent vectors X and Y to M
      
    ω(X,JX)≥κ_1 X_g^2  and  |ω(X,Y)|≤κ_2X_gY_g

      for some positive constants κ_1 and κ_2.
  
  * the injectivity radius of (W,g) is bounded away from zero and the sectional curvature of (W,g) is bounded from above.



Obviously, closed symplectic manifolds are geometrically bounded; a product of two geometrically bounded sympletic manifolds is such a manifold. For a more detailed discussion of this concept please refer to Chapters V (by J.-C. Sikorav) and X (by M. Audin, F. Lalonde and L. Polterovich) in <cit.>. One can easily check that manifolds convex at infinity, e.g., (ℝ^2m,∑_i=1^2mdx_i∧ dy_i) and the cotangent bundle (T^*M,ω_0), are geometrically bounded. It is also well known that every twisted cotangent bundle (T^*M,ω_σ) admits abundant almost complex structures such that it is geometrically bounded (see <cit.>). In general J_g∉𝒥(ω_σ). However, Proposition 4.1 in <cit.> implies that there exists a constant ε_0=ε_0(g)>0 such that for all r≥ε_0|σ|_g,

    𝒥(ω_σ)∩ B_J_g(r)≠∅,

 where B_J_g(r) denotes the open ball of radius r about J_g in 𝒥(T^*M). In fact, Lu in <cit.> shows that for r≥ε_0|σ|_g we can find an almost complex structure in 𝒥(ω_σ)∩ B_J_g(r) such that (T^*M,ω_σ) for the natural metric G_g is geometrically bounded; see also <cit.>. This will be very useful in the proof of Theorem <ref> and Theorem <ref> since we will use it to choose suitable almost complex structures to estimate the energy of Floer cylinders and to ensure the necessary compactness of moduli spaces of solutions of Floer equations on twisted cotangent bundles.

Finally let us note that the first Chern class satisfies c_1(T^*M,J)=0 for each J∈𝒥(ω_σ) since the twisted cotangent bundle (T^*M,ω_σ) admits a Lagrangian distribution T^vT^*M (see <cit.>).






 §.§ The definition of filtered Floer homology

Let σ∈𝒫(M), and let α∈ [S^1,M] be a σ-atoroidal class. We denote ℋ as the space of smooth compactly supported Hamiltonian functions on S^1× D_RT^*M. For c>0 we denote by ℋ_c the subspace of all Hamiltonian functions H∈ℋ satisfying sup_S^1× MH≤ -c. Let 𝔏_α T^*M be the set of all 1-periodic loops x whose projections to M belong to 𝔏_α M. Fix a reference loop q_α∈𝔏_α M.
We define the action functional 𝒜_H,σ:𝔏_α T^*M→ℝ by

    𝒜_H,σ(x)=∫_S^1x^*λ-∫_[0,1]× S^1 w^*σ-∫^1_0H(t,x)dt,

where w:[0,1]× S^1→ M is any smooth map such that

    w(0,t)=q_α(t)  and   w(1,t)=π(x(t)).

Since α is a σ-atoroidal class,

    𝒜_σ(q):=∫_[0,1]× S^1 w^*σ  ∀ q∈𝔏_α M

is independent of the choice of w, and therefore  𝒜_H,σ is well defined. It is easy to check that the set Crit𝒜_H,σ of critical points of 𝒜_H,σ equals 𝒫_α(H,σ). The set of values of 𝒜_H,σ on 𝒫_α(H,σ) is called the action spectrum with respect to α, and we denote it by

    𝒮_α(H,σ)={𝒜_H,σ(x)|
    x∈𝒫_α(H,σ)}.

Consider the space ℋ with the strong Whitney C^∞-topology. Note that the action spectrum 𝒮_α(H,σ) is a compact and measure zero subset of ℝ for any H∈ℋ, and is lower semicontinuous as a multivalued function of H∈ℋ (see <cit.>).
For a,b∈ℝ∪{±∞}, if a,b∉𝒮_α(H,σ), we set

    𝒫^(a,b)_α(H,σ)={x∈𝒫_α(H,σ)|a<𝒜_H,σ(x)<b}.

In order to define the filtered Floer homology associated to H,σ and α we need the following nondegeneracy condition:

  
  * (C)  Every element x∈𝒫^(a,b)_α(H,σ) is non-degenerate, that is, the linear map dϕ_1^H,σ(x(0)) does not have 1 as an eigenvalue, where ϕ_1^H,σ is the time-one map of the flow of X_H,σ.

For any pair a<b we consider the class of admissible Hamiltonians by

    ℋ_σ;α^a,b={H∈ℋ|a,b∉𝒮_α(H,σ)}.

If α=0, then every point in the complement of ⋃_t∈ S^1 supp(H_t) is a degenerate 1-periodic orbit of X_H,σ with zero action of 𝒜_H,σ. To avoid these trivial periodic orbits, we require that 0∉[a,b] whenever α=0.

Given H∈ℋ_σ;α^a,b satisfying nondegeneracy condition (C), for every x=(q(t),p(t))∈𝒫^(a,b)_α(H,σ) we define the index μ(x)=-μ_CZ(x)+ν(x) following the paper by Weber <cit.>. Here μ_CZ(x) denotes the Conley-Zehnder index of x (see <cit.>), and ν(x):=0 if the pullback bundle q^*TM over S^1 is trivial and ν(x):=1 otherwise. Consider the ℤ_2-vector space CF^(a,b)_α(H,σ)[2][2]We use the convention that the complex generated by the empty set is zero. defined by

    CF^(a,b)_α(H,σ):= CF^b_α(H,σ)/ CF^a_α(H,σ),   CF^a_α(H,σ):=⊕_x∈𝒫^(-∞,a)
    _α(H,σ)ℤ_2x

graded by the index μ.

The Floer boundary operator  is defined as follows. Let J_gb be an almost complex structure such that (T^*M,ω_σ) is geometrically bounded. Denote by 𝒥 the set of smooth time-dependent ω_σ-tame almost complex structures on T^*M that are compatible with ω_σ near supp(H) and equal to J_gb outside some compact set. Every J_t∈𝒥 give rises to a positive-definite bilinear form on 𝔏_α T^*M.
Given x_±∈ CF^(a,b)_α(H,σ) we denote by ℳ^α(x_-,x_+,H,J,σ) the moduli space of smooth solution u:ℝ× S^1→ T^*M of the Floer differential equation

    ∂_su+J_t(u)(∂_tu-X_H,σ(u))=0

with the asymptotic boundary conditions

    lim_s→±∞u(s,t)=x_±  and  lim_s→±∞∂_su(s,t)=0

uniformly in t∈ S^1. For every solution of (<ref>) and
(<ref>) we have the energy identity

    E(u):=∫^∞_-∞∫_0^1ω_σ(∂_su,
    J∂_su)dsdt=𝒜_H,σ(x_-)-𝒜_H,σ(x_+).

Now we observe:

  
  (i) The moduli space ℳ^α(x_-,x_+,H,J,σ) is uniformly C^0-bounded. This is because H is compactly supported, and (T^*M,ω_σ) with J_gb is geometrically bounded; see Chapter V in <cit.> or <cit.>.
  
  (ii) Since ω_σ is symplectically aspherical, no bubbling off of holomorphic spheres can occur in T^*M. From this fact, the energy identity (<ref>) and (i) we deduce that the moduli space  ℳ^α(x_-,x_+,H,J,σ) is compact with respect to C^∞-convergence on compact sets.
  
  (iii) For a dense subset 𝒥_reg(H,σ)⊂𝒥, the linearized operator for equation (<ref>) is surjective for each finite-energy solution of (<ref>) in the homotopy class α (see <cit.>).



For each H satisfying (C), each J∈𝒥_reg(H,σ) and each pair x_±∈ CF^(a,b)_α(H,σ) the moduli space ℳ^α(x_-,x_+,H,J,σ) is an empty set or a smooth manifold of dimension μ(x_+)-μ(x_-)=μ_CZ(x_+)-μ_CZ(x_-); see <cit.>. As usual, the Floer boundary operator ∂=∂^H,J_σ;α on CF^(a,b)_α(H,σ) is defined by


    ∂ x_-=∑_x_+∈ CF^(a,b)_α(H,σ)
    
    μ(x_-)-μ(x_+)=1n(x_-,x_+)x_+.

Here n(x_-,x_+) stands for the number (mod 2) of elements in the set ℳ^α(x_-,x_+,H,J,σ)/ℝ (modulo time shift). The operator ∂ satisfies ∂∘∂=0, and the resulting Floer homology groups

    HF^(a,b)_α(H,σ)= ker∂/ im∂

are independent of the choice of J∈𝒥_reg(H,σ).


It is unclear whether or not the Floer homology HF^(a,b)_α(H,σ) is always independent of the choice of J_gb. It is independent of the choice of J_gb whenever the set of almost complex structures for which (T^*M,ω_σ) is geometrically bounded is connected.



 Assume that a<b<c, 0∉ [a,c] and a,b,c∉𝒮_α(H,σ). Then we have the exact sequence of complexes
 
    0→ CF^(a,b)_α(H,σ)→ CF^(a,c)_α(H,σ)→ CF^(b,c)_α(H,σ)→ 0.

This induces the exact sequence at the homology level

    ⋯→ HF^(a,b)_α*(H,σ)→ HF^(a,c)_α*(H,σ)→ HF^(b,c)_α*(H,σ)→ HF^(a,b)_α(*-1)(H,σ)→⋯.




 §.§ Homotopic invariance

Suppose that H^±∈ℋ_σ;α^a,b satisfy (C) and x_±∈𝒫_α(H^±,σ). Let H^s:ℝ→ℋ be a smooth homotopy connecting H^- and H^+ such that H^s=H^- for s≤ 0 and H^s=H^+ for s≥ 1. Consider the parameter-dependent Floer equation

    ∂_su+J^s_t(u)(∂_tu-X_H^s_t,σ(u))=0

which satisfies uniformly in t∈ S^1 the asymptotic boundary conditions

    lim_s→±∞u(s,t)=x_±  and  lim_s→±∞∂_su(s,t)=0.

Here J^s_t:ℝ→𝒥 is a regular homotopy of smooth families of almost complex structures satisfying

  
  * J^s_t=J^-_t∈𝒥_reg(H^-,σ) for s≤0.
  
  * J^s_t=J^+_t∈𝒥_reg(H^+,σ) for s≥1.
  
  * J^s_t is constant and and equals to J_gb outside some compact set of D_RT^*M.
  
  * The linearized operator for equation (<ref>) is surjective for each finite-energy solution of (<ref>) in the homotopy class α.

The moduli space ℳ^α(x_-,x_+,H^s,J^s,σ) of smooth solutions of (<ref>) satisfying the boundary conditions (<ref>) is C_loc^∞-compact. A crucial ingredient for the proof of the compactness is the following energy identity:

    E(u):   =   ∫^∞_-∞∫_0^1ω_σ(∂_su,
    J^s_t∂_su)dsdt
       =   𝒜_H^-,σ(x_-)-𝒜_H^+,σ(x_+)
    -∫^∞_-∞∫^1_0(∂_s H^s)(t,u(s,t))dsdt.

For |H^+-H^-| small enough we can define a chain map (being similar to the argument in <cit.>)

    Ψ^σ_H^+H^-: CF^(a,b)_α(H^-,σ)→ CF^(a,b)_α(H^+,σ)

which induces an isomorphism

    Ψ^σ_H^+H^-: HF^(a,b)_α(H^-,σ)→ HF^(a,b)_α(H^+,σ).

The isomorphism Ψ^σ_H^+H^- is independent of the choice of the homotopy H^s and J^s by a homotopy of homotopies argument; see <cit.>. As a result, we can define the Floer homology groups
HF^(a,b)_α(H,σ) for any H∈ℋ_σ;α^a,b by a small perturbation since the Hamiltonians satisfying (C) for a<b are dense in ℋ_σ;α^a,b.




 §.§ Monotone homotopies

Let H,K∈ℋ_σ;α^a,b be two functions with H(t,x)≤ K(t,x) for all (t,x)∈ S^1× D_RT^*M. Choose a monotone homotopy s→ H^s∈ℋ from H to K such that ∂_s H^s≥0 everywhere (Here we do not require H^s to be in ℋ_σ;α^a,b for every s∈ [0,1]). From the energy identity (<ref>) we deduce that such a homotopy induces a natural homomorphism, which is called monotone homomorphism

    Ψ_KH^σ: HF^(a,b)_α(H,σ)→ HF^(a,b)_α(K,σ).

It is well known that these monotone homomorphisms are independent of the choice of the monotone homotopy of Hamiltonians and satisfy the following properties
(see, e.g., <cit.>):


    Ψ^σ_HH= id  ∀ H∈ℋ_σ;α^a,b,
    Ψ^σ_KH∘Ψ^σ_HG=Ψ^σ_KG

whenever G,H,K∈ℋ_σ;α^a,b satisfy G≤ H≤ K.


If K^s is a monotone homotopy from H to K such that
K^s∈ℋ_σ;α^a,b for every s∈[0,1], then Ψ^σ_KH is an isomorphism.




To simplify the notation, we omit the mark 0 whenever σ=0, and abbreviate, for example, 𝒜_H,0, ℋ_0;α^a,b,
HF^(a,b)_α(H,0) and Ψ^0_H^+H^- by 𝒜_H, ℋ_α^a,b, HF^(a,b)_α(H) and Ψ_H^+H^- respectively.
All the above arguments in this section go through word for word whenever the closed two-form σ vanishes identically . In this case, there is no restriction on the free homotopy class α∈[S^1,M], and the Floer homology for H∈ℋ_α^a,b defined here is exactly as that in Weber's paper <cit.>. Indeed, J_g∈𝒥(ω_0) is an almost complex structure such that (T^*M,ω_0) is convex at infinity, and hence is geometrically bounded for G_g. Using  <cit.>, Weber defines the filtered Floer homology of a broader class of admissible Hamiltonians:


    𝒦_R;α^a,b:=   {   H∈ C^∞(S^1× T^*M)|∃τ≥0 ∃ c∈ℝ such that H_t(q,p)=τp_g+c if 
          p_g≥ R with {a,b}∩𝒮_α(H)=∅, and τ∉Λ_α or c∉[a,b]}

which satisfies 𝒦_R;α^a,b⊇ℋ_α^a,b.
Here we emphasize that Lemma <ref> also holds for monotone homotopies in 𝒦_R;α^a,b; see <cit.>. This will be very useful to compute Floer homology in Section <ref>.








§ SYMPLECTIC DEFORMATIONS OF FLOER HOMOLOGY


Floer's work <cit.> tells us that Floer homology is a topological invariant of a closed symplectically aspherical  manifold on which different symplectic forms give rise to the same Floer homology (up to isomorphisms). A direct proof of such a fact can be found in the paper by Viterbo <cit.>. In this section, by following the idea of Bae and Frauenfelder <cit.>, we discuss the continuation homomorphisms for symplectic deformations under additional hypotheses concerning the sympletic forms. For related results about Floer homology under symplectic perturbations, we refer to the paper by Ritter <cit.>.



 §.§ Quadratic isoperimetric inequality


Assume that σ∈𝒫(M) and α∈[S^1,M] is a σ-atoroidal class. Then for every q∈𝔏_α M we have

    |𝒜_σ(q)|≤ϵ_0(
    ∫^1_0q̇(t)_gdt)^2+ϵ_1,

where ϵ_0=ϵ_0(M,g,σ)>0 and ϵ_1=ϵ_1(M,g,σ,α)>0 are some constants.


This lemma is based on Lemma 2.7 in <cit.>; for a proof of it we refer to <cit.>.






























It is easy to check that 𝒫(M) is a linear space. Moreover, if σ∈𝒫(M), for any δ∈ℝ it holds that

    |𝒜_δσ(q)|≤ |δ|ϵ_0(
    ∫^1_0q̇(t)_gdt)^2+|δ|ϵ_1,

where the constants ϵ_0 and ϵ_1 are given as in Lemma <ref>. By Remark <ref>, for every closed Riemannian manifold (M,g) of negative curvature, the constants ϵ_0(M,g,σ) and ϵ_1(M,g,σ,α) in Lemma <ref> converge to zero as |σ|_g→ 0.






 §.§ Gap estimates for action spectrum



Let H∈ℋ.
Suppose that a∈ℝ is not in the action spectrum of 𝒜_H. Then there exist some constants ε_0=ε_0(H,g,σ,a,α)>0 and δ_0=δ_0(H,g,σ,a,α)>0 such that if |δ|<δ_0, then
[a-ε_0,a+ε_0]∩𝒮_α(H,δσ)
=∅.


 Arguing by contradiction, suppose that there is a sequence of number {δ_k}_k∈ℕ⊆ℝ such that |δ_k|<1/(k|σ|_g) (if σ=0 let δ_k=0 for all k), and for every σ_k:=δ_kσ there exists some x_k∈𝒫_α(H,σ_k), that is,
 
    ẋ_k=X_H,σ_k(t,x_k)  with   x_k(0)=x_k(1),

such that 𝒜_H,σ_k(x_k)=a_k∈(a-1/k,a+1/k).
Since 
    ι(X_H)ω_0=dH_t=ι(X_H,σ_k)ω_σ_k
    =ι(X_H,σ_k)ω_0+ι(X_H,σ_k)π^*σ_k,

we deduce that

    ω_0(X_H-X_H,σ_k,J_g(X_H-X_H,σ_k))
    =π^*σ_k(X_H,σ_k,J_g(X_H-X_H,σ_k)),

where J_g is defined as in (<ref>). Then we have

    X_H-X_H,σ_k^2_G_g≤σ_k_g
    X_H,σ_k_G_gX_H-X_H,σ_k_G_g

which implies that

    X_H-X_H,σ_k_G_g≤σ_k_gX_H,σ_k_G_g  and  X_H,σ_k_G_g≤X_H_G_g/1-σ_k_g

for |σ_k|_g<1. Combining this with the fact that H is compactly supported in S^1× D_RT^*M we deduce that X_H,σ_k is uniformly bounded, and hence x_k(t) is equicontinuous. Then, by Arzela-Ascoli theorem, passing to a subsequence, x_k(t) converges to some x_0(t) in D_RT^*M. We claim that x_0(t) is a Hamiltonian periodic orbit of H for ω_0. For k large enough, we may assume without loss of generality that H is defined on ℝ^2n (by using some local  coordinate near x_0(t)). Now we only need to prove

    x_0(t)-x_0(0)=∫^t_0X_H(s,x_0(s))ds.

Note that 
    x_0(t)-x_0(0)=lim_k→∞(x_k(t)-x_k(0))
    =lim_k→∞∫^t_0ẋ_k(s)ds.
 We compute

    x_0(t)-x_0(0)-∫^t_0X_H(s,x_0(s))ds   =   lim_k→∞∫^t_0(ẋ_k(s)-X_H(s,x_0(s)))ds
       =   lim_k→∞∫^t_0(ẋ_k(s)-X_H,σ_k(s,x_k(s)))ds
          +lim_k→∞∫^t_0(X_H,σ_k(s,x_k(s))-X_H(s,x_k(s)))ds
          +lim_k→∞∫^t_0(X_H(s,x_k(s))-X_H(s,x_0(s)))ds.

In the last equality, the first term is zero due to (<ref>), the second term is zero since X_H,σ_k uniformly converges to X_H by (<ref>) and the compactness of supp(H), and the third term is zero since x_k(t)  uniformly tends to x_0(t). Let q_k(t)=π(x_k(t)). By lemma <ref>, we have

    |𝒜_σ_k(q_k)|   ≤   δ_kϵ_0(
    ∫^1_0q̇_k(t)_gdt)^2+δ_kϵ_1
       ≤   δ_kϵ_0(
    ∫^1_0X_H,σ_k_G_gdt)^2+δ_kϵ_1→0  as k→0.

It follows that

    a=lim_k→∞𝒜_H,σ_k(x_k)=lim_k→∞𝒜_H(x_k)
    -lim_k→∞𝒜_σ_k(q_k)=𝒜_H(x_0)

which contradicts our assumption that a is not in the action spectrum of 𝒜_H.


By Remark <ref>, the proof of Lemma <ref> implies


Let (M,g) be a closed Riemannian manifold of non-positive curvature, and let H∈ℋ.
Suppose that a∈ℝ is not in the action spectrum of 𝒜_H, and that σ is any closed 2-form on M. Then there exists some constants δ_0=δ_0(H,g,a,α)>0 and ε_0=ε_0(H,g,a,α)>0 such that if |σ|_g<δ_0, then [a-ε_0,a+ε_0]∩𝒮_α(H,σ)=∅.



Under the hypotheses of  Lemma <ref>, if, moreover,  {H_k}_k∈ℕ⊆ℋ converges to H in the C^∞-topology, then we conclude that there exists a positive integer k_0>0 such that [a-ε_0,a+ε_0]∩𝒮_α(H_k,δσ)
=∅ for every δ satisfying |δ|<δ_0(H,g,σ,a,α) and every k≥ k_0.
Similarly, under the hypotheses of  Lemma <ref>, if ℋ∋ H_k C^∞⟶H, then there exists a positive integer k_0>0 such that if k≥ k_0 and |σ|_g<δ_0(H,g,a,α), then [a-ε_0,a+ε_0]∩𝒮_α(H_k,σ)=∅.








 §.§ Invariance of Floer homology for symplectic deformations



Assume that H∈ℋ_α^a,b. Then the following holds.

  
  (1) There exists a constant δ_0=δ_0(H,g,σ,a,b,α)>0 such that if |δ|<δ_0, then there is a continuation chain map

    Ψ_ω_0^ω_δσ: CF^(a,b)_α(H)→ CF^(a,b)_α(H,δσ)

which induces an isomorphism

    Ψ_ω_0^ω_δσ: HF^(a,b)_α(H)→ HF^(a,b)_α(H,δσ).

  
  (2) If K∈ℋ_α^a,b is another Hamiltonian function satisfying 
    H(t,x)≤ K(t,x)  ∀ (t,x)∈ [0,1]× D_RT^*M,


then for |δ|<min{δ_0(H,g,σ,a,b,α),
δ_0(K,g,σ,a,b,α)} the following diagram commutes:


    HF^(a,b)_α(H) @>Ψ_KH>>  HF^(a,b)_α(K)
    
    @VΨ_ω_0^ω_δσVV  @VVΨ_ω_0^ω_δσV 
     HF^(a,b)_α(H,δσ) @>Ψ^δσ_KH>>  HF^(a,b)_α(K,δσ)







In what follows, we always assume that |σ|_g≠0 (nothing needs to be proved if σ=0).
Consider H^i∈ℋ_α^a,b, i=0,1, which satisfies

    H^0(t,x)≤ H^1(t,x)  ∀ (t,x)∈ [0,1]× D_RT^*M.

By Lemma <ref>, there exist some constants ε^i=ε^i(H^i,g,σ,a,b,α)>0 and δ̂^i=δ̂(H^i,g,σ,a,b,α)>0, i=0,1, such that for every δ^i∈(-δ̂^i,δ̂^i) it holds that

    𝒮_α(H^i,δ^iσ)∩[a-2ε^i,a+2ε^i]=∅  and  𝒮_α(H^i,δ^iσ)∩[b-2ε^i,b+2ε^i]=∅.

By a perturbation argument, we may assume that without loss of generality H^0 and H^1 satisfy the nondegeneracy condition (C) for ω_δ^0σ and ω_δ^1σ respectively. We will show that there is a Floer chain map from
CF^(a,b)_α(H^0,δ^0σ) to CF^(a,b)_α (H^1,δ^1σ) which induces a homomorphism

    Ψ^δ^1δ^0_H^1H^0: HF^(a,b)_α(H^0,δ^0σ)→ HF^(a,b)_α(H^1,δ^1σ)

whenever |δ^i| (i=0,1) is small enough.

Let β:ℝ→ [0,1] be a smooth cut-off function such that
β=0 for s≤0, β(s)=1 for s≥ 1 and 0≤β'(s)≤1.
Set δ^s=(1-β(s))δ^0+β(s)δ^1 and
H^s=(1-β(s))H^0+β(s)H^1. Let ω^s=ω_0+δ^sσ, and let X^ω^s_H_t^s be the Hamiltonian vector field such that 
    dH_t^s=ι_X^ω^s_H_t^sω^s.

Then by (<ref>) we have

    |X^ω^s_H_t^s|_G_g≤|X^ω_0_H_t^s|_G_g/1-|δ^s||σ|_g≤2min{|X^ω_0_H_t^0|_G_g,|X^ω_0_H_t^1|_G_g}

for |δ^0|,|δ^1|≤1/(2|σ|_g) (σ≠ 0).
Let s→ J^s∈𝒥(ω_δ^sσ)∩
B_J_g(ε_0|δ^sσ|_g) (recall that ε_0 is a constant given on page 10)
be a homotopy of one-periodic almost complex structures such that J_t^s=J_t^-∈𝒥_reg(H^0,δ^0σ) for s≤0, and J_t^s=J_t^+∈𝒥_reg(H^1,δ^1σ) for s≥1. Such a choice of J^s is possible because δ^s is constant outside of [0,1] and Proposition 4.1 in <cit.> can actually be extended to a parametric version for a family of twisted symplectic forms ω_σ^s with s belonging to some compact interval since those almost complex structures constructed there are canonical with respect to a choice of metric. For every (s,t)∈ℝ× S^1 and every X,Y∈ TT^*M, we have

    ω^s(X,J^s_tX)   =   ω_0(X,J_gX)+ω_0(X,(J^s_t-J_g)X)
    +δ^sπ^*σ(X,J_t^sX)
       ≥   X^2_G_g-|J^s_t-J_g|_G_gX^2_G_g-
    |δ^s||σ|_g|J^s_t|_G_gX^2_G_g
       ≥   (1-ε_0|δ^s||σ|_g-(1+
    ε_0|δ^s||σ|_g)|δ^s||σ|_g)
    X^2_G_g
       ≥   1/2X^2_G_g

provided |δ^0|,|δ^1|≤ (6ε_0+4)^-1|σ|_g^-1, and it holds that

    |ω^s(X,Y)|≤5/3X_G_gY_G_g.

Therefore J^s is a 1-periodic almost complex structure for which  (T^*M,ω^s) for the natural metric G_g is geometrically bounded for every s∈ℝ and every t∈ S^1.

Given x∈𝒫_α(H^0,δ^0σ) and y∈𝒫_α(H^1,δ^1σ),
consider u:ℝ× S^1→ T^*M satisfying the following equation

    ∂_su+J^s_t(u)(∂_tu-X^ω^s_H_t^s(u))=0

with the asymptotic boundary conditions

    lim_s→-∞u(s,t)=x,  lim_s→+∞u(s,t)=y  and  lim_s→±∞∂_su(s,t)=0.

Here we emphasize that {J^s_t} is also chosen such that solutions of (<ref>) and (<ref>) are transverse (the associated linearized operators are surjective) by a perturbation argument; see <cit.>.
Now we can define a map

    Ψ^δ^1δ^0_H^1H^0: CF^(a,b)_α(H^0,δ^0
    σ)→ CF^(a,b)_α(H^1,δ^1
    σ)

given by

    Ψ^δ^1δ^0_H^1H^0(x)
    =∑_μ(x)=μ(y)#_2
    ℳ^α(x,y,H^s,J^s,ω^s)y,

where the space ℳ^α(x,y,H^s,J^s,ω^s) consists of solutions of (<ref>) and (<ref>), and #_2 denotes the number of elements modulo two. Since ω_s is symplectically aspherical for every s∈ℝ, there is no bubbling.  Then in order to obtain the compactness of ℳ^α(x,y,H^s,J^s,ω^s), we need certain energy estimations. Write

    𝒜_H,δ(s)σ(u(s,·))=∫_S^1u(s,·)^*
    λ-𝒜_δ(s)σ(q(s,·))-∫^1_0H(t,u(s,t))dt,

where q(s,t) is the projection of u(s,t)∈ T^*M to M for every (s,t)∈ℝ× S^1. Then we compute

    𝒜_H^1,δ^1σ(y)-𝒜_H^0,δ^0σ(x)   =   ∫^∞_-∞d/ds𝒜_H,δ(s)σ
    (u(s,·))ds
       =   -∫^∞_-∞∫_0^1ω_s(∂_s u,J_t^s∂_s u)ds dt-∫^∞_-∞∫_0^1∂_sH^s_t(u(s,t))dsdt.
          -∫^∞_-∞(δ^1-δ^0)β^'(s)𝒜
    _σ(q(s,·))ds.

By Lemma <ref> we estimate

    |𝒜
    _σ(q(s,·))|   ≤   ϵ_0(∫^1_0
    ∂_tq(s,t)_gdt)^2+ϵ_1
       ≤   ϵ_0∫^1_0∂_tq(s,t)^2_gdt+ϵ_1
       ≤   ϵ_0∫^1_0∂_tu(s,t)^2_G_gdt+ϵ_1

for some positive constants ϵ_0=ϵ_0(g,σ) and ϵ_1=ϵ_1(g,σ,α).  By (<ref>) and (<ref>) we have

    ∂_tu^2_G_g   =   J^s_t(u)∂_su+X_H_t^s^ω^s(u)
    ^2_G_g
       ≤   2(J^s_t(u)∂_su^2_G_g+X_H_t^s^ω_s(u)
    ^2_G_g)
       ≤   2(|J^s_t|_G_g∂_su^2_G_g+C_0)
       ≤   2(4/3∂_su^2_G_g+C_0)

for |δ^0|,|δ^1|≤ (6ε_0+4)^-1|σ|_g^-1.
Here C_0:=C_0(H^0,H^1,g)=4min{|X^ω_0_H^0
|^2_G_g,|X^ω_0_H^1|^2_G_g}.
Plugging (<ref>) into (<ref>), we arrive at

    ∂_tu^2_G_g   ≤   16/3ω^s
    (∂_su,J^s_t∂_su)+2C_0.

Combining (<ref>) and (<ref>) leads to

    |𝒜
    _σ(q(s,·))|≤16ϵ_0/3∫^1_0ω_s(∂_su,J^s_t∂_su)dt+2ϵ_0C_0+ϵ_1.

Hence,

    |∫^∞_-∞β^'(s)𝒜
    _σ(q(s,·))ds|   =   |∫^1_0β^'(s)𝒜
    _σ(q(s,·))ds|
       ≤   C_1∫^∞_-∞∫^1_0ω_s(∂_su,J^s_t∂_su)dsdt
    +C_2,

where C_1=16ϵ_0/3 and
C_2=2ϵ_0C_0+ϵ_1. Then by (<ref>) we obtain

    (1-δ C_1)∫^∞_-∞∫^1_0ω_s
    (∂_su,J^s_t∂_su)dsdt   ≤   𝒜_H^0,δ^0σ(x)-𝒜_H^1,δ^1σ(y)
    +δ C_2
          -∫^∞_-∞∫_0^1∂_sH^s_t(u(s,t))dsdt,

with δ=(|δ^0|+|δ^1|).

Denote ε:=min{ε^0,ε^1}, and set

    δ_0:=δ_0(H^0,H^1,g,σ,a,b,α):=min{δ̂^0, δ̂^1,1/(6ε_0+4)|σ|_g,
    1/2C_1,ε/2C_2}.

For brevity, hereafter we drop the explicit dependence on a,b in the notation δ_0. 
Then for |δ^i|<δ_0 the Floer map from CF_α(H^0,δ^0σ) to CF_α(H^1,δ^1σ) defined by the solutions of (<ref>) preserves the subcomplexes CF^a_α and CF^b_α (since ∂_sH^s=β̇(s)(H^1-H^0)≥0).
Therefore, the solutions of (<ref>) give rise to the continuation map Ψ^δ^1δ^0_H^1H^0. By a standard gluing argument in Floer homology theory, Ψ^δ^1δ^0_H^1H^0 commutes with the boundary operators. Hence Ψ^δ^1δ^0_H^1H^0 is a chain map which induces a homomorphism

    Ψ^δ^1δ^0_H^1H^0: HF^(a,b)_α(H^0,δ^0σ)→ HF^(a,b)_α(H^1,δ^1σ).

Suppose that

    Ψ^δ^2δ^1_H^2H^1: HF^(a,b)_α(H^1,δ^1σ)→ HF^(a,b)_α(H^2,δ^2σ)

is another homomorphism defined as above if H^1≤ H^2 and
|δ^1|,|δ^2|<δ_0(H^1,H^2,g, σ,α). By a homotopy-of-homotopies argument, we have the following commutative diagram

    HF^(a,b)_α(H^0,δ^0σ)
    [rr]^Ψ^δ^2δ^0_H^2H^0[dr]_Ψ^δ^1δ^0_H^1H^0       HF^(a,b)_α(H^2,δ^2σ)
        HF^(a,b)_α(H^1,δ^1σ)[ur]_Ψ^δ^2δ^1_H^2H^1

for |δ^0|,|δ^1|,|δ^2|<min{δ_0(H^0,H^1,g,σ,α)
,δ_0(H^1,H^2,g,σ,α),δ_0(H^0,H^2, g,σ,α)}.

If H^s≡ H∈ℋ_α^a,b, then for |δ^1|,|δ^2|<δ_0(H,H,g,σ,α)
 we can define the map

    Ψ^δ^0δ^1_HH: HF^(a,b)_α(H,δ^1σ)→ HF^(a,b)_α(H,δ^0σ).

Since Ψ^δδ_HH is an isomorphism for every δ∈ℝ and every
H∈ℋ_δσ;α^a,b, we deduce from (<ref>) with δ_0=δ_2 that the homomorphism

    Ψ^δ^1δ^0_HH: HF^(a,b)_α(H,δ^0σ)→ HF^(a,b)_α(H,δ^1σ)

is an isomorphism with inverse Ψ^δ^0δ^1_HH for |δ^0|,|δ^1|<δ_0(H,H,g,σ,α).

Now we are in a position to prove Theorem <ref>. Denote 
    δ_0(H,g,σ,α):=δ_0(H,H,g,σ,α).

Let δ^0=0 and δ^1=δ with |δ|<δ_0(H,g,σ,α) for every H∈ℋ_α^a,b, and denote 
    Ψ_ω_0^ω_δσ:=Ψ^δ^1δ^0_HH
    : HF^(a,b)_α(H)→ HF^(a,b)_α(H,δσ).

Then statement (1) follows from the isomorphism (<ref>) immediately. Notice that

    min{δ_0(H,g,σ,α), δ_0(K,g,σ,α)}≤δ_0(H,K,g,σ,α).

For |δ|<min{δ_0(H,g,σ,α),
δ_0(K,g,σ,α)} we deduce from (<ref>) that the homomorphism

    Ψ^δ 0_KH: HF^(a,b)_α(H)→ HF^(a,b)_α(K,δσ)

satisfies Ψ^δ 0_KH=Ψ^δσ_KH∘Ψ_ω_0^ω_δσ with (H^0,δ^0)=(H,0), (H^1,δ^1)=(H,δ) and (H^2,δ^2)=(K,δ), and 
Ψ^δ 0_KH=Ψ_ω_0
^ω_δσ∘Ψ_KH with (H^0,δ^0)=(H,0), (H^1,δ^1)=(K,0) and (H^2,δ^2)=(K,δ). So Ψ^δσ_KH∘Ψ_ω_0^ω_δσ=Ψ_ω_0
^ω_δσ∘Ψ_KH. The proof of statement (2) is completed.



The quadratic isoperimetric inequality in Lemma <ref> plays an essential role in the proof of the above theorem. A careful inspection of the proof of Theorem <ref> shows that one could obtain the desired various estimations whenever the term 𝒜_σ associated to σ∈𝒫(M) is well controlled. In fact, when the underlying closed manifold M admits a metric g of negative curvature, Theorem <ref> can be upgraded to the following theorem.





Let (M,g) be a closed Riemannian manifold of negative curvature, and let α∈[S^1,M] be a free homotopy class.
Assume that H∈ℋ_α^a,b and that σ is any closed 2-form on M. Then there exists a constant δ_0=δ_0(H,g,a,b,α)>0 such that if |σ|_g<δ_0, then there is a continuation chain map

    Ψ_ω_0^ω_σ: CF^(a,b)_α(H)→ CF^(a,b)_α(H,σ)

which induces an isomorphism

    Ψ_ω_0^ω_σ: HF^(a,b)_α(H)→ HF^(a,b)_α(H,σ).

Moreover, if K∈ℋ_α^a,b is another Hamiltonian function satisfying 
    H(t,x)≤ K(t,x)  ∀ (t,x)∈ [0,1]× D_RT^*M,
  then for |σ|_g<min{δ_0(H,g,a,b,α),
δ_0(K,g,a,b,α)} the following diagram commutes:


    HF^(a,b)_α(H) @>Ψ_KH>>  HF^(a,b)_α(K)
    
    @VΨ_ω_0^ω_σVV  @VVΨ_ω_0^ω_σV 
     HF^(a,b)_α(H,σ) @>Ψ^σ_KH>>  HF^(a,b)_α(K,σ)






Firstly, Remark <ref> implies that under the hypotheses of Theorem <ref> the filtered Floer homology HF^(a,b)_α(H,σ) can be defined as in Subsection <ref>. As Remark <ref> points out, the proof of Theorem <ref> for σ-atoroidal class α goes through verbatim with only a minor modification. Here we just mention that in order to obtain the corresponding energy estimations we need to replace Lemma <ref> by Lemma <ref> and use the fact that the constants ϵ_0 and ϵ_1 in Lemma <ref> converge to zero as |σ|_g→ 0 (see Remark <ref>).









§ COMPUTATIONS OF FLOER HOMOLOGY


In this section we will closely follow the paper by Weber <cit.> to construct two sequences of Hamiltonian functions compactly supported in D_RT^*M and compute their Floer homologies for T^*M endowed  with the canonical symplectic form ω_0=-dλ.





 §.§ Radial Hamiltonians


 Let H^f:T^*M→ℝ be an autonomous function of the form

    H^f(q,p)=f(p_g)  ∀ (q,p)∈ T^*M,

where f:ℝ→ℝ is a smooth function such that f(r)=f(-r). Then the set of the critical points of 𝒜_H^f is given by

    𝒫_α(H^f):=   {    z=(q,p)∈ C^∞(S^1, T^*M)|
    q(t) is a geodesic in the class α, 
          l:=q̇_g,  p(t)=±r/lq̇(t), where r>0 satisfies  f^'(r)=± l}.

Moreover, for each z∈𝒫_α(H^f) it holds that 𝒜_H^f(z)=f^'(r)r-f(r), that is, the value of the action functional 𝒜_H^f at z is equal to the minus y-intercept of the tangential line of the graph y=f(x) at x=r.




 §.§ The action functional on free loop space

The action functional ℰ on ℒ_α M is defined by

    ℰ(q)=1/2∫^1_0ẋ(t)_g^2dt.

It is not hard to check that a loop q∈ℒ_α M is a critical point of ℰ if and only if q is a 1-periodic geodesic representing α. Given a∈ℝ, denote

    ℒ_α^aM:={q∈ℒ_α M|ℰ(q)≤ a}.

For a≤ b the natural inclusion

    ι_a^b:ℒ_α^aM↪ℒ_α^bM

induces the homomorphism

    [ι_a^b]: H_*(ℒ_α^aM)→ H_*(ℒ_α^bM).

Here H_*(ℒ_α^aM) denotes the singular homology with ℤ_2-coefficients of the sublevel set ℒ_α^aM.

The homomorphism [ι_a^b] is nonzero whenever l_α≤ a≤ b.





 §.§ Two families of profile functions


Fix c>0 and pick a∈(0,c] with a/R∉Λ_α.
Since the marked length spectrum Λ_α⊆ℝ is a closed and nowhere dense subset, there is a dense subset Δ of (0,a/c) such that for every η∈Δ it holds that

    ν_η:=1/R[a/η-(c-a)]∈ (a/R,∞)∖Λ_α.

Fix η∈Δ,  using the conventions sup∅=0 and inf∅=∞, we define

    [ l_0:=inf(Λ_α∩(a/ R,∞)),   l_-=l_-(η):=sup((0,ν_η)∩Λ_α),;  l_+=l_+(η):=inf((ν_η,∞)∩Λ_α),   l_1=l_1(η):=ν_η+l_-/2. ]

Clearly, we have

    (a/R,l_0)∩Λ_α=∅,  
    (l_-,l_+)∩Λ_α=∅  and   0≤ l_-<l_1<ν_η<l_+≤∞.

Denote

    [    r_k1:=(k-1)R/k+3/16k(R-a/l_0),  
    r_k2:=R-3/16k(R-a/l_0),; λ_k(x):=akx-R+(3/(16k))(R-a/l_0)/R-(3/8)(R-a/l_0),     k∈ℕ. ]

Here let us remark that ν_η,l_+(η),l_1(η)→+∞ as η→0, and r_k1,r_k2→ R as k→∞.






Sequence A of profile functions: Choose a sequence of smooth functions f_k (see Figure 1) by smoothing out a sequence of piecewise linear functions f̂_k, which are given by

    f̂_k(x):={[         λ_k(r_k1/2)    if x∈[0,r_k1/2),;              λ_k(x) if x∈[r_k1/2,r_k2),;                   0     if x∈[r_k2,+∞). ].

Each f_k is required to coincide with f̂_k away from sufficiently small neighbourhoods of r_k1/2 and r_k2. In particular,  graphf_k equals graphf̂_k in the region that lies below the line x→ ax/R-a and above the line x→ -a (grey region in Figure 1). Besides, we require that f_k^'≥ 0 everywhere, f_k^''≥ 0 near r_k1/2, f_k^''≤ 0 near r_k2, and f_k^''=0 elsewhere. Since a/R<a/r_k2<l_0, the slope of the unique tangent of the graph of f_k through the point (0,-a) lies in the interval (a/R,l_0). It follows that a∉𝒮_α(H^f_k).








Sequence B of profile functions: Let η_k∈Δ be a sequence of numbers such that η_k→ 0 as k→∞. Denote 
    T_k:=max{0,l_1(η_k)R-a},  τ_k:=a/η_kR  and   r_k:=η_kR(T_k+c)/a.

Note that T_k→+∞ and τ_k→+∞ as k→∞, and that 0≤ r_k≤ R. Consider the piecewise linear curve Γ in ℝ^2:


    ∀ Γ∋(x,y)={[    (x,τ_kx-c) if x∈[0,r_k],;       (x,T_k) if x∈[r_k,R],;         (R,y) if y∈[0,T_k]. ].

Smoothing out this piecewise linear curve near its corners we obtain a sequence of smooth functions h_k (see Figure 2). Here every h_k is also required to satisfy h_k^''≥ 0 near the point (0,-c) and h_k^'(0)=h_k^'(1)=0. We claim that a∉𝒮_α(H^h_k). In fact, the action of 𝒜_h_k at some one-periodic orbit is positive if and only if the y-intercept of the tangential line of graph h_k at the  corresponding point is negative. This happens in two clusters. The slope of the tangent of any point in one of those clusters passing through the point (0,-a) lies in the interval (l_-,l_+). The other cluster is located near the point (0,-c), at which the y-intercepts of the tangential lines of graphf_k are less than -c. So (l_-,l_+)∩Λ_α=∅ implies that a∉𝒮_α(H^h_k).




Fix 0<a≤ c with a/R∉Λ_α, and choose η_k∈Δ satisfying η_k→ 0 as k→ 0. Let {f_k}_k∈ℕ and {h_k}_k∈ℕ be two sequences of those functions constructed above. Choose k∈ℕ sufficiently large so that
f_k(0)≤-c and f_k≤ h_k. Set μ_k:=ν_η_k. Then there exist natural isomorphisms

    Θ_f_k: HF^(a,+∞)_α(H^f_k)→ H_*(ℒ_α^a^2/(2R^2)M)

and

    Θ_h_k: HF^(a,+∞)_α(H^h_k)→ H_*(ℒ_α^μ_k^2/2M)

such that the following diagram commutes:

    HF^(a,+∞)_α(H^f_k) @>Ψ_h_k f_k>>  HF^(a,+∞)_α(H^h_k)
    
    @VΘ_f_kV≃ V  @V≃ VΘ_h_kV 
     H_*(ℒ_α^a^2/(2R^2)M) @>[ι^μ_k^2/2_a^2/(2R^2)]>>  H_*(ℒ_α^μ_k^2/2M)








The proof of proposition <ref> is standard, which can be completed by mimicking that of <cit.>. Indeed, in Weber's paper <cit.> he gives the above
result with R=1 but by a rescaling argument one gets the result for general R>0. 
The basic idea is to deform f_k and h_k by monotone homotopies to convex radial functions so that Theorem <cit.> can be applied. Full details can be found in Appendix <ref>.





§ PROOFS OF THE MAIN THEOREMS AND REMARKS.














 §.§ Proof of Theorem <ref>



Let c=-sup_S^1× MH. Then c> Rl_α.
Choose a∈[Rl_α,c] so that a/R∉Λ_α. Then we can construct two sequences of functions {f_k}_k∈ℕ and {h_k}_k∈ℕ as in Section <ref>. For sufficiently large k it follows that

    H^f_k≤ H≤ H^h_k

as illustrated in Figure <ref>.
Fix such an integer k=k(H). Then by Proposition <ref> we have the commutative diagram (<ref>).

So the monotone homomorphism Ψ_h_k f_k is nonzero due to our assumption that μ_k≥ a/R≥ l_α; see Remark <ref>. Applying Theorem <ref> to the Hamiltonians  H^f_k and H^h_k, there exist positive constants δ_1=δ_1(H^f_k,g,σ,a,α) and δ_2=δ_2(H^h_k,g,σ,a,α) such that if |δ|<δ_0:=min{δ_1, δ_2}, then we have the following commutative diagram


    HF^(a,+∞)_α(H^f_k) @>Ψ_h_kf_k>>  HF^(a,+∞)_α(H^h_k)
    
    @VΨ_ω_0^ω_δσVV  @VVΨ_ω_0^ω_δσV 
     HF^(a,+∞)_α(H^f_k,δσ) @>Ψ^δσ_h_kf_k>>  HF^(a,+∞)_α(H^h_k,δσ)

Combining (<ref>) and (<ref>) we deduce that  for |δ|<δ_0 the monotone homomorphism
Ψ^δσ_h_kf_k is also nonzero. Given δ∈(-δ_0,δ_0),  choose a sequence of Hamiltonian functions H_i ∈ℋ such that H_i satisfy the non-degeneracy condition (C) and converge to H in C^∞ topology, H^f_k≤ H_i≤ H^h_k, a∉𝒮_α(H_i,δσ) and sup_S^1× MH_i<- Rl_α. Then, by Lemma <ref>, the non-trivial  homomorphism
Ψ^δσ_h_kf_k factors through the Floer homology group HF^(a,+∞)_α(H_i,δσ). Hence there exists a sequence of periodic orbits x_i∈𝒫_α(H_i,δσ) such that
𝒜_H_i,δσ(x_i)> a. Passing to a converging subsequence, we get a periodic orbit x∈𝒫_α(H,δσ) with 𝒜_H,δσ(x)≥ a.











 §.§ Proof of Theorem <ref>




Consider the Hamiltonian function defined by

    H̅(t,x):=-H(-t,x)  ∀ (t,x)∈ S^1× D_RT^*M.

Let C(α)=Rl_α. Obviously, x(t) is a periodic orbit of H representing -α if and only if x(-t) is a periodic orbit of H̅ representing α, and
it holds that

    sup_S^1× MH̅≤ -C(α).

Imitating the proof of Theorem <ref> with replacing Theorem <ref> by Theorem <ref> concludes the proof of Theorem <ref>.










 §.§ Proofs of Theorem <ref> and Theorem <ref>

The idea of the proof of Theorem <ref> is the same as that of Theorem <ref>, while the difference between the two proofs is that one can squeeze uniformly a class of functions (whose graphs lie in the grey region in Figure <ref> and on the line y=0,x≥ R-ρ) from above and below in the proof of Theorem <ref>. The proof of Theorem <ref> is similar to that of Theorem <ref>. Here we only show the latter. 





Let ϵ>0 be an arbitrary number in (0,A-Rl_α) and set c=Rl_α+ϵ. Pick a=a(ϵ)∈[Rl_α,c] so that a/R∉Λ_α. Construct two sequences of functions {f_k}_k∈ℕ and {h_k}_k∈ℕ as in Section <ref>. Then there exists an integer k_0=k_0(g,σ,α, V,A,ρ,ϵ)>0 such that

    H^f_k_0≤ H≤ H^h_k_0

for every H∈ℋ_c(U_R,U_R-ρ,V,A).
Following the proof of Theorem <ref> shows that
there exists a constant

    δ_0:=δ_0(g,σ,α, V,A,ρ,ϵ)>0
 such that if |δ|<δ_0 then every H∈ℋ_c(U_R,U_R-ρ,V,A)  admits a 1-periodic Hamiltonian orbit with respect to the twisted symplectic form ω_δσ whose projection to M represents α. Therefore, by definition, for each δ∈(-δ_0,δ_0) we have ĉ_ BPS(U_R,U_R-ρ,V,A;δσ,α)≤ Rl_α+ϵ.











 §.§ Proofs of Theorem <ref> and Theorem <ref>

The key to the proofs of Theorem <ref> and Theorem <ref> is the following monotonicity of the restricted BPS capacity.

Let W_1,W_2 be open subsets of T^*M containing M, V_2⊂ V_1⊂ U_1⊂ U_2, W_1⊂ W_2 and 0<A_1≤ A_2, where W_i,U_i,V_i, i=1,2 are open subsets of T^*M and U_i have compact closure U̅_i⊂ W_i. Then it holds that

    ĉ_ BPS(W_1,U_1,V_1,A_1;σ,α)≤ĉ_ BPS(W_2,U_2,V_2,A_2;σ,α).



Our proof of Theorem <ref> is an adaption of the standard almost existence theorem (see <cit.> or <cit.>).
For the sake of completeness we shall give a proof of Theorem <ref>. The proof of Theorem <ref> is nearly identical to that of Theorem <ref>, we omit it here.

Along the lines of <cit.>, we proceed in 3 steps.

Step 1.
By our assumption, the sublevel {H<s} is contained in U_r(s), where r:ℝ→ (0,∞) is a nondecreasing function. Consider the monotone functions ĉ_δ:[d,T]→[0,∞] defined by

    ĉ_δ(s):=ĉ_ BPS(U_r(T+ρ)+ρ,{H<s},V,A;δσ,α).

Theorem <ref> implies that there exists a constant
δ_0:=δ_0(g,σ,α, V,A,T,ρ)>0 such that if |δ|<δ_0 then

    ĉ_ BPS(U_r(T+ρ)+ρ,U_r(T+ρ),V,A;δσ,α)< A

provided A>(r(T+ρ)+ρ)l_α.
Then we deduce from Proposition <ref> that

    ĉ_δ(s)≤ĉ_δ(T+ρ)≤ĉ_ BPS(U_r(T+ρ)+ρ,U_r(T+ρ),V,A;δσ,α)< A  ∀ s∈[d,T+ρ],

where |δ|<δ_0(g,σ,α, V,A,T,ρ). Fix δ∈(-δ_0,δ_0).
So Lebesgue's last theorem implies that the function ĉ_δ is
differentiable at almost every point in the sense of measure theory. Suppose that s_0∈ [d,T] is a regular value of H and ĉ_δ is Lipschitz continuous at s_0. Then, by the
implicit function theorem, S_0:=H^-1(s_0) is a hypersurface in T^*M,  which is compact and bounds
the sublevel set B_0:={H<s_0} since H is proper and bounded below. By the implicit function theorem again and compactness of
S_0, one can find a parameterized family of hypersurfaces S_ε in T^*M with S_ε=H^-1(s_0+ε) such that

    c(ε)≤ c(0)+Lε,   c(ε):=ĉ_δ(s_0+ε)

for 0≤ε≤η, where L and η are positive constants. By choosing η smaller, let us require that η<min{ρ,(A-c(0))/(2L)} (A>c(0) by our assumption). For  0<ε≤η, let us denote B_ε:={H<s_0+ε}, then
B_ε⊂{H<T+ρ}⊂ U_r(T+ρ).
Fixing ς∈(0,η], we define a smooth function f:ℝ→ [-2Lς,0] by

    [    f(s)=-2Lς      if s≤ 0;       f(s)=0     if s≥ς/2; 0<f^'(s)≤ 8L  if 0<s<ς/2. ]

By the definition of the restricted BPS capacity ĉ_δ(s_0)=c(0), there exists a Hamiltonian function G∈ℋ(U_r(T+ρ)+ρ,B_0,A) such that

    -c(0)<sup_VG≤ -(c(0)-Lς)
      and  𝒫_α(G,δσ;τ)=∅  ∀  0<τ≤1.

Otherwise, c(0)=ĉ_δ(s_0)≤ c(0)-Lς which contradicts with the fact that Lς≥ 0.
Now we constructed a new function G by cutting off G without creating new fast periodic orbits.
More precisely, let χ:[-c(0)-ϵ,-c(0)+ϵ]→ℝ be a function with 0≤χ^'≤ 1 such that χ(t)=-c(0) for t near the left endpoint of the interval and χ(t)=t for t near the right endpoint of the interval, where ϵ>0 is sufficiently small
so that -c(0)+ϵ<sup_VG. Set

    G(x):={[                     -c(0)         if G(x)≤ -c(0)-ϵ,;                   χ(G(x)) if -c(0)-ϵ≤ G(x)≤-c(0)+ϵ,;                      G(x)         if -c(0)+ϵ≤ G(x). ].

The new non-positive function G compactly supported in B_0 satisfies 
    𝒫_α(G,δσ;τ)
    =∅  ∀  0<τ≤1
 since |χ^'|≤ 1. Furthermore, sup_VG≤ -(c(0)-Lς) and inf_B_0G≥ -c(0).
Consider the compactly supported Hamiltonian function K∈ C^∞(B_ς) defined by

    [     K(x)=G(x)-2Lς         if x∈ B_0;         K(x)=f(ε) if x∈ S_ε,  0≤ε<ς;            K(x)=0        if x∉ B_ς. ]

The function K satisfies

    sup_VK=sup_VG-2Lς≤ -(c(0)-Lς)-2Lς≤ -c(ς)  and


    inf_B_ςK= inf_B_0G-2Lς
    > -c(0)-2Lη> -A.

The definition of the restricted BPS capacity c(ς) shows that
K has a fast periodic orbit x with respect to ω_δσ whose projection to M represents α. We claim that x cannot intersect B_0. Indeed, if x intersects B_0 then it stays completely inside B_0 since B_0 is invariant under the flow of K. This is impossible because the flows of K and G on B_0 coincide and
G does not have fast periodic orbits. Since α≠ 0, the fast periodic orbit x whose projection represents α is nontrivial.
As a consequence, x must be contained in B_ς∖B_0, and hence it lies on
S_ε for some 0< ε<ς.

Step 2. Step 1 works for every ς∈(0,η]. Choosing a sequence ς_j→ 0, one can find sequences K_j and ε_j, and a corresponding sequence x_j(t) of periodic orbits of X_K_j,δσ having periods 0<τ_j≤ 1 and lying on
S_ε_j with ε_j→ 0. Consider the Hamiltonian H on the set

    U=⋃_ε∈(-η,η)S_ε.

Obviously,  if x∈ S_ε then H(x)=s_0+ε and K_j(x)=f_j(ε)=f_j(H(x)-s_0). By construction, the periodic orbits x_j solve the equations

    [ ẋ_j(t)=f^'_j(ε_j)X_H,δσ(x_j(t));                   x_j(0)=x(τ_j) ]

with the periods 0≤τ_j≤1. Normalizing the periods to 1 we define the functions

    y_j(t)=x_j(τ_jt)  ∀ t∈[0,1]

which solve the Hamiltonian equations

    ẏ_j(t)=f^'_j(ε_j)τ_jX_H,δσ(y_j(t))  and   H(y_j(t))=ε_j.


Step 3. By construction, f^'_j≤ 8L and hence f^'_j(ε_j)τ_j are bounded. This observation is very useful for us to obtain a periodic orbit on S_0. Indeed, we first note that the hypersurfaces S_ε_j are contained in the compact set B_η, hence the functions x_j are uniformly bounded. For all t∈ S^1 and all j∈ℕ we estimate

    ẏ_j(t)_G_g=|f^'_j(ε_j)τ_j|·X_H,δσ(x_j(t))_G_g≤ 8L sup_x∈B_ηX_H,δσ_G_g.

Then, by Arzela-Ascoli theorem, passing to subsequences, f^'_j(ε_j)τ_j converges to some τ≥0 and y_j(t) converges in C^0-topology, and,  by making use of the equations, even converges in C^∞-topology to a smooth 1-periodic solution y of the equation

    ẏ(t)=τ X_H,δσ(y(t)),   y(t)⊂ S_0.

We claim that τ≠ 0. Otherwise, y(t)=y^* for some point y^*∈ S_0. This contradicts with the fact that
the projection of y on M represents the non-trivial class α since [π (y_j)]=α and π (y_j) converges to π(y) in C^∞-topology. Reparametrizing time we obtain
the τ-periodic solution x(t):=y(t/τ) of the equation

    ẋ(t)=X_H,δσ(x(t))

whose projection on M represents α. The proof of Theorem <ref> is completed.





 §.§ Concluding remarks




  §.§.§ Hamiltonian flows without non-contractible closed trajectories

The following proposition shows that the constant δ_0(H,g,σ,a,b,α) in Theorem <ref> depends on H. As a consequence, one can not extend Theorem <ref> (resp. Theorem <ref>) to the case that δ_0(H,g,σ,a,b,α) (resp. δ_0(H,g,a,b,α)) is arbitrarily large.
 
Let M be a closed Riemanian surface endowed with a metric g of constant curvature K=-1. Then there exists a sequence of compactly supported Hamiltonians {H_n}_n∈ℕ⊂ C^∞(D_1T^*M) with inf_MH_n>n and a sequence of numbers {δ_n}_n∈ℕ converging to 0 such that the periodic orbits of the Hamiltonian flow of H_n with respect to ω_n=ω_0+δ_nπ^*σ are all contractible.


This proposition is due to Niche <cit.>. For the sake of completeness, we outline a proof of Proposition <ref> below. The following proof is different from Niche's proof and is based on a theorem of Ginzburg <cit.>.


We start with ω=ω_0+π^*dA where dA is the area form on (M,g). Further, let F be the standard kinetic energy Hamiltonian.
Then <cit.> shows that on every level {F=c} with c<1/2 all integral curves of F are closed and contractible, and there is no closed orbit on {F=1/2}  (on which the Hamiltonian flow is the horocycle flow, cf. <cit.>). Let V={F<1/2}. Let χ:[0,1/2]→ [0,C] be a “one-sided bump" function with χ^'≤ 0 such that χ(t)=C  near t=0 and χ(t)=0 near t=1/2. Here C is a constant and can be made arbitrarily large. Let H=χ∘ F. Then H is supported in V. Moreover, the flow of H is essentially a reparametrization of the flow of H on V and hence all orbits are closed and contractible.

Let us now replace the area form dA by a new magnetic field δ dA where 0<δ≤ 1. The region V will shrink to V_δ or in other words the threshold level {F=1/2} will get closer to M getting replaced by {F=δ^2 /2}. But everything else remains the same. Indeed, by a rescaling argument, the existence of closed trajectories of ω=ω_0+δπ^*dA on the energy level {F=δ^2 /2} is equivalent to the existence of closed trajectories of ω=δω_0+δπ^*dA on the energy level {F=1/2}. The flow of the later is a reparametrization of the flow of F with respect to ω=ω_0+π^*dA on {F=1/2}, and hence has no non-contractible periodic obits.

Finally, by taking some sequence {δ _n}_n=1^∞ satisfying δ _n→ 0 and a sequence of Hamiltonians H_n obtained by composing the same F with more and more narrow bump functions, we obtain the desired result.






  §.§.§ Counterexample

The condition {H<d}⊃ M in Theorem <ref> and Theorem <ref> cannot be dropped. The following example is given by Salomão and Weber <cit.>.

Let M=S^1=ℝ/ℤ. Consider the function H:T^*M=S^1×ℝ→ℝ given by

    H(q,p)=1/2p_g^2+V(q),

where V(q)=1+cos2π q. Then {H<1} does not contain M, and for any energy s∈[1, 2) the level set H=s does not carry non-contractible periodic orbits.






§ APPENDIX. THE PROOF OF PROPOSITION <REF>

To prove Proposition <ref>, we need a theorem of Weber in <cit.> that computes Floer homology of convex radial Hamiltonians.


	Let f:ℝ→ℝ be a smooth symmetric function with f^''≥0. If τ∈ℝ^+∖Λ_α and f^'(r)=τ for some r>0, then there is a natural isomorphism
	
    Φ_f^τ: HF^(-∞,b_f,τ)_α(H^f)→ H_*(ℒ_α^τ^2/2M),   b_f,τ:=rf^'(r)-f(r).

	If H^h is another such Hamiltonian, then there exists an isomorphism Ψ_hf^τ such that the following diagram commutes:
	
    HF^(-∞,b_f,τ)_α(H^f)
    		[rr]_≃^Ψ_hf^τ[dr]^≃_Φ_f^τ       HF^(-∞,b_h,τ)_α(H^h)[dl]_≃^Φ_h^τ
        H_*(ℒ_α^τ^2/2M)

	If ρ∈(0,τ]∖Λ_α and f^'(s)=ρ for some s>0, then we have the following communicative diagram whose top horizonal row is the natural inclusion ι^F:
	
    HF^(-∞,b_f,ρ)_α(H^f) @>ι^F>>  HF^(-∞,b_f,τ)_α(H^f)
    
    	@VΦ_f^ρV≃ V  @V≃ VΦ_f^τV 
     H_*(ℒ_α^ρ^2/2M) @>[ι^τ^2/2ρ^2/2]>>  H_*(ℒ_α^τ^2/2M)

	

Here the Floer homology HF^(a,b)_α(H^f) is well defined for every H^f∈𝒦_R;α^a,b; see Remark <ref>.









	The basic idea is to deform f_k and h_k by monotone homotopies to convex functions so that Theorem <ref> can be applied. To show the isomorphism (<ref>), first, we follow the graph of f_k until the slope of it becomes a for the second time at a point, saying p, near r_k2, then continue linearly with slope a, we obtain a function which is of C^1-class at p.
	Then smoothing out such a function near p yields a smooth function  f_k^'∈𝒦_R;α^-∞,a (see Figure <ref>). The monotone homotopy between f_k and f_k^', as showed in Figure <ref>, provides the monotone isomorphism
	
    Ψ_f_k^' f_k: HF^(a,+∞)_α(H^f_k)→ HF^(a,+∞)_α(H^f^'_k).

	This is the consequence of the fact that the y-intercepts of the tangential line at all points that do not remain constant during the homotopy are strictly larger than -a.
	
	
	
	
	
	Second, we define the function f_k^'' obtained by following the the graph f_k^' until the slope of it becomes a for the first time at some point q; then continue linearly with slope a (see Figure <ref>). The result function is of C^1-class at q. Smoothing near q we obtain a smooth function f_k^''∈𝒦_R;α^-∞,a.
	Note that all tangential lines of the graphs during the monotone homotopy, as indicated in Figure <ref>, which pass through the point (0,-a) lie strictly between the lines x↦ ax/R-a and x↦ l_0x-a. It follows that the homotopy between f_k^' and f_k^'' induces the monotone isomorphism
	
    Ψ_f_k^''f_k^': HF^(a,+∞)_α(H^f^'_k)→ HF^(a,+∞)_α(H^f_k^'').

	Since the y-intercept of the tangential line of graph f_k^'' at any point is less than -a, the action of
	𝒜_H^f_k^'' at each 1-periodic orbit
	is larger than a. Then the exact sequence (<ref>) yields isomorphisms
	
    HF^(a,+∞)_α(H^f_k^'') HF^(-∞,+∞)_α(H^f_k^'')
    	→ HF^(-∞,B)_α(H^f_k^'').

	Here B=b_f_k^'',a/R is defined in Theorem <ref> (the fact that a/R∉Λ_α is used). Then by Theorem <ref> we have the natural isomorphism
	
    Φ_f_k^''^a/R: HF^(-∞,B)_α(H^f_k^'')→ H_*(ℒ_α^a^2/(2R^2)M).

	By composing (<ref>)-(<ref>) we arrive at the desired isomorphism (<ref>).
	
	
	
	
	
	To obtain the isomorphism (<ref>), we deform h_k initially by the monotone homotopy as showed in Figure <ref> (a) to a smooth function h_k^', then keep on deforming h_k^' by the monotone homotopy (see Figure <ref> (b)) to a smooth function h_k^''. Here if x≥ R+ε for some sufficiently small positive constant ε, the graph of h_k^'' turns into a ray with slope μ_k which is very close to the line x→μ_kx-a. Note that all points on members of the homotopy whose tangential lines pass through the point (0,-a) lie strictly between the lines l_-x-a and l_+-a ; see Figure <ref>. Therefore we obtain the monotone isomorphisms
	
    HF^(a,+∞)_α(H^h_k) HF^(a,+∞)_α(H^h^'_k)
    	 HF^(a,+∞)_α(H^h^''_k).

	
	
	
	
	
	Next, consider function h̃_k given by following the graph of h_k^'' until it takes on slope μ_k for the first time (near the point (0,-c)); then continue linearly with slope μ_k (see Figure <ref>). Smoothing near (0,-c) yields h̃_k. Since the y-intercepts of the tangential lines
	of the graphs during the homotopy as showed in <ref> are less than -a, we obtain the monotone isomorphism
	
    Ψ_h̃_kh_k^'': HF^(a,+∞)_α(H^h_k^'')→ HF^(a,+∞)_α(H^h̃_k).

	Again, using the fact that the y-intercepts of the tangential lines of the graph h̃_k are less than -a,
	the exact sequence (<ref>) implies the following isomorphisms:
	
    HF^(a,+∞)_α(H^h̃_k)→ HF^(-∞,+∞)_α(H^h̃_k)
    	→ HF^(-∞,B^')_α(H^h̃_k)

	where B^'=b_h̃_k,μ_k is as in (<ref>). Then Theorem <ref> gives the isomorphsim
	
    Φ_h̃_k^μ_k: HF^(-∞,B^')_α(H^h̃_k)→ H_*(ℒ_α^μ_k^2/2M).

	Composing  (<ref>) - (<ref>) yields the desired isomorphism (<ref>).
	
	The proof of (<ref>)
	is identical to that of the communicative diagram (53) in <cit.> without any essential changes; see p.563 and p. 564, which is an easy application of Theorem <ref>, we omit it here.











SK

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