Another look at the Hofer-Zehnder conjecture
aa r X i v : . [ m a t h . S G ] S e p ANOTHER LOOK AT THE HOFER–ZEHNDER CONJECTURE
ERMAN ÇİNELİ, VIKTOR L. GINZBURG, AND BAŞAK Z. GÜREL
Abstract.
We give a different and simpler proof of a slightly modified (andweaker) variant of a recent theorem of Shelukhin extending Franks’ “two-or-infinitely-many” theorem to Hamiltonian diffeomorphisms in higher dimensionsand establishing a sufficiently general case of the Hofer–Zehnder conjecture. Afew ingredients of our proof are common with Shelukhin’s original argument,the key of which is Seidel’s equivariant pair-of-pants product, but the new proofhighlights a different aspect of the periodic orbit dynamics of Hamiltoniandiffeomorphisms.
Dedicated to Claude Viterbo on the occasion of his 60th birthday
Contents
1. Introduction and main results 11.1. Introduction 11.2. Shelukhin’s theorem 32. Preliminaries 52.1. Conventions and notation 52.2. Equivariant Floer cohomology and the pair-of-pants product 73. Floer graphs 93.1. Main result 93.2. Implications and the proof of Theorem 1.1 104. A few words about the shortest bar 115. Proof of theorem 3.1 and further remarks 135.1. Proof of theorem 3.1 135.2. Degenerate case 16References 191.
Introduction and main results
Introduction.
In this paper we give a different and simpler proof of a slightlymodified and weaker version of a recent theorem of Shelukhin, [Sh19a], extendingFranks’ “two-or-infinitely-many” theorem, [Fr92, Fr96], to higher dimensions.This celebrated theorem of Franks asserts that every area preserving diffeomor-phism of S has either exactly two or infinitely many periodic points. (Moreover, Date : September 29, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Periodic orbits, Hamiltonian diffeomorphisms, Frank’s theorem, equi-variant Floer cohomology, pseudo-rotations.The work is partially supported by NSF CAREER award DMS-1454342 (BG) and by SimonsFoundation Collaboration Grant 581382 (VG). in the setting of Franks’ theorem, there are also strong growth rate results; see, e.g,[FH, LeC, Ke].) A generalization of Franks’ theorem conjectured in [HZ, p. 263]is that a Hamiltonian diffeomorphism ϕ of a closed symplectic manifold has infin-itely many periodic points whenever ϕ has “more than absolutely necessary” fixedpoints. (Hence, the title of [Sh19a] and of this paper.) The vaguely stated lowerbound “more than absolutely necessary” is usually interpreted as a lower boundarising from some version of the Arnold conjecture, e.g., as the sum of the Bettinumbers. For CP n , the expected threshold is n + 1 regardless of the non-degeneracyassumption and, in particular, it is for S = CP as in Franks’ theorem. A slightlydifferent interpretation of the conjecture, not directly involving the count of fixedpoints, is that the presence of a fixed or periodic point that is unnecessary from ahomological or geometrical perspective is already sufficient to force the existenceof infinitely many periodic points. We refer the reader to [GG14, Gü13, Gü14] forsome results in this direction.We note that whenever ϕ has finitely many periodic points, by passing to aniterate one can assume them to be fixed points. Furthermore, ϕ has infinitely manyperiodic points if and only if it has infinitely many simple, i.e., un-iterated, periodicorbits and the results are often stated in these terms. It is also worth keeping inmind that all known Hamiltonian diffeomorphisms ϕ with finitely many periodicorbits are strongly non-degenerate, i.e., ϕ k is non-degenerate for all k ∈ N .Volume preserving diffeomorphisms or flows with finitely many simple periodicorbits play an important role in dynamics; see, e.g., [FK] and references therein.In the Hamiltonian setting they are sometimes referred to as pseudo-rotations. Re-cently, symplectic topological methods have been employed to study the dynamicsof pseudo-rotations and its connections with symplectic topological properties ofthe underlying manifold in all dimensions; see [AS, Ban, Br15b, Br15a, ÇGG19,ÇGG20, GG18a, LRS, Sh19b, Sh19c].The original proof of Franks’ theorem utilized methods from low-dimensionaldynamics, and the first purely symplectic topological proof was given in [CKRTZ].However, that proof and also a different approach from [BH] were still strictlylow-dimensional, and Shelukhin’s theorem, [Sh19a, Thm. A], is the first sufficientlygeneral higher-dimensional variant of Franks’ theorem. (Strictly speaking, [Sh19a,Thm. A] and our Theorem 1.1 and Corollary 1.2, which are overall slightly weaker,still fall short of completely reproving Franks’ theorem in dimension two; we willdiscuss and compare these results in Section 1.2.) Similarly to [Sh19a], the keyingredient of our proof is Seidel’s Z -equivariant pair-of-pants product, [Se]. (Whilewe use the original version of the product, [Sh19a] relies on its Z p -equivariantversion from [ShZa].) Our proof also uses several simple ingredients from persistenthomology theory in the form developed in [UZ] (see also [PS]), although to a muchlesser degree than [Sh19a].Finally, it is worth pointing out that Hamiltonian pseudo-rotations are extremelyrare and most of the manifolds do not admit such maps. This statement is known asthe Conley conjecture. The state of the art result is that the Conley conjecture holdsfor a manifold ( M, ω ) unless there exists A ∈ π ( M ) such that h c ( T M ) , A i > and h ω, A i > ; see [Çi, GG17]. For example, the Conley conjecture holds when c ( T M ) | π ( M ) = 0 or when M is negative monotone. For many manifolds theconjecture is also known to hold C ∞ -generically (see [GG09]); we refer the readerto [GG15] for a detailed survey and further references. NOTHER LOOK AT THE HOFER–ZEHNDER CONJECTURE 3
Shelukhin’s theorem.
Let ϕ be a Hamiltonian diffeomorphism of a closedmonotone symplectic manifold M . We view ϕ as the time-one map in a time-dependent Hamiltonian flow and denote by P k ( ϕ ) the set of its k -periodic points,arising from contractible k -periodic orbits. The Hamiltonian diffeomorphism ϕ issaid to be k -perfect if P k ( ϕ ) = P ( ϕ ) and perfect if ϕ is k -perfect for all k ∈ N . (We refer the reader to Sections 2 for further notation and definitions usedhere.) We call ϕ a non-degenerate pseudo-rotation over a field F if it is non-degenerate, perfect and the differential in the Floer complex of ϕ over F vanishes.This condition is independent of the choice of an almost complex structure and,by Arnold’s conjecture, equivalent to that the number of 1-periodic orbits |P ( ϕ ) | is equal to the sum of Betti numbers of M over F . Denote by β ( ϕ ) the barcodenorm of ϕ over F , i.e., the length of the maximal finite bar in the barcode of ϕ ; seeSection 4.One of the goals of this paper is to give a simple proof to the following theoremproved in a slightly different form in [Sh19a]. Theorem 1.1 (Shelukhin’s Theorem, [Sh19a]) . Assume that ϕ is strongly non-degenerate and perfect and that β ( ψ ) over F := Z is bounded from above for allHamiltonian diffeomorphisms ψ of M or at least for all iterates ψ = ϕ k (e.g., M = CP n ). Then ϕ is a pseudo-rotation. Applying this to the iterates ϕ k we obtain Corollary 1.2 ([Sh19a]) . Assume that ϕ is strongly non-degenerate, β (cid:0) ϕ k (cid:1) over F is bounded from above (e.g., M = CP n ), and |P ( ϕ ) | is strictly greater than thesum of Betti numbers of M over F . Then (cid:12)(cid:12) P k ( ϕ ) (cid:12)(cid:12) → ∞ as k → ∞ . This theorem is proved in Section 3.2 as an easy consequence of Theorem 3.1,a new result in this paper. (However, at least on the conceptual level, our proofof that theorem is also a subset of Shelukhin’s argument, although the inclusion israther implicit.)In the rest of this section we discuss the conditions of Theorem 1.1 and alsosome of the differences between Corollary 1.2 and the original Shelukhin’s theorem,[Sh19a, Thm. A], which is in several ways more general and more precise.First of all, the coefficient field in [Sh19a, Thm. A] is Q rather than F andthe assertion is that P p ( ϕ ) contains a simple periodic orbit for every large prime p . As a consequence, one obtains the growth of order at least O ( k/ log k ) for thenumber of simple periodic orbits of period up to k . This difference stems from thefact that the main tool used in [Sh19a] is the Z p -equivariant pair-of-pants productintroduced in [ShZa] while we rely on a somewhat simpler Z -equivariant pair-of-pants product defined in [Se]. We touch upon the p -iterated analogues of Theorem1.1 and Corollary 1.2 in Remark 5.5.Secondly, [Sh19a, Thm. A] allows for some degeneracy of ϕ . Namely, in thesetting of Corollary 1.2, the number of 1-periodic orbits |P ( ϕ ) | in the conditionthat |P ( ϕ ) | is strictly greater than the sum of Betti numbers is replaced by X x ∈P ( ϕ ) dim F HF( x ; F ) , (1.1)where HF( x ; F ) is the local Floer (co)homology of x with coefficients in a field F (see, e.g., [GG10]); F = Q in [Sh19a]. Note that, as a consequence, Corollary 1.2 ERMAN ÇİNELİ, VIKTOR GINZBURG, AND BAŞAK GÜREL still holds without the non-degeneracy assumption, provided that the number of1-periodic orbits with
HF( x ; F ) = 0 is greater than the sum of Betti numbers. Inthe setting of this paper, one should take F = F and we will further discuss thedegenerate case of Theorem 1.1 and Corollary 1.2 in Section 5.2. Overall, the roleof the condition that HF( x ; F ) = 0 is unclear to us beyond the case of S . Franks’theorem has an analogue for a certain class of symplectomorphisms of surfaces andthen, interestingly, this condition becomes essential; see [Bat, GG09].However, from our perspective, the most important difference lies in the proofs,which highlight different aspects of the dynamics and Floer theory of ϕ . Our prooffocuses on the behavior of the shortest bar β min in the barcode of ϕ (rather thanthe longest bar β ≥ β min ) or, to be more precise, of the shortest Floer arrow underthe iteration from ϕ to ϕ ; see Section 3.1. In particular, we show in Theorem3.1 that when ϕ is 2-perfect the shortest arrow persists under such an iteration,although it may migrate into the equivariant domain for ϕ , and the length of thearrow doubles. The shortest non-equivariant arrow for ϕ is at least as long asthe equivariant one. Hence β min (cid:0) ϕ (cid:1) ≥ β min ( ϕ ) , and Theorem 1.1 readily followsfrom Theorem 3.1 applied to a sequence of period doubling iterations; see Section3.2. The key ingredient in the proof of Theorem 3.1 is the equivariant pair-of-pants product, introduced in [Se], having a very strong non-vanishing property alsoproved therein (see Proposition 2.3).Finally, a few words are due on the requirement in Theorem 1.1 and Corollary1.2 that β ( ψ ) is bounded from above. First of all, note that while it would besufficient to only have an upper bound on β min ( ψ ) where ψ = ϕ k or, as in [Sh19a,Thm. A], on β ( ψ ) where ψ = ϕ p , all relevant results proved to date are more robustand give an upper bound on β ( ψ ) for all ψ . (This is the curse (and the blessing)of symplectic topological methods in dynamics: they are very robust and general,but not particularly discriminating; they often tell the same thing about all maps.There are, however, exceptions.)The simplest manifold for which such an a priori bound is established is CP n for any coefficient field (suppressed in the notation), and the result essentially goesback to [EP]. The argument is roughly as follows. (We use here the notation andconventions from Section 2.1.) First recall that β ( ψ ) ≤ γ ( ψ ) . (1.2)Here γ ( ψ ) is the γ -norm of ψ defined, using cohomology, as γ ( ψ ) = − (cid:0) c ( ψ ) + c ( ψ − ) (cid:1) , where c α ( ψ ) is the spectral invariant associated with a quantum cohomology class α ∈ HQ( M ) and is the unit in the ordinary cohomology H( M ) of M . (We suppressthe grading in the cohomology notation when it is irrelevant.) The upper bound(1.2) holds for any closed monotone symplectic manifold and its proof is similar tothe proof in [Us] of the upper bound for β by the Hofer norm, but with continuationmaps replaced by the multiplications by the image of in HF( ψ ) and HF (cid:0) ψ − (cid:1) .(We refer the reader to [KS] for some further results along these lines.) Applyingthe Poincaré duality in Floer cohomology (see [EP]), it is not hard to show that c ( ψ − ) = − c ̟ ( ψ ) when N ≥ n + 1 , where ̟ is the generator of H n ( M ) and N is the minimal Chern number of M n . In particular, this is true for M = CP n since then N = n + 1 . By construction, for any two classes α and ζ in HQ( M ) thespectral invariants satisfy the Lusternik–Schnirelmann inequality c α ∗ ζ ( ψ ) ≥ c α ( ψ ) . NOTHER LOOK AT THE HOFER–ZEHNDER CONJECTURE 5
Thus, from the identity ̟ ∗ ζ = q where ζ is the generator of HQ ( CP n ) , weconclude that c ( ψ ) ≤ c ̟ ( ψ ) ≤ c ( ψ ) + π . These inequalities, combined with (1.2),show that β ( ψ ) ≤ γ ( ψ ) ≤ π for any Hamiltonian diffeomorphism ψ of CP n .A similar upper bound on β holds for all closed monotone manifolds M suchthat HQ even ( M ; F ) for some field F is semi-simple, i.e., splits as an algebra into adirect sum of fields. This is [Sh19a, Thm. B] and, interestingly, this result bypassesthe upper bound (1.2) in its original form. In fact, HQ( S × S ; Q ) is semi-simple,but γ is not bounded from above for S × S ; see [Sh19a, Rmk. 7] and also [RP,Thm. 6.2.6]. We are not aware of any algebraic criteria for an a priori bound onthe γ -norm. Nor do we know how large the class of monotone symplectic manifoldswith semi-simple HQ even ( M ; F ) is. In addition to CP n (with any F ), the complexGrassmannians, S × S , and the one point blow-up of CP with standard monotonesymplectic structures are in this class when c har F = 0 (see [EP] and referencestherein); but S × S is not for F = F . Acknowledgements.
The authors are grateful to Egor Shelukhin for useful dis-cussions. 2.
Preliminaries
Conventions and notation.
For the reader’s convenience we set here ourconventions and notation and briefly recall some basic definitions. The reader maywant to consult this section only as needed.Throughout this paper, the underlying symplectic manifold ( M, ω ) is assumedto be closed and strictly monotone, i.e., [ ω ] | π ( M ) = λc ( T M ) | π ( M ) = 0 for some λ > . The minimal Chern number of M is the positive generator N of the subgroup h c ( T M ) , π ( M ) i ⊂ Z and the rationality constant is the positive generator λ =2 N λ of the group h ω, π ( M ) i ⊂ R .A Hamiltonian diffeomorphism ϕ = ϕ H = ϕ H is the time-one map of the time-dependent flow ϕ t = ϕ tH of a 1-periodic in time Hamiltonian H : S × M → R , where S = R / Z . The Hamiltonian vector field X H of H is defined by i X H ω = − dH . Inwhat follows, it will be convenient to view Hamiltonian diffeomorphisms togetherwith the path ϕ tH , t ∈ [0 , , up to homotopy with fixed end points, i.e., as elementsof the universal covering of the group of Hamiltonian diffeomorphisms.Let x : S → M be a contractible loop. A capping of x is an equivalence classof maps A : D → M such that A | S = x . Two cappings of x are equivalent if theintegral of ω (or of c ( T M ) since M is strictly monotone) over the sphere obtainedby clutching the cappings is equal to zero. A capped closed curve ¯ x is, by definition,a closed curve x equipped with an equivalence class of cappings, and the presenceof capping is indicated by a bar.The action of a Hamiltonian H on a capped closed curve ¯ x = ( x, A ) is A (¯ x ) = − Z A ω + Z S H t ( x ( t )) dt. The space of capped closed curves is a covering space of the space of contractibleloops, and the critical points of A H on this space are exactly the capped 1-periodicorbits of X H . ERMAN ÇİNELİ, VIKTOR GINZBURG, AND BAŞAK GÜREL
The k -periodic points of ϕ are in one-to-one correspondence with the k -periodic orbits of H , i.e., of the time-dependent flow ϕ t . Recall also that a k -periodic orbitof H is called simple if it is not iterated. A k -periodic orbit x of H is said tobe non-degenerate if the linearized return map Dϕ k : T x (0) M → T x (0) M has noeigenvalues equal to one. A Hamiltonian H is non-degenerate if all its 1-periodicorbits are non-degenerate. We denote the collection of capped k -periodic orbits of H by ¯ P k ( ϕ ) .Let ¯ x be a non-degenerate capped periodic orbit. The Conley–Zehnder index µ (¯ x ) ∈ Z is defined, up to a sign, as in [Sa, SZ]. In this paper, we normalize µ sothat µ (¯ x ) = n when x is a non-degenerate maximum (with trivial capping) of anautonomous Hamiltonian with small Hessian.Fixing an almost complex structure, which will be suppressed in the notation,we denote by (CF( ϕ ) , d Fl ) and HF( ϕ ) the Floer complex and cohomology of ϕ over F = Z ; see, e.g., [MS, Sa]. (Throughout this paper, all complexes andcohomology groups are over F .) The complex CF( ϕ ) is generated by the capped1-periodic orbits ¯ x of H , graded by the Conley–Zehnder index, and filtered by theaction. The filtration level (or the action) of a chain ξ ∈ CF( ϕ ) is defined by A ( ξ ) = min {A (¯ x i ) } , where ξ = X ¯ x i . (2.1)(Note that the filtration depends on H , not just on ϕ , making of the notation CF( ϕ ) somewhat misleading.) The differential d Fl is the upward Floer differential:it increases the action and also the index by one. The Floer complex CF( ϕ ) is alsoa finite-dimensional free module over the Novikov ring Λ . There are several choicesof Λ ; see, e.g., [MS]. For our purposes, it is convenient to take the field of Laurentseries F ((q)) with | q | = 2 N as Λ . With this choice, Λ naturally acts on CF( ϕ ) byrecapping, and multiplication by q corresponds to the recapping by A ∈ π ( M ) with h c ( T M ) , A i = N . Furthermore, CF( ϕ ) is a finite-dimensional vector space over Λ with a preferred basis formed by 1-periodic orbits with arbitrarily fixed capping.Notationally, it is convenient to equip CF( ϕ ) with a non-degenerate F -valuedpairing h , i for which ¯ P ( ϕ ) is an orthogonal basis: h ¯ x, ¯ y i = δ ¯ x ¯ y . Then, essentiallyby definition, d Fl ¯ x = X h d Fl ¯ x, ¯ y i ¯ y. There is a canonical, grading-preserving isomorphism
HF( ϕ ) ∼ = −→ HQ( M )[ − n ] where HQ( M ) is the quantum cohomology of M ; see, e.g., [Sa, MS] and referencestherein. (Depending on the context, this is the PSS-isomorphism or the continua-tion map or a combination of the two.) The cohomology groups HQ( M ) and HF( ϕ ) are also modules over a Novikov ring Λ , and HQ( M ) ∼ = H( M ) ⊗ Λ ∼ = HF( ϕ ) (as amodule).The Floer complex carries a pairing CF( ϕ ) ⊗ CF( ϕ ) → CF (cid:0) ϕ (cid:1) [ n ] descending, on the level of cohomology, to the so-called pair-of-pants product HF( ϕ ) ⊗ HF( ϕ ) → HF (cid:0) ϕ (cid:1) [ n ] , which we denote by ∗ . Thus with our conventions | α ∗ β | = | α | + | β | + n . In quantumcohomology, this product corresponds to the quantum product , also denoted by ∗ ,which makes it into a graded-commutative algebra over Λ with unit . This productis a deformation (in q ) of the cup product: α ∗ β = α ∪ β + O (q) . NOTHER LOOK AT THE HOFER–ZEHNDER CONJECTURE 7
Equivariant Floer cohomology and the pair-of-pants product.
Equivariant Floer cohomology: a brief introduction.
The equivariant Floercohomology HF eq (cid:0) ϕ (cid:1) , introduced in [Se], is the homology of a certain complex (cid:0) CF eq (cid:0) ϕ (cid:1) , d eq (cid:1) called the equivariant Floer complex . As a graded F -vector spaceor as a Λ -module, CF eq (cid:0) ϕ (cid:1) = CF (cid:0) ϕ (cid:1) [h] where | h | = 1 , and the differential d eq has the form d eq = d Fl + h d + h d + . . . = d Fl + O (h) . This differential is Λ[ h ] -linear and non-strictly action-increasing. It is roughlyspeaking defined as follows, mimicking Borel’s construction of the Z -equivariantMorse cohomology.Fix a family ˜ J of 2-periodic in t almost complex structures on M parametrizedby the unit infinite-dimensional sphere S ∞ ⊂ R ∞ . Here R ∞ is the direct sum ofinfinitely many copies of R , i.e., its elements ξ = ( ξ , ξ , . . . ) have only finitelymany non-zero components, and S ∞ = {k ξ k = 1 } with k ξ k = P k | ξ k | . Thealmost complex structure ˜ J is required to satisfy the symmetry condition ˜ J − ξ = ˜ J ′ ξ ,where ˜ J ′ ξ is obtained from ˜ J ξ by the time-shift t t + 1 . Consider the self-indexingquadratic form f ( ξ ) = P k k | ξ k | on S ∞ and an antipodally symmetric metricsuch that the natural equatorial embedding S ∞ → S ∞ given by ( ξ , ξ , . . . ) (0 , ξ , . . . ) is an isometry. (Note also that the pull back of f by this embedding is f + 1 .) The almost complex structure ˜ J must furthermore be constant in ξ near thecritical points of f , invariant under the equatorial embedding, and satisfy a certainregularity requirement. Denote by w ± k the critical points of f of index k .Next, consider the hybrid Morse-Floer complex of A + f with respect to ˜ J andthe metric on S ∞ . This complex has pairs (¯ x, w ± k ) with ¯ x ∈ ¯ P ( ϕ ) as generators andcarries a natural Z -action, free on the generators, sending (¯ x, w ± k ) to (¯ x ′ , w −± k ) ,where ¯ x ′ is the time-shift of ¯ x . It is easy to see that the homology of this hybridcomplex is equal to HF( ϕ ) . By definition, CF eq (cid:0) ϕ (cid:1) is the Z -invariant part ofthis hybrid complex, where we write ¯ x h k for (¯ x, w + k ) + (¯ x ′ , w − k ) . The fact thatthe differential is h -linear follows from the requirement that f (up to a constant)and the auxiliary data are invariant under the equatorial embedding. Thus, inself-explanatory notation, d k ¯ x = X (cid:10) d k ¯ x, h k ¯ y (cid:11) ¯ y, where µ (¯ y ) = µ (¯ x ) + 1 − k and (cid:10) d k ¯ x, h k ¯ y (cid:11) counts mod 2 the total number of continuation Floer trajectoriesfrom ¯ x to ¯ y along gradient lines of f connecting w +0 to w + k and from ¯ x to ¯ y ′ alonggradient lines of f connecting w +0 to w − k . Clearly, the complex (and hence itscohomology) is filtered by the action A in addition to the filtration by A + f . Onthe level of (co)chains the filtration is defined similarly to (2.1), but with the powersof h ignored: A ( ξ ) = min {A (¯ x i ) } , where ξ = X h m i ¯ x i . The equivariant complex and the cohomology has natural continuation properties;see [Se].
Example . Assume that ϕ is 2-perfect and ϕ admits a regular 1-periodic almostcomplex structure J , i.e., for every pair ¯ x and ¯ y of 2-periodic orbits the space of ERMAN ÇİNELİ, VIKTOR GINZBURG, AND BAŞAK GÜREL
Floer trajectories connecting ¯ x to ¯ y has dimension µ (¯ y ) − µ (¯ x ) . In particular, thisspace is empty when µ (¯ y ) ≤ µ (¯ x ) , except when ¯ y = ¯ x and the space comprises oneconstant trajectory. Set ˜ J = J to be a constant (i.e., independent of ξ ) almostcomplex structure. Then ˜ J is also regular and d j = 0 for j ≥ since continua-tion trajectories for a constant homotopy are just Floer trajectories. Thus, in thiscase, HF eq (cid:0) ϕ (cid:1) = HF( ϕ )[h] for any interval of action. These conditions are met,for instance, when ϕ = ϕ H is generated by a C -small autonomous Hamiltonian H . As a consequence, for any ϕ the global cohomology HF eq (cid:0) ϕ (cid:1) is not a partic-ularly interesting object: it is simply isomorphic to HQ( M )[h] via the equivariantcontinuation (or the PSS map); see [Wi18a, Wi18b] for further details. Remark . One difference between ourdefinition of CF eq (cid:0) ϕ (cid:1) and the one in [Se] is that there CF eq (cid:0) ϕ (cid:1) = CF (cid:0) ϕ (cid:1) [[h]] ;for in that setting the expansion d eq ¯ x = P k h k d k ¯ x may have infinitely many non-vanishing terms. However, as already pointed out in [Se, Sect. 7], when M isstrictly monotone this expansion is necessarily finite. Indeed, otherwise it wouldinvolve capped orbits ¯ y ∈ ¯ P ( ϕ ) with arbitrarily small index µ (¯ y ) . However, dueto monotonicity and since P ( ϕ ) is finite, such orbits would eventually have actionstrictly smaller than that of ¯ x , which is impossible. This difference is essential forour proof as at some point in the argument we evaluate the elements of CF eq (cid:0) ϕ (cid:1) at h = 1 .2.2.2. Equivariant pair-of-pants product.
For our purposes, the most important fea-ture of the equivariant Floer complex is that it is the target space of the equivariantpair-of-pants product , also defined in [Se]. On the level of complexes this productis a chain map ℘ : C (cid:0) Z ; CF( ϕ ) ⊗ CF( ϕ ) (cid:1) → CF eq (cid:0) ϕ (cid:1) . The domain of ℘ is the group cochain complex C (cid:0) Z ; CF( ϕ ) ⊗ CF( ϕ ) (cid:1) := CF( ϕ ) ⊗ CF( ϕ )[h] with the differential d Z = d Fl + h( i d + τ ) . Here τ is the involution τ (¯ x ⊗ ¯ y ) = ¯ y ⊗ ¯ x and the first term is induced by the Floerdifferential on CF( ϕ ) ⊗ CF( ϕ ) . Note also that in these formulas and throughoutthe paper, all tensor products are over F unless specified otherwise. Furthermore,we distinguish between F and Z : the former is a field and the latter is a group.The equivariant pair-of-pants product is bilinear over Λ[h] and respects the actionfiltration. In particular, it can also be defined for a fixed action interval [ a, b ] inthe domain and [2 a, b ] in the target, but here we will not need the filtered versionof this construction. The map ℘ is a perturbation of the ordinary pair-of-pantsproduct: ℘ (¯ x ⊗ ¯ y ) = ¯ x ∗ ¯ y + O (h) , (2.2)and the O (h) part is again polynomial in h involving only finitely many terms(depending on ¯ x and ¯ y ).The cohomology of the domain of ℘ is the group cohomology H (cid:0) Z ; CF( ϕ ) ⊗ CF( ϕ ) (cid:1) of Z with coefficients in CF( ϕ ) ⊗ CF( ϕ ) . Thus, on the level of cohomology,the equivariant pair-of-pants product turns into a homomorphism H (cid:0) Z ; CF( ϕ ) ⊗ CF( ϕ ) (cid:1) ∼ = H (cid:0) Z ; HF( ϕ ) ⊗ HF( ϕ ) (cid:1) → HF eq (cid:0) ϕ (cid:1) . (2.3) NOTHER LOOK AT THE HOFER–ZEHNDER CONJECTURE 9 (The first isomorphism is a consequence of the fact that
CF( ϕ ) ⊗ CF( ϕ ) and HF( ϕ ) ⊗ HF( ϕ ) are equivariantly quasi-isomorphic.) The map (2.3) obviously killsthe h -torsion in the domain; it is a deformation in h of the standard pair-of-pantsproduct due to (2.2) and is closely related to a quantum deformation of the Steen-rod squares; see [Se, Wi18a, Wi18b] and also [ÇGG20] for a short introduction.The map (2.3) is a monomorphism modulo h -torsion; [Sh19a]. For symplecticallyaspherical manifolds, but not in the strictly monotone case, (2.3) is also onto andhence an isomorphism modulo h -torsion i.e., the kernel and the cokernel are torsionmodules; see [Se].On the level of complexes ℘ has the following extremely important feature: Proposition 2.3 (Seidel’s non-vanishing theorem; [Se], Prop. 6.7) . For every ¯ x ∈ ¯ P ( ϕ ) , we have ℘ (¯ x ⊗ ¯ x ) = h m ¯ x + . . . , (2.4) where ¯ x ∈ ¯ P ( ϕ ) is the second iterate of ¯ x and m = 2 µ (¯ x ) − µ (cid:0) ¯ x (cid:1) + n and thedots stand for a sum of capped orbits with action strictly greater than A (¯ x ) . This non-vanishing property points to a stark difference between the equivariantand non-equivariant pair-of-pants products: ¯ x ∗ ¯ x = ¯ x + . . . only when µ (cid:0) ¯ x (cid:1) =2 µ (¯ x ) + n , i.e., m = 0 in (2.4); cf. [ÇGG19]. Remark . A generalization of the equivariant pair-of-pants product to the p -thiterates ϕ p , where p is a prime, replacing Z by Z p and F by F p is constructed in[ShZa]. This construction and the analogue of Seidel’s non-vanishing theorem forthe p -th iterate plays a crucial role in the original proof of Shelukhin’s theorem in[Sh19a]; cf. Remark 5.5. 3. Floer graphs
Main result.
The key to the statement of our main result is the followingadmittedly naive and obvious construction which has been used, at least on aninformal level, for quite some time.Let ϕ be a non-degenerate Hamiltonian diffeomorphism of a closed monotonesymplectic manifold M . Consider the directed graph Γ( ϕ ) whose vertices are cappedfixed points of ϕ , and two vertices ¯ x and ¯ y are connected by an arrow (from ¯ x to ¯ y ) if and only if µ (¯ y ) = µ (¯ x ) + 1 and there is an odd number of Floer trajectoriesfrom ¯ x to ¯ y , i.e., h d Fl ¯ x, ¯ y i = 1 . The length of an arrow is the difference of actionsof ¯ y and ¯ x . We call Γ( ϕ ) the Floer graph of ϕ .When M is strictly monotone as is always assumed in this paper, the group Z actsfreely on Γ( ϕ ) by simultaneous recapping, preserving the arrow length. Sometimesit is convenient to consider the reduced Floer graph ˜Γ( ϕ ) := Γ( ϕ ) / Z . The lengthof an arrow in ˜Γ( ϕ ) is still well-defined. Note that, unless M is symplecticallyaspherical, both Γ( ϕ ) and ˜Γ( ϕ ) are infinite, but the latter has finitely many arrows.In particular, if d Fl = 0 , there exists a shortest arrow. Such an arrow might notbe unique, although it is unique for a generic ϕ , but obviously all shortest arrowshave the same length.The equivariant Floer graph Γ eq (cid:0) ϕ (cid:1) of ϕ is defined in a similar fashion. (Weare assuming that ϕ is non-degenerate, and hence ϕ is also non-degenerate.) Itsvertices are capped two-periodic orbits of ϕ . The vertices ¯ x and ¯ y are connectedby an arrow if and only if ¯ y enters d eq (¯ x ) with non-zero coefficient. In other words, now we do not require the index difference to be 1, and ¯ x and ¯ y are connected byan arrow if and only if ¯ x and h m ¯ y , where m = µ (¯ x ) − µ (¯ y ) + 1 , are connected by anodd number of equivariant Floer trajectories. The length of an arrow is again thedifference of actions. As in the non-equivariant case, the reduced equivariant Floergraph ˜Γ eq (cid:0) ϕ (cid:1) := ˜Γ eq (cid:0) ϕ (cid:1) / Z has only finitely many arrows, and hence the shortestarrows exist.We note that Γ (cid:0) ϕ (cid:1) and Γ eq (cid:0) ϕ (cid:1) (and their reduced counterparts) have the samevertices. Furthermore, since d eq = d Fl + O (h) , every arrow in Γ (cid:0) ϕ (cid:1) is also an arrowin Γ eq (cid:0) ϕ (cid:1) , i.e., the equivariant Floer graph is obtained from its non-equivariantcounterpart by adding arrows. Note that in the process the shortest arrow lengthcan only get shorter or remain the same. Also, observe that there is a naturalone-to-one map from the vertices of ˜Γ( ϕ ) to the vertices of ˜Γ (cid:0) ϕ (cid:1) sending ¯ x to ¯ x ;likewise for un-reduced graphs. However, even when ϕ is 2-perfect, this map is notonto unless M is symplectically aspherical.The main new result of the paper is the following theorem which relates theFloer graphs for ϕ and its second iterate ϕ . Theorem 3.1.
Assume that ϕ is 2-perfect and ϕ is non-degenerate. Then ¯ x and ¯ y are connected by one of the shortest arrows in Γ( ϕ ) if and only ¯ x and ¯ y areconnected by one of the shortest arrows in Γ eq (cid:0) ϕ (cid:1) . This theorem is proved in Section 5.1 after we recall in Section 4 a few relevantfacts about barcodes.
Remark . The Floer graph of ϕ de-pends on the choice of an almost complex structure J , and hence should ratherbe denoted by Γ( ϕ, J ) . Likewise, the equivariant Floer graph depends on theparametrized almost complex structure. However, in both cases, the collectionof shortest arrows is independent of this choice. This fact implicitly follows fromTheorem 3.1 or can be proved directly by a continuation argument.Note also that Floer graphs are stable under small perturbations of ϕ and J . Tobe more precise, Γ( ϕ, J ) = Γ( ˜ ϕ, ˜ J ) whenever ˜ ϕ is sufficiently close to ϕ and ˜ J isclose to J . The same is true in the equivariant setting.3.2. Implications and the proof of Theorem 1.1.
Theorem 3.1 shows thatwhen ϕ is perfect, the shortest arrow (or, to be more precise, every shortest arrow)persists from ϕ to ϕ , although in the process it might move to the equivariantdomain. This happens exactly when the difference of indices changes: µ (¯ y ) − µ (¯ x ) = 1 but µ (cid:0) ¯ y (cid:1) − µ (cid:0) ¯ x (cid:1) = 1 . Moreover, in this case, we necessarily have µ (cid:0) ¯ y (cid:1) − µ (cid:0) ¯ x (cid:1) < . On the other hand, if the difference of indices remains equalto one, the orbits continue to be connected by one of the shortest non-equivariantarrows.Denote by β min ( ϕ ) = A (¯ y ) − A (¯ x ) the length of a shortest arrow. As followsfrom Proposition 4.3, β min ( ϕ ) is exactly equal to the shortest bar in the barcodeof ϕ . Since every non-equivariant arrow for ϕ is also an equivariant arrow, theshortest equivariant arrow length β eq min (cid:0) ϕ (cid:1) for ϕ does not exceed β min (cid:0) ϕ (cid:1) , i.e., β eq min (cid:0) ϕ (cid:1) ≤ β min (cid:0) ϕ (cid:1) . In the setting of Theorem 3.1, β eq min (cid:0) ϕ (cid:1) = A (cid:0) ¯ y (cid:1) − A (cid:0) ¯ x (cid:1) = 2 β min ( ϕ ) . NOTHER LOOK AT THE HOFER–ZEHNDER CONJECTURE 11
We conclude that β min (cid:0) ϕ k (cid:1) ≤ β min (cid:0) ϕ k +1 (cid:1) as long as the iterates of ϕ remain perfect and non-degenerate, and hence k β min ( ϕ ) ≤ β min (cid:0) ϕ k (cid:1) . In particular, when ϕ is perfect, the longest finite bar β ( ϕ ) (and even the shortestbar) in the barcode cannot be bounded from above for the iterates of ϕ . This provesTheorem 1.1. Remark . An interesting question that arises from Theorem 3.1 is if a shortestarrow could persist in the non-equivariant domain for all iterates ϕ k , assumingthat ϕ is perfect. As discussed above, this would be the case if and only if µ (cid:0) ¯ y k (cid:1) − µ (cid:0) ¯ x k (cid:1) = 1 for all k ∈ N . Using a slightly simplified version of the index divisibilitytheorem from [GG18b] one can show that this is impossible when ϕ is replaced bya suitable iterate ϕ m . (This is non-obvious.) Passing to an iterate is apparentlyessential because there exist pairs of strongly non-degenerate elements A and B in f Sp(2 n ) such that µ (cid:0) A k (cid:1) − µ (cid:0) B k (cid:1) = 1 for all k = 0 , , , . . . .4. A few words about the shortest bar
In this section we recall a few facts, well-known to experts, about persistenthomology in the context of Hamiltonian Floer theory. All results discussed hereare contained in, e.g., [UZ], although in some instances implicitly and usually ina much more general setting. A reader sufficiently familiar with the material caneasily skip this section. There are, however, two points the reader might wantto keep in mind. Namely, our emphasis here is on the shortest bar rather thanthe longest finite bar (aka the boundary depth) which is more frequently used inapplications to dynamics. Secondly, our sign conventions are different from thosein [UZ] due to the fact that we are working with Floer cohomology.Consider the Floer complex C := CF( ϕ ) of a non-degenerate Hamiltonian diffeo-morphism ϕ of a strictly monotone symplectic manifold, equipped with the stan-dard action filtration. Clearly, C is a finite-dimensional vector space over Λ and thecollection of 1-periodic orbits of ϕ with fixed capping forms a basis of C .A finite set of vectors ξ i ∈ C is said to be orthogonal if for any collection ofcoefficients λ i ∈ Λ we have A (cid:0) X λ i ξ i (cid:1) = min A ( λ i ξ i ) . (Recall that with our conventions, A ( ξ ) := min A (¯ x i ) when ξ = X ¯ x i ; see (2.1).) It is not hard to show that an orthogonal set is necessarily linearlyindependent over Λ . Example . Assume that all capped 1-periodic orbits of ϕ have distinct actions.Write ξ i = ¯ x i + . . . , where the dots stand for the orbits with action strictly greaterthan ¯ x i . Then it is easy to see that the set ξ i is orthogonal if and only if the cappedorbits ¯ x i are distinct. Definition 4.2.
A basis B = { α i , η j , γ j } of C over Λ is said to be a singulardecomposition if • d Fl α i = 0 , • d Fl η j = γ j , • B is orthogonal.It is shown in [UZ, Sections 2 and 3] that C admits a singular decomposition.For the sake of brevity we omit the proof of this fact. In what follows we will orderthe pairs ( η j , γ j ) so that A ( γ ) − A ( η ) ≤ A ( γ ) − A ( η ) ≤ . . . . (4.1)This increasing sequence is usually referred to as the barcode of ϕ (or to be moreprecise the collection of finite bars). The maximal entry in the sequence is called the barcode norm β ( ϕ ) or the boundary depth, [Us]. The barcode is independent of thechoice of a singular decomposition (see, e.g., [UZ]), but here we do not use this fact.Instead, we need the following characterization of the shortest bar β min = β min ( ϕ ) : Proposition 4.3 ([UZ]) . Set β min := A ( γ ) − A ( η ) . Then β min = inf {A (¯ y ) − A (¯ x ) | h d Fl ¯ x, ¯ y i = 1 } (4.2) = inf {A ( d Fl ξ ) − A ( ξ ) | ξ ∈ C , ξ = 0 } . (4.3) Here, in the first equality, the infimum is taken over all capped 1-periodic orbits ¯ x and ¯ y such that ¯ y enters d Fl ¯ x with non-zero coefficient and, in the second, over allnon-zero ξ ∈ C . In particular, β min ( ϕ ) is the shortest arrow length in Γ( ϕ ) . Note that the infimums in (4.2) and (4.3) are actually attained and thus can bereplaced by minima, and that the proposition can be thought of as an analoguefor C of the Courant-Fischer minimax theorem giving a variational interpretationof the eigenvalues of a quadratic form. For the sake of completeness we include aproof of Proposition 4.3. Proof.
Let us denote the right-hand sides in (4.2) and (4.2) by β ′ min and, respec-tively, β ′′ min . We claim that β ′ min = β ′′ min . Indeed, setting ξ = ¯ x , in (4.3), it is easyto see that β ′′ min ≤ β ′ min . On the other hand, writing ξ = ¯ x + ¯ x + . . . in the orderof increasing action and d Fl ξ = P d Fl ¯ x i = ¯ y + . . . , we observe that h ¯ y, d Fl ¯ x i i = 1 for some i . Then A ( d Fl ξ ) − A ( ξ ) = A (¯ y ) − A (¯ x ) ≥ A (¯ y ) − A (¯ x i ) ≥ β ′ min , and thus β ′′ min ≥ β ′ min .Next, clearly, β min ≥ β ′′ min . Therefore, it remains to show that β min ≤ β ′′ min . Tothis end, let us decompose ξ in the basis B over Λ : ξ = X λ j η j + X λ ′ j γ j + X λ ′′ i α i . Then d Fl ξ = X λ j γ j . By orthogonality, A ( d Fl ξ ) = min A ( λ j γ j ) = A ( λ k γ k ) NOTHER LOOK AT THE HOFER–ZEHNDER CONJECTURE 13 for some k , and, again by orthogonality, A ( ξ ) ≤ min A ( λ j η j ) ≤ A ( λ k η k ) . Therefore, A ( d Fl ξ ) − A ( ξ ) ≥ A ( λ k γ k ) − A ( λ k η k )= A ( γ k ) − A ( η k ) ≥ A ( γ ) − A ( η ) = β min . As a consequence, β min ≤ β ′′ min , which finishes the proof of the proposition. (cid:3) Remark . In conclusion, we point out that all results in this section are purelyalgebraic and extend in a straightforward way to any un-graded finite-dimensionalcomplex over Λ with an “action filtration” having expected properties; see [UZ].5. Proof of theorem 3.1 and further remarks
Proof of theorem 3.1.
We begin by proving the theorem under the addi-tional background assumption that all actions and action differences for ϕ and ϕ are distinct modulo the rationality constant λ . Then, in the last step of the proof,we will show how to remove this extra assumption. Note that in particular thisassumption guarantees that the shortest arrow is unique for Γ( ϕ ) and Γ eq (cid:0) ϕ (cid:1) . Remark . It is worth pointing out that while this background assumption issatisfied C ∞ -generically, it is not quite innocuous in the context of pseudo-rotationsor perfect Hamiltonian diffeomorphisms. Indeed, in this case one can expect certain“resonance relations” between actions or actions and mean indices to hold; see[GK, GG09].The proof is carried out in three steps. Step 1: The shortest arrow for ϕ . In this step we simply apply the machinery fromSection 4 to
CF( ϕ ) . Let B = { α i , η j , γ j } be a singular decomposition for CF( ϕ ) over Λ ; see Definition 4.2. Due to the background assumption, the inequalities in(4.1) are strict: A ( γ ) − A ( η ) < A ( γ ) − A ( η ) < . . . . (5.1)Let us write γ = ¯ y ∗ + . . . and η = ¯ x ∗ + . . . , where dots stand for higher action terms, and ¯ x ∗ and ¯ y ∗ are unique by the back-ground assumption. Then, by definition, A ( γ ) = A (¯ y ∗ ) and A ( η ) = A (¯ x ∗ ) , and hence β min := A ( γ ) − A ( η ) = A (¯ y ∗ ) − A (¯ x ∗ ) . We claim that h d Fl ¯ x ∗ , ¯ y ∗ i = 1 . (5.2)Indeed, h d Fl ¯ x, ¯ y ∗ i = 1 for some ¯ x entering η . Then β min = A (¯ y ∗ ) − A (¯ x ∗ ) ≥ A (¯ y ∗ ) − A (¯ x ) ≥ β min . It follows that the first inequality is in fact an equality and ¯ x = ¯ x ∗ due to thebackground assumption. Therefore, by Proposition 4.3 and (5.2), ¯ x ∗ and ¯ y ∗ are connected by the shortestarrow in Γ( ϕ ) . Step 2: The shortest arrow for ϕ . In the previous step we have shown that ¯ x ∗ and ¯ y ∗ are connected by the shortest arrow in CF( ϕ ) . Our goal now is to prove thefollowing key fact. Lemma 5.2.
The iterated orbits ¯ x ∗ and ¯ y ∗ are connected by the shortest arrow in Γ eq (cid:0) ϕ (cid:1) . Since under the background assumption the shortest arrows in ˜Γ( ϕ ) and Γ eq (cid:0) ϕ (cid:1) are unique, this will establish the theorem. Proof of Lemma 5.2.
In the notation from Section 2.2, set ˆ α i = ℘ ( α i ⊗ α i ) , ˆ η j = h ℘ ( η j ⊗ η j ) + ℘ ( η j ⊗ γ j ) , ˆ γ j = ℘ ( γ j ⊗ γ j ) . Then, by Seidel’s non-vanishing theorem (Proposition 2.3), ˆ η = h m ¯ x ∗ + . . . and ˆ γ = h m ′ ¯ y ∗ + . . . for some m ≥ and m ′ ≥ , where the dots again stand for higher action terms.Since ℘ is a chain map, i.e., ℘ ◦ d Z = d eq ◦ ℘ , we have d eq ˆ α i = 0 and d eq ˆ η j = h ℘ ( γ j ⊗ η j ) + h ℘ ( η j ⊗ γ j )+ ℘ (h η j ⊗ γ j + h γ j ⊗ η j )+ ℘ ( γ j ⊗ γ j )= ˆ γ j . This indicates that the collection ˆ B := { ˆ α i , ˆ η j , ˆ γ j } can be thought of as a singulardecomposition of CF eq (cid:0) ϕ (cid:1) with the minimal bar given by A (ˆ γ ) − A (ˆ η ) = A (cid:0) ¯ y ∗ (cid:1) − A (cid:0) ¯ x ∗ (cid:1) , and, arguing similarly to Step 1, we should be able to show that ¯ x ∗ and ¯ y ∗ areconnected by the shortest arrow. A minor technical difficulty that arises at thisstage is that CF eq (cid:0) ϕ (cid:1) does not fit in with the algebraic framework of Section 4 or[UZ]. Namely, CF eq (cid:0) ϕ (cid:1) is not finite-dimensional over Λ ; it is finite-dimensionalover Λ[h] , but the latter is not a field. We circumvent this difficulty by a trickwhich essentially amounts to setting h = 1 . (This is the point where our choice ofworking with polynomials in h rather than formal power series as in [Se] is essential;cf. Remark 2.2.)Consider the ungraded complex ˜ C defined as follows: ˜ C := CF (cid:0) ϕ (cid:1) ⊂ CF eq (cid:0) ϕ (cid:1) as a vector space over Λ with the differential ˜ dα := d eq α | h=1 for α ∈ ˜ C . Since d eq is h -linear, we have ˜ d = 0 . More formally, ˜ C is the quotient complex in the shortexact sequence of ungraded complexes −→ CF eq (cid:0) ϕ (cid:1) −→ CF eq (cid:0) ϕ (cid:1) π −→ ˜ C −→ NOTHER LOOK AT THE HOFER–ZEHNDER CONJECTURE 15 over Λ , where π is the h = 1 evaluation map. Remark . This exact sequence, for any action interval, gives rise to the exacttriangle in Floer cohomology relating
H( ˜ C ) and HF eq (cid:0) ϕ (cid:1) via multiplication by .As any map of the form i d + O (h) , this multiplication map in Floer cohomology isone-to-one, and thus H( ˜ C ) ∼ = HF eq (cid:0) ϕ (cid:1) / (1 + h) HF eq (cid:0) ϕ (cid:1) , and hence dim F H( ˜ C ) = rk F [h] HF eq (cid:0) ϕ (cid:1) , for any action interval. For global co-homology, H( ˜ C ) ∼ = HF (cid:0) ϕ (cid:1) as ungraded Λ -modules by the continuation argumentand Example 2.1.Since, by construction, ˜ C is a finite-dimensional vector space over Λ , now themachinery from [UZ] applies literally; see Remark 4.4. In self-explanatory notation, h d eq ¯ z, h m ¯ z ′ i 6 = 0 where m = µ (¯ z ) − µ (¯ z ′ ) + 1 ⇐⇒ (cid:10) ˜ d ¯ z, ¯ z ′ (cid:11) = 0 for ¯ z and ¯ z ′ in ¯ P ( ϕ ) . Furthermore, we can also form the Floer graph for ˜ C andthis graph is identical to the equivariant Floer graph Γ eq (cid:0) ϕ (cid:1) . Claim 5.4.
The subset ˜ B := π ( ˆ B ) in ˜ C formed by ˜ α i := π (ˆ α i ) and ˜ η j := π (ˆ η j ) and ˜ γ j := π (ˆ γ j ) is a singular decomposition for ˜ C . Putting aside the proof of the claim, let us first show how Lemma 5.2 followsfrom it. Observe that A (˜ γ j ) − A (˜ η j ) = 2 (cid:0) A ( γ j ) − A ( η j ) (cid:1) . (5.3)Indeed, set η j = ¯ x j + . . . ,γ j = ¯ y j + . . . , where as usual the dots stand for strictly higher action terms. (Thus ¯ x ∗ = ¯ x and ¯ y ∗ = ¯ y .) By Seidel’s non-vanishing theorem (Proposition 2.3), we have ˆ η j = h m j ¯ x j + . . . , ˆ γ j = h m ′ j ¯ y j + . . . for some m j ≥ and m ′ j ≥ , and hence ˜ η j = ¯ x j + . . . , ˜ γ j = ¯ y j + . . . . Therefore, A (˜ γ j ) − A (˜ η j ) = A (cid:0) ¯ y j (cid:1) − A (cid:0) ¯ x j (cid:1) = 2 (cid:0) A (¯ y j ) − A (¯ x j ) (cid:1) = 2 (cid:0) A ( γ j ) − A ( η j ) (cid:1) , which proves (5.3).In particular, similarly to (5.1), we have A (˜ γ ) − A (˜ η ) < A (˜ γ ) − A (˜ η ) < . . . . Therefore, β min ( ˜ C ) := A (˜ γ ) − A (˜ η ) = A (cid:0) ¯ y ∗ (cid:1) − A (cid:0) ¯ x ∗ (cid:1) is the shortest bar for ˜ C . As in Step 1, we infer that (cid:10) ˜ d ¯ x ∗ , ¯ y ∗ (cid:11) = 1 . Hence there is an arrow connecting these two orbits in the Floer graph for ˜ C andthis is the shortest arrow. The Floer graph for ˜ C is defined similarly and in factidentical to the equivariant Floer graph Γ eq (cid:0) ϕ (cid:1) . Therefore, this arrow is also theshortest arrow in Γ eq (cid:0) ϕ (cid:1) , completing the proof of Lemma 5.2 modulo Claim 5.4. Proof of Claim 5.4.
Since π is a homomorphism of complexes, we have ˜ d ˜ α i = 0 and ˜ d ˜ η j = ˜ γ j . Therefore, we only need to show that ˜ B is an orthogonal basis. Forthis we do not need to distinguish between different types of elements of B . Write B = { ξ i } , where ξ i = ¯ z i + . . . with the dots denoting the entries of strictly higheraction. Then, by the definition of ˆ B and Seidel’s non-vanishing theorem, ˜ B = { ˜ ξ i } comprises the elements ˜ ξ i := π ( ˆ ξ i ) = ¯ z i + . . . . Now, as in Example 4.1, the orthogonality for B is equivalent to that the orbits ¯ z i are distinct. Similarly, the orthogonality for ˜ B is equivalent to that the orbits ¯ z i are again distinct. It follows that ˜ B is orthogonal if (in fact, iff) B is orthogonalwhich is a part of its definition. As a consequence, ˜ B is linearly independent over Λ .Finally, since ˜ C = CF (cid:0) ϕ (cid:1) as Λ -modules and ϕ is 2-perfect, we have dim Λ ˜ C = dim Λ CF (cid:0) ϕ (cid:1) = dim Λ CF( ϕ ) = |B| = | ˜ B| , and ˜ B is a basis. (cid:3) This concludes the proof of Lemma 5.2. (cid:3)
Step 3: Removing the background assumption.
Recall that the Floer graphs Γ( ϕ ) and Γ eq (cid:0) ϕ (cid:1) are stable under small perturbations of ϕ . With this in mind, we canreplace ϕ by a C ∞ -small perturbation ˜ ϕ meeting the background assumption, sincethe latter is a C ∞ -generic condition. More precisely, one can change the action ofa single orbit by a small amount (positive or negative) using a localized C ∞ -smallperturbation ˜ ϕ . Hence, given any arrow in the Floer graphs ˜Γ( ϕ ) and ˜Γ eq (cid:0) ϕ (cid:1) ,pick some small ǫ > . Then one can apply local perturbations at the two endsto shorten its length by ǫ while not changing the lengths of the remaining arrowsmore than ǫ . It follows that every shortest arrow in the Floer graphs ˜Γ( ϕ ) and ˜Γ eq (cid:0) ϕ (cid:1) can be perturbed into the unique shortest arrow. Now, Theorem 3.1 for ϕ follows from that theorem for ˜ ϕ . (cid:3) Remark Z p -equivariant analogue) . This argument extends with only veryminor changes to the p th iterates ϕ p , where p is a prime, proving the analogue ofTheorem 3.1 for Z p -equivariant cohomology of ϕ p over F p and relying on the resultsfrom [ShZa]; cf. Remark 2.4. As a consequence, as in the proof of Theorem 1.1, if ϕ is strongly non-degenerate, β is a priori bounded from above and |P ( ϕ ) | is greaterthan the sum of Betti numbers of M over Q , then there exists a simple p -periodicorbit for every sufficiently large prime p as is shown in [Sh19a].5.2. Degenerate case.
Perhaps, the simplest way to extend our arguments and, inparticular, Theorem 1.1 and Corollary 1.2 to include some degenerate Hamiltoniandiffeomorphisms as in [Sh19a] is by bypassing Theorem 3.1 and using a somewhatless precise argument. Below we outline the key steps of this generalization, some ofwhich again overlap with [Sh19a]. The account is deliberately brief. The main new
NOTHER LOOK AT THE HOFER–ZEHNDER CONJECTURE 17 point here is the construction of the (equivariant) Floer graph in the degeneratecase.Assume that ϕ is 2-perfect and that the second iteration is admissible: − is notan eigenvalue of Dϕ x for any x ∈ P ( ϕ ) . (The latter requirement is satisfied once ϕ is replaced by its sufficiently high iterate ϕ k .) Then, as shown in [GG10], forevery ¯ x ∈ ¯ P ( ϕ ) we have a canonical isomorphism in local Floer cohomology: HF(¯ x ) ∼ = −→ HF (cid:0) ¯ x (cid:1) (5.4)up to a shift of grading. By the Smith inequality in local Floer cohomology, whichcan be proved by exactly the same argument as in [Se] (see also [ÇG, Sh19a]),we have HF eq (cid:0) ¯ x (cid:1) ∼ = HF (cid:0) ¯ x (cid:1) [h] , where, strictly speaking, on the left we have thegraded module associated with the h -adic filtration of HF eq (cid:0) ¯ x (cid:1) . (We expect thatin this situation d eq = d Fl , and hence HF eq (cid:0) ¯ x (cid:1) ∼ = HF (cid:0) ¯ x (cid:1) [h] literally, withoutpassing to graded modules, but we have not been able to prove this.)For every ¯ x ∈ ¯ P ( ϕ ) , fix a basis ξ i, ¯ x in HF(¯ x ) so that this system of bases isrecapping-invariant. Applying (5.4) to this system, we obtain bases ξ ′ i, ¯ x in HF (cid:0) ¯ x (cid:1) with ¯ x ∈ ¯ P ( ϕ ) , and this system extends to a recapping-invariant system over theentire ¯ P ( ϕ ) .We also have a recapping-invariant system of bases in HF eq (cid:0) ¯ x (cid:1) arising from ℘ ( ξ i, ¯ x ⊗ ξ i, ¯ x ) ∈ HF eq (cid:0) ¯ x (cid:1) . To be more precise, it is convenient to replace theequivariant cohomology (local or global) by the homology of the ungraded complex ˜ C obtained by setting h = 1 as in the proof of Theorem 3.1. For the sake of brevity,we keep the notation HF eq for this cohomology suppressing the projection π in thenotation. Set ξ eq ¯ x,i := ℘ ( ξ i, ¯ x ⊗ ξ i, ¯ x ) . We claim that this is a basis in HF eq (cid:0) ¯ x (cid:1) whichis now just a vector space over F . Then, extending, we get a recapping invariantfamily of bases over ¯ P ( ϕ ) .To show that { ξ eq ¯ x,i } is indeed a basis, we first recall that, without changing Dϕ x and the local cohomology, ϕ can be deformed near x to the direct product ofdegenerate and totally non-degenerate maps; see [GG10, Sect. 4.5]. This essentiallyreduces the question to the case, which for the sake of brevity we will focus on,where x is totally degenerate, i.e., all eigenvalues of Dϕ x are equal to 1 and inparticular ϕ can be made C -close to the identity. Furthermore, recall that HF(¯ x ) ∼ =HF( ϕ f ) ∼ = HM( f ) by [Gi, Sect. 3.3 and 6], where HM stands for the local Morsecohomology, f is the generating function of ϕ and ϕ f is the germ of the Hamiltoniandiffeomorphism generated by f . These isomorphisms come from continuation mapsand there are similar isomorphisms (equivariant and non-equivariant) for ¯ x and ϕ f = ϕ f , where we can replace the generating function for ϕ by f ; see [GG10,Sect. 4.3]. Now, as in Example 2.1 and Remark 5.3, we arrive at the continuationmap identifications HF eq (cid:0) ¯ x (cid:1) ∼ = HF (cid:0) ¯ x (cid:1) ∼ = HF(¯ x ) ∼ = H( Y f ) , (5.5)where Y f is a certain topological space (the Conley index) associated with thecritical point x of f . Furthermore, the map α ℘ ( α ⊗ α ) turns into the Steenrodsquare Sq on H( Y f ) ; see [Wi18a]. Thus, with these identifications in mind, ξ ¯ x,i = ξ ′ ¯ x,i and ξ eq ¯ x,i = Sq( ξ ¯ x,i ) = ξ ¯ x,i + . . . , (5.6) where the dots stand for the terms of higher degree in H( Y f ) . It follows that thevectors ξ eq ¯ x,i are linearly independent and, since dim F HF eq (cid:0) ¯ x (cid:1) = dim F HF(¯ x ) by(5.5), this system is a basis.The action filtration spectral sequence in Floer cohomology has E = L ¯ x HF(¯ x ) and converges to HF( ϕ ) . With bases fixed, we can canonically collapse this spectralsequence into one complex with the same features as the ordinary Floer complexincluding the action filtration and cohomology equal to HF( ϕ ) ; cf. [GG19, Sect.2.1.3 and 2.5]. This data is sufficient to define the Floer graph Γ( ϕ ) of ϕ withvertices ξ ¯ x,i . (Note that the orbits with HF(¯ x ) = 0 do not contribute to Γ( ϕ ) andthe graph depends on the choice of the bases { ξ ¯ x,i } .) It is also worth keeping inmind that even in the non-degenerate case this graph and the complex might differfrom the Floer graph as defined in Section 3 and from the Floer complex. However,they have the same formal properties as CF( ϕ ) and the original graph, and theresulting homology is isomorphic to the Floer cohomology HF( ϕ ) ; cf. [GG19].A similar construction applies to ϕ in the ordinary and equivariant settings and ξ ′ ¯ x,i ↔ ξ eq ¯ x,i gives rise to an action-preserving one-to-one correspondence betweenthe vertices of Γ (cid:0) ϕ (cid:1) and Γ eq (cid:0) ϕ (cid:1) . The condition that the sum (1.1) with F = F is strictly greater than the sum of Betti numbers guarantees that the graph Γ( ϕ ) ,and hence Γ (cid:0) ϕ (cid:1) and Γ eq (cid:0) ϕ (cid:1) , have at least one arrow.Denote by β min the length of the shortest arrows in a Floer graph. Our goal isto show that ϕ cannot be k -perfect, where k is sufficiently large, assuming an apriori upper bound on β min (cid:0) ϕ k (cid:1) as in Theorem 1.1. (Note that in contrast withthe non-degenerate case the Floer graphs are now sensitive to small perturbationsof ϕ and we usually cannot make the shortest arrow unique without changing thegraph unless dim F HF( x ) = 1 for all x ∈ P ( ϕ ) .)The equivariant pair-of-pants product ℘ extends to the complexes we have con-structed, and Seidel’s non-vanishing theorem takes the form ℘ ( ξ ¯ x,i ⊗ ξ ¯ x,i ) = ξ eq ¯ x,i + . . . , (5.7)where now the dots stand for terms with action greater than or equal to the actionof ξ eq ¯ x,i , but with the provision that the first term enters the whole sum with non-zero coefficient. (This is a consequence of (5.6) and Seidel’s non-vanishing theoremapplied to the non-degenerate part in the splitting of ϕ at x .)Pick one of the shortest arrows, say v , in Γ eq (cid:0) ϕ (cid:1) . After recapping, we can ensurethat the beginning of v has the form ξ eq ¯ x,i . Using (5.7) and the facts that ℘ is a chainmap and v is a shortest arrow, it is not hard to see that ξ ¯ x,i is the beginning of anarrow in Γ( ϕ ) whose length is at most β eq min (cid:0) ϕ (cid:1) / . Hence, β min ( ϕ ) ≤ β eq min (cid:0) ϕ (cid:1) . (5.8)(This proves a somewhat weaker version of Theorem 3.1: every shortest equivariantarrow comes from an arrow for ϕ .)On the other hand, β eq min (cid:0) ϕ (cid:1) ≤ β min (cid:0) ϕ (cid:1) . (5.9)Indeed, dim F HF I (cid:0) ϕ (cid:1) ≥ rk F [h] HF I eq (cid:0) ϕ (cid:1) for any action interval I , as is easy tosee from the h -adic filtration spectral sequence. Applying this to an interval tightlyenclosing one of the shortest arrows in Γ eq (cid:0) ϕ (cid:1) we obtain (5.9). In fact, we expectthat, as in the non-degenerate case, Γ eq (cid:0) ϕ (cid:1) incorporates all arrows of Γ (cid:0) ϕ (cid:1) (and, NOTHER LOOK AT THE HOFER–ZEHNDER CONJECTURE 19 perhaps, more). 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EÇ and VG: Department of Mathematics, UC Santa Cruz, Santa Cruz, CA 95064,USA
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