On the Hofer-Zehnder conjecture for non-contractible periodic orbits in Hamiltonian dynamics
aa r X i v : . [ m a t h . S G ] F e b ON THE HOFER-ZEHNDER CONJECTURE FORNON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIANDYNAMICS
YOSHIHIRO SUGIMOTO
Abstract.
In this paper, we treat an open problem related to the numberof periodic orbits of Hamiltonian diffeomorphisms on closed symplectic man-ifolds. Hofer-Zehnder conjecture states that a Hamiltonian diffeomorphismhas infinitely many periodic orbits if it has ”homologically unnecessary peri-odic orbits”. For example, non-contractible periodic orbits are homologicallyunnecessary periodic orbits because Floer homology of non-contractible peri-odic orbits is trivial. We prove Hofer-Zehnder conjecture for non-contractibleperiodic orbits for very wide classes of symplectic manifolds. Introduction and main results
In this section, we briefly explain the main theme of this paper. The precisedefinitions and notations are given in the next section. Roughly speaking, Hofer-Zehnder conjecture states that any Hamiltonian diffeomorphism on any closed sym-plectic manifold has infinitely many simple periodic orbits if it has more than ”ho-mologically necessary” number of simple periodic orbits ([19] p. 263). We have toclarify the meaning of ”homologically necessary periodic orbits”. Floer homologygroup of contractible periodic orbits is generated by contractible periodic orbitsand it is isomorphic to the singular homology of the underlying manifold. So, ho-mologically necessary number of contractible periodic orbits is the sum of the Bettinumbers. On the other hand, Floer homology group of non-contractible periodicorbits is zero. This implies that homologically necessary number of non-contractibleperiodic orbits is zero. In summary, Hofer-Zehnder conjecture is stated in the fol-lowing form.
Conjecture 1 (Hofer-Zehnder conjecture) . (1) Let φ ∈ Ham ( M, ω ) be a Hamil-tonian diffeomorphism with simple contractible periodic orbits more thanthe sum of Betti numbers. Then, φ has infinitely many simple contractibleperiodic orbits. (2) Let φ ∈ Ham ( M, ω ) be a Hamiltonian diffeomorphism which has at leastone non-contractible periodic orbit. Then φ has infinitely many simple non-contractible periodic orbits. This Conjecture 1 (1) is a generalization of Frank’s theorem [6, 7], which statesthat any area preserving homeomorphism on S with more than two fixed pointshas infinitely many periodic points. Shelukhin proved Conjecture 1 (1) for spher-ically monotone symplectic manifolds with semi-simple quantum cohomology ring([27]). Readers may consider the relationship between Conley conjecture and Hofer-Zehnder conjecture. Roughly speaking, Conjecture 1 (1) states any Hamiltonian diffeomorphism with finitely many contractible periodic orbits is so-called pseudo-rotation. Pseudo-rotation is a Hamiltonian diffeomorphism with minimum numberof contractible periodic orbits. Conley conjecture and Hofer-Zehnder conjectureimply that it is important to know the necessary and sufficient conditions that asymplectic manifold has a pseudo-rotation. Conley conjecture implies a symplecticmanifold with pseudo-rotation is very rare. In [3, 4, 15, 16, 28, 29], the importanceof the existence of non-trivial pseudo-holomorphic curve was pointed out. Theyproved that the quantum Steenrod square is deformed if the symplectic manifold ismonotone and has a pseudo-rotation. This is a strong evidence of McDuff-Chanceconjecture which states that Conley conjecture holds if some Gromov-Witten in-variants are vanishing or quantum cohomology ring is undeformed ([12, 13]).Compared to the contractible periodic orbits case, very little is known about theexistence of non-contractible periodic orbits. The main reason of this is that non-contractible periodic orbits may be empty and Floer homology of non-contractibleperiodic orbits is always trivial. So, it is very difficult to know information ofnon-contractible periodic orbits from Floer homology theory. However, G¨urel,Ginzburg-G¨urel and Orita proved that non-contractible Floer homology theory ispartially applicable and they proved Conjecture 1 (2) for some cases [11, 17, 23, 24].In this paper, we apply equivariant Floer theory [26, 30] and prove Conjecture 1(2) for very wide classes of symplectic manifolds.The statement of our main result is stated as follows. A symplectic manifold( M, ω ) is called weakly monotone symplectic manifold if it satisfies one of thefollowing conditions. We explain the precise meaning of terminologies in the nextsection.(1) (
M, ω ) is monotone(2) c ( A ) = 0 for every A ∈ π ( M )(3) The minimum Chern number N > n − Theorem 1.1.
Let ( M, ω ) be a closed and weakly monotone symplectic manifold.Let φ ∈ Ham ( M, ω ) be a Hamiltonian diffeomorphism such that -periodic orbitin the class γ = 0 ∈ H ( M : Z ) / Tor is finite, not empty and at least local Floerhomology HF loc ( φ, x ) of at least one of them is not zero. Then, for sufficientlylarge prime p , φ has p -periodic or p ′ -periodic simple orbit in the class p · γ . Here p ′ is the first prime number greater than p . In particular, there are infinitely manysimple periodic orbits of φ in the class N · γ . The assumption ”weakly monotone” is a purely technical assumption. Our proofis based on the Z p -equivariant Floer homology theory and we need Z p -coefficientFloer theory. For general closed symplectic manifolds, we need so-called virtualtechnique to define Floer theory ([10, 20]), but virtual technique works on Q -coefficient in general (see also [9]). We are not sure we can construct an equivarianttheory on general closed symplectic manifolds. N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS3
Acknowledgement
This work was carried out during my stay as a research fellow in National Centerfor Theoretical Sciences. The author thanks NCTS for a great research atmosphereand many supports. He also gratefully acknowledges his teacher Kaoru Ono forcontinuous supports. 2.
Preliminaries
In this section, we explain notations and terminologies used in this paper.2.1.
Elementary notations.
Let (
M, ω ) be a symplectic manifold, so M is afinite dimensional C ∞ -manifold and ω ∈ Ω ( M ) is a symplectic form on M . In thispaper, we always assume that M is a closed manifold.For any C ∞ -function H ∈ C ∞ ( M ), we define the Hamiltonian vector field X H by the following relation. ω ( X H , · ) = − dH We can also consider S -dependent (=1-periodic) Hamiltonian function H andHamiltonian vector field X H in the same formula. The time 1 flow of X H is called aHamiltonian diffeomorphism generated by H . We denote this flow by φ H . The setof all Hamiltonian diffeoomorphisms is called Hamiltonian diffeomorphism groupand we denote the Hamiltonian diffeomorphism group of ( M, ω ) by Ham(
M, ω ).Ham(
M, ω ) = { φ H | H ∈ C ∞ ( S × M ) } We also consider ”iterations” of H and φ H . For any integer k ∈ N , we define H ( k ) as follows. H ( k ) = kH ( kt, x )It is straightforward to see that φ H ( k ) = ( φ H ) k . Let P l ( H ) be the space of l -periodicperiodic orbits of X H . P l ( H ) = { x : S l → M | ˙ x ( t ) = X H t ( x ( t )) } S l = R /l · Z It is also straightforward to see that there is one to one correspondence between P k ( H ) and P ( H ( k ) ). We abbreviate P ( H ) to P ( H ). A l -periodic orbits x ∈ P l ( H )is called simple if there is no l ′ -periodic orbits y ∈ P l ′ ( H ) which satisfies the fol-lowing conditions. l = l ′ · m ( l ′ , m ∈ N ) x ( t ) = y ( π l,l ′ ( t ))Here π l,l ′ : S l → S l ′ is the natural projection. So a periodic orbit is simple if andonly if it is not iterated periodic orbit of lower period.Next, we explain the definition of the minimum Chern number N . A symplecticmanifold ( M, ω ) becomes an almost complex manifold, and its tangent bundle hasa natural first Chern class c ( T M ) ∈ H ( M : Z ). The minimum Chern number N ∈ N ∪ { + ∞} is the positive generator of c ( T M ) | π ( M ) . Note that if the image is YOSHIHIRO SUGIMOTO zero, N is defined by N = + ∞ . A symplectic manifold ( M, ω ) is called monotoneif the cohomology class of the symplectic form [ ω ] over π ( M ) is a non-negativemultiple of the first Chern class. In other words, there is a constant λ ≥ Z S v ∗ ω = λ Z S v ∗ c holds for any v : S → M . As we mentioned in the previous section, a weaklymonotone symplectic manifold is defined as follows. Definition 2.1 (weakly monotone symplectic manifold) . A symplectic manifold ( M, ω ) is called weakly monotone if and only if it satisfies one of the followingconditions. (1) ( M, ω ) is a monotone symplectic manifold. (2) c ( A ) = 0 holds for any A ∈ π ( M ) . (3) The minimum Chern number N is greater than or equal to n − . Floer homology theory.
In this subsection, we explain Floer homology fornon-contractible periodic orbits. Essentially, there is nothing new in the non-contractible case, but in this paper we need non-contractible Floer chain complexover the universal Novikov ring. Let K be the ground field (In this pape, we con-sider the case of K = F p where F p is a field of characteristic p .). We also assumethat ( M, ω ) is a weakly monotone symplectic manifold. The universal Novikov ringΛ is defined as follows.Λ = (cid:26) l X i =1 a i · T λ i (cid:12)(cid:12)(cid:12)(cid:12) a i ∈ K , λ i ∈ R , λ i → + ∞ (if l = + ∞ ) (cid:27) We need the subring Λ ⊂ Λ as follows.Λ = (cid:26) X a i · T λ i ∈ Λ (cid:12)(cid:12)(cid:12)(cid:12) λ i ≥ (cid:27) We need non-contractible Floer theory over Λ and Λ . We fix γ = 0 ∈ H ( M : Z ) / Torand we denote the set of 1-periodic orbit of H ∈ C ∞ ( S × M ) in γ by P ( H, γ ). P ( H, γ ) = { x ∈ P ( H ) | [ x ] = γ } Floer chain complex over Λ and Λ is defined as follows. We assume that anyperiodic orbit in P ( H, γ ) is non-degenerate. CF ( H, γ : Λ) = M x ∈ P ( H,γ ) Λ · xCF ( H, γ : Λ ) = M x ∈ P ( H,γ ) Λ · x Note that above CF ( H, γ : Λ) and CF ( H, γ : Λ ) are not graded over Z (It ispossible to give a grading over Z .). The differential operator d F is defined asfollows. d F ( x ) = X y ∈ P ( H,γ ) ,λ ≥ n λ ( x, y ) T λ · y N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS5 n λ ( x, y ) ∈ K is the number of the solutions of the following Floer equation mod-ulo the natural R -action. u : R × S −→ M∂ s u ( s, t ) + J t ( u ( s, t ))( ∂ t u ( s, t ) − X H t ( u ( s, t ))) = 0lim s →−∞ u ( s, t ) = x ( t ) , lim s → + ∞ u ( s, t ) = y ( t ) λ = Z R × S u ∗ ω + Z H ( t, x ( t )) − H ( t, y ( t )) dt Floer homology HF ( H, γ : Λ) is the homology of ( CF ( H, γ : Λ) , d F ) and HF ( H, γ : Λ )is the homology of ( CF ( H, γ : Λ ) , d F ). It is straightforward to see that HF ( H, γ : Λ)does not depend on the Hamiltonian function H and it always equals to zero. How-ever, HF ( H, γ : Λ ) does not equal to zero in general. So, there is a sequence0 < β ≤ · · · ≤ β k such that HF ( H, γ : Λ ) ∼ = k M i =1 Λ /T β i · Λ holds. We have the following filtration on ( CF ( H, γ : Λ) , d F ). For any c ∈ R , wehave the following subcomplex of ( CF ( H, γ : Λ) , d F ). CF c ( H, γ : Λ) = (cid:26) X x ∈ P ( H : γ ) (cid:16) X λ i ≥ c a i · T λ i (cid:17) · x ∈ CF ( H, γ : Λ) (cid:27)
Then for any c < d , we can define the Floer homology with action window [ c, d ) asfollows. HF [ c,d ) ( H, γ : Λ) = H (( CF c ( H, γ : Λ) /CF d ( H, γ : Λ) , d F ))2.3. Local Floer homology.
We need local theory of Floer homology theory, so-called local Floer homology (see [14]). Let x ∈ P ( H, γ ) be an isolated periodic orbitof a Hamiltonian function H ∈ C ∞ ( S × M ). In this subsection, we explain thelocal Floer homology HF loc ( H, x ). Note that x is not necessarily non-degenerateperiodic orbit. Let U ⊂ S × M be a sufficiently small open neighborhood of theembedded circle ( t, x ( t )) ⊂ S × M and let e H be a non-degenerate C ∞ -small per-turbation of H . Then x splits into non-degenerate periodic orbits { x i , · · · , x k } of e H in U . The local Floer chain complex CF loc ( H, x ) is generated by these perturbedperiodic orbits. CF loc ( H, x ) = k M i =1 K · x i The boundary operator d locF on CF loc ( H, x ) is defined by counting solutions of theFloer equation in U . So, d locF can be written by d locF ( x i ) = P n ( x i , x j ) x j wherecoefficient n ( x i , x j ) ∈ K is determined by the number of solutions of the followingequation modulo the natural R -action. YOSHIHIRO SUGIMOTO u : R × S −→ M∂ s u ( s, t ) + J t ( u ( s, t ))( ∂ t u ( s, t ) − X e H t ( u ( s, t ))) = 0lim s →−∞ u ( s, t ) = x i ( t ) , lim s → + ∞ u ( s, t ) = x j ( t )( t, u ( s, t )) ∈ U The local Floer homology HF loc ( H, x ) is the homology of the chain complex ( CF loc ( H ) , d locF ).The following properties hold (see [14]).(1) If x is non-degenerate periodic orbit of H , then HF loc ( H, x ) ∼ = K holds.(2) HF loc ( H, x ) does not depend on the choice of the perturbation e H .(3) Let { β , · · · , β l } ⊂ S \ { } be the set of eigenvalues of the differential map dφ H : T x (0) M −→ T x (0) M on S \ { } ⊂ C . An integer k ∈ N is called admissible if β ki = 1 holds forany 1 ≤ k ≤ l . If k is admissible, the local Floer homology of ( H, x ) and( H ( k ) , x ( k ) ) are isomorphic. In other words, HF loc ( H ( k ) , x ( k ) ) ∼ = HF loc ( H, x )holds.Note that the local Floer chain complex ( CF loc ( H, x ) , d locF ) is a chain complexover a finite dimensional vector space over K (not over Λ nor Λ ).3. Z p -equivariant Floer theory Our proof of Theorem 1.1 is an application of the Z p -equivariant Floer homol-ogy. In this section, we briefly review constructions and basic properties of theequivariant theory which we will use in our proof of Theorem 1.1. The readers canfind the detailed constructions of the equivariant Floer theory in [30] (see also [26]).The first attempt to this direction is due to Seidel in [26], where he constructed Z -equivariant Floer theory. After that, Seidel’s construction was generalized to Z p -equivariant Floer theory by Shelukhin-Zhao and Shelukhin in [30, 27]. In thesepapers, they gave a various applications of the equivariant Floer homology. Inparticular, the equivariant Floer theory is very useful when we study the relation-ship between HF ( φ ) and iterated Floer homology HF ( φ p ). We will focus on thebehavior of the torsion of HF ( φ p ) as p becomes bigger and bigger.We fix a prime number p and we assume that H ∈ C ∞ ( S × M ) is a Hamiltonianfunction such that H ( p ) is non-degenerate. In this section, we assume that ( M, ω )is a monotone symplectic manifold. We will give a construction of the equivarianttheory for weakly monotone symplectic manifolds in the last section, where weexplain what is the technical difficulty in the weakly monotone case and how wecan overcome this problem. The equivariant Floer homology is a mixture of Floertheory on (
M, ω ) and Morse theory on the classifying space.Let C ∞ be the infinite-dimensional complex vector space C ∞ = { z = { z k } k ∈ Z ≥ | z k ∈ C , z k = 0 for sufficiently large k } and let S ∞ ⊂ C ∞ be the infinite-dimensional sphere defined by S ∞ = { z ∈ C ∞ | X | z k | = 1 } . N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS7
There is a natural Z p -action and a shift operator τ on S ∞ as follows. { m · z } k = exp (cid:26) πimp (cid:27) · z k ( m ∈ Z p ) { τ ( z ) } k = ( k = 0) z k − ( k ≥ f : S ∞ → R be the following Bott-Morse function. f ( z ) = ∞ X k =0 k | z k | This f satisfies τ ∗ f = f + 1 and f ( z ) = f ( m · z ) for ( m ∈ Z p ). The critical sub-manifolds are S l = { z ∈ S ∞ | | z k | = 1 } and their index are 2 l . Next, we perturb f to a Morse function with above properties. For example, we can use the followingexplicit perturbation (see [30]). e F = f ( z ) + ǫ · X k Re( z pk )Then e F is a Mose function and S l splits into critical points { Z l , · · · , Z p − l } and { Z l +1 , · · · , Z p − l +1 } . The Morse index of Z mj is j . It is straightforward to see that τ ∗ e F = e F + 1 and e F ( z ) = e F ( m · z ) hold. We also fix a Riemannian metric e g suchthat e g is invariant under Z p -action and τ and ( e F , e g ) is a Morse-Smale pair.Mose boundary operator d M can be written in the following form. d M ( Z l ) = Z l +1 − Z l +1 d M ( Z l +1 ) = Z l +2 + Z l +2 + · · · + Z p − l +2 This d M is equivariant under the Z p action m · Z ji = Z j + m (mod p ) i . Let ( H ( p ) , J t ) bea pair of Hamiltonian function H ( p ) and S -dependent almost complex structureon ( M, ω ). We want to extend J t to a family of S -dependent almost complexstructures parametrized by w ∈ S ∞ . So, we consider a family of almost complexstructures { J w,t } w ∈ S ∞ ,t ∈ S which satisfies the following conditions. • (local constant at critical points) In a small neighborhood of Z mi , J w,t = J t − mp holds. • ( Z p -equivariance) J m · w,t = J w,t − mp holds for any m ∈ Z p and w ∈ S ∞ . • (invariance under the shift τ ) J τ ( w ) ,t = J z,t holds.We consider the following equation, which is a mixture of the Floer equation andthe Morse equation for x, y ∈ P ( H ( p ) ), m ∈ Z p , λ ≥ α ∈ { , } , i ∈ Z ≥ . YOSHIHIRO SUGIMOTO ( u, v ) ∈ C ∞ ( R × S , M ) × C ∞ ( R , S ∞ ) ∂ s u ( s, t ) + J v ( s ) ,t ( u ( s, t ))( ∂ t u ( s, t ) − X H ( p ) ( u ( s, t ))) = 0 dds v ( s ) − grad e F = 0lim s →−∞ v ( s ) = Z α , lim s → + ∞ v ( s ) = Z mi , lim s →−∞ u ( s, t ) = x ( t ) , lim s → + ∞ u ( s, t ) = y ( t − mp ) Z R × S u ∗ ω + Z H ( p ) ( t, x ( t )) − H ( p ) ( t, y ( t )) dt = λ We denote the space of solutions of this equation by M λα,i,m ( x, y ). M λα,i,m ( x, y ) = { ( u, v ) | ( u, v ) satisfies the above equation } / ∼ The equivalence relation ∼ is given by the natural R -action on the solution space.We define d i,mα ( α ∈ { , } , i ∈ Z ≥ , m ∈ Z p ) as follows. d i,mα : CF ( H ( p ) , γ : Λ ) −→ CF ( H ( p ) , γ : Λ ) x X x ∈ P ( H ( p ) ) ,λ ≥ ♯ M λα,i,m ( x, y ) · T λ y Let d iα be the sum d iα = P m ∈ Z p d i,mα . Note that d = d = d F holds. We define the Z p -equivariant Floer chain complex as follows. CF Z p ( H ( p ) , γ ) = CF ( H ( p ) , γ : Λ ) ⊗ Λ [ u − , u ]] h θ i Here, h θ i is the exterior algebra on the formal variable θ and Λ [ u − , u ]] is the ringof Laurant polynomials of formal variable u . So any element of CF Z p ( H ( p ) , γ ) iswritten in the following form for some k ∈ Z .( x k + y k θ ) u k + ∞ X i = k +1 ( x i + y i θ ) u i ( x j , y j ∈ CF ( H ( p ) , γ : Λ ))The equivariant Floer boundary operator d eq is a Λ [ u − , u ]]-linear map anddefined in the following formula. d eq ( x ⊗
1) = ∞ X i =0 d i ( x ) ⊗ u i + ∞ X i =0 d i +10 ( x ) ⊗ u i θd eq ( x ⊗ θ ) = ∞ X i =0 d i +11 ( x ) ⊗ u i θ + ∞ X i =1 d i ( x ) ⊗ u i d = d = d F implies that d eq is a sum of d F and higher terms. The Z p -equivariantFloer homology HF Z p ( H ( p ) , γ ) is defined by HF Z p ( H ( p ) , γ ) = H (( CF Z p ( H ( p ) , γ ) , d eq )) . Remark 3.1. d eq is defined on Laurant polynomial ring Λ [ u − , u ]] , but we candefine d eq over the formal power series Λ [[ u ]] . We defined d eq over Laurant polyno-mial ring because the Z p -equivariant pair of pants product gives a local isomorphismbetween local Floer homology and local equivariant Floer homology over Λ [ u − , u ]] N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS9 (see [26, 30] ). We will explain this in the latter sections and we apply this localisomorphism. Local (equivariant) Floer homology and homologicalperturbation theory
In the proof of Theorem 1.1, we have to treat Hamiltonian diffeomorphism withfinite periodic orbits, but not necessarily non-degenerate. One natural way to over-come this difficulty is to just perturb the original Hamiltonian diffeomorphsm φ toa non-degenerate Hamiltonian diffeomorphism e φ and consider HF ( e φ : Λ ) insteadof HF ( φ : Λ ). However, it is not sufficient because the structure of HF ( e φ : Λ )does depend on the perturbation e φ . We have to construct a chain complex anda homology theory in a homologically canonical way. In this section, we explaina construction of Floer thoery for not necessarily non-degenerate Hamiltonian dif-feomorphism (see also [27]). This is an application of the perturbation theory in[21].Assume that H is not necessarily non-degenerate Hamiltonian function and P ( H, γ ) is finite for γ = 0 ∈ H ( M : Z ) / Tor. Let e H be a small perturbation of H so that φ e H is non-degenerate. Then every element x i ∈ P ( H, γ ) splits into afinite non-degenerate periodic orbits { x i , · · · , x l i i } ⊂ P ( e H, γ ). First, we deformthe complex ( CF ( e H, γ : Λ ) , d F ) in a canonical way. For any x i ∈ P ( H, γ ) and x ji ∈ P ( e H, γ ), we fix a sufficiently small connecting cylinder between x i and x ji asfollows. v ji : [0 , × S −→ Mv ji (0 , t ) = x i ( t ) , v ji (1 , t ) = x ji ( t )The ”gap” of the action functional c ( x i , x ji ) = Z [0 , × S ( v ji ) ∗ ω + Z H ( t, x i ( t )) − e H ( t, x ji ( t )) dt does not depend on the choice of v ji . We define a correction map τ as follows. τ : CF ( e H, γ : Λ) → CF ( e H, γ : Λ) x ji T c ( x i ,x ji ) · x ji Note that τ is defined over Λ (not over Λ ). Then we can define the modifieddifferential operator f d F by f d F = τ − ◦ d F ◦ τ . It is easy to see that f d F is alsodefined over Λ . Note that τ gives an isomorphism between Floer chain complexesover Λ (but not over Λ ) as follows. τ e H : ( CF ( e H, γ : Λ) , f d F ) −→ ( CF ( e H, γ : Λ) , d F ) x τ ( x )Let e H ′ be another non-degenerate perturbation of H and we also denote themodified differential operator on CF ( e H ′ , γ : Λ) by f d F . It is straightforward to seethat the natural continuation map between Floer chain complexesΦ : ( CF ( e H, γ : Λ) , d F ) −→ ( CF ( e H ′ , γ : Λ) , d F ) descends to a continuation map over Λ (not only over Λ) as follows. e Φ : ( CF ( e H, γ : Λ ) , f d F ) −→ ( CF ( e H ′ , γ : Λ ) , f d F ) x ( τ e H ′ ) − ◦ Φ ◦ τ e H The continuation map of the inverse directionΨ : ( CF ( e H ′ , γ : Λ) , d F ) −→ ( CF ( e H, γ : Λ) , d F )also descends to a continuation map e Ψ : ( CF ( e H ′ , γ : Λ ) , f d F ) −→ ( CF ( e H, γ : Λ ) , f d F )in the same way. The chain homotopy maps between the identity and Ψ ◦ Φ, Φ ◦ Ψalso descend to chain homotopy maps between the identity and e Ψ ◦ e Φ, e Φ ◦ e Ψ. Thisimplies that the perturbed and modified Floer chain complex ( CF ( e H, γ : Λ ) , f d F )is unique up to chain homotopy equivalence. So we can define the Floer homologyof H by the homology of this chain complex and this is well defined. HF ( H, γ : Λ ) = H ( CF ( e H, γ : Λ ) , f d F )However, this construction is not sufficient for our purpose. The chain complex( CF ( e H, γ : Λ ) , f d F ) is not strict in the following sense. ”Strict” means that for any z ∈ CF ( e H, γ : Λ ), there is ǫ > f d F ( z ) ∈ CF ǫ ( e H, γ : Λ )holds. Note that we can identify CF ( e H, γ : Λ ) and L x ∈ P ( H,γ ) CF loc ( H, x ) ⊗ Λ .Then f d F can be decomposed into the sum of f d F = d locF + D where D is a higherterm. D ( CF ( e H, γ : Λ )) ⊂ CF ǫ ( e H, γ : Λ ) ( ǫ > CF ( e H, γ : Λ ) , f d F ) is strict if and only if d locF = 0 holds. Our aim of therest of this section is to construct a homologically canonical boundary operatoron L x ∈ P ( H,γ ) HF loc ( H, x ) ⊗ Λ instead on L x ∈ P ( H,γ ) CF loc ( H, x ) ⊗ Λ and provethat the chain complex is chain homotopy equivalent to the original chain complex.We defined the Floer chain complex of H by CF ( H, γ : Λ ) = M x ∈ P ( H,γ ) HF loc ( H, x ) ⊗ Λ . We choose a basis { X ai , Y bi , Z ci } of CF loc ( H, x i ) = L F p · x ji which satisfies thefollowing relations. d locF ( X ai ) = 0 (1 ≤ a ≤ dim F p HF loc ( H, x i )) d locF ( Y bi ) = 0 (1 ≤ b ≤
12 ( l i − dim F p HF loc ( H, x i ))) d locF ( Z ci ) = Y ci (1 ≤ c ≤
12 ( l i − dim F p HF loc ( H, x i )))Then HF loc ( H, x i ) can be identified with span F p h X i , · · · , X dim F p HF loc ( H,x i ) i i . Wedefine two operators Π i and Θ i on CF loc ( H, x i ) as follows. N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS11 Π i ( X α a X ai + X β b Y bi + X γ c Z ci ) = X α a X ai Θ i ( X α a X ai + X β b Y bi + X γ c Z ci ) = X β c Z ci Let Π = P Π i and Θ = P Θ i be the sums of the above operators. Next, we applythe perturbation theory in [21]. First we explain what is the perturbation theoryand how we can apply it to our case.Let M = ( M, d M ) be a chain complex with a decreasing filtration { F p M } p ∈ Z ≥ . M = F M ⊂ F M ⊂ F M ⊂ · · · We assume that this filtration is complete. In other words, M is complete withrespect to the F p -adic topology and a infinite sum P p m p such that m p ∈ F p M holds converges uniquely to an element of M . Let ( N, d N ) be another chain complexwith a complete and decreasing filtration { F p N } . Morphisms between filteredchain complexes are maps that preserve filtrations. Let f : M → N be a morphism.Another morphism g : M → N is called a perturbations of f if( f − g ) F p M ⊂ F p +1 N holds for any p ∈ Z ≥ . Perturbations of boundary operators d M and d N are definedin the same way. In [21], Markl studied the following problem. Let F : ( M, d M ) −→ ( N, d N ) G : ( N, d N ) −→ ( M, d M )be chain maps that preserve filtrations and assume that we also have chain homo-topies between the identity and GF , F G as follows. H : M → M, L : N → NGF − Id M = d M H + Hd M F G − Id N = d N L + Ld N Next, we perturb the original boundary operator d M to a new boundary operator f d M = d M + D M ( D M ( F p M ) ⊂ F p +1 M holds for any p ). Then, can we perturb d N , F , G , H and L so that the above equations hold? Markl gave a complete answer tothis problem. It is always possible if the ”obstruction class” vanishes (Ideal pertur-bation lemma). However, we do not need the full generality of Markl’s theorem. Weonly treat the simplest case of this perturbation problems. We treat the case that( N, d N ) is the strong deformation retract of ( M, d M ). Let ( M, d M ) and ( N, d N ) betwo chain complexes with complete and decreasing filtrations. Assume that chainmaps F : ( M, d M ) −→ ( N, d N ) G : ( N, d N ) −→ ( M, d M )preserve filtrations and there is a morphism H : M −→ N which satisfies the fol-lowing conditions. GF − Id M = d M H + Hd M F G = Id N HH = 0 , HG = 0 , F H = 0 (annihilation properties) Let f d M = d M + D M be a perturbation of d M . Then Basic perturbation lemma([21]) states that there are perturbations f d N , e F , e G and e H which satisfy the followingconditions. e G e F − Id M = f d M e H + e H f d M e F e G = Id N So we can perturb (
N, d N ) so that ( M, f d M ) and ( N, f d N ) are also chain homotopyequivalent. We also have explicit formulas for these perturbations as follows. f d N = d N + F X l ≥ ( D M H ) l D M G e F = F X l ≥ ( D M H ) l e G = X l ≥ ( HD M ) l G e H = H X l ≥ ( D M H ) l Next, we apply this basic perturbation lemma to construct a strict Floer chaincomplex of H . In our case, ( M, d M ), ( N, d N ) and f d M correspond to the followingthings. ( M, d M ) = ( M s ∈ P ( H,γ ) CF loc ( H, x ) ⊗ Λ , d locF )( N, d N ) = ( M s ∈ P ( H,γ ) HF loc ( H, x ) ⊗ Λ , f d M = f d F = d locF + DF = Π , H = Θ , The ”inclusion” G : N −→ M is defined as follows. HF loc ( H, x i ) −→ CF loc ( H, x i ) (cid:20) X α a X ai (cid:21) X α a X ai Then according to the basic perturbation lemma and the explicit constructionof f d N , we can define the boundary operator d F : CF ( H, γ : Λ ) → CF ( H, γ : Λ )by the following formula. d F = Π ◦ ∞ X l =0 ( D Θ) l D Note that we identify HF loc ( H, x i ) with span F p h X i , · · · , X dim F p HF loc ( H,x i ) i i in thisformula. Then, ( CF ( H, γ : Λ ) , d F ) is a strict chain complex which is also chainhomotopy equivalent to the original chain complex ( CF ( e H, γ : Λ ) , f d F ). So wedefined the Floer chain complex ( CF ( H, γ : Λ ) , d F ) in a homologically canonicalway. N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS13
Next, assume that P ( H ( p ) , γ ) is finite. Then, we can construct a strict boundaryoperator d eq on the Z p -equivariant Floer chain complex CF Z p ( H ( p ) , γ ) = M x ∈ P ( H ( p ) ,γ ) HF loc Z p ( H ( p ) , x ) ⊗ Λ by the same formula as in the definition of ( CF ( H, γ : Λ ) , d F ). So we can alsodefine the Z p -equivariant Floer chain complex for not necessarily non-degenerateHamiltonian diffeomorphisms.5. Hofer-Zehnder conjecture for non-contractible periodic orbits
In this section, we prove Theorem 1.1. We divide the proof in two parts, mono-tone case and weakly monotone case. The reason of this is that we have notconstructed Z p -equivariant Floer theory for weakly monotone symplectic manifoldsyet. As we mentioned before, we have to overcome some technical difficulties inweakly monotone case. Once we establish Z p -equivariant Floer theory on weaklymonotone symplectic manifolds, the rest of the proof is almost the same as in themonotone case. So essential part of our proof is given in the monotone case.5.1. Monotone case.
In this subsection, we prove Theorem 1.1 for monotone sym-plectic manifolds. Let (
M, ω ) be a closed monotone symplectic manifold. We fix H ∈ C ∞ ( S × M ) and γ = 0 ∈ H ( M : Z ) / Tor. We also assume that P ( H, γ ) = { x , · · · , x k } and P ( H ( p ) , pγ ) = { y , · · · , y k } and y i = x ( p ) i holds. In other words, every p -periodicorbits of H in pγ is not simple. Our purpose is to prove that there is a simple p ′ -periodic orbit in pγ .In the previous section, we defined the Z p -equivariant Floer chain complex( CF ( H ( p ) , pγ ) , d eq ) and the Z p -equivariant Floer homology HF Z p ( H ( p ) , pγ ). Re-call that we applied the perturbation theory to make ( CF ( H ( p ) , pγ ) , d eq ) a strictdifferential complex.Let e H be a perturbation of H such that e H ( p ) is non-degenerate. In the definitionof d eq , we have seen that d eq is written as d F + higer terms if H ( p ) is non-degenerateand we do not have to apply perturbation lemma. Next we prove that this is alsotrue when H ( p ) is not non-degenerate and we apply perturbation lemma.In this case, each y i ∈ P ( H ( p ) , pγ ) splits into non-degenerate periodic orbits { y i , · · · , y l i i } ⊂ P ( e H ( p ) , pγ ). We prove the following lemma. Lemma 5.1.
We can choose a basis { e X ai , e Y bi , e Z ci , e X ai,θ , e Y bi,θ , e Z ci,θ } of CF loc Z p ( H ( p ) , y i ) = CF loc ( H ( p ) , y i ) ⊗ F p [ u − , u ]] h θ i over F p [ u − , u ]] and a basis { X ai , Y bi , Z ci } of CF loc ( H ( p ) , y i ) which satisfies the fol-lowing conditions. e X ai = X ai ⊗ a ( a )1 ⊗ θ + ∞ X k =1 X k ′ =0 , a ( a )2 k + k ′ ⊗ u k θ k ′ ) e Y bi = Y bi ⊗ b ( b )1 ⊗ θ + ∞ X k =1 X k ′ =0 , b ( b )2 k + k ′ ⊗ u k θ k ′ ) e Z ci = Z ci ⊗ c ( c )1 ⊗ θ + ∞ X k =1 X k ′ =0 , c ( c )2 k + k ′ ⊗ u k θ k ′ ) e X ai,θ = X ai ⊗ θ + ∞ X k =1 X k ′ =0 , d ( a )2 k + k ′ ⊗ u k θ k ′ e Y bi,θ = Y bi ⊗ θ + ∞ X k =1 X k ′ =0 , e ( b )2 k + k ′ ⊗ u k θ k ′ e Z ci,θ = Z ci ⊗ θ + ∞ X k =1 X k ′ =0 , f ( c )2 k + k ′ ⊗ u k θ k ′ d locF ( X ai ) = d locF ( Y bi ) = 0 , d locF ( Z ci ) = Y ci , d loceq ( e X ai ) = d loceq ( e Y bi ) = 0 , d loceq ( e Z ci ) = e Y ci d loceq ( e X ai,θ ) = d loceq ( e Y bi,θ ) = 0 , d loceq ( e Z ci,θ ) = e Y ci,θ ≤ a ≤ dim F p HF loc ( H ( p ) , y i ) , ≤ b, c ≤
12 ( l i − dim F p HF loc ( H ( p ) , y i )) { a ( a ) ∗ , b ( b ) ∗ , c ( c ) ∗ , d ( a ) ∗ , e ( b ) ∗ , f ( c ) ∗ } ⊂ CF loc ( H ( p ) , y i )Note that CF loc ( H ( p ) , y i ) = ⊕ j F p · y ji holds. The existence of a basis { X ai , Y bi , Z ci } is trivial and we can define e Y bi , e Z ci , e Y bi,θ , e Z ci,θ by e Z ci = Z ci ⊗ e Z ci,θ = , Z ci ⊗ θ , e Y bi = d loceq ( e Z bi )and e Y bi,θ = d loceq ( e Z bi,θ ). So what we have to prove is the existence of e X ai and e X ai,θ .Let C be the set of all cycles in ( CF loc Z p ( H ( p ) , y i ) , d loceq ). We consider two projec-tions π , π : C −→ span h X i , · · · , X di i ( d = dim F p HF loc ( H ( p ) , y i )) . For z = ( a k ⊗ b k ⊗ θ ) u k + P l = k +1 ( a l ⊗ b l ⊗ θ ) u l , we define π ( z ) and π ( z )by the following formula. Let Π i : CF ( H ( p ) , y i ) → span h X i , · · · , X di i be the pro-jection as before. π ( z ) = Π i ( a k ) π ( z ) = ( Π i ( b k ) a k = 00 a k = 0Assume that Im( π ) is generated by f V j and Im( π ) is generated by f W j .Im( π ) = span F p h f V , · · · , f V α i Im( π ) = span F p h f W , · · · , g W β i N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS15
We choose { V , · · · , V α , W , · · · , W β } ⊂ C so that π ( V j ) = f V j and π ( W j ) = f W j hold. Then it straightforward to see that { V , · · · , V α , W , · · · , W β } ⊂ C generates HF loc Z p ( H ( p ) , y i ) over F p [ u − , u ]]. Note that there is an isomoprhism between localFloer homology and local Z p -equivariant Floer homology ([30]). HF loc Z p ( H ( p ) , y i ) ∼ = HF loc ( H ( p ) , y i ) ⊗ F p [ u − , u ]] h θ i Here we use the fact that p is admissible and HF loc ( H ( p ) , y i ) ∼ = HF loc ( H, x i ) holds.This implies that α = β = dim F p HF loc ( H ( p ) , y i ) and Im( π i ) = span F p h X , · · · , X d i .So we can choose e X ai and e X ai,θ and we proved Lemma 5.1. (cid:3) Our next purpose is to calculate HF Z p ( H ( p ) , pγ ). For this purpose, we introduce Z p -equivariant Tate homology for ( H, γ ) (see [30]). The Z p -equivariant Tate chaincomplex is defined as follows. C T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) = CF ( H, γ : Λ ) ⊗ p ⊗ Λ [ u − , u ]] h θ i The Floer differential d F on CF ( H, γ : Λ ) naturally extends to a differential d ( p ) F on CF ( H, γ : Λ ) ⊗ p . There is a natural Z p action τ on CF ( H, γ : Λ ) ⊗ p . τ ( x ⊗ x ⊗ · · · ⊗ x p − ) = ( − | x p − | ( | x | + ··· + | x p − | ) x p − ⊗ x ⊗ · · · x p − and let N be the sum N = 1 + τ + τ · · · + τ p − . Then the Tate differential d T ate is a Λ [ u − , u ]]-linear map and defined as follows. d T ate ( x ⊗
1) = d ( p ) F ( x ) ⊗ − τ )( x ) ⊗ θd T ate ( x ⊗ θ ) = d ( p ) F ( x ) ⊗ θ + N ( x ) ⊗ uθ The Tate homology is defined by the homology of this complex. H T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) = H ( C T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) , d T ate ) H T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) is determined by HF ( H, γ : Λ ) from the followinglemma. Lemma 5.2 (Lemma 17 in [27]) . There is so-called quasi-Frobenius isomorphismas follows. r ∗ p HF ( H, γ : Λ ) ⊗ Λ [ u − , u ]] h θ i ∼ = H T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) Here, r p : Λ → Λ is a map induced by T → T p . Assume that there is an isomorphism HF ( H, γ : Λ ) ∼ = m M i =1 Λ /T β j Λ ( β j > . Then, Lemma 5.2 implies that there is the following isomoprhism. H T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) ∼ = (cid:16) m M i =1 Λ /T pβ j Λ (cid:17) ⊗ Λ [ u − , u ]] h θ i So, the module structure of the Tate homology is completely determined by HF ( H, γ : Λ ). We also have one more important operation, so-called Z p -equivariant pair ofpants product. P : H T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) −→ HF Z p ( H ( p ) , pγ )The construction of P is just counting the solutions of Floer equations on p -branched cover of the cylinder R × S ( p -legged pants) parametrized by S ∞ (seesection 8 in [30], see also [26] for Z -equivariant case). The detailed constructionof P is not needed here. The important point is that P gives a local isomorphismbetween their local homologies in the following sense ([30]).We define an action filtration on C T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) and CF Z p ( H ( p ) , pγ ).We fix a sufficiently small positive real number ǫ >
0. We define the filtration F q C T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) and F q CF Z p ( H ( p ) , pγ ) ( q ∈ Z ≥ ) as follows. F q C T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) = T qǫ C T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) F q CF Z p ( H ( p ) , pγ ) = T qǫ CF Z p ( H ( p ) , pγ )If we divide d T ate into d locT ate + D T ate , D T ate ( F q C T ate ( Z p , CF ( H, γ : Λ ) ⊗ p )) ⊂ F q +1 C T ate ( Z p , CF ( H, γ : Λ ) ⊗ p )holds for any q ∈ Z ≥ . So E -term of the associated spectral sequences of F q C T ate and F q CF Z p are given by local homologies. This fact and local isomorphism theo-rem of P implies that P gives an isomorphism between their E -pages of spectralsequences. So, in order to prove that P is an isomoprhism, it suffices to prove thefollowing strong convergences of each spectral sequences (Theorem 3.2 in Chapter14 in [2]). Lemma 5.3 (strong convergence) . Let ( A, d ) be a filtered differential module sothat ( A, d ) = ( C T ate , d
T ate ) or ( A, d ) = ( CF Z p , d eq ) holds. (1) (weakly convergence) For each q ∈ Z ≥ , the intersection of the images ofthe homomorphisms H ( F q A/F q + r A ) −→ H ( F q +1 A ) r ≥ z ] [ dz ] is zero. (2) The natural map u : H ( A ) −→ lim ←− H ( A ) /F q H ( A ) is an isomorphism. Here, F q H ( A ) is the image of H ( F q A ) −→ H ( A ) . Proof (Lemma 5.3)(1) Let K be another Hamiltonian function and let ( B, d ) be either the Tatechain complex ( C T ate ( Z p , CF ( K, γ : Λ ) ⊗ p , d T ate ) or CF Z p ( K ( p ) , pγ : Λ ) respec-tively. Then, there is a continuation homomorphism over Λ F : ( A ⊗ Λ , d ) −→ ( B ⊗ Λ , d ) G : ( B ⊗ Λ , d ) −→ ( A ⊗ Λ , d ) N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS17 and a chain homotopy H between Id and GF as follows. Let D be the constant D = p × || H − K || = p × Z n max( H t − K t ) − min( H t − K t ) o dt. Then, H ( A α ) ⊂ A α − D holds for any α ∈ R ( A α = T α A ). If K is a C ∞ -small func-tion, P ( K, γ ) = P ( K ( p ) , pγ ) = ∅ and ( B, d ) = (0 ,
0) hold. This implies that thereis a map L : A ⊗ Λ → A ⊗ Λsuch that Id A ⊗ Λ = d L + L d L ( A α ) ⊂ A α − p || H || This implies that for r > p || H || ǫ + 1 and any cycle C ∈ F q + r A , C ′ = L ( C ) satisfies C ′ ∈ F q +1 A and d ( C ′ ) = C . So the map H ( F q A/F q + r A ) −→ H ( F q +1 A )[ z ] [ dz ]is zero in this case. So we proved (1).(2) The above arguments imply that F q H ( A ) is zero for q > p || H || ǫ . So, u is aninjection. It suffices to prove that u is a surjection. We fix an element { [ a q ] } q ≥ ∈ lim ←− H ( A ) /F q H ( A )where a q ∈ A are cycles and we construct a cycle b in A such that u ([ b ]) = { [ a q ] } holds. For this purpose, we construct a sequence { b q } q ∈ Z ≥ ⊂ A such that db q = 0 b q +1 ≡ b q mod F q A [ b q ] = [ a q ] in H ( A ) /F q H ( A )hold. Assume that we have constructed b , · · · , b l . Then[ b l ] = [ a l ] = [ a l +1 ] in H ( A ) /F l H ( A )holds. So, there is a cycle γ l ∈ F l A such that [ b l + γ l ] = [ a l +1 ] holds in H ( A ). Wedefine b l +1 by b l +1 = b l + γ l . Then b l +1 satisfies b l +1 ≡ b l mod F l A [ b l +1 ] = [ a l +1 ] in H ( A ) /F l +1 H ( A ) . So we can extend to b l +1 and hence we can construct a sequence { b q } . This sequencesatisfies b q − b q + r ∈ F q A ( r >
0) and satisfies Cauchy condition. This sequenceconverges to a cycle b ∈ A . b − b q ∈ F q A implies that u ( b ) = { [ a q ] } holds. So weproved (2). (cid:3) So we proved that r ∗ p HF ( H, γ : Λ ) ⊗ Λ [ u − , u ]] h θ i ∼ = H T ate ( Z p , CF ( H, γ : Λ ) ⊗ p ) ∼ = HF Z p ( H ( p ) , pγ )holds. Remark 5.1.
In the above proof, we do not need to assume that every periodicorbit in P ( H ( p ) , pγ ) is an iteration of an periodic orbit in P ( H, γ ) . We only needto assume that P ( H, γ ) and P ( H ( p ) , pγ ) are finite. Next, we prove the following lemma (see also section 3.2 in [5]).
Lemma 5.4.
Assume that there is an isomorphism HF ( H, γ : Λ ) ∼ = m M j =1 Λ /T β j Λ < β ≤ β · · · ≤ β m and there is an isomorphism HF ( H ( p ) , pγ : Λ ) ∼ = m ′ M j =1 Λ /T δ j Λ < δ ≤ δ · · · ≤ δ m ′ . Then, δ ≥ pβ holds. Proof (Lemma 5.4)The above isomorphism implies that the following isomorphism holds. HF Z p ( H ( p ) , pγ ) ∼ = ( m M j =1 Λ /T pβ j Λ ) ⊗ Λ [ u − , u ]] h θ i We define spectral number σ ( z ) and τ ( z ) for z ∈ CF ( H ( p ) , pγ : Λ ) as follows. σ ( z ) = sup { α ∈ R | z ∈ T α CF ( H ( p ) , pγ : Λ ) } τ ( z ) = σ ( d F ( z )) − σ ( z )We also define σ eq ( z ) and τ eq ( z ) for z ∈ CF Z p ( H ( p ) , pγ ) as follows. σ eq ( z ) = sup { α ∈ R | z ∈ T α CF Z p ( H ( p ) , pγ ) } τ eq ( z ) = σ eq ( d eq ( z )) − σ eq ( z )First, we prove the following claim. Claim 5.1.
Let β and δ be positive real numbers defined in the statement ofLemma 5.4. Then the following equalities hold. δ = inf { τ ( z ) | z ∈ CF ( H ( p ) , pγ : Λ ) } pβ = inf { τ eq ( z ) | CF Z p ( H ( p ) , pγ ) } We prove only the first equality (the proof for the second equality is the same).We can choose a cycle z ∈ CF ( H ( p ) , pγ : Λ ) such that σ ( z ) = 0 holds and T δ z isa boundary. So we can choose w ∈ CF ( H ( p ) , pγ : Λ ) so that d F ( w ) = T δ z holds.This implies that τ ( w ) ≤ δ and δ ≥ inf { τ ( z ) | z ∈ CF ( H ( p ) , pγ : Λ ) } holds. Assume that δ > RHS holds. Then there is z ∈ CF ( H ( p ) , pγ : Λ ) such that σ ( z ) = 0 , α = τ ( z ) = σ ( d F ( z )) < δ
1N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS19 holds. So, w = T − α d F ( z ) satisfies σ ( w ) = 0 and T α w = d F ( z ) is a boundary. Let { C , · · · , C m ′ } be the generators of HF ( H ( p ) , pγ : Λ ) ∼ = m ′ M j =1 Λ /T δ j Λ . [ T α w ] = 0 ∈ HF ( H ( p ) , pγ : Λ ) implies that [ w ] ∈ HF ( H ( p ) , pγ : Λ ) is written inthe following form.[ w ] = X δ − α ≤ λ ,j <δ a ,λ ,j T λ ,j [ C ] + · · · + X δ m ′ − α ≤ λ m ′ ,j <δ m ′ a m ′ ,λ m ′ ,j T λ m ′ ,j [ C m ′ ]Note that [ w ] = 0 holds because our chain complex is strict and any boundary b ∈ CF ( H ( p ) , pγ : Λ ) satisfies σ ( b ) >
0. We choose a chain v ∈ CF ( H ( p ) , pγ : Λ )so that C = X δ − α ≤ λ ,j <δ a ,λ ,j T λ ,j C + · · · + X δ m ′ − α ≤ λ m ′ ,j <δ m ′ a m ′ ,λ m ′ ,j T λ m ′ ,j C m ′ C = w + dv holds. However σ ( dv ) > < δ − α ≤ σ ( C ) = σ ( w + dv ) = σ ( w ) = 0holds. This is a contradiction and we proved Claim 5.1.In Lemma 5.1 we proved the existence of ”nice” basises for the local equivariantFloer homology and Floer homology. This implies that d eq ( x ⊗
1) = d F ( x ) ⊗ x ⊗ θ + X k =1 X k ′ =0 , x k + k ′ ⊗ u k θ k ′ holds for x ∈ CF ( H ( p ) , pγ : Λ ). This implies that σ ( d F ( x )) ≥ σ eq ( d eq ( x ⊗ τ ( x ) ≥ τ eq ( x ⊗
1) hold. So δ = inf { τ ( x ) | x ∈ CF ( H ( p ) , pγ : Λ ) } ≥ inf { τ eq ( z ) | z ∈ CF Z p ( H ( p ) , pγ ) } = pβ holds and we finished the proof of Lemma 5.4. (cid:3) Next we apply Lemma 5.4 to prove Theorem 1.1. Recall that we assumed P ( H, γ ) = { x , · · · , x k } P ( H ( p ) , pγ ) = { x ( p )1 , · · · , x ( p ) k } holds and p is admissible in the beginning of this subsection. Our purpose is toprove there is a simple p ′ periodic orbit in pγ . This is equivalent to prove that P ( H ( p ′ ) , pγ ) = ∅ because any periodic orbit in P ( H ( p ′ ) , pγ ) is simple if p (hence p ′ )is sufficiently large. We fix C > p so that HF ( H, γ : Λ F p ) ∼ = m M j =1 Λ F p /T β p,j Λ F p (0 < β p, ≤ · · · ≤ β p,m ) C < β p,
10 YOSHIHIRO SUGIMOTO holds. Here Λ F p is the universal Novikov ring of ground field F p . This is alwayspossible because β p, is greater than the minimum energy of the solution of Floerequation (=positive constant independent of p ). Lemma 5.4 implies that τ ( z ) ≥ pβ p, ≥ pC holds for any z ∈ CF ( H ( p ) , pγ : Λ F p ). This implies that any element x = 0 ∈ HF loc ( H ( p ) , x ( p ) i )determines a cycle in CF [ − pC,ǫ ) ( H ( p ) , pγ : Λ F p ) and CF [ − ǫ,pC ) ( H ( p ) , pγ : Λ F p ) andthey are not boundaries ( ǫ > τ ( z ) ≥ pC holds forany z . This implies that the natural homomorphism ι : HF [ − ǫ,pC ) ( H ( p ) , pγ : Λ F p ) −→ HF [ − pC,ǫ ) ( H ( p ) , pγ : Λ F p )is not zero ( ι ([ x ]) = 0). Let p ′ be the first prime number greater than p . We assume p is sufficiently large prime so that2( p ′ − p ) || H || < pC holds. This is possible because p ′ − p = o ( p ) holds (see [1]). We have two continu-ation homomorphism as follows. F : HF [ − ǫ,pC ) ( H ( p ) , pγ : Λ F p ) −→ HF [ − ǫ − ( p ′ − p ) || H || ,pC − ( p ′ − p ) || H || ) ( H ( p ′ ) , pγ : Λ F p ) G : HF [ − ǫ − ( p ′ − p ) || H || ,pC − ( p ′ − p ) || H || ) ( H ( p ′ ) , pγ : Λ F p ) −→ HF [ − pC,ǫ ) ( H ( p ) , pγ : Λ F p )The composition of F and G satisfies GF = ι = 0. So, P ( H ( p ′ ) , pγ ) = ∅ holds andwe proved the theorem.5.2. Weakly monotone case.
In this subsection, we prove Theorem 1.1 for weaklymonotone symplectic manifolds. The difference between monotone case and weaklymonotone case is that we have not constructed Z p -equivariant Floer homology and Z p -equivariant pair of pants product in weakly monotone case. So in order to proveTheorem 1.1 for weakly monotone case, it suffices to construct these theories. Therest of the proof is totally the same as in the monotone case.In the monotone case, we can exclude sphere bubble easily because Maslov indexof a holomorphic sphere is greater than or equal to 2. In weakly monotone case,we have to exclude sphere bubbles much more carefully. Recall the constructionof Floer homology theory for weakly monotone symplectic manifolds [18, 22]. Forgeneric choice of almost complex structure J , there is no holomorphic spheres withnegative Chern numbers. Let H be a Hamiltonian function. The pair ( H, J ) isnot necessary Floer regular pair. However we can perturb H to e H so that ( e H, J )is Floer regular and sphere bubbles do not appear in the definition of the Floerboundary operator d F .Recall that in the definition of Z p -equivariant Floer homology for monotonesymplectic manifolds, we considered a family of almost complex structures { J w,t } parametrized by S ∞ and S while fixing a Hamiltonian function H ( p ) . So one possi-ble modification for the weakly monotone case is to consider a family of Hamiltonianfunction parametrized by S ∞ and S while fixing an almost complex structure J .Let K ∈ C ∞ ( S × M ) be a perturbation of H ( p ) so that ( K, J ) is a Floer regularpair. Note that K is not necessarily p -periodic Hamiltonian function. We alsoconsider a family of Hamiltonian functions K w,t parametrized by ( w, t ) ∈ S ∞ × S which satisfies the following conditions (compare it to the definition of J w,t in sec-tion 3). N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS21 • (local constant at critical points) In a small neighborhood of Z mi ∈ S ∞ , K w,t = K t − mp holds. • ( Z p -equivariance) K m · w,t = K w,t − mp holds for any m ∈ Z p and w ∈ S ∞ . • (invariance under the shift τ ) K τ ( w ) ,t = K w,t holds.We consider the following equation for x, y ∈ P ( K ), m ∈ Z p and i ∈ Z .( u, v ) ∈ C ∞ ( R × S , M ) × C ∞ ( R , S ∞ ) ∂ s u ( s, t ) + J ( u ( s, t ))( ∂ t u ( s, t ) − X K v ( s ) ,t ( u ( s, t ))) = 0 dds v ( s ) − grad e F = 0lim s →−∞ v ( s ) = Z α , lim s → + ∞ v ( s ) = Z mi , lim s →−∞ u ( s, t ) = x ( t ) , lim s → + ∞ u ( s, t ) = y ( t − mp )One might try to define d i,mα : CF ( K, γ : Λ ) −→ CF ( K, γ : Λ )by counting above solutions and define d eq . However, this attempt contains thefollowing difficulty. The action gap of the solution ( u, v ) of the above equation Z R × S u ∗ ω + Z K ( t, x ( t )) − K ( t, y ( t )) dt is not necessarily non-negative. This problem happens when i ∈ Z becomes suf-ficiently large. As i becomes bigger and bigger, the effect of the perturbation H ( p ) t → K w,t becomes bigger and we cannot define a differential operator d eq overΛ . What we can do is to fix some N ∈ Z and define a finite operators { d i,mα } for i ≤ N .This difficulty is very similar to the difficulty in [8]. In [8], Fukaya, Oh, Ohtaand Ono constructed an A ∞ -algebra associated to a Lagrangian submanifold in asymplectic manifold. They constructed A ∞ -operators { m k } k ∈ Z ≥ by using mod-uli spaces of holomorphic discs bounding the Lagrangian submanifold. In orderto determine { m k } k ∈ Z ≥ , they had to achieve transversality of infinitely manyinterrelated moduli spaces by perturbing multisections of Kuranishi structures.Unfortunately, this is impossible because perturbations of lower order operatorsinfuluence perturbations of higher order operators and the higher perturbationsbecomes bigger and bigger. So what they could do is to construct finite opera-tors { m , , · · · , m n,K } ( A n,K -algebra) by one perturbation. They constructed A ∞ -operators { m k } k ∈ Z ≥ by ”gluing” infinitely many A n,K -algebras (( n, K ) → (+ ∞ , + ∞ ))by applying homological algebra developed in [8]. Our situation is much more sim-pler because we do not have to consider higher algebraic operators (we only needdifferential operator) and we do not have to consider the space of infinitely manysingular chains. So we can mimic the construction of Lagrangian A ∞ -algebra byapplying algebraic machineries developed in [8]. In the rest of this subsection, weexplain how we can apply [8] in our cases.We consider the following situation. Let C be a Λ -module. We want to constructa family of operators { d i : C → C } i ∈ Z ≥ so that P i + j = k d i d j = 0 holds for any k ∈ Z ≥ . Then the infinite sum d = d + d u + d u + d u · · · becomes a differential on C ⊗ Λ [ u − , u ]] as follows. d : C ⊗ Λ [ u − , u ]] −→ C ⊗ Λ [ u − , u ]] x ⊗ u m X i ≥ d i ( x ) ⊗ u m + i Assume that a family of operators { d i : C → C } ki =0 satisfies P i + j = m d i d j = 0for 0 ≤ m ≤ k . In this section, we call ( C, { d i } ki =0 ) a X k -module (note that this isa temporary terminology used only in this paper). Let ( D, { δ i } ki =0 ) be another X k -module. We call a family of homomorphisms { f i = C → D } ki =0 X k -homomorphismif it satisfies the following relations. X i + j = m δ i f j = X i ′ + j ′ = m f i ′ d j ′ (0 ≤ m ≤ k )Note that this is equivalent to that( k X i =0 δ i u i )( k X j =0 f j u j ) ≡ ( k X i ′ =0 f i ′ u i ′ )( k X j ′ =0 d j ′ u j ′ ) mod( u k +1 )is satisfied and f = P ki =0 f i u i is a ”chain map modulo u k +1 ”. Let { f i } ki =0 and { g i } ki =0 be two X k -homomorphisms from ( C, { d i } ki =0 ) to ( D, { δ i } ki =0 ). We say { f i } ki =0 and { g i } ki =0 are X k -homotopic if there is a family of operators {H i : C → D } ki =0 which satisfies the following relations. f m − g m = X i + j = m H i d j + X i ′ + j ′ = m δ i ′ H j ′ A composition of two X k -homomorphism { f i : C → D } and { g j : D → E } is definedby ( f ◦ g ) i = P j + l = i f j g l . We call a X k -homomorphism F : C → D X k -homotopyequivalence if there is a X k -homomorphism G : D → C such that F G and GF are X k -homotopic to the identity.Let ( C, { d i } ki =0 ) be a X k -module and ( C, { d ′ i } ∞ i =0 ) be a X ∞ -module. We say( C, { d ′ i } ∞ i =0 ) is a promotion of ( C, { d i } ki =0 ) if d ′ i = d i holds for 0 ≤ i ≤ k . A promo-tion of X k -homomorphism is defined in the same manner (see Definition 7.2.182 in[8]). We treat the following situation. Let { ( C, { d ( k ) i } ki =0 ) } k ∈ Z and { ( D, { δ ( k ) i } ki =0 ) } k ∈ Z be families of X k -modules ( k = 0 , , , · · · ) and I k : ( C k , { d ( k ) i } ki =0 ) −→ ( C k +1 , { d ( k +1) i } k +1 i =0 ) I ′ k : ( D k , { δ ( k ) i } ki =0 ) −→ ( D k +1 , { δ ( k +1) i } k +1 i =0 )be families of X k -homomorphisms which gives X k -homotopy equivalence betweenthem. Let F k : ( C k , { d ( k ) i } ki =0 ) −→ ( D k , { δ ( k ) i } ki =0 )be a family of X k -morphisms such that I ′ k F k and F ′ k +1 I k are X k -homotopic. So wehave the following diagram. N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS23 I k − −−−−→ ( C k , { d ( k ) i } ki =0 ) I k −−−−→ ( C k +1 , { d ( k +1) i } k +1 i =0 ) I k +1 −−−−→ ( C k +2 , { d ( k +2) i } k +2 i =0 ) I k +2 −−−−→ F k y F k +1 y F k +2 y I ′ k − −−−−→ ( D k , { δ ( k ) i } ki =0 ) I ′ k −−−−→ ( D k +1 , { δ ( k +1) i } k +1 i =0 ) I ′ k +1 −−−−→ ( D k +2 , { δ ( k +2) i } k +2 i =0 ) I ′ k +2 −−−−→ We have the following Claim.
Lemma 5.5 (Lemma 7.2.184 in [8]) . Let { ( C k , { d ( k ) i } ki =0 ) } , { ( D k , { δ ( k ) i } ki =0 ) } , { I k } , { I ′ k } and { F k } be a directed system as above. (1) We can promote each C k and D k to X ∞ -modules and promote I k and I ′ k to X ∞ -homomorphism which give X ∞ -homotopy equivalences. (2) We can promote each F k to X ∞ -homomorphisms between promoted C k and D k so that the above diagram is commutative up to X ∞ -homotopy. So we can construct X ∞ -modules and X ∞ -homomorphisms from directed sys-tems of X k -modules and X k -homomorphisms.We apply these machineries to the following situation. Let { H k ∈ C ∞ ( S × M ) } ∞ k =1 be a family of Hamiltonian functions which satisfies the following conditions. • H k −→ H in C ∞ -topology • ( H k , J ) is a Floer regular pairAssume that P ( H, γ ) is a finite set { x , · · · , x l } . For fixed 1 ≤ i ≤ l , x i splits into { x i , · · · , x l i i } ⊂ P ( H k , γ ). As in section 4, we slightly modify the Floer differentialoperator of ( H k , J ) as follows.Let v ji : [0 , × S → M be a small cylinder connecting v ji (0 , t ) = x i ( t ) and v ji (1 , t ) = x ji ( t ).Let c ( x i , x ji ) ∈ R be the action gap c ( x i , x ji ) = Z [0 , × S ( v ji ) ∗ ω + Z H ( t, x i ( t )) − H k ( t, x ji ( t )) dt. and let τ be a correction map defined as follows. τ : CH ( H k , γ : Λ) −→ CH ( H k , γ : Λ) x ji T c ( x i ,x ji ) x ji The modified differential operator f d F was defined by f d F = τ − d F τ By using the modified Floer differential f d F , we can define a modified Tate differ-ential ] d T ate : CF ( H k , γ : Λ ) ⊗ p ⊗ Λ [ u − , u ]] h θ i −→ CF ( H k , γ : Λ ) ⊗ p ⊗ Λ [ u − , u ]] h θ i as follows. ] d T ate ( x ⊗
1) = f d F ( x ) ⊗ − τ ) ⊗ θ ] d T ate ( x ⊗ θ ) = f d F ( x ) ⊗ θ + N ( x ) ⊗ uθ Then ] d T ate determines a X ∞ -module structure on C k = CF ( H k , γ : Λ ) ⊗ p ⊗ Λ h θ i . Note that ( C T ate ( Z p , CF ( H k , γ : Λ )) , ] d T ate ) and ( C T ate ( Z p , CF ( H k ′ , γ : Λ )) , ] d T ate )are chain homotopy equivalence (hence X ∞ -homotopy equivalence) for any k and k ′ .Next we consider a directed system for Z p -equivariant Floer homology. Let { G k } ∞ k =1 be a family of Hamiltonian functions in C ∞ ( S × M ) which satisfies thefollowing conditions. • G k → H ( p ) in C ∞ -topology • ( G k , J ) is Floer regularLet G ( k ) w,t be a family of Hamiltonian functions parametrized by ( w, t ) ∈ S k +1 ⊗ S ⊂ S ∞ ⊗ S which satisfies the following conditions. • (local constant at critical points) In a small neighborhood of Z mi ∈ S k +1 , G ( k ) w,t = G t − mp holds. • ( Z p -equivariance) G m · w,t = G w,t − mp • (invariance under the shift τ ) G τ ( w ) ,t = G w,t holds.As in the definition of the Z p -equivariant Floer differential operator for mono-tone symplectic manifolds, we consider the following equation for x, y ∈ P ( G k , pγ ), m ∈ Z p , λ ≥ α ∈ { , } and 0 ≤ i ≤ k + 1. Assume that e x, e y ∈ P ( H ( p ) , pγ ) areperiodic orbits which split into { x, · · · } ⊂ P ( G k , pγ ) and { y, · · · } ⊂ P ( G k , pγ ).( u, v ) ∈ C ∞ ( R × S , M ) × C ∞ ( R , S k +1 ) ∂ s u ( s, t ) + J ( u ( s, t ))( ∂ t u ( s, t ) − X G v ( s ) ,t ) = 0 dds v ( s ) − grad( e F ) = 0lim s →−∞ v ( s ) = Z α , lim s → + ∞ v ( s ) = Z mi , lim s →−∞ u ( s, t ) = x ( t ) , lim s → + ∞ u ( s, t ) = y ( t − mp ) Z R × S u ∗ ω + Z H ( p ) ( t, e x ( t )) − H ( p ) ( t, e y ( t )) dt = λ We denote the space of solutions by N λ, ( k ) α,i,m ( x, y ). We define d i,m, ( k ) α as follows. d i,m, ( k ) α : CF ( G k , pγ : Λ ) −→ CF ( G k , pγ : Λ ) x X λ ≥ ,x ∈ P ( G k ) ♯ N λ, ( k ) α,i,m ( x, y ) · T λ y Let d i, ( k ) α be the sum d i, ( k ) α = P m ∈ Z p d i,m, ( k ) α . We define X k -module structure on D k = CF ( G k , pγ : Λ ) ⊗ Λ h θ i . We determine { δ ( k ) i : D k → D k } ki =0 as follows. δ ( k ) i ( x ⊗
1) = d i, ( k )0 ( x ) ⊗ d i +1 , ( k )0 ( x ) ⊗ θδ ( k ) i ( x ⊗ θ ) = d i, ( k )1 ( x ) ⊗ d i +1 , ( k )1 ( x ) ⊗ θ There is a natural connecting X k -homomorphism between ( D k , { δ ( k ) i } ) and ( D k +1 , { δ ( k +1) i } ). I ′ k : ( D k , { δ ( k ) i } ) −→ ( D k +1 , { δ ( k +1) i } ) N THE HOFER-ZEHNDER CONJECTURE FOR NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIAN DYNAMICS25 I ′ k is a X k -homotopy equivalence. We can modify naturally the Z p -equivariant pairof pants equation to define X k -homomorphism between ( C k , { d ( k ) i } ) and ( D k , { δ ( k ) i } )in a natural manner. We denote this Z p -equivariant pair of pants product up toorder k by P k . So we have the following diagram. I k − −−−−→ ( C k , { d ( k ) i } ki =0 ) I k −−−−→ ( C k +1 , { d ( k +1) i } k +1 i =0 ) I k +1 −−−−→ P k y P k +1 y I ′ k − −−−−→ ( D k , { δ ( k ) i } ki =0 ) I ′ k −−−−→ ( D k +1 , { δ ( k +1) i } k +1 i =0 ) I ′ k +1 −−−−→ This diagram is commutative up to X k -homotopy. By applying Lemma 5.5,we can promote each ( C k , { d ( k ) i } ) and ( D, { δ ( k ) i } ) to X ∞ -modules and promoteeach P k to X ∞ -homomorphisms so that the above diagram is commutative up to X ∞ -homotopy. So we can define Z p -equivariant chain complex (not necessarilystrict) and also Z p -equivariant pair of pants product between Tate complex and Z p -equivariant Floer chain complex. The uniqueness of the above promotions upto X ∞ -homotopies also follows from homotopy of homotopy theory (see section7.2.12 and section 7.2.13 in [8]). So we constructed Z p -equivariant Floer homologyand Z p -equivariant pair of pants product in a homologically canonical way. Notethat Z p -equivariant pair of pants product gives an isomorphism of E -pages of theassociated spectral sequences as in the monotone case because for sufficiently large k , the promotions of P k and ( D k , { δ ( k ) i } ) does not influence the local theory in thefollowing sense (of course this also follows from the above homotopy of homotopytheory). Let {P k,i } ∞ i =0 and ( D k , { δ i } ∞ i =0 ) be promotions of P k and ( D k , { δ ( k ) i } ). δ ( k ) i : D k −→ D k P k,i : C k −→ D k Assume that k is sufficiently large and ǫ > δ ( k ) j ( D k ) ⊂ T ǫ D k P k,j ( C k ) ⊂ T ǫ D k hold for j > k by degree reasons (locally we can define a grading of periodic orbitsand local equivariant Floer differential and local equivariant pair of pants productare finite sums). So our promotions does not influence local theories and the localisomorphism theorem of Z p -equivariant pair of pants product ([30]) also holds inour case. This implies that the promoted Z p -pair of pants product also gives anisomorphism of E -pages of associated spectral sequences. The rest of the proof ofTheorem 1.1 for weakly monotone case is the same as in the monotone case. (cid:3) References [1] R.C. Baker, G.Harman, J. Pintz.
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