Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds
aa r X i v : . [ m a t h . S G ] M a r NON-CONTRACTIBLE PERIODIC ORBITS IN HAMILTONIANDYNAMICS ON CLOSED SYMPLECTIC MANIFOLDS
VIKTOR L. GINZBURG AND BAŞAK Z. GÜREL
Abstract.
We study Hamiltonian diffeomorphisms of closed symplectic man-ifolds with non-contractible periodic orbits. In a variety of settings, we showthat the presence of one non-contractible periodic orbit of a Hamiltonian dif-feomorphism of a closed toroidally monotone or toroidally negative monotonesymplectic manifold implies the existence of infinitely many non-contractibleperiodic orbits in a specific collection of free homotopy classes. The main newingredient in the proofs of these results is a filtration of Floer homology by theso-called augmented action. This action is independent of capping, and, underfavorable conditions, the augmented action filtration for toroidally (negative)monotone manifolds can play the same role as the ordinary action filtrationfor atoroidal manifolds.
Contents
1. Introduction 12. Main results 32.1. Conventions and notation 32.2. Results 53. Augmented action filtration 93.1. Preliminaries: iterated Hamiltonians 93.2. Floer homology for non-contractible periodic orbits 103.3. Filtration 103.4. Homotopy and continuation 134. Proofs 144.1. Proofs of Theorems 2.2 and 2.4 144.2. Proof of Theorem 2.1 184.3. Proof of Theorem 2.7 21References 221.
Introduction
In this paper, focusing on closed symplectic manifolds, we study Hamiltoniandiffeomorphisms with non-contractible periodic orbits. We show that in a variety
Date : May 2, 2019.2000
Mathematics Subject Classification.
Key words and phrases.
Non-contractible periodic orbits, Hamiltonian flows, Floer homology,augmented action.The work is partially supported by the NSF grants DMS-1414685 (BG) and DMS-1308501(VG). of settings the presence of one non-contractible one-periodic orbit of a Hamiltoniandiffeomorphism of a closed symplectic manifold guarantees the existence of infinitelymany non-contractible periodic orbits.More specifically, we concentrate on closed symplectic manifolds ( M n , ω ) whichare toroidally monotone or toroidally negative monotone and Hamiltonians with atleast one non-contractible one-periodic orbit x . Then we prove that under minorassumptions on the free homotopy class f of x and under certain dynamical andFloer theoretic conditions on x and M , the Hamiltonian has infinitely many simpleperiodic orbits in the collection of free homotopy classes f N := { f k | k ∈ N } ; seeTheorems 2.1 and 2.2. The conditions on f are automatically satisfied when thehomology class [ f ] is non-zero in H ( M ; Z ) / Tor or when f = 1 and π ( M ) is hyper-bolic and torsion free. These theorems partially extend the main result of [Gü13]from atoroidal symplectic manifolds to toroidally monotone or negative monotonemanifolds. (We show in the next section that there are numerous manifolds andHamiltonians meeting the requirements of the theorems.) The phenomenon weconsider here is C ∞ -generic. To be more precise, we show in Theorem 2.7 thatthe presence of one non-contractible periodic orbit x in a class f such that f N implies C ∞ -generically the existence of infinitely many non-contractible periodicorbits in f N . Finally, we also refine the results from [Gü13] for atoroidal symplecticmanifolds; see Theorem 2.4.All these results are manifestations of the same underlying phenomenon thatthe presence of a periodic orbit which is geometrically or homologically unneces-sary forces a Hamiltonian system to have infinitely many periodic orbits. (On aclosed symplectic manifold non-contractible periodic orbits are clearly unnecessary.For example, a C -small autonomous Hamiltonian has only constant one-periodicorbits.) This phenomenon of “forced existence” of infinitely many periodic orbits isvery general and has been observed in a variety of other settings. For instance, thecelebrated theorem of Franks, [Fr92, Fr96], asserting that a Hamiltonian diffeomor-phism (or, even, an area preserving homeomorphism) of S with at least three fixedpoints must have infinitely many periodic orbits is a prototypical result along theselines; see also [LeC] for further refinements and [CKRTZ, Ker] for a symplectictopological proof. Another instance is a theorem from [GG14] that for a certainclass of closed monotone symplectic manifolds including CP n any Hamiltonian dif-feomorphism with a hyperbolic fixed point must necessarily have infinitely manyperiodic orbits. Yet, the specific question of forced existence for non-contractibleperiodic orbits considered here is largely unexplored except for [Gü13] focusing onsymplectically atoroidal manifolds. We refer the reader to, e.g., [Gü14] for someother related results and to [GG15] for a detailed discussion of the phenomenonand further references.The proofs of our main theorems rely on the machinery of Floer homology fornon-contractible periodic orbits. In the course of the last two decades, this versionof Floer homology has been studied and used in a number of papers, but usually ina more topological context and focusing on Hamiltonians on open manifolds suchas twisted or ordinary cotangent bundles; see, e.g., [BPS, BH, GaL, Le, Ni, SW,We, Xu]. (The recent works [Ba15b, PS] are closer to the setting considered inthis paper which on the conceptual level can be thought of as a continuation of[Gü13].) The main difficulty in applying the technique to closed manifolds is thatthe global Floer homology for non-contractible orbits vanish and, moreover, already ON-CONTRACTIBLE PERIODIC ORBITS 3 for closed surfaces, a Hamiltonian may have no non-contractible periodic orbits ofany period. (As a consequence, the Floer complex can be trivial for all iterations.)Thus to infer that a Hamiltonian has a number of periodic orbits, e.g., infinitelymany, an additional input is required. In our case, this is one non-contractibleperiodic orbit of the Hamiltonian, which serves as a seed generating infinitely manyoffsprings. The new periodic orbits are detected by analyzing the change in certainfiltered Floer homology groups under the iteration of the Hamiltonian and using the“stability of the filtered homology”. This argument shares many common elementswith the reasoning in, e.g., [GG10, Gü13, Gü14]. We feel that it can also be cast inthe framework of the barcode and persistent homology theory for Floer homology(cf. [PS, UZ]) and it would be interesting to see if a systematic use of this theorywould lead to new results in this class of questions.There are three key ingredients to the proofs in this paper.The main new component is the observation, perhaps of independent interest,that under favorable circumstances the Floer homology for a toroidally monotoneor toroidally negative monotone manifold is filtered by the so-called augmentedaction. The augmented action ˜ A H is the difference A H − ( λ/ H between thestandard symplectic action A H and the (renormalized) mean index ∆ H of an orbit,where λ is the monotonicity constant; cf. [GG09a, Sect. 1.4]. The key feature ofthe augmented action is that it is independent of capping and hence is assignedto the orbit itself. When the augmented action gap is sufficiently large, the Floerdifferential does not increase the augmented action, and the augmented actionfiltration is defined. In this case, the filtration behaves similarly to the ordinaryaction filtration in the aspherical or atoroidal case and can be used in the sameway. (One essential difference, which is a source of several complications, is thatthe augmented action filtration is not strict: in general, the Floer differential canconnect orbits with equal augmented action even if the gap is large.)The other two ingredients are the stability of the filtered homology already men-tioned above and the ball-crossing theorem [GG14, Thm. 3.1] used in one of theproofs. This theorem gives an iteration-independent lower bound on the energy ofa Floer trajectory asymptotic to a hyperbolic orbit.The paper is organized as follows. In Section 2, we set our conventions andnotation, introduce the necessary notions, state the main results of the paper, anddiscuss in detail the classes of manifolds and Hamiltonians these results apply to. InSection 3, we define the augmented action filtration and establish its key properties.Finally, in Section 4, we prove the main theorems of the paper. Acknowledgements.
The authors are grateful to Alberto Abbondandolo, AndrésKoropecki, Patrice Le Calvez, Denis Osin for useful discussions and suggestions.2.
Main results
Conventions and notation.
To state the main results of the paper, let usfirst introduce some relevant definitions and set our conventions and notation.Throughout the paper, we assume that ( M n , ω ) is a closed toroidally mono-tone or toroidally negative monotone symplectic manifold unless specifically statedotherwise. To be more specific, recall that a cohomology class w is atoroidal if forevery map v : T → M the integral of w over v vanishes: h w, [ v ] i = 0 . A symplectic VIKTOR GINZBURG AND BAŞAK GÜREL manifold ( M, ω ) is said to be toroidally monotone (resp., toroidally negative mono-tone) when for some constant λ ≥ (resp., λ < ) the class w = [ ω ] − λc ( T M ) is atoroidal. The constant λ is referred to as the toroidal monotonicity constant .Note that the case λ = 0 corresponds to an atoroidal class [ ω ] . Toroidally (negative)monotone manifolds are automatically spherically (negative) monotone. We referthe reader to Section 2.2 for examples of toroidally monotone or negative mono-tone manifolds. We call the positive generator N T of the group generated by theintegrals h c ( T M ) , [ v ] i for all tori the minimal toroidal first Chern number of M .We set N T = ∞ when this group is { } , i.e., c ( T M ) is atoroidal. For a toroidallymonotone or negative monotone manifold, this implies that [ ω ] is also atoroidal.We denote by ˜ π ( M ) the set of homotopy classes of free loops in M . The freehomotopy class of a loop x and its integer homology class (modulo torsion) aredenoted by J x K and, respectively, by [ x ] . Likewise, we write [ f ] ∈ H ( M ; Z ) / Tor forthe homology class modulo torsion of a free homotopy class f ∈ ˜ π ( M ) .All Hamiltonians H are assumed to be one-periodic in time, i.e., H : S × M → R ,and we set H t = H ( t, · ) for t ∈ S = R / Z . The Hamiltonian vector field X H of H is defined by i X H ω = − dH . The (time-dependent) flow of X H is denoted by ϕ tH and its time-one map by ϕ H . Such time-one maps are called Hamiltoniandiffeomorphisms . For the sake of brevity, we will refer to the periodic orbits of ϕ H or, equivalently, the periodic orbits of ϕ tH with integer period as the periodic orbitsof H . For a Hamiltonian H and a collection of free homotopy classes c ⊂ ˜ π ( M ) ,we set P k ( H, c ) to be the set of k -periodic orbits of H in c . For instance, P k ( H, γ ) ,where γ ∈ H ( M ; Z ) / Tor , comprises the k -periodic orbits of H in the homologyclass γ .For a class f ∈ ˜ π ( M ) , let us fix a reference loop z ∈ f . A capping of x : S → M with free homotopy class f is a cylinder (i.e., a homotopy) Π : [0 , × S → M connecting x and z taken up to a certain equivalence relation. Namely, two cappings Π and Π ′ are equivalent if the integral of c ( T M ) , and hence of ω , over the torusobtained by attaching Π ′ to Π is equal to zero.The action of H on a capped loop ¯ x = ( x, Π) is A H (¯ x ) = − Z Π ω + Z S H t ( x ( t )) dt. Clearly, A H (¯ x ) is well defined. Moreover, the critical points of A H are exactlythe capped one-periodic orbits of H in the homotopy class f . The action spectrum S ( H, f ) is the set of critical values of A H . It has zero measure; see, e.g., [HZ].Furthermore, let us fix a trivialization of T M along the reference loop z . Then, toa capped one-periodic orbit ¯ x with x ∈ f , one can associate the mean index ∆ H (¯ x ) in a standard way. Namely, we extend the trivialization of T M | z to the cappingof x and then use the resulting trivialization of T M | x to turn the linearized flow dϕ tH | x along x into a path in the group Sp(2 n ) . The mean index ∆ H (¯ x ) is bydefinition the mean index of the resulting path; see, e.g., [Lo, SZ]. The mean indexmeasures, roughly speaking, the total rotation number of certain unit eigenvaluesof the linearized flow along x . The mean index ∆ H ( x ) of a non-capped orbit x iswell defined as an element of R / N T Z .By analogy with the case of contractible orbits (see [GG09a]), we define the augmented action of a one-periodic orbit x to be ˜ A H ( x ) = A H (¯ x ) − λ H (¯ x ) . ON-CONTRACTIBLE PERIODIC ORBITS 5
The action and the mean index change under recapping in the same way, up to thefactor λ/ , and hence the augmented action of x is well defined, i.e., independentof the capping. Note, however, that the augmented action depends on the choicesof the reference loop z and the trivialization. When λ = 0 , i.e., [ ω ] is atoroidal theaugmented action turns into the ordinary action.The augmented action spectrum ˜ S ( H, f ) is the collection of the augmented actionvalues for all one-periodic orbits of H in the class f , i.e., ˜ S ( H, f ) = { ˜ A H ( x ) | x ∈ P ( H, f ) } . In contrast with the ordinary action spectrum, ˜ S ( H, f ) need not have zero measure– in fact, it can contain whole intervals – unless, of course, H has finitely manyperiodic orbits in f . However, ˜ S ( H, f ) is a compact set, which depends continuouslyon H . To be more precise, as is easy to see, for any open set U ⊃ ˜ S ( H, f ) , we have U ⊃ ˜ S ( K, f ) when K is sufficiently C -close to H .Assume now that all one-periodic orbits of H in the class f with augmentedaction in an open interval I are isolated. Then we set χ ( H, I, f ) to be the sum of thePoincaré-Hopf indices of their return maps. This definition extends by continuityto all Hamiltonians H with possibly non-isolated orbits as long as the end pointsof I are outside ˜ S ( H, f ) . For instance, χ ( H, I, f ) = 0 when I ∩ ˜ S ( H, f ) contains onlyone point a and H is non-degenerate and has an odd number of one-periodic orbits(e.g., one) in the class f with augmented action a .The above definitions generalize in an obvious way to the setting where one-periodic orbits are replaced by k -periodic orbits or where one class f ∈ ˜ π ( M ) isreplaced by a collection of classes c ⊂ ˜ π ( M ) . In the latter case, a reference loop z and a trivialization of T M | z must be fixed for every f ∈ c .2.2. Results.
Now we are in a position to state our main results. We recall that all manifolds considered in this paper are assumed to be toroidally monotone ortoroidally negative monotone unless specifically stated otherwise . Furthermore, sincethe proofs heavily depend on Hamiltonian Floer homology, we need to either assumethroughout the paper that M is weakly monotone or rely on the machinery of virtualcycles. To be more precise, recall that a manifold M n is said to be weakly mono-tone if one of the following conditions is satisfied: M is (not necessarily strictly)monotone, i.e., [ ω ] | π ( M ) = λc ( T M ) | π ( M ) with λ ≥ , or c ( T M ) | π ( M ) = 0 , or N ≥ n − where N is the minimal Chern number. Under any of these conditions,the Floer homology is defined and has standard properties; see [HS, McDS, On]and references therein. We refer the reader to, e.g., [FO, HWZ, LT, Pa] for variousincarnations of the technique of virtual cycles. In all but one of our results (The-orem 2.7) the ambient manifold M is automatically monotone (not strictly), andhence weakly monotone.Our first result asserts that under certain additional assumptions on M thepresence of one non-contractible hyperbolic one-periodic orbit implies the existenceof infinitely many non-contractible periodic orbits; cf. [GG14]. Theorem 2.1.
Assume that N T ≥ n/ , where n = dim M , and that a Hamil-tonian H on M has a hyperbolic one-periodic orbit x such that all homotopy classesin the set J x K N = { J x K k | k ∈ N } are distinct and nontrivial. (This is the case, e.g.,when [ x ] = 0 in H ( M ; Z ) / Tor .) Then H has infinitely many simple periodic orbitswith homotopy class in J x K N . In particular, if all such orbits are isolated, there aresimple non-contractible periodic orbits of arbitrarily large period. VIKTOR GINZBURG AND BAŞAK GÜREL
The condition that N T ≥ n/ appears to be purely technical even thoughit plays an essential role in the proof. As we will show below, there are numeroussymplectic manifolds M and Hamiltonians H meeting the requirements of the the-orem. For instance, the theorem applies to M = Σ g × CP m , where Σ g is a closedsurface of genus g ≥ .The second result is more accurate and covers a broader range of manifolds andmaps, although it still relies on some additional topological assumptions about theflow of H ; cf. [Gü13]. Theorem 2.2.
Assume that P ( H, [ f ]) is finite and χ ( H, I, f ) = 0 for some interval I with end points outside ˜ S ( H, f ) , where f ∈ ˜ π ( M ) and [ f ] = 0 in H ( M ; Z ) / Tor .Then, for every sufficiently large prime p , the Hamiltonian H has a simple periodicorbit in the homotopy class f p and with period either p or p ′ , where p ′ is the firstprime greater than p . Moreover, when π ( M ) is hyperbolic and torsion free, thecondition [ f ] = 0 can be replaced by f = 1 and no finiteness requirement is needed.Remark . We emphasize that in these theorems we impose no non-degeneracyconditions on H . An interesting new point in the second part of Theorem 2.2 isthat, in contrast with many other results of this type, we do not need to require P ( H, f ) to be finite to have simple periodic orbits with arbitrarily large period. Itis immediate to see that, if H is non-degenerate, χ ( H, I, f ) = 0 for a short interval I centered at a ∈ ˜ S ( H, f ) when the one-periodic orbits of H in the class f havedifferent augmented action values.Also note that by passing to an iteration one can replace in both of the theoremsone-periodic orbits by k -periodic. Then the first theorem remains correct as stated.In the second theorem, after replacing H by the iterated Hamiltonian H ♮k (seeSection 3.1), we can only conclude that H has infinitely many simple periodicorbits in f N . (Of course, the theorem still applies literally to H ♮k in place of H , butsimple periodic orbits of H ♮k are not necessarily simple as periodic orbits of H .)Let us now further discuss the conditions of Theorems 2.1 and 2.2, beginningwith those concerning the manifold and then moving on to the Hamiltonian.The manifolds M meeting the requirements of these theorems exist in abun-dance. To construct specific examples, let us start with a symplectically atoroidalmanifold ( M , ω ) , i.e., a closed symplectic manifold such that [ ω ] and c ( T M ) are atoroidal. Among these are, for instance, surfaces of genus g ≥ and, moregenerally, all Kähler manifolds with negative sectional curvature. (See, e.g., [Gü13]for further references and a discussion of such manifolds.) Next, let ( M , ω ) bea closed spherically monotone or negative monotone symplectic manifold. Thereare numerous examples of such manifolds including, in the negative monotone case,those with arbitrarily large minimal (spherical) Chern number N S . For instance,let M be a smooth complete intersection in CP m + k given by k homogeneous poly-nomials of degrees d , . . . , d k . Then M is monotone or negative monotone with N S = | m + k − d | , where d = d + . . . + d k , unless N S = 0 (see, e.g., [LM, p. 88]).To be more precise, M is monotone when m + k − d > , negative monotone when m + k − d < , and c ( T M ) = 0 when m + k − d = 0 . Now it is easy to seethat M = M × M is toroidally monotone or toroidally negative monotone (when N S = 0 ) with N T = N S .Furthermore, π ( M ) = π ( M ) × π ( M ) . In particular, H ( M ; Z ) / Tor = 0 when H ( M ; Z ) / Tor = 0 . Moreover, when M is a complete intersection and M ON-CONTRACTIBLE PERIODIC ORBITS 7 is a Kähler manifold with negative sectional curvature, π ( M ) is hyperbolic andtorsion free. Indeed, in this case, π ( M ) = π ( M ) since complete intersections aresimply connected (see, e.g., [Sh, Chap. IX, Sect. 4.1]).Next, note that for any symplectic manifold M and f ∈ ˜ π ( M ) there exists aHamiltonian H : S × M → R with a hyperbolic one-periodic orbit in the class f ; see,e.g., [Ba15a, Prop. 1.3]. (In fact, one can prescribe arbitrarily a periodic orbit andthe linearization of the flow along it.) Thus, whenever M satisfies the conditionsof Theorem 2.1 and H ( M ; Z ) / Tor = 0 or π ( M ) is hyperbolic and torsion free,there exists a C ∞ -open, non-empty set of Hamiltonians this theorem applies to.Furthermore, in the setting of Theorem 2.2, the collection of Hamiltonians withone-periodic orbits in f is non-empty and, as one can easily see, has a locally non-empty interior, i.e., the intersection of this collection with a neighborhood of its anypoint has a non-empty interior. It is also not hard to show that the Hamiltoniansmeeting the requirements of the theorem form a C ∞ -open and dense subset inthis collection. (Indeed, non-degenerate Hamiltonians form a C ∞ -open and densesubset, and one can further ensure by a C ∞ -small perturbation of H that all one-periodic orbits have distinct augmented actions.)It is worth pointing out that it is not clear how large this open subset is, i.e.,how common the Hamiltonians with at least one non-contractible periodic orbitare. This is an interesting question and to the best of our understanding very littleis known about this problem. Consider, for instance, time-dependent Hamiltonianson a closed surface M of positive genus. Then a bump function with small supportor, as Paul Seidel pointed out to us, a self-indexing Morse function are examples ofHamiltonians without non-contractible periodic orbits of any period. Furthermore,a simple KAM argument shows that for a C ∞ -generic autonomous Hamiltonian H on M = T such that one of the components of the level { H = 0 } is a parallel, noHamiltonian sufficiently C ∞ -close to H has periodic orbits in the free homotopyclass collinear to the class of the meridian. (This observation is due to LeonidPolterovich.) However, as far as we know there are no counterexamples to theconjecture that a C -generic (or even C ∞ -generic) Hamiltonian on M has a non-contractible periodic orbit. In fact, as has been pointed out to us by Patrice LeCalvez and Andrés Koropecki, the conjecture holds for M = T for C ∞ -genericHamiltonians, [LeCT, Prop. J]. (See also [Ta] for some possibly relevant results.)Our next theorem is a minor refinement of [Gü13, Thm. 1.1]. This is a resultstronger than Theorems 2.1 and 2.2, but applicable to a much more narrow classof manifolds. Theorem 2.4.
Assume that the class [ ω ] is atoroidal and let H be a Hamiltonianhaving a non-degenerate one-periodic orbit x with homotopy class f such that [ f ] = 0 in H ( M ; Z ) / Tor and P ( H, [ f ]) is finite. Then, for every sufficiently large prime p , the Hamiltonian H has a simple periodic orbit in the homotopy class f p and withperiod either p or p ′ , where p ′ is the first prime greater than p . Moreover, when π ( M ) is hyperbolic and torsion free, the condition [ f ] = 0 can be replaced by f = 1 .Remark . Here, as in [Gü13, Thm. 3.1], the requirement that x is non-degeneratecan be replaced by a much less restrictive condition that x is isolated and has non-trivial local Floer homology; see Theorem 4.1. The key difference between Theorems2.4 and 2.2 is that a non-degenerate Hamiltonian H can, at least hypothetically,have several one-periodic orbits in the class f , yet χ ( H, I, f ) = 0 for any interval I . (The reason why in Theorem 2.2, in contrast with Theorem 2.4 or [Gü13], it VIKTOR GINZBURG AND BAŞAK GÜREL is not sufficient to assume that H has a non-contractible periodic orbit is that, ashas already been mentioned in the introduction, the augmented action filtrationis not strict. We will come back to this issue in Section 3.3.) The new points ofTheorem 2.4 as compared with the results from [Gü13] are the “moreover” part ofthe theorem and the control of the homotopy classes of the orbits rather than thehomology classes.Among symplectic manifolds with atoroidal class [ ω ] are the Kähler manifoldswith negative sectional curvature mentioned above and also some other classes ofsymplectic manifolds; see, e.g., [BK, Kę]. Remark . In the settings of Theorems 2.2 and 2.4, the number ofsimple non-contractible periodic orbits with period less than or equal to k , or thenumber of distinct homotopy classes represented by such orbits, is bounded frombelow by c onst · k/ ln k . An immediate consequence of the theorems is that H has infinitely many simple periodic orbits with homology class in N [ f ] regardless ofwhether or not the set of one-periodic orbits (in the class [ f ] ) is finite.The simplest manifold the above theorems do not apply to is the standard sym-plectic torus T n with n ≥ . In dimension two, it is easy to see that for stronglynon-degenerate Hamiltonian diffeomorphisms the presence of one non-contractibleorbit in a homotopy class f implies the existence of infinitely many simple peri-odic orbits with homotopy class in f N . (Recall that a diffeomorphism is said to bestrongly non-degenerate if all its periodic orbits are non-degenerate.) The proof ofthis fact amounts to the observation that in dimension two the mean index deter-mines the Conley–Zehnder index and is similar to the proofs of [Ab, Thm. 5.1.9] or[GG09b, Thm. 1.7]. However, somewhat surprisingly, it is not clear at all whetherthe non-degeneracy condition here can be replaced as in, e.g., [Gü13, Gü14] by therequirement that the orbit has non-vanishing local Floer homology. When n ≥ ,no results along these lines have been established for T n .It is interesting to contrast the previous theorems with the following, admittedlysuperficial, C ∞ -generic existence result. Namely, C ∞ -generically, the existence ofone non-contractible one-periodic orbit is sufficient to infer the existence of infinitelymany simple non-contractible periodic orbits under no conditions on M and withonly very minor assumptions about f ; cf. [GG09b]. To be more precise, denoteby F f the collection of strongly non-degenerate Hamiltonian diffeomorphisms witha one-periodic orbit in f . Clearly, F f is C ∞ -open in the group of Hamiltoniandiffeomorphisms. As has been pointed out above, this set is non-empty by, e.g.,[Ba15a, Prop. 1.3]. Recall also that a subset is called residual, or second Bairecategory, when it is the intersection of a countable collection of open and densesubsets. Theorem 2.7.
Assume that f k = 1 for all k ∈ N . Then the subset F ∞ f of F f formed by Hamiltonian diffeomorphisms with infinitely many simple periodic orbitsin f N is C ∞ -residual. It is essential that in this theorem the ambient manifold M is not required to betoroidally monotone or negative monotone. In fact, no conditions on M , other thancompactness, is necessary. However, as in the case of other results of this paper, theproof makes use of the Hamiltonian Floer homology, and we need to either assumethat M is weakly monotone or rely on the machinery of virtual cycles. ON-CONTRACTIBLE PERIODIC ORBITS 9
A consequence of Theorem 2.7 in the spirit of an observation in [PS] is thatnon-autonomous Hamiltonian diffeomorphisms (i.e., Hamiltonian diffeomorphismsthat cannot be generated by autonomous Hamiltonians) form a C ∞ -residual subsetof F f . In fact, every ϕ ∈ F ∞ f is necessarily non-autonomous. Indeed, when k > ,simple k -periodic orbits of an autonomous Hamiltonian diffeomorphism are neverisolated, and hence, in particular, never non-degenerate.3. Augmented action filtration
Our goal in this section is to show that when the augmented action gap is suf-ficiently large the Floer homology for non-contractible periodic orbits is filtered bythe augmented action, and to analyze the behavior of this filtration under contin-uation maps. As in the rest of the paper, we assume that ( M n , ω ) is a closed,toroidally monotone or toroidally negative monotone symplectic manifold. How-ever, the construction of the Floer homology (but not of the augmented actionfiltration) goes through in general for any compact manifold M , at least when M is weakly monotone or via the technique of virtual cycles.3.1. Preliminaries: iterated Hamiltonians.
Let H : S × M → R be a one-periodic in time Hamiltonian on M . The augmented action of H is homogeneousunder the iterations of ϕ H . To make this more precise, let us recall a few standarddefinitions.Let K and H be two one-periodic Hamiltonians. The “composition” K♮H is, bydefinition, the Hamiltonian ( K♮H ) t = K t + H t ◦ ( ϕ tK ) − , and the flow of K♮H is ϕ tK ◦ ϕ tH . We set H ♮k = H♮ . . . ♮H ( k times). Abusingterminology, we will refer to H ♮k as the k th iteration of H . (Note that the flow ϕ tH ♮k = ( ϕ tH ) k , t ∈ [0 , , is homotopic with fixed end-points to the flow ϕ tH , t ∈ [0 , k ] .)In general, H ♮k is not one-periodic, even when H is. However, H ♮k becomesone-periodic when, for example, H ≡ ≡ H . The latter condition can alwaysbe met by reparametrizing the Hamiltonian as a function of time without changingthe time-one map. This procedure does not affect the Hofer norm, and actions andindices of the periodic orbits. Thus, in what follows, we usually treat H ♮k as aone-periodic Hamiltonian. Alternatively, the Hamiltonian diffeomorphism ϕ kH canbe obtained as the time- k flow of H . Thus, in some instances such as the proof ofLemma 4.2, it is more convenient to treat H ♮k as the k -periodic Hamiltonian H t with t ∈ R /k Z . We will always state specifically when this is the case. Clearly,these two Hamiltonians, both denoted by H ♮k , have canonically isomorphic filteredFloer homology.The k th iteration of a one-periodic orbit x of H is denoted by x k . More specifi-cally, x k is the k -periodic orbit x ( t ) , t ∈ [0 , k ] , of H . There is an action– and meanindex–preserving one-to-one correspondence between the one-periodic orbits of H ♮k and the k -periodic orbits of H . Thus, we can also think of x k as the one-periodicorbit x k ( t ) = ϕ tH ♮k ( x (0)) of H ♮k .Assume now that all iterated homotopy classes f k , k ∈ N , are distinct and non-trivial. As above, we have a reference loop z ∈ f fixed together with a trivializationof T M | z . Let us chose the iterated loop z k with the “iterated trivialization” as the reference loop for f k . Then the action and the mean index are both homogeneouswith respect to the iteration and, as a consequence, ˜ A H ♮k ( x k ) = k ˜ A H ( x ) . Floer homology for non-contractible periodic orbits.
The key tool usedin the proofs of Theorems 2.1 and 2.2 is the Floer homology for non-contractibleperiodic orbits of Hamiltonian diffeomorphisms. Various flavors of Floer homologyin this case for both open and closed manifolds have been considered in severalother works; see, e.g., [BPS, GaL, GG14, Gü13, Le, Ni, We]. Below we assumethat M n is closed and toroidally monotone or toroidally negative monotone. Inthe latter case, to have the Floer homology defined, one must either rely on themachinery of multivalued perturbations (and set the coefficient ring to be Q ) orrequire in addition that N S ≥ n to ensure that M is weakly monotone, where N S is the minimal spherical Chern number; cf. [LO].Let us now briefly describe the elements of the construction of the Floer homologyrelevant to our argument. Fix f ∈ ˜ π ( M ) . Let H be a Hamiltonian such that all one-periodic orbits of H in f are non-degenerate. (Here x is said to be non-degenerateif the linearized return map dϕ H : T x (0) M → T x (0) M does not have one as aneigenvalue.) The Floer complex CF( H, f ) is generated, over some fixed coefficientring, by these orbits. The Floer differential is defined in the standard way. Withthis definition, the complex CF( H, f ) is neither graded nor does it carry an actionfiltration. The homology HF( H, f ) of CF( H, f ) is equal to zero when f = 1 . Indeed,by the standard continuation argument HF( H, f ) is independent of H (cf. Section3.4) and, since all one-periodic orbits of a C -small autonomous Hamiltonian H arecontractible, we have HF( H, f ) = 0 . As is well known, HF( H,
1) = H ∗ ( M ) at leastover Q ; see, e.g., [McDS] for further references.To give the complex CF( H, f ) some more structure, let us fix a reference loop z ∈ f and a trivialization of T M | z . Using this trivialization, we can define the Conley–Zehnder index µ CZ ( H, ¯ x ) ∈ Z of a capped non-degenerate orbit ¯ x as in,e.g., [McDS, SZ]. For future reference, note that | ∆ H (¯ x ) − µ CZ (¯ x ) | ≤ n. (3.1)Similarly to the contractible case, the Conley-Zehnder µ CZ ( H, x ) of an orbitwithout capping is defined only modulo N T . As a result, we obtain a Z N T -grading of the complex CF( H, f ) and of the homology HF( H, f ) , and, in particular,a Z -grading. Replacing the one-periodic orbits of H by the capped one-periodicorbits, one could define the Floer complex and the homology of H as a moduleover a suitably chosen Novikov ring and, as in the contractible case, this complexand the homology would be Z -graded and filtered by the action. However, forour purposes it is more convenient to work with the homology HF( H, f ) and thecomplex CF( H, f ) generated by the non-capped orbits and defined as above.The constructions from this section readily carry over to the case where a singlefree homotopy class f is replaced by a collection of free homotopy classes. For in-stance, one can specify the collection of free homotopy classes of loops by prescribinga homology class.3.3. Filtration.
Let, as above, M n be a toroidally monotone or toroidally nega-tive monotone closed symplectic manifold with monotonicity constant λ . In whatfollows, we have a free homotopy class f or a collection of such classes together with ON-CONTRACTIBLE PERIODIC ORBITS 11 the reference loops and trivializations fixed and suppressed in the notation. Thuswe write
CF( H ) for CF( H, f ) , etc. With these auxiliary data fixed, the augmentedaction spectrum ˜ S ( H ) := ˜ S ( H, f ) is defined for any Hamiltonian H on M .The augmented action gap is the infimum of the distance between two distinctpoints in the augmented action spectrum ˜ S ( H ) , i.e., gap( H ) = inf | s − s ′ | ∈ [0 , ∞ ] , where s and s ′ = s are in ˜ S ( H ) .We emphasize that gap( H ) is defined even when H is degenerate. It is also worthpointing out that gap( H ) is neither upper nor lower semicontinuous in H .Set c ( M ) = | λ | n ± , (3.2)where the sign ± is sign( λ ) . We say that the gap condition is satisfied whenever gap( H ) > c ( M ) . (3.3) Proposition 3.1.
Assume that all one-periodic orbits of H in f are non-degenerateand (3.3) holds. Then the complex CF( H ) , and hence the homology HF( H ) , isfiltered by the augmented action. In other words, ˜ A H ( y ) ≤ ˜ A H ( x ) (3.4) whenever y enters ∂x = P a y y with non-zero coefficient.Remark . In contrast with the standard action filtration, the augmented actionfiltration is not necessarily strict, i.e., equality in (3.4) can occur. Note also that inthis proposition it suffices to have a non-strict inequality in the gap condition (3.3).
Proof.
Throughout the proof, let us assume that λ ≥ , i.e., M is toroidally mono-tone. The negative monotone case is dealt with by a similar (up to some signs)calculation.To establish (3.4), let us fix a capping of x . Then an orbit y entering ∂x withnon-zero coefficient inherits a capping from ¯ x . We have ˜ A H ( y ) = A H (¯ y ) − λ H (¯ y ) < A H (¯ x ) − λ (cid:0) µ CZ (¯ y ) − n (cid:1) = A H (¯ x ) − λ (cid:0) µ CZ (¯ x ) − n − (cid:1) ≤ A H (¯ x ) − λ (cid:0) ∆ H (¯ x ) − n − (cid:1) = ˜ A H ( x ) + c ( M ) . Here we used (3.1) and the facts that µ CZ (¯ y ) = µ CZ (¯ x ) − and that ∂ is actiondecreasing. Thus we have shown that ∂ does not increase the augmented action bymore than c ( M ) . Now the required inequality (3.4) follows once the augmentedaction gap is greater than c ( M ) , i.e., when the gap condition (3.3) holds. (cid:3) With Proposition 3.1 in mind, we can define the augmented action filtration onthe homology exactly in the same way as for the ordinary action. Thus, assumethat (3.3) is satisfied and a ˜ S ( H ) and denote by f CF ( −∞ , a ) ( H ) the subcomplex of CF( H ) generated by the orbits with augmented action below a . Let f HF ( −∞ , a ) ( H ) be the homology of this subcomplex. Furthermore, when I = ( a, b ) is an intervalwith end points outside ˜ S ( H ) , we set f CF I ( H ) = f CF ( −∞ , b ) ( H ) / f CF ( −∞ , a ) ( H ) . In other words, f CF I ( H ) is the complex generated by the orbits with augmented ac-tion in I , equipped with the naturally defined differential. We denote the homologyof this complex by f HF I ( H ) . (The role of the tilde here is to emphasize that we usethe augmented action rather than the ordinary action and that I is an augmentedaction range.) We have the long exact sequence · · · → f HF ( −∞ , a ) ( H ) → f HF ( −∞ , b ) ( H ) → f HF I ( H ) → · · · and a similar exact sequence for three intervals · · · → f HF ( c, a ) ( H ) → f HF ( c, b ) ( H ) → f HF ( a, b ) ( H ) → · · · . (3.5)Our next goal is to show that the construction of the augmented action fil-tered Floer homology extends by continuity to all, not necessarily non-degenerate,Hamiltonians. Proposition 3.3.
Let H be a Hamiltonian on M , not necessarily non-degenerate,such that the gap condition (3.3) is satisfied and let a ˜ S ( H ) . Then for any non-degenerate Hamiltonian K sufficiently C -close to H , the subspace f CF ( −∞ , a ) ( K ) ⊂ CF( K ) is a subcomplex. This result is not an immediate consequence of Proposition 3.1. Since the aug-mented action gap is not lower semicontinuous in the Hamiltonian, we cannot guar-antee that (3.3) holds for K if it holds for H , and thus a priori Proposition 3.1need not apply to K . Proof.
Let x be a one-periodic orbit of K with ˜ A K ( x ) < a and let y be an orbitentering ∂x with non-zero coefficient. We need to show that ˜ A K ( y ) < a .The orbits x and y are C -small perturbations of one-periodic orbits x ′ and y ′ of H with augmented actions close to those of x and y . By continuity of theaugmented action spectrum, we necessarily have ˜ A H ( x ′ ) < a when K is C -closeto H .If ˜ A H ( y ′ ) > ˜ A H ( x ′ ) , we have ˜ A H ( y ′ ) − ˜ A H ( x ′ ) > c ( M ) by (3.3), and therefore ˜ A K ( y ) − ˜ A K ( x ) > c ( M ) . This is impossible because, as we have seen from theproof of Proposition 3.1, the differential cannot increase the augmented action bymore than c ( M ) . Thus ˜ A H ( y ′ ) ≤ ˜ A H ( x ′ ) . Then ˜ A K ( y ) ≈ ˜ A H ( y ′ ) ≤ ˜ A H ( x ′ ) < a, and hence ˜ A K ( y ) < a when K is C -close to H . (cid:3) Now, for any Hamiltonian H , when the end-points of an interval I are outside ˜ S ( H ) and (3.3) holds, we can, utilizing Proposition 3.3, set f HF I ( H ) := f HF I ( K ) ,where K is a C -small non-degenerate perturbation of H . Using standard contin-uation arguments (cf. Section 3.4), it is easy to see that the resulting homology iswell defined, i.e., independent of K . Example . The setting we are interested in where the gap condition (3.3) issatisfied is when H is a high prime order iteration of some Hamiltonian F , i.e., H = F ♮k and k is a large prime. In this case, either F has simple k -periodic ON-CONTRACTIBLE PERIODIC ORBITS 13 orbits or gap( H ) = k gap( F ) . Thus, either new periodic orbits are created or thegap grows linearly under the iterations of F and eventually becomes greater than c ( M ) .It is clear that all these constructions respect the Z N T -grading (and hence the Z -grading) of the complexes and the homology. Thus, for instance, (3.5) is anexact sequence of graded complexes and the connecting map has degree − .For degenerate Hamiltonians with isolated one-periodic orbits, one can, similarlyto the case of the standard action filtration, view the local Floer homology as build-ing blocks for the Floer homology filtered by the augmented action. For instance,assume that ˜ S ( H ) ∩ I = { c } and all one-periodic orbits x with augmented action c are isolated. Then it is not hard to see that there exists a spectral sequence with E = L x HF( x ) converging to f HF I ( H ) , where HF( x ) stands for the local Floerhomology of x . (We refer the reader to, e.g., [Gi10, GG10, McL] for the definitionand a discussion of the local Floer homology.) In contrast with the case of theordinary action filtration, we do not necessarily have E = f HF I ( H ) even when H is non-degenerate. The reason is that the augmented action filtration is not strictand the Floer differential, or more generally Floer trajectories, can connect orbitswith equal augmented action.However, as is easy to see, for any interval I with end points outside ˜ S ( H ) , wehave χ ( H, I ) = ( − n (cid:2) dim f HF I even ( H ) − dim f HF I odd ( H ) (cid:3) . (3.6)In particular, f HF I ( H ) = 0 if χ ( H, I ) = 0 . (Here, as everywhere in this section, wehave suppressed the class f in the notation.)3.4. Homotopy and continuation.
The behavior of the augmented action underhomotopies is similar to that of the ordinary action. Namely, recall that a continu-ation map shifts the action filtration upward by a certain constant; see, e.g., [Gi07,Sect. 3.2.2]. This is still true for the augmented action, although the size of theshift is slightly different. Furthermore, when the homotopy is monotone decreas-ing, the action shift is zero, and the induced map in homology preserves the actionfiltration. This fact does not have a direct analogue for the augmented action, butthe augmented action filtration is preserved when the augmented action gaps forthe Hamiltonians are large enough.To be more precise, consider a homotopy H s from a Hamiltonian H to a Hamil-tonian H on M , and set c a ( H s ) = Z ∞−∞ Z S max M ∂ s H s dt ds. For instance, c a ( H s ) = Z S max M ( H − H ) dt when H s is a linear homotopy from H to H . The augmented action shift isgoverned by the constant c h ( H s ) = max (cid:8) , c a ( H s ) (cid:9) + | λ | n ≥ . (3.7) Proposition 3.5.
Assume that both Hamiltonians H and H satisfy (3.3) , i.e., gap( H ) > c ( M ) and gap( H ) > c ( M ) . (3.8) Then a homotopy H s from H to H induces a map in the Floer homology shiftingthe action filtration upward by c h ( H s ) : f HF I ( H ) → f HF c h ( H s )+ I ( H ) , where c h ( H s ) + I stands for the interval I moved to the right by c h ( H s ) . Further-more, if gap( H ) > c h ( H s ) and gap( H ) > c h ( H s ) in addition to (3.8) , the map induced by the homotopy preserves the augmentedaction filtration, i.e., we have f HF I ( H ) → f HF I ( H ) . Note that here the shift c h ( H s ) can be replaced by any constant a > c h ( H s ) .The proof of the proposition is standard and we omit it. Here we only mentionthat the first term in (3.7) is the maximal action shift induced by the homotopy(see, e.g., again [Gi07, Sect. 3.2.2]) and the second term is the maximal mean indexshift, as can be seen from an argument similar to the proof of Proposition 3.1. Remark . The arguments from this section carry over to contractible periodicorbits, i.e., to the case where f = 1 , with some simplifications and straightforwardmodifications. Namely, in this case, it is enough to assume that M is monotone ornegative monotone to have the augmented action defined; see [GG09a]. An analogueof (3.3) is still sufficient to ensure that the Floer complex and the homology arefiltered by the augmented action and Propositions 3.1, 3.3 and 3.5 still hold.4. Proofs
With the action filtration introduced, we are now in a position to prove the mainresults of the paper.4.1.
Proofs of Theorems 2.2 and 2.4.
Proof of Theorem 2.2: the case [ f ] = 0 . Since P ( H, [ f ]) is finite, only finitely manydistinct free homotopy classes f i ∈ ˜ π ( M ) occur as the free homotopy classes of one-periodic orbits of H in the homology class [ f ] . We claim that then, for a sufficientlylarge prime p , the classes f pi are also distinct.To see this, first note that for any two elements g = h in any group there isat most one prime p such that g p = h p . Indeed, assume that there are two suchdistinct primes p and q . Then, since p and q are relatively prime, ap + bq = 1 forsome integers a and b . Hence, g = ( g p ) a ( g q ) b = ( h p ) a ( h q ) b = h, which is impossible since g = h . Clearly, the same is true for conjugacy classes.As a consequence, for any finite collection of distinct conjugacy classes, their largeprime powers are also distinct.Throughout the proof we will always require p to be a sufficiently large primeto satisfy the above condition for the collection f i . (Later we will need to imposeadditional lower bounds on p .)Let us assume that H has no simple p -periodic orbits in the class f p . Our goalis to show that it has a simple p ′ -periodic orbit, where p ′ is the first prime greaterthan p , in the homotopy class f p . ON-CONTRACTIBLE PERIODIC ORBITS 15
Then all p -periodic orbits in f p are the p th iterations of one-periodic orbits, since p is prime. Furthermore, by the above requirement on p , these one-periodic orbitsare necessarily in the free homotopy class f . Thus we have ˜ S ( H ♮p , f p ) = p ˜ S ( H, f ) (4.1)with respect to the p th iteration of the reference loop z ∈ f and of the trivializationof T M | z . As a consequence, gap( H ♮p , f p ) = p gap( H, f ) , (4.2)and the augmented action filtration on the Floer homology f HF( H ♮p , f p ) is definedonce p is so large that p gap( H, f ) > c ( M ) ; see (3.2) and Section 3.3. We also have ˜ S ( H ♮p , f p ) ∩ pI = p ( ˜ S ( H, f ) ∩ I ) . (4.3)Here pI = ( pa, pb ) for I = ( a, b ) .Next we claim that, when p is sufficiently large, χ ( H ♮p , pI, f p ) = χ ( H, I, f ) . (4.4)To see this, denote by x i the one-periodic orbits of H in the class f and withaugmented action in I . This is a finite collection of orbits since P ( H, [ f ]) is finite.Then all sufficiently large primes p are admissible in the sense of [GG10] for all orbits x i , i.e., 1 has the same multiplicity as a generalized eigenvalue of the linearizedreturn maps dϕ H and dϕ pH at x i and the two maps have the same number ofeigenvalues in ( − , . (Indeed, it suffices to require p to be larger than andlarger than the degree of any root of unity among the eigenvalues of dϕ H at x i .)By the Shub–Sullivan theorem (see [CMPY, SS]), the orbits x i and x pi have thesame Poincaré–Hopf index. The orbits x pi are the only p -periodic orbits of H in f p with augmented action in pI , and (4.4) follows. Alternatively, one can argue as inthe proof of the case f = 1 of the theorem; see below.By (3.6) and since χ ( H, I, f ) = 0 , we conclude that f HF pI ( H ♮p , f p ) = 0 . Now we are in a position to show that H must have at least one p ′ -periodicorbit in the class f p , where p ′ is the first prime greater than p , provided again that p is sufficiently large. Then, as the last step of the proof, we will show that this p ′ -periodic orbit is necessarily simple.Arguing by contradiction, assume that there are no such orbits. Then gap( H ♮p ′ , f p ) = ∞ , and, obviously, the augmented action filtration is defined on the Floer homologyfor H ♮p ′ and f p . (Of course, the resulting complex and the homology is zero forany augmented action interval, but this is not essential at this point.) By roughlyfollowing the line of reasoning from [Gü13, Gü14] and relying on the fact thatthe filtered homology is defined, we will show that the homology is non-trivial fora certain augmented action interval and thus arrive at a contradiction with theassumption that H has no p ′ -periodic orbits in the class f p .Set e + = max (cid:26)Z S max M H t dt, (cid:27) and e − = max (cid:26) − Z S min M H t dt, (cid:27) . Then a ± := ( p ′ − p ) e ± + | λ | n ≥ c ± , where the constants c ± = c h ( H s ) are defined by (3.7) for the linear homotopiesfrom H ♮p to H ♮p ′ and from H ♮p ′ to H ♮p .Furthermore, recall that p ′ − p = o ( p ) as p → ∞ ; see, e.g., [BHP]. Thus, when p is sufficiently large, we have gap( H ♮p , f p ) = p gap( H, f ) > a ± ≥ c ± . Hence, the conditions of Proposition 3.5 are satisfied, and the continuation maps f HF pI ( H ♮p , f p ) → f HF pI + a + ( H ♮p ′ , f p ) and f HF pI + a + ( H ♮p ′ , f p ) → f HF pI + a + + a − ( H ♮p , f p ) are defined.Consider now the following commutative diagram: f HF pI (cid:0) H ♮p , f p (cid:1) (cid:15) (cid:15) ∼ = * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ f HF pI + a + (cid:0) H ♮p ′ , f p (cid:1) / / f HF pI + a + + a − (cid:0) H ♮p , f p (cid:1) Here the diagonal map is an isomorphism. To see this, denote by δ > the distancefrom the end points of I to ˜ S ( H, f ) . Then the distance from the end points of pI to ˜ S ( H ♮p , f p ) is pδ and, when p is large, pδ > a + + a − again because p ′ − p = o ( p ) . Hence, the intervals ( pI + a + + a − ) \ pI and pI \ ( pI + a + + a − ) contain no points of ˜ S ( H ♮p , f p ) , and the diagonal map is indeedan isomorphism. (In fact, this argument shows that, as in the second part ofProposition 3.5, one can eliminate the shifts a ± and a + + a − in the continuationmaps when p is sufficiently large.)Moreover, as we have shown above, f HF pI ( H ♮p , f p ) = 0 . Therefore, the middlegroup f HF pI + a + ( H ♮p ′ , f p ) in the diagram is also non-trivial, and thus H must havea p ′ -periodic orbit in the homotopy class f p .It remains to show that this orbit is necessarily simple. However, otherwise, itwould be the p ′ th iteration of a one-periodic orbit in the homology class p [ f ] /p ′ .This is impossible because p [ f ] /p ′ is not an integer homology class when p , andhence p ′ , are large since [ f ] = 0 . This completes the proof of the case [ f ] = 0 of thetheorem. (cid:3) Proof of Theorem 2.2: the case f = 1 . The proof follows the same path as in thecase where [ f ] = 0 , and here we only indicate the necessary changes in the argument.The key to the proof is the fact that when π ( M ) is a torsion free hyperbolicgroup and f = 1 in ˜ π ( M ) there exists a constant r ( f ) ∈ N such that the equation f p = h q ON-CONTRACTIBLE PERIODIC ORBITS 17 in ˜ π ( M ) , where p and q are primes greater than r ( f ) and h ∈ ˜ π ( M ) , is satisfiedonly when h = f and p = q .To see this, we first note that it is sufficient to prove this fact for π ( M ) ratherthan ˜ π ( M ) . In other words, given f ∈ π ( M ) , f = 1 , we need to show that f p = h q for sufficiently large primes p and q (depending on f ) only when h = f and p = q .To this end, recall that for any hyperbolic group G , every element f ∈ G of infiniteorder is contained in a unique maximal virtually cyclic subgroup E ( f ) and E ( f ) = { g ∈ G | g − f l g = f ± l for some l ∈ N } ; see [Ol] (and also [Gr]). Applying this to f = 1 in G = π ( M ) , which is alsoassumed to be torsion free, and using the fact that a torsion free and virtuallycyclic group is cyclic, we conclude that E ( f ) is infinite cyclic, i.e., Z . Furthermore, h ∈ E ( f ) as a consequence of the condition f p = h q . Indeed, h − f p h = h − h q h = h q = f p . This reduces the question to the case where f and h belong to the infinite cyclicgroup E ( f ) , and in this case the result is obvious. From now on, we require that p ≥ r ( f ) . (Later we will need to introduce addi-tional lower bounds for p .) As in the proof of the first case of the theorem, assumethat H has no simple p -periodic orbits in the class f p . Then every p -periodic orbit isthe p th iteration of a one-periodic orbit and, by the above observation (with p = q ),this one-periodic orbit must also be in the class f . Clearly, (4.1), (4.2), and (4.3)still hold.Furthermore, (4.4) also holds, i.e., χ ( H ♮p , pI, f p ) = χ ( H, I, f ) , although now thereason is slightly different. Consider the set F of the initial conditions x (0) forall one-periodic orbits of H in the class f and with augmented action in I . Sincethe end points of I are outside ˜ S ( H, f ) , the set F is closed. Under a small non-degenerate perturbation ˜ H of H , the set F splits into a finite collection of the initialconditions of the orbits ˜ x i of ˜ H in f with augmented action in I , and χ ( H, I, f ) isthe sum of the Poincaré–Hopf indices of the orbits ˜ x i . We can furthermore ensurethat there are no roots of unity among the Floquet multipliers of these orbits. Byour assumptions, F is also the set of the initial conditions for all p -periodic orbitsof H in the class f p with augmented action in pI . If ˜ H is sufficiently close to H ,the only p -periodic orbits of ˜ H in f p with augmented action in pI are ˜ x pi . Hence, χ ( H ♮p , pI, f p ) is the sum of the Poincaré–Hopf indices of the orbits ˜ x pi . When p > ,the orbits ˜ x i and ˜ x pi have the same Poincaré–Hopf index due to the assumption thatnone of the Floquet multipliers is a root of unity. As a consequence, we have (4.4).The rest of the proof is identical to the argument in the case where [ f ] = 0 exceptfor the very last step. Thus we have proved the existence of a p ′ -periodic orbit x inthe class f p and now need to show that this orbit is simple. Assume the contrary.Then, since p ′ is prime, x is necessarily the p ′ th iteration of a one-periodic orbit insome class h . We have f p = h p ′ , and hence p ′ = p when p > r ( f ) . This is impossible since p ′ is the first prime greaterthan p . (cid:3) Turning to Theorem 2.4, note that, as in [Gü13], a more general result holds.Namely, recall that the local Floer homology
HF( x ) is associated to an isolated The authors are grateful to Denis Osin for this argument. periodic orbit x of H . The group HF( x ) , already mentioned in Section 3.3, isroughly speaking the homology of the Floer complex generated by the orbits which x splits into under a non-degenerate perturbation; see, e.g., [GG10] for more details.In particular, when x is non-degenerate, HF( x ) is equal to the ground ring andconcentrated in degree µ CZ ( x ) . We have the following generalization of Theorem 2.4: Theorem 4.1.
Assume that the class [ ω ] is atoroidal and let H be a Hamiltonianhaving an isolated one-periodic orbit x with homotopy class f such that HF( x ) = 0 and that [ f ] = 0 in H ( M ; Z ) / Tor and P ( H, [ f ]) is finite. Then, for every suffi-ciently large prime p , the Hamiltonian H has a simple periodic orbit in the homotopyclass f p and with period either p or p ′ , where p ′ is the first prime greater than p .Moreover, when π ( M ) is hyperbolic and torsion free, the condition [ f ] = 0 can bereplaced by f = 1 . The main new point here is the “moreover” part of the theorem. The case of thetheorem where [ f ] = 0 , proved in [Gü13], is included for the sake of completeness.The proof of the “moreover” part is a combination of the proof of [Gü13, Thm. 3.1]and the proof of the case f = 1 of Theorem 2.2. Here we only briefly touch uponthis argument. On the proof of Theorem 4.1.
The key difference between the settings of Theorems2.4 and 4.1 and that of Theorem 2.2 is that now the class [ ω ] is atoroidal andthus we have the standard action filtration HF I ( H ; f ) on the Floer homology of H rather than the augmented action filtration. On the level of complexes, the actionfiltration is strictly monotone, i.e., the differential is strictly action decreasing. (Incontrast, the augmented action is only non-increasing; see the discussion in Section3.3.) As a consequence, HF I ( H ; f ) = 0 when I is a small interval centered at theaction A H ( x ) whenever x and A H ( x ) are isolated and HF( x ) = 0 . Furthermore,when H has no simple p -periodic orbits in the class f p , we have HF pI ( H ♮p ; f p ) = 0 by the persistence of the local Floer homology results from [GG09a]. With this inmind, one argues essentially word-for-word as in the proof of Theorem 2.2, withsome straightforward simplifications. We omit the details. (cid:3) Proof of Theorem 2.1.
Arguing by contradiction, assume that H has onlyfinitely many simple periodic orbits with homotopy class in f N = { f k | k ∈ N } ,where f = J x K . We denote these orbits by x j and let k j be the period of x j . Set F = H ♮ k and y j = x k /k j j , where k is the least common multiple of the periods k j . The orbits y j are the one-periodic orbits of F .We claim that for all k ∈ N every k -periodic orbit z of F in the collection of thehomotopy classes f N is the k th iteration of one of the orbits y j . Indeed, then z isalso a kk -periodic orbit of H in f N . Hence, z = x kk /k j j = y kj for some j .Thus we have a collection of free homotopy classes f N generated by f ∈ ˜ π ( M ) ,and a Hamiltonian F with finitely many one-periodic orbits y j in f N and no othersimple periodic orbits in f N . One of these orbits, h , is an even iteration of theoriginal hyperbolic orbit. Hence, h is hyperbolic with an even number of eigenvaluesin ( − , .From now on we focus on the Hamiltonian F and its periodic orbits. Set h = J h K ;clearly, h N ⊂ f N .Let us fix reference curves and trivializations for the collection h N . Namely, itis convenient to take h as a reference curve for h . Then the reference trivialization ON-CONTRACTIBLE PERIODIC ORBITS 19 is fixed by the condition that ∆ F (¯ h ) = 0 , where ¯ h stands for h equipped withthe “identity” capping. (Such a trivialization exists since h is hyperbolic and hasan even number of eigenvalues in ( − , , and hence the mean index of h withrespect to any trivialization is an even integer.) The class h k is then given theiterated reference curve h k and the “iterated” trivialization. We fix cappings of theorbits y j , suppressed in the notation, and equip the iterated orbits with “iterated”cappings. As a consequence, the action, the mean index, and the augmented actionare homogeneous under iterations for periodic orbits in h N . (It is essential here thatall free homotopy classes in h N are distinct and nontrivial, and hence the referencecurve and the trivialization are well defined.)Without loss of generality, by adding if necessary a constant to F , we can ensurethat ˜ A F ( h ) = 0 .By our assumptions, we have ˜ S ( F ♮k , h k ) = k ˜ S ( F, h ) and gap( F ♮k , h k ) = k gap( F, h ) . It follows that the augmented action filtered Floer homology of F ♮k is defined when k is large enough.Furthermore, let I be an interval such that A F ( h ) is the only point in ˜ S ( F, h ) ∩ I , and the end points of I are not in ˜ S ( F, h ) . Then we also have ˜ S ( F ♮k , h k ) ∩ kI = { } (4.5)and the end points of kI are outside ˜ S ( F ♮k , h k ) . Lemma 4.2. f HF k i I ( F ♮k i , h k i ) = 0 for some sequence k i → ∞ . Assuming the lemma, let us finish the proof of the theorem. Similarly to theproof of Theorem 2.2, set a + = max (cid:26)Z S max M F t dt, (cid:27) + | λ | n and a − = max (cid:26) − Z S min M F t dt, (cid:27) + | λ | n. These constants are greater than or equal to the constants c h given by (3.7) for thelinear homotopies from F ♮k to F ♮ ( k +1) and from F ♮ ( k +1) to F ♮k , and denoted againby c ± . Thus, when k is sufficiently large, gap( F ♮k , h k ) = k gap( F, h ) > a ± ≥ c ± . Hence, by Proposition 3.5, the continuation maps f HF kI ( F ♮k , h k ) → f HF kI + a + ( F ♮ ( k +1) , h k ) and f HF kI + a + ( F ♮ ( k +1) , h k ) → f HF kI + a + + a − ( F ♮k , h k ) are defined.Let δ > be the distance from the end points of I to ˜ S ( F, h ) . Then the distancefrom the end points of kI to ˜ S ( H ♮k , h k ) is kδ . When k is large, kδ > a + + a − , and the intervals ( kI + a + + a − ) \ kI and kI \ ( kI + a + + a − ) contain no points of ˜ S ( F ♮k , h k ) . Thus the natural quotient-inclusion map f HF kI ( F ♮k , h k ) → f HF kI + a + + a − ( F ♮k , h k ) is an isomorphism.Let now k be one of the sufficiently large entries in the sequence k i from Lemma4.2. Consider the following commutative diagram: f HF kI (cid:0) F ♮k , h k (cid:1) (cid:15) (cid:15) ∼ = + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ f HF kI + a + (cid:0) F ♮ ( k +1) , h k (cid:1) / / f HF kI + a + + a − (cid:0) F ♮k , h k (cid:1) Here the diagonal map is an isomorphism and, by Lemma 4.2, f HF kI ( F ♮k , h k ) = 0 .Therefore, the middle group f HF kI + a + ( F ♮ ( k +1) , h k ) in the diagram is also non-trivial,and F has a ( k + 1) -periodic orbit z in the homotopy class h k .We have z = y k +1 j for some j . Furthermore, h = f a and J y j K = f b for some a and b in N . Thus f ak = f b ( k +1) . Since all homotopy classes in f N are distinct, we inferthat ak = b ( k + 1) , where a is independent of k . This is clearly impossible for k ≥ a since k and k + 1 are relatively prime, and we have arrived at a contradiction. Tofinish the proof of the theorem, it remains to prove the lemma. Proof of Lemma 4.2.
Throughout the proof, it is convenient to interpret the iter-ated Hamiltonian F ♮k as the k -periodic Hamiltonian F t with t ∈ R /k Z . Further-more, without loss of generality, we can ensure that λ/ by rescaling ω , andthus ˜ A F = A F − ∆ F . Since ˜ A F ( h ) = 0 and ∆ F (¯ h ) = 0 , we also have A F (¯ h ) = 0 = ∆ F (¯ h ) . (4.6)Recall that, by our assumptions, is the only point of the action spectrum ˜ S ( F ♮k , h k ) in the interval kI (see (4.5)) and that y kj are the only k -periodic orbits of F .Fix a neighborhood U of h which does not intersect any of the other orbits y j and a small parameter ǫ > , depending on U , to be specified later. There exists asequence k i → ∞ such that for all j and k i , we have (cid:13)(cid:13) ∆ F ♮ki ( y k i j ) (cid:13)(cid:13) N T < ǫ, (4.7)where k a k N T stands for the distance from a ∈ S N T = R / N T Z to or, equiva-lently, from a ∈ R to the nearest point in the lattice N T Z ⊂ R . Here we can treatthe mean index ∆ F ♮ki ( y k i j ) = k i ∆ F ( y j ) as a real number when y j is capped or as a point in S N T when the capping isdiscarded.To prove (4.7), consider the torus T m = ( S N T ) m where m is the number of theorbits y j and set ∆ = (cid:0) ∆ F ( y ) , . . . , ∆ F ( y m ) (cid:1) ∈ T m . The closure Γ of the orbit { k ∆ | k ∈ Z } is a subgroup in T m and, for every k , theset { k ∆ | k > k } is dense in Γ . Hence, the point k ∆ is within the ǫ -neighborhoodof ∈ Γ for infinitely many values of k .From now on k will stand for one of the entries in the sequence k i . ON-CONTRACTIBLE PERIODIC ORBITS 21
Let G be a C ∞ -small, non-degenerate perturbation of F ♮k equal to F ♮k on theneighborhood U . The orbits y kj , other than h k , split into a finite collection of non-degenerate orbits of G in the class h k . Among these we are interested exclusively inthe orbits with augmented action in kI . These orbits can only come from the orbits y kj with action in kI , i.e., by (4.5), from the orbits y j with ˜ A F ( y j ) = 0 . We denotethe resulting orbits of G by z j . (The number of the orbits z j may be different fromthe number of the orbits y j .) It is clear that the orbits z j do not enter U and that (cid:12)(cid:12) ˜ A G ( z j ) (cid:12)(cid:12) ≤ η, (4.8)where η = O (cid:0) k F ♮k − G k C (cid:1) .It suffices now to show that when ǫ > is sufficiently small the orbit h k of G is closed (i.e., a cycle), but not exact, in the Floer complex f CF kI ( G, h k ) . To thisend, we will prove that h k cannot be connected to any of the orbits z j by a Floertrajectory of relative index ± .By (4.6), we have ∆ G (¯ h k ) = k ∆ F (¯ h ) = 0 and A G (¯ h k ) = k A F (¯ h ) = 0 In particular, µ CZ ( h k ) = ∆ G ( h k ) = 0 .Let now ¯ z be one of the capped orbits ¯ z j . Our goal is to show that every Floertrajectory u connecting the capped orbits ¯ z and ¯ h k has relative index different from ± . Since µ CZ (¯ h k ) = 0 and by (3.1), it is enough to prove that | ∆ G (¯ z ) | > n + 1 . (4.9)The orbit z does not enter U . Thus, by [GG14, Thm. 3.1], there exists a constant e > , depending on U , but not on k , such that the energy of u is bounded frombelow by e . In other words, using the fact that A G (¯ h k ) = 0 , we have |A G (¯ z ) | > e. Set ǫ < e .By (4.7), ∆ G (¯ z ) ∈ ( ℓ − ǫ, ℓ + ǫ ) for some ℓ ∈ Z . If ℓ = 0 , and hence | ∆ G (¯ z ) | < ǫ ,we also have |A G (¯ z ) | < ǫ + η by (4.8). This is impossible when η > is smaller than | e − ǫ | , i.e., when G issufficiently C -close to F ♮k , since ǫ < e . Thus ℓ = 0 , and therefore | ∆ G (¯ z ) | > N T − ǫ. Recall that N T ≥ n/ by the assumptions of the theorem. Hence, when ǫ < ,we have | ∆ G (¯ z ) | ≥ n + 2 − ǫ > n + 1 . This proves (4.9), completing the proof of the lemma and of the theorem. (cid:3)
Proof of Theorem 2.7.
The argument is quite standard; it follows the sameline of reasoning as the proof of [GG09b, Prop. 1.6], which, in turn, has a lot ofsimilarities with, e.g., an argument in [Hi].
Proof.
It suffices to show that a Hamiltonian diffeomorphism ϕ H ∈ F f has a non-hyperbolic periodic orbit in some homotopy class f k or ϕ H has infinitely manyperiodic orbits in f N . Indeed, the presence of a non-hyperbolic periodic orbit x implies, by the Birkhoff–Lewis–Moser fixed-point theorem (see [Mo]), the existence of infinitely many periodic orbits in a tubular neighborhood of x C ∞ -genericallyfor Hamiltonian diffeomorphisms close to ϕ H .To this end, observe that since HF( H, f ) = 0 due to the condition f = 1 , theHamiltonian H necessarily has one-periodic orbits in the class f with odd and witheven Conley–Zehnder indices. As a consequence, it has either a non-hyperbolic one-periodic orbit or a hyperbolic orbit with an odd number of real Floquet multipliersin the interval ( − , . In the former case the proof is finished.In the latter case, let us apply this argument to ϕ H . By the assumptions of thetheorem f = 1 and thus HF( H ♮ , f ) = 0 . Hence, H has two-periodic orbits in theclass f with odd and with even Conley–Zehnder indices. The second iterations ofone-periodic orbits from the class f are necessarily positive hyperbolic. Therefore,there exists a simple two-periodic orbit in f , which is either non-hyperbolic orhyperbolic with an odd number of real Floquet multipliers in the interval ( − , .In the former case, the proof is finished, and in the latter we repeat this process for ϕ and so forth.As a result, we will either find a non-hyperbolic orbit in some class f k or constructa sequence of simple periodic orbits in f N . (cid:3) Remark . As is clear from the proof, it is sufficient to assume only that f k = 1 when k is a power of . In this case, the result asserts the generic existence ofinfinitely many periodic orbits in the set of homotopy classes f N , although theseorbits may now be contractible. References [Ab] A. Abbondandolo,
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