aa r X i v : . [ m a t h . S G ] F e b ON THE GENERIC CONLEY CONJECTURE
YOSHIHIRO SUGIMOTO
Abstract.
In this paper, we treat an open problem related to the numberof periodic orbits of Hamiltonian diffeomorphisms on closed symplectic mani-folds, so-called (generic) Conley conjecture. Generic Conley conjecture statesthat generically Hamiltonian diffeomorphisms have infinitely many simple con-tractible periodic orbits. We prove generic Conley conjecture for very wideclasses 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. The information of periodicorbits of Hamiltonian diffeomorphisms is very important in Hamiltonian dynamics.Conley conjecture was originally stated for Hamiltonian diffeomorphisms on thestandard torus ( T n , ω ) ([2]). It states that any Hamiltonian diffeomorphism on( T n , ω ) has infinitely many simple contractible periodic orbits (Simple means it isnot iterated periodic orbit of lower period.). It is easy to see that this conjecture cannot be generalized to any closed symplectic manifolds. For example, an irrationalrotation on the standard sphere S ⊂ R has only two contractible periodic orbits,the north pole and the south pole.However, Conley conjecture was proved for wide classes of closed symplectic man-ifolds. For example, Conley conjecture holds on symplectically aspherical manifolds,negatively monotone symplectic manifolds and symplectic manifolds with vanishingspherical Chern class([4, 6, 7, 8, 10, 13]). So, today’s Conley conjecture is a conjec-ture that every Hamiltonian diffeomorphism has infinitely many simple contractibleperiodic orbits on ”almost all” closed symplectic manifolds.Another variant of above Conley conjecture is so-called generic Conley conjec-ture ([5, 7, 8]). Generic conley conjecture states that ”almost all” Hamiltoniandiffeomorphisms have infinitely many simple contractible periodic orbits on everyclosed symplectic manifold. Conley conjecture and generic Conley conjecture statethat Hamiltonian diffeomorphims with finitely many simple periodic orbits (likethe irrational rotation on the sphere S ) are very rare. In summary, we have thefollowing two conjectures. Conjecture 1 ((generic) Conley conjecture) . (1) On ”almost all” closed sym-plectic manifolds, every Hamiltonian diffeomorphism has infinitely manysimple contractible periodic orbits. (2)
On every closed symplectic manifolds, almost all Hamiltonian diffeomor-phisms have infinitely many simple contractible periodic orbits.
In this paper, we study Conjecture 1 (2), the generic Conley conjecture. Thestatement of our main result is stated as follows.
Theorem 1.1.
Let ( M, ω ) be a n -dimensional closed symplectic manifold andlet N ∈ N ∪ {∞} be the minimum Chern number of ( M, ω ) . Assume that ( M, ω ) satisfies at least one of the following conditions. (1) n is odd. (2) H odd ( M : Q ) = 0(3) N > Then, there is a C ∞ -dense and C ∞ -residual ( = contains a countable intersectionof C ∞ -open dense subsets) subset U ⊂
Ham ( M, ω ) such that any element of U hasinfinitely many simple contractible periodic orbits. Note that the above conditions (1), (2) and (3) cover almost all closed symplecticmanifolds.
Remark 1.1.
The case (2) of Theorem 1.1 was also proved in Proposition 1.6 in [5] . The case (3) is a generalization of Theorem 1.2 in [5] where Ginzburg andG¨urel proved generic Conley conjecture for N ≥ n + 1 . The proof of Proposition1.6 in [5] was an application of Birkhoff-Moser fixed point theorem and the proof ofTheorem 1.2 in [5] was an application of ”resonance relation” proved in [9] . Ourproof of Theorem 1.1 is a modification of the former proof. 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 and Victor L. Ginzburg for checking the draft and giving mecomments and advices. 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 by 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 ) N THE GENERIC CONLEY CONJECTURE 3
It is straightforward to see that φ H ( k ) = ( φ H ) k . Let P l ( H ) be the space of l -periodiccontractible periodic orbits of X H . P l ( H ) = { x : S l → M | ˙ x ( t ) = X H t ( x ( t )) , x : contractible } 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 imageis zero, N is defined by N = + ∞ .2.2. Floer homology and degrees of periodic orbits.
In this subsectin, weexplain basic notations of Floer homology theory and Conley-Zehnder index ofperiodic orbit. Let H be a 1-periodic Hamiltonian function. We call H is non-degenerate if the differential map dφ H : T M x → T M x does not has 1 as an eigen-value for any fixed point x ∈ Fix( φ H ). Note that H is non-degenerate if and onlyif graph( φ H ) ⊂ M × M is transverse to the diagonal ∆ M ⊂ M × M .We construct Novikov covering of P ( H ) as follows. ^ P ( H ) = { ( u, x ) | x ∈ P ( H ) , u : D → M, ∂u = x } / ∼ where D is the two dimensional disc D ⊂ R and the equivalence relation ∼ isdefined as follows. ( u, x ) ∼ ( v, y ) ⇐⇒ x = yω ( u♯v ) = 0 c ( u♯v ) = 0Here v is the disc with the opposite orientation on the domain and u♯v is theglued sphere. Each [ u, x ] ∈ ^ P ( H ) has a Conley-Zehnder index µ CZ ([ u, x ]) ∈ Z . Wenormalize µ CZ so that Conley-Zehnder index of a local maximum of a C -smallMorse function is equal to n . Conley-Zehnder index gives a grading of Floer chaincomplex and Floer homology. We also have the action functional A H on ^ P ( H ) asfollows. A H ([ u, x ]) = − Z D u ∗ ω + Z H ( t, x ( t )) dt Then Floer chain complex CF ∗ ( H ) is defined as follows. YOSHIHIRO SUGIMOTO CF ∗ ( H ) = (cid:26) X z ∈ ^ P ( H ) a z · z (cid:12)(cid:12)(cid:12)(cid:12) a z ∈ Q , ∀ C ∈ R , ♯ { z ∈ e P ( H ) | a z = 0 , A H ( z ) > C } < ∞ (cid:27) The boundary operator d F has the following form. d F ( z ) = X w ∈ ^ P ( H ) n ( z, w ) w The coefficient n ( z, w ) ∈ Q is the number of solutions of the following Floer equationmodulo the natural R -action ([3, 11]). Let J t be an almost complex structure on M parametrized by t ∈ S . z = [ v − , x − ] , w = [ v + , x + ] 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 ) = x + ( t ) , ( v − ♯u, x + ) ∼ ( v + , x + )Floer homology HF ∗ ( H ) is the homology of the chain complex ( CF ∗ ( H ) , d F ).We introduce the notion of Novikov ring of ( M, ω ). We define an abelian groug Γby Γ = π ( M )Ker ω ∩ Ker c where ω : π ( M ) → R is the integration of the symplectic form ω and c : π ( M ) → Z is the integration of the first Chern class. We define the degree of u ∈ Γ by − c ( u ).Novikov ring Λ ( M,ω ) is defined by the set of possibly infinite sums of Γ with suitableconvergence as follows.Λ ( M,ω ) = (cid:26) X u ∈ Γ a u · u (cid:12)(cid:12)(cid:12)(cid:12) a u ∈ Q , ∀ C ∈ R , ♯ { u ∈ Γ | a u = 0 , ω ( u ) < C } < ∞ (cid:27) Then Floer homology is isomorphic to the singular homology group with Novikovring coefficient ([3, 11]). HF ∗ ( H ) ∼ = H ∗− n ( M : Q ) ⊗ Λ ( M,ω ) Generic Conley conjecture
We prove Theorem 1.1 in this section. Throughout this section, we assumethat (
M, ω ) is a 2 n -dimensional closed symplectic manifold with minimum Chernnumber N and it also satisfies at least one of the following conditions.(1) n is odd.(2) H odd ( M : Q ) = 0(3) N > X ⊂ Ham(
M, ω ) and a familyof subsets { Y k ⊂ Ham(
M, ω ) } (1 ≤ k < + ∞ ) which satisfy the following conditions. • X ⊂ Ham(
M, ω ) is a C ∞ -dense subset. • Y k ⊂ Ham(
M, ω ) are C ∞ -open dense subsets. N THE GENERIC CONLEY CONJECTURE 5 • Any element of X has infinitely many simple contractible periodic orbits. • X = T ∞ k =1 Y k holds.The above conditions imply that X is a C ∞ -residual subset of Ham( M, ω ) andgenerically Hamiltonian diffeomorphisms have infinitely many simple contractibleperiodic orbits.As in [5], our proof is based on applications of Birkhoff-Moser fixed point theo-rem (local theory) and Floer homology theory (global theory). Roughly speaking,Birkhoff-Moser fixed point theorem guarantees infinitely many periodic orbits of asymplectic map near non-hyperbolic fixed points which satisfies some generic con-ditions. For the reader’s convenience, we briefly recall the statement and propertiesof Birkhoff-Moser fixed point theorem.
Theorem 3.1 (Birkhoff-Moser fixed point theorem [12]) . Let φ be s symplectic mapdefined in an open neighborhood of the origin ( = p ) in ( R n , ω ) and the origin is afixed point of φ . Here ω is the standard symplectic form P ni =1 x i ∧ y i on R n . Let λ , · · · , λ m , λ − , · · · , λ − m be the all eigenvalues of the differential map dφ p : T p M −→ T p M on the unit circle in C . Assume that φ satisfies the following conditions. (1) m ≥ Q mk =1 λ j k k = 1 for ≤ P mk =1 | j k | ≤ The Taylor coefficient of φ up to order satisfies a non-degenerate condi-tion.Then φ possesses infinitely many periodic orbits in any neighborhood of p . The meaning of ”non-degenerate” in (3) is difficult to state briefly because itsmeaning becomes clear in the proof of the theorem. We just introduce an exampleof ”non-degenerate” condition.
Example 3.1 (non-degeneracy condition [12]) . Let φ be a symplectic map definedin a open neighborhood of the origin in ( R n , ω ) and the origin is a fixed point of φ . Assume that φ can be written in the following form . φ (( x , · · · , x n , y , · · · , y n )) = ( x (1)1 , · · · , x (1) n , y (1)1 , · · · , y (1) n ) x (1) k = x k cos Φ k − y k sin Φ k + f k y (1) k = x k sin Φ k + y k cos Φ k + f k + n Φ k = α k + P nl =1 β kl ( x l + y l ) The error terms f k are assumed to have vanishing derivatives up to order 3 at theorigin. Then non-degeneracy means that the matrix ( β kl ) is non-singular. Moser first proved Birkhoff-Moser fixed point theorem for the above special case.Then he proved that general cases can be reduced to this special case. So roughlyspeaking, ”non-degenerate” means that it can be reduced to the above form so thatthe matrix ( β kl ) is non-singular.Birkhoff-Moser fixed point theorem (and its proof in [12]) implies the followingfact. Let x ∈ P ( H ) be a non-degenerate contractible periodic orbit of a Hamiltonianfunction H ∈ C ∞ ( S × M ). We also assume that there is at least one eigenvalueof the differential map dφ H : T x (0) M −→ T x (0) YOSHIHIRO SUGIMOTO on the unit circle and all eigenvalues on the unit circle are pairwise distinct.Thenwe can perturb H to e H near x so that it satisfies the all required conditions inthe statement of Birkhoff-Moser fixed point theorem. Moreover, these conditionsare satisfied in a sufficiently small open neighborhood of φ e H and hence all of thempossesses infinitely many simple contractible periodic orbits.We apply this observation to our proof of Theorem 1.1. Let H sn ⊂ Ham(
M, ω )be the set of strongly non-degenerate Hamiltonian diffeomorphisms (strongly non-degenerate means any iteration of it is non-degenerate). We divide H sn into thefollowing three pairwise disjoint subsets. H (1) sn = ( φ ∈ H sn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) the number of simple contractible periodic orbits is finiteand all contractible periodic orbits are hyperbolic ) H (2) sn = ( φ ∈ H sn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) the number of simple contractible periodic orbits is finiteand at least one contractible periodic orbit is non-hyperbolic ) H (3) sn = ( φ ∈ H sn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ possesses infinitely many simple contractible periodic orbits ) First we prove that H (1) sn is empty. We fix φ ∈ H (1) sn and let { x i , · · · , x l } be theset of all simple contractible periodic orbits of φ and let p , · · · , p l ∈ N be theirperiods. We also choose a common multiple k of p , · · · , p l . Then all periodicorbits of ψ ′ = φ ( k ) are 1-periodic orbits and all of them are hyperbolic. For anycapped periodic orbit ¯ z , we have the equation µ CZ (¯ z ) = ∆ ψ ′ (¯ z )where ∆ ψ ′ (¯ z ) is the mean index [13]. This implies that µ CZ (¯ z m ) = mµ CZ (¯ z )holds for any k ∈ N . For the iteration ψ = ψ ′ (2 N ) and any capped periodic orbit of ψ , the same equation µ CZ (¯ z ) = ∆ ψ (¯ z ) holds. Let { y , · · · , y l } be all contractible pe-riodic orbits of ψ . Note that they are 2 N -times iterations of { x ( kp )1 , · · · , x ( kpl ) l } . Thismeans that any capped periodic orbit ¯ z of { y , · · · , y l } has mean index ∆ ψ (¯ z ) = 2 N m ( m ∈ Z ). So we can choose a capping ¯ y i of y i ( i = 1 , · · · , l ) so that µ CZ ( ¯ y i ) = 0is satisfied. This implies that Conley-Zehnder index of any capped periodic orbit isdivided by 2 N and HF odd ( ψ ) = 0 holds. If n is an odd integer, this is a contradictionbecause HF n ( ψ ) = 0 holds. So, H (1) sn is empty if n is odd.Next, assume that H odd ( M : Q ) = 0. Without loss of generality, we assumethat n is an even integer. Then the isomorphism HF ∗ ( ψ ) ∼ = H ∗− n ( M : Q ) ⊗ Λ ( M,ω ) implies that there is at least one capped periodic orbit of ψ whose Conley-Zehnderindex is odd. This is a contradiction. So H (1) sn is empty if H odd ( M : Q ) = 0 holds.Assume that N > n is even.Note that H n +2 ( M : Q ) = 0 holds in this case. This implies HF ( ψ ) = 0, but thisis impossible because Conley-Zehnder index of any capped periodic periodic orbitcan be divided by 2 N . So we have proved that H (1) sn is empty in all cases. N THE GENERIC CONLEY CONJECTURE 7
Next we fix φ ∈ H (2) sn . Let { x , · · · , x l } be the set of all simple periodic orbitsand let p , · · · , p l be their periods. We divide { x , · · · , x l } into hyperbolic periodicorbits and non-hyperbolic periodic orbits. Let { x , · · · , x l ′ } be the set of all non-hyperbolic periodic orbits. We perturb φ to e φ so that all differential maps( dφ ) p i : T x i (0) M −→ T x i (0) M have 2 n pairwise distinct eigenvalues (see the arguments in the proof of Lemma7.1.5 in [1]). The perturbed e φ may not be strongly non-degenerate and perturbedperiodic orbits { f x , · · · , f x l ′ } may not be non-hyperbolic. We prove that at leastone e x i is non-hyperbolic. Note that e φ may have simple contractible periodic orbitsmore than l , but we can assume that the period of ”new” periodic orbit is muchgreater than 2 N k . So the existence of ”new” periodic orbits does not influence ourarguments. Assume that all e x i are hyperbolic. As in the proof of H (1) sn = ∅ , we fix ψ = e φ (2 N × k ) where k is a common multiple of p , · · · , p l . Then each periodic orbit e x i (2 N × kpi ) of ψ has a capping ¯ z i so that µ CZ ( ¯ z i ) = 0. This is a contradiction as inthe proof of H (1) sn = ∅ . So at least one of e x i is non-hyperbolic periodic orbit.Let f x be a non-hyperbolic periodic orbit. We can perturb e φ so that f x satis-fies all required conditions in the statement of Birkhoff-Moser fixed point theorem.These arguments imply that we can choose a sequence { φ k } k ∈ N and open neigh-borhoods W k of φ k which satisfy the following conditions. • φ k −→ φ in C ∞ -topology • Any element of W k satisfies all required conditions in the statement ofBirkhoff-Moser fixed point theorem and hence possesses infinitely manysimple contractible periodic orbits.We define V ( φ ) and V ( k ) ( φ ) ( k ∈ N ) for φ ∈ H (2) sn as follows. V ( φ ) = ∞ [ i =1 W i V ( k ) ( φ ) = V ( φ )Next we fix φ ∈ H (3) sn . There are the following two possibilities.(1) There is an open neighborhood U of φ (in Ham( M, ω )) such that U ∩ H sn ⊂ H (3) sn holds.(2) There is no open neighborhood U as above. In other words, we can choosea sequence { φ k } k ∈ N ⊂ H (2) sn such that φ k → φ holds.In the case of (2), we define V ( φ ) and V ( k ) ( φ ) ( k ∈ N ) as follows. V ( φ ) = ∞ [ k =1 V ( φ k ) V ( k ) ( φ ) = V ( φ )In the case of (1), we define V ( φ ) and V ( k ) ( φ ) ( k ∈ N ) as follows. YOSHIHIRO SUGIMOTO V ( φ ) = U ∩ H sn V ( k ) ( φ ) = { ψ ∈ U | ψ, · · · , ψ k are non-degenerate } Then V ( k ) ( φ ) ⊂ U is open dense and V ( φ ) = T ∞ k =1 V ( k ) ( φ ) holds. We can define X and Y k as follows. X = [ φ ∈H sn V ( φ ) Y k = [ φ ∈H sn V ( k ) ( φ ) X and { Y k } satisfy the required conditions and we proved Theorem 1.1. (cid:3) References [1] M. Audin, M. Damian.
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