PFH spectral invariants on the two-sphere and the large scale geometry of Hofer's metric
Dan Cristofaro-Gardiner, Vincent Humilière, Sobhan Seyfaddini
PPFH spectral invariants on the two-sphere and thelarge scale geometry of Hofer’s metric.
Daniel Cristofaro-Gardiner, Vincent Humili`ere, Sobhan SeyfaddiniFebruary 9, 2021
Abstract
We resolve three longstanding questions related to the large scalegeometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer’s metric. Namely: (1) we resolve theKapovich-Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernelof Calabi over any proper open subset is unbounded; and (3) we showthat the group of area and orientation preserving homeomorphisms ofthe two-sphere is not a simple group. Central to all of our proofs arenew sequences of spectral invariants over the two-sphere, defined viaperiodic Floer homology.
Contents ( S , ω ) . . . . . . . . . . . . . . . . 51.3 New spectral invariants . . . . . . . . . . . . . . . . . . . . . 61.4 Relationship with previous work . . . . . . . . . . . . . . . . 9 a r X i v : . [ m a t h . S G ] F e b Non-simplicity of
Homeo ( S , ω ) It is a remarkable fact that the group of Hamiltonian diffeomorphisms of asymplectic manifold admits a bi-invariant Finsler metric, known as
Hofer’smetric . The existence of such a metric on an infinite dimensional Lie groupis highly unusual, due to the lack of compactness, and stands in contrastto the fact that a simple finite dimensional Lie group admits a bi-invariantFinsler metric only if it is compact; see [PR14, Prop. 1.3.15].The theme of this article is the large-scale geometry of Hofer’s metric,on Ham( S , ω ), the Hamiltonian diffeomorphisms of the 2–sphere. Our firstresult, Theorem 1.4, settles two longstanding questions, presented below,about the quasi-isometry type of Ham( S , ω ). The first of the two questionswas posed by Kapovich and Polterovich in 2006. Question 1.1. Is Ham( S , ω ) quasi-isometric to the real line R ? The second question is due to Polterovich and dates back to the 2000s.To state it, consider a connected, proper open set U ⊂ S and denote byHam U ( S , ω ) the subgroup of Ham( S , ω ) consisting of Hamiltonian diffeo-morphisms supported in U . This subgroup carries a well-known group ho-momorphism called the Calabi homomorphism :Cal : Ham U ( S , ω ) → R , whose definition we recall in Section 2.1; see Equation (7). Question 1.2.
Suppose that Area( U ) ≤ Area( S ) . Is the kernel of Cal :Ham U ( S , ω ) → R an unbounded subset of Ham( S , ω ) ? The Hofer geometry of the two-sphere has long remained mysterious,and these two basic questions have received much attention over the pastyears. This is especially the case for Question 1.1, which appears as Prob-lem 21 on the list of open problems of McDuff-Salamon [MS17, Sec. 14.2];it is mentioned as one of the motivations behind the influential article ofPolterovich and Shelukhin [PS16, Sec. 1.3]; and it is highlighted in severalarticles such as [Py08, EPP12, KS18, BS17]. Ham(
M, ω ) is simple for closed M , by a theorem of Banyaga [Ban78]. If Area( U ) > Area( S ), then the question is known to have an affirmative answer byPolterovich [Pol98].
2e also continue the direction of research initiated in our recent arti-cle [CGHS20]. In particular, we answer the following question from the1980 article of Fathi [Fat80] on the algebraic structure of Homeo ( S , ω ),the connected component of the identity in the group of area preservinghomeomorphisms of the two-sphere. Question 1.3.
Is the group
Homeo ( S , ω ) simple? Although at first glance this question might appear unrelated to Hofer’sgeometry, we will see that the large scale geometry of Hofer’s metric playsa crucial role in the solution. The two-sphere is the only closed manifoldfor which the question of simplicity of the component of the identity inthe group of volume-preserving homeomorphisms remained open; for otherclosed manifolds this was settled by Fathi in the late 1970s.
Let d H denote the Hofer metric on Ham( M, ω ), the group of Hamiltoniandiffeomorphisms of a closed and connected symplectic manifold (
M, ω ); wewill review the definition of d H , and other basic notions from symplecticgeometry, in Section 2.1.A fundamental notion in large-scale geometry is that of quasi-isometry ,which we now recall. A quasi-isometric embedding is a mapping Φ :( X , d ) → ( X , d ) of metric spaces for which there exist constants A ≥ , B ≥ A d ( x, y ) − B (cid:54) d (Φ( x ) , Φ( y )) (cid:54) A d ( x, y ) + B. (1)The map Φ, satisfying the above, is said to be a quasi-isometry if it is quasi-surjective , i.e. if there exists a constant C > X is within distance C of the image Φ( X ).The large-scale geometry of Hofer’s metric, on general symplectic mani-folds, has been studied extensively ever since Hofer’s discovery of the metricin 1990 [Hof90]; see for example [Ost03, EP03, Ush13, Py08, Kha09, Hum12,Sey14, Kha16, PS16, AGKK + , d H ) is a “large” metricspace. For example, it is conjectured to be always unbounded, and thishas been proven for many manifolds [LM95b, Pol98, Sch00, Ost03, EP03,McD10, Ush13]. Moreover, Usher [Ush13] has proven that, for a large classof manifolds, including closed surfaces of positive genus, it admits a quasi-isometric embedding of infinite-dimensional normed vector spaces; see alsoPy’s article [Py08]. For any transformation group, the simplicity question is only interesting for the com-ponent of the identity because it forms a normal subgroup of the larger group. As observed in [Py08], such results for surfaces of posoitive genus can be deduced fromthe arguments in [LM95b, Pol98]. S , ω ),and the subgroup Ham U ( S , ω ), are unbounded and admit a quasi-isometricembedding of the real line R ; this was proven by Polterovich [Pol98]. Asfor the kernel of Cal : Ham U ( S , ω ) → R , with Area( U ) ≤ Area( S ), it isnot even know if it is unbounded, i.e. whether it is quasi-isometric to thepoint. It is our understanding that when Question 1.1 and Question 1.2were posed, there were not even clear conjectures about what their answersshould be.Our first point in the present work is that the kernel of Calabi is indeedrather big, which we illustrate in two different ways. Theorem 1.4.
Let U ⊂ S , with U (cid:54) = S . Then:(a) For any n ∈ N , there exists a quasi-isometric embedding of R n into (Ham( S , ω ) , d H ) whose image is included in the kernel of the Calabihomomorphism Cal : Ham U ( S , ω ) → R .(b) The kernel of Cal : Ham U ( S , ω ) → R is not coarsely proper. To review the terminology here, recall that a metric space (
X, d ) is saidto be coarsely proper if there exists R > X, d ) can be covered by finitely many balls of radius R ; see[CdlH16, Definition 3.D.10]. Examples of coarsely proper spaces includethe Euclidean space R n or any bounded spaces — in particular, part (b)of Theorem 1.4 resolves Question 1.1 and Question 1.2 — but on the otherhand, an infinite-dimensional Banach space is not coarsely proper. Recallalso that a quasi-flat in a metric space ( X, d ) is the image of a quasi-isometric embedding of R n ; moreover, the quasi-flat rank of a metric space( X, d ) is the supremum, over all n , such that there exists a quasi-isometricembedding of R n into X . Thus, part (a) of Theorem 1.4 is equivalent tothe statement that the metric space (Ham( S , ω ) , d H ) and the subset givenby the kernel of the Calabi homomorphism Cal : Ham U ( S , ω ) → R haveinfinite quasi-flat rank. Now, it is known that the quasi-flat rank of R n is n and so we see that part (a) of Theorem 1.4 also answers Questions 1.1and 1.2. In fact, we will see in Example 1.5 below that Theorem 1.4 tellsus quite a bit more about the quasi-isometry type of the metric spaces inquestion. Example 1.5.
Let (
G, d ) be a finite dimensional connected Lie group, witha left invariant Finsler metric induced from a norm on its Lie algebra; we callsuch a d a compatible metric. As was explained above, the existence ofHofer’s metric dramatically contrasts the situation for finite dimensional Liegroups; one might hope that the large-scale geometry also sees this. Indeed itis known that any such (
G, d ) both has finite quasi-flat rank, and is coarselyproper. So, our main theorem precludes this as a quasi-isometry type for4Ham( S ) , d ) or for the kernel of Calabi. Similarly, any finitely generatedgroup, or more generally, any locally compact and compactly generatedgroup (here we refer the reader to [CdlH16] for the precise definition) iscoarsely proper, see [CdlH16, Proposition 3.D.29]. It would be interestingto understand to what degree the quasi-isometry type of Ham( S ) is unique,for example whether it differs from that of Ham( S ) for other surfaces S . (cid:74) Remark 1.6.
Contemporaneously with our work, Polterovich-Shelukhinhave shown [PS], using very different methods, that there is an isometricembedding of the space of even compactly supported functions on ( − , )into Ham( S , ω ) . This clearly answers the Kapovich-Polterovich questionand, moreover, implies that Ham( S , ω ) is neither coarsely proper nor offinite quasi-flat rank. It would be very interesting to relate our methodshere to the methods in [PS]. (cid:74) Homeo ( S , ω ) We turn now to continuous symplectic geometry.In our recent article [CGHS20], we proved that the group of com-pactly supported area-preserving homeomorphisms of the disc is not sim-ple. Our next theorem settles the simplicity question for the sphere. Recallthat Homeo ( S , ω ) denotes the identity component in the group of area-preserving homeomorphisms of the two-sphere. Theorem 1.7.
Homeo ( S , ω ) is not simple. In fact, as in our previous article [CGHS20], this theorem implies astronger statement by appealing to a beautiful argument of Epstein andHigman [Eps70, Hig54]. Recall that a group is perfect if it is equal to itscommutator subgroup.
Corollary 1.8.
Homeo ( S , ω ) is not perfect. Theorem 1.7 answers a question of Fathi from the 70s [Fat80, AppendixA.6], whose history we now briefly review. The question of simplicity ofgroups of homeomorphisms and diffeomorphisms was studied extensivelyin the 50s, 60s, and 70s and is fairly well-understood in most scenarios.However, area-preserving homeomorphisms of surfaces have remained mys-terious. For example, in the case of closed manifolds, the simplicity questionhad been answered by the late 70s for all of the following groups: homeo-morphisms, diffeomorphisms , volume-preserving diffeomorphisms and sym-plectomorphisms. And in the case of volume preserving homeomorphisms it We have learned in recent conversation with Polterovich that this question is wideopen. We are considering C ∞ diffeomorphims here. For C k diffeomorphisms, simplicity isknown for all k except when k = dim(M) + 1 which remains open to this date. ( S , ω ) is surprising as it standsin dramatic contrast to the fact that on closed simply connected manifolds,such as spheres of dimension greater than one, this is the only example ofthe “usual” transformation groups known to be non-simple. For example,it is known that for simply connected manifolds the identity component inany of the groups mentioned in the previous paragraph is simple except, ofcourse, in our case of area-preserving homeomorphisms of the sphere.The simplicity of the aforementioned groups was established through theworks of a long list of mathematicians who studied the question from the30s to the late 70s. For a summary of the long history of the simplicityquestion, we refer the interested reader to [CGHS20, Sec. 1]. We now discuss the main tools that we use and develop here for provingthe above theorems. We henceforth view S as the unit sphere in standard R and equip it with the symplectic form ω := π dθ ∧ dz, where ( θ, z ) arecylindrical coordinates. Note that this gives the sphere a total area of 1. Periodic Floer homology and spectral invariants
To prove our results we use a version of Floer homology for area-preservingdiffeomorphisms called periodic Floer homology (PFH) which was intro-duced by Hutchings [HS05]; we will review PFH in Section 2.3. As will bereviewed in Section 3, one can use PFH to define a collection of invariantsof Hamiltonians on the sphere c d,k : C ∞ ( S × S ) → R which are indexed by d ∈ N and k ∈ Z with k having the same parity as d .We show in Section 3 that these invariants have various useful properties;see Proposition 3.2. In particular, we show that they can be used to defineinvariants c d,k : (cid:93) Ham( S , ω ) → R ,c d : (cid:93) Ham( S , ω ) → R , where c d := c d, − d , which are well-defined on the universal cover ofHam( S , ω ). Moreover, we show in Proposition 3.5, that if d is even then c d,k : (cid:93) Ham( S , ω ) → R descends to Ham( S , ω ) and so in particular we obtain c d : Ham( S , ω ) → R , defined for even d . 6 omogenization As is evident from the works of Entov-Polterovich [EP03, EP09], for thepurposes of applications to Hofer’s geometry, it is often beneficial to homog-enize spectral invariants. This is true in our work as well and, in fact, weprove Theorem 1.4 using the homogenizations of the invariants c d which wenow introduce. More precisely, we can define for ϕ ∈ Ham( S , ω ), and forall d ∈ N , µ d ( ϕ ) := lim sup n →∞ c d ( ˜ ϕ n ) n , (2)where ˜ ϕ ∈ (cid:93) Ham( S , ω ) is any lift of ϕ ; we show in Proposition 3.6 that theabove lim sup is well defined and that µ d ( ϕ ) does not depend on the choiceof ˜ ϕ ∈ (cid:93) Ham( S , ω ). We also define the related invariant ζ d : C ∞ ( S ) → R by ζ d ( H ) := lim sup n →∞ c d ( nH ) n . (3)We will see that these two homogenized invariants are related by the formula µ d ( ϕ H ) = ζ d ( H ) − d (cid:90) S H ω.
A useful property of any µ d is that it coincides with (a multiple of)the Calabi invariant for Hamiltonian diffeomorphisms with small supports.More precisely, suppose that supp( ϕ ), the support of ϕ ∈ Ham( S , ω ), iscontained in a topological disc D with Area( D ) < d +1 . Then, the followingequality holds 1 d µ d ( ϕ ) = − Cal( ϕ ) . (4)The above properties of µ d , ζ d will be proven in Section 3. Remark 1.9.
The properties of the µ d are reminiscent of the Calabi quasi-morphism of Entov-Polterovich [EP03]. It is an open question whetherHam( S , ω ) admits any Hofer continuous (homogeneous) quasimorphismsother than the one constructed by Entov-Polterovich. We plan to investi-gate in future work whether the invariants µ d are quasi-morphisms. (cid:74) The Hofer Lipschitz property and monotone twists
A critical fact which we will show, and which is at the heart of all applicationsto Hofer’s geometry is the
Hofer Lipschitz property. For the invariants µ d this means that the following holds: | µ d ( ϕ ) − µ d ( ψ ) | (cid:54) C d d H ( ϕ, ψ )7or all ϕ, ψ ∈ Ham( S , ω ). The Lipschitz constant is C d = d . In particular,these invariants can be used to bound the Hofer distance from below.In view of the Hofer Lipschitz property, to prove our results, we will haveto produce examples of Hamiltonian diffeomorphisms whose invariants wecan compute. This will be done by studying monotone twist Hamilto-nians , that is autonomous Hamiltonians H : S → R of the form H ( z, θ ) = 12 h ( z ) , where h (cid:48) ≥ , h (cid:48)(cid:48) ≥ , h ( −
1) = h (cid:48) ( −
1) = 0; we developed a combinatorialmodel in our previous work [CGHS20] which can be used to compute the c d for Hamiltonians like this under the additional technical assumption that h (cid:48) (1) ∈ N . For monotone twist Hamiltonians, the invariant ζ d has a beautifulexpression. Proposition 1.10.
For any Hamiltonian H as above we have ζ d ( H ) = d (cid:88) i =1 h (cid:18) − id + 1 (cid:19) . In other words, ζ d is the sum of the values of H on d equally dis-tributed horizontal circles. We learn from the above proposition that ζ d ( H ) is at least as large as the value H takes on each of the d circles C i = { ( z, θ ) : z = − id +1 } , where i ∈ { , . . . , d } . This bears some resem-blance to the notion of heaviness of equators introduced in the works ofEntov-Polterovich [EP09]. What is surprising is that the circles C i are alldisplaceable for d ≥
2, while heaviness of a set, as defined in [EP09], impliesthat the set is not diplaceable by Hamiltonian diffeomorphisms. Sensitivityto the displaceable circles C i is the distinguishing feature of our invariant µ d , ζ d which powers our applications to the Hofer geometry of the kernel ofCalabi. C continuity and non-simplicity of Homeo ( S , ω ) To prove Theorem 1.7, we need invariants which are continuous with respectto the C topology. The invariants c d and µ d , while useful, are not in general C continuous. We remedy this by taking certain linear combinations of the c d to define C continuous invariants η d : Ham( S , ω ) → R . Not only are these invariants C continuous, but also they extend con-tinuously to Homeo ( S , ω ). Moreover, they are also Hofer Lipschitz. Wesummarize the properties of the η d in Proposition 3.9.8 .4 Relationship with previous work As mentioned above, in our previous work we used PFH to define spec-tral invariants for compactly supported area-preserving diffeomorphisms andhomeomorphisms of the two-disc. For all of the applications discussed here,we need to rework this theory over the two-sphere. In the disc case, wecould assume that the maps were generated by a Hamiltonian that vanishesnear the boundary of the disc. This is no longer possible, so new ideas areneeded.One idea here, familiar to specialists, see for example [EP03, Oh05,Sch00], is to attempt to work with mean-normalized Hamiltonians. A care-ful analysis shows that this gives invariants which are well-defined on (cid:93)
Ham;then, after homogenization as in the previous section, we can obtain invari-ants of Ham. These invariants would be enough to prove the theorems in1.1. However, as stated above, they are not C continuous, and so can notbe used to study the algebraic structure of the homeomorphism group. Thisis where the η d , defined by taking a difference of spectral invariants, comein. Acknowledgments
We thank Yves de Cornulier, Bertrand R´emy, and Rich Schwartz for veryhelpful correspondence concerning the beautiful subject of large-scale geom-etry. We also thank Leonid Polterovich and Egor Shelukhin for very helpfulcorrespondence concerning their work [PS], see Remark 1.6. We thank Mo-hammed Abouzaid and Fr´ed´eric Le Roux for their comments on an earlierversion of the article.This article was written while DCG was at the Institute for AdvancedStudy, supported in part by the Minerva Research Foundation and the Na-tional Science Foundation. DCG is extremely grateful to the institute forproviding such a fantastic environment for conducing this research. DCGalso thanks the NSF for their support under agreement DMS 1711976.This project is an outgrowth of research that started in the summer of2018 when DCG was an “FSMP Distinguished Professor” at the InstitutMath´ematiques de Jussieu-Paris Rive Gauche (IMJ-PRG). DCG is gratefulto the Fondation Sciences Math´ematiques de Paris (FSMP) and IMJ-PRGfor their support.This project has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovationprogram (grant agreement No. 851701) and from Agence Nationale de laRecherche (ANR project “Microlocal” ANR-15-CE40-0007).9
Preliminaries
In this section we fix our notation and introduce the necessary backgroundon symplectic geometry and periodic Floer homology.
Here we recall some basic facts about symplectic geometry and the Hoferdistance.Let (
M, ω ) be a symplectic manifold. Let H ∈ C ∞ ( S × M ) be a Hamil-tonian; if M happens to be non-compact, then we consider only compactlysupported Hamiltonians. We can think of such H as a family of functions H t on M , depending on time; we think of S as parametrized by 0 ≤ t ≤ H gives rise to a possibly time-varying vector field X H t on M , calledthe Hamiltonian vector field , defined by ω ( X H t , · ) = dH t . The flow of X H t is called the Hamiltonian flow and is denoted ϕ tH . Theset of time-1 maps of Hamiltonian flows is called the set of Hamiltoniandiffeomorphisms of M and denoted Ham( M, ω ); it forms a subgroup ofthe symplectomorphisms of (
M, ω ). We can define the
Hofer norm || ϕ || ofany ϕ ∈ Ham(
M, ω ) as follows. First, to a Hamiltonian H ∈ C ∞ ( S × M ),we associate the norm (cid:107) H (cid:107) , ∞ := (cid:90) max M ( H t ) − min M ( H t ) dt. We then define (cid:107) ϕ (cid:107) := inf {(cid:107) H (cid:107) , ∞ : ϕ = ϕ H } . The above quantity is invariant under conjugation, i.e. (cid:107) ψ − ϕψ (cid:107) = (cid:107) ϕ (cid:107) .This follows from the fact that ϕ tH ◦ ψ = ψ − ϕ tH ψ ; see [HZ94, Sec. 5.1, Prop.1], for example.Finally, we can define a metric on Ham( M, ω ), the
Hofer metric , by d H ( ϕ, ψ ) = (cid:107) ϕ − ◦ ψ (cid:107) . As mentioned above, this yields a nondegenerate, bi-invariant metric, whichis quite remarkable given the noncompactness of Ham. Non-degeneracy iswhat is difficult to prove and it was established by Hofer for R n [Hof90],by Polterovich for rational symplectic manifolds [Pol93], and by Lalonde-McDuff in full generality [LM95a].The bi-invariance of Hofer’s distance also implies the following identities d H ( ϕ ϕ , ψ ψ ) (cid:54) d H ( ϕ , ψ ) + d H ( ϕ , ψ ) , (5) d H ( ϕ, ψ − ϕψ ) (cid:54) d H ( ψ, Id) . (6)10ndeed (5) follows from (9) below and (6) is proved as follows: d H ( ϕ, ψ − ϕψ ) = (cid:107) ϕ − ψ − ϕψ (cid:107) (cid:54) (cid:107) ϕ − ψ − ϕ (cid:107) + (cid:107) ψ (cid:107) = 2 d H ( ψ, Id) . Now let M = S = { ( x, y, z ) ∈ R : x + y + z = 1 } . This has a sym-plectic form ω := π dθ ∧ dz, where ( θ, z ) are cylindrical coordinates. We letDiff( S , ω ) denote the set of smooth diffeomorphisms ϕ , such that ϕ ∗ ω = ω .In fact, Diff( S , ω ) = Ham( S , ω ) . The Hofer geometry of Diff( S , ω ), withthis identification implied, will be the topic of study in the present work.We recall, for later use, that the fundamental group of Ham( S , ω ) is Z / Z and is generated by Rot, the full rotation around the North-South axis ofthe sphere; for a proof of this see, for example, [Pol01]; the Hamiltonian H ( θ, z ) = z generates this full rotation.We will denote the universal cover of Ham( S , ω ) by (cid:93) Ham( S , ω ). Thiscan be described as the set of Hamiltonian paths, considered up to homotopyrelative to endpoints; here, by a Hamiltonian path , we mean a path ofHamiltonian diffeomorphisms { ϕ t , ≤ t ≤ } . This is a two-fold covering,by the discussion in the previous paragraph.We next recall the displacement energy of a subset A ⊂ S . This isby definition the quantity e ( A ) := inf {(cid:107) φ (cid:107) : φ ( A ) ∩ A = ∅} . It is known that for a disjoint union of closed discs, each with area a andwhose union covers less than half the area of the sphere, the displacementenergy is a . We will need the following lemma in Section 4.3. Lemma 2.1.
Let
D, D (cid:48) ⊂ S be two disjoint closed discs of equal area.Then, inf {(cid:107) φ (cid:107) : φ ( D ) = D (cid:48) } = Area( D ) . Proof.
Let us denote a := Area( D ) and E := inf {(cid:107) φ (cid:107) : φ ( D ) = D (cid:48) } . Itfollows from the above discussion on displacement energy that E (cid:62) a .For the reverse inequality, note that the same discussion also impliesthat for any ε >
0, there exists ψ ∈ Ham( S , ω ), with (cid:107) ψ (cid:107) < a + ε and ψ ( D ) ∩ D = ∅ . Since ψ ( D ) and D (cid:48) have the same area and are both includedin S \ D , there exists a Hamiltonian diffeomorphism χ , supported in S \ D which maps ψ ( D ) onto D (cid:48) . The assumption on the support implies that χ − ( D ) = D . We now pick φ = χψχ − . We see that φ ( D ) = D (cid:48) and byconjugation invariance of the Hofer norm we have (cid:107) φ (cid:107) = (cid:107) ψ (cid:107) < a + ε . Sincesuch a diffeomorphism φ may be found for any ε >
0, this shows the reverseinequality E (cid:54) a .Next, we review the definition of the Calabi homomorphism
Cal : Ham U ( S , ω ) → R , U ⊂ S , we denote byHam U ( S , ω ) the subgroup of Ham( S , ω ) consisting of Hamiltonian diffeo-morphisms which are supported in U . Given ϕ ∈ Ham U ( S , ω ), defineCal( ϕ ) = (cid:90) S (cid:90) S H ( t, · ) ω dt, (7)where H ∈ C ∞ ( S × S ) is any Hamiltonian supported in U whose time–1flow is ϕ . It is well-known that Cal( ϕ ) does not depend on the choice of H and, moreover, Cal : Ham U ( S , ω ) → R is a group homomorphism; see[Cal70, MS17] for further details.In Section 5, we will also want to consider the group Homeo ( S , ω )of area and orientation preserving homeomorphisms of S . This isdefined to be the group of homeomorphisms of S , preserving the measureinduced by ω , in the component of the identity. It has a distance d C ,called the C distance, defined by picking a Riemannian metric d on S , anddefining d C ( ϕ, ψ ) = sup x ∈ M d ( ϕ ( x ) , ψ ( x )) . We remark for later use that Diff( S , ω ) sits densely in the C distance inHomeo ( S , ω ). We now recall the action spectrum , defined in [CGHS20, Section 2.5]. Let H ∈ C ∞ ( S × S ) . Recall the action functional associated to H A H ( z, u ) = (cid:90) H ( t ( z ( t )) dt + (cid:90) D u ∗ ω, (8)defined for capped loops ( z, u ). The critical points of A H are pairs ( z, u ),where z is a 1-periodic orbit of ϕ tH , and the set of associated critical values iscalled the action spectrum Spec( H ) of H . The forthcoming PFH spectralinvariants will take values in the order d action spectrum of H , defined bySpec d ( H ) := ∪ k + ... + k j = d Spec( H k ) + . . . + Spec( H k j ) , where H k denotes the k -fold composition of H with itself. Here, the com-position is defined by H G ( t, x ) := H ( t, x ) + G ( t, ( ϕ tH ) − ( x )) . The Hamil-tonian flow of H G satisfies the identity ϕ tH G = ϕ tH ◦ ϕ tG ; see [HZ94, Sec.5.1, Prop. 1], for example. For future reference, we also define a relatedoperation on Hamiltonians, the join , by( G (cid:5) H )( t, x ) = (cid:40) ρ (cid:48) (2 t ) H ρ (2 t ) ( x ) , if t ∈ [0 , ] , ρ (cid:48) (2 t − G ρ (2 t − ( x ) , if t ∈ [ , , ρ : [0 , → [0 ,
1] is a fixed non-decreasing smooth function whichis equal to 0 near 0 and equal to 1 near 1. Note that we do not need H and G to be one-periodic to define the join, and even if they are not one-periodic, G (cid:5) H will still be, since it is zero for t close to 0 and 1. As withthe composition, the time 1-map of G (cid:5) H is ϕ G ◦ ϕ H . Note that for anyHamiltonians G , G , H , H , we have (cid:107) G (cid:5) H − G (cid:5) H (cid:107) , ∞ = (cid:107) G − G (cid:107) , ∞ + (cid:107) H − H (cid:107) , ∞ . (9)We state here some of the properties of the order d action spectrumwhich will be used in the following sections. Recall that H ∈ C ∞ ( S × S )is said to be mean-normalized if (cid:82) S H ( t, · ) ω = 0 for all t ∈ S . TwoHamiltonians H , H are said to be homotopic if there exists a smoothpath of Hamiltonians connecting H to H such that ϕ H = ϕ H s = ϕ H forall s ∈ [0 , { ϕ tH } and { ϕ tH } , for0 (cid:54) t (cid:54)
1, coincide as elements of the universal cover (cid:93)
Ham( S , ω ). Here is alist of properties of Spec d which will be needed.(i) Symplectic invariance:
Spec d ( H ◦ ψ ) = Spec d ( H ), for all H ∈ C ∞ ( S × S ) and ψ ∈ Ham( S , ω ).(ii) Homotopy invariance: If H , H are mean-normalized and homo-topic, then Spec d ( H ) = Spec d ( H ).(iii) Measure zero:
Spec d ( H ) is of measure zero.The above properties are well-known in the case of Spec( H ), that is when d = 1; see for example [Oh05]. It is not difficult to see that the two initialproperties follow from the case d = 1: Symplectic invariance follows fromthe identity ( H ◦ ψ ) k = H k ◦ ψ , for any k ∈ N , and Homotopy invariance is aconsequence of the fact that H k , H k are mean-normalized and homotopic, forany k ∈ N , if H and H are. As we will now explain, the third property alsofollows from the d = 1 case. As a consequence of the definition of Spec d ( H ),it is sufficient to prove that the set Spec( H k )+ . . . +Spec( H k j ) is of measurezero, for any choice of k , . . . , k j with the property that k + . . . + k j = d . Tothat end, let ( M, ω ⊕ . . . ⊕ ω ) be the symplectic manifold obtained by takingthe j − fold product of ( S , ω ) and consider the Hamiltonian F : S × M → R defined by F ( t, x , . . . , x j ) = H k ( t, x ) + . . . + H k j ( t, x j ) . We conclude that Spec( H k )+ . . . +Spec( H k j ) has measure zero by observingthat it coincides with the set Spec( F ) which we know has measure zero.13 .3 Definition of PFH We now recall the definition of periodic Floer homology (PFH), from forexample [HS05], which is a tool that will be central in our work. While PFHcan be defined over any surface, for simplicity we consider the case whereour surface is S , which is the only case that is relevant for the present work.We start with some preliminaries. Let ϕ ∈ Diff( S , ω ). Given ϕ , we candefine the mapping torus Y ϕ := S × [0 , t / ∼ , ( x, ∼ ( ϕ ( x ) , . This has a natural vector field R := ∂ t , a natural one form dt , and a naturaltwo-form ω ϕ induced from the area form ω . The pair ( dt, ω ϕ ) is a sta-ble Hamiltonian structure in the sense of for example [BEH +
03, CM05,HT09b, Wen]. The manifold Y ϕ has a plane field ξ defined to be the verticaltangent bundle for the fibration π : Y ϕ → S .We will be interested in closed integral curves α : R /T Z → Y ϕ , of R , modulo reparametrization of the domain, which we call closed orbits ;we can identify an embedded closed orbit with its image. A closed orbit α has an integral degree d ( α ) := π ∗ [ α ] ∈ H ( S ) = Z . The linearized returnmap P α for a closed orbit α is defined for any p ∈ α as the linearizationof the time T flow of R on ξ | p . A closed orbit is called nondegenerate if1 is not an eigenvalue of the linearized return map; a nondegenerate closedorbit is called hyperbolic if the eigenvalues of P α are real and elliptic ifthe eigenvalues lie on the unit circle; these definitions do not depend on thechoice of p .Define an orbit set α := { ( α i , m i ) } to be a finite set, where the α i are distinct embedded closed orbits of R , and the m i are positive integers.The degree of the orbit set α is the sum of the degrees of the α i . Themap ϕ is d -nondegenerate if every closed orbit with degree at most d isnondegenerate; this is a generic condition. A degree d orbit set for a d -nondegenerate ϕ is called admissible if m i = 1 whenever α i is hyperbolic.Let X = R s × Y ϕ . This has a natural symplectic form ω = ds ∧ dt + ω ϕ . The pair (
X, ω ) is called the symplectization of Y ϕ . Recall that an almostcomplex structure on X is a smooth bundle map J : T X → T X suchthat J = −
1. A J -holomorphic curve in X is a map u : (Σ , j ) → ( X, J ),satisfying the equation du ◦ j = J ◦ du. Here, Σ is a closed (possibly disconnected) Riemann surface, minus a finitenumber of punctures, and the map u is assumed asymptotic to Reeb orbitsnear the punctures, see for example [Hut14] for the precise definition.14he periodic Floer homology P F H ( S , ϕ, d ) is the homology of achain complex P F C ( S , ϕ, d ). The chain complex P F C ( S , ϕ, d ) is freelygenerated over Z by admissible orbit sets α of degree d >
0. The chaincomplex differential ∂ counts J -holomorphic curves in X , for generic ad-missible J ; here, an almost complex structure is called admissible if itpreserves ξ , is R -invariant, sends ∂ s to R , and its restriction to ξ is tamedby ω ϕ . More precisely, (cid:104) ∂α, β (cid:105) = M I =1 J ( α, β ) , where I denotes the ECH index, defined below, we are considering curves in X up to equivalence of currents and modulo translation in the R direction,and that ∂ = 0, so the homology is well-defined; it is shown in [LT12] that it agrees with a version of Seiberg-WittenFloer cohomology and in particular is independent of ϕ .To define spectral invariants, we will want to use a twisted version ofPFH, denoted (cid:94) P F H ( S , ϕ, d ); as we will see in 3.1.1, the twisted PFH carriesa natural action filtration which we will use to define the spectral invariants.To define twisted PFH, let γ be any degree 1 cycle in Y ϕ , transverse to ξ ;choose a homotopy class of trivializations τ on ξ | γ . The twisted PFH chaincomplex (cid:94) P F C is generated by pairs ( α, Z ), called twisted PFH gener-ators , where α is a degree d admissible orbit set, and Z ∈ H ( Y ϕ , α, dγ ).The differential counts I = 1 curves C from ( α, Z ) to ( β, Z (cid:48) ), namely curves C ∈ M I =1 J ( α, β ), such that [ C ] + Z (cid:48) = Z. For each d , there is a grading, defined below, which we call the k -grading .The homology is an invariant, and so can be computed, with the result thatfor d ≥ (cid:94) P F H ∗ ( S , ϕ, d ) = (cid:40) Z , if ∗ = d mod 2 , ϕ is takento be an irrational rotation of the sphere; for more details see, for example,[CGHS20, Sec. 3.3]. We now define the ECH index I , and the grading k .The ECH index I depends only on the relative homology class A ∈ H ( Y ϕ , α, β ) between two orbit sets. We have I ( A ) = c τ ( A ) + Q τ ( A ) + CZ Iτ ( A ) , (11) More precisely, [HT09a] proves that the differential in embedded contact homologysquares to zero. As pointed out in [HT07] and [LT12] this proof carries over, nearlyverbatim, to our setting. τ denotes a homotopy class of trivializations of ξ over all Reeb orbits, c τ ( A ) denotes the relative Chern class of ξ restricted to A , Q τ ( A ) denotesthe relative self-intersection , and CZ Iτ denotes the total Conley-Zehnderindex. We will not need the precise definitions of these terms in the presentwork, so we omit them for brevity, referring the reader to [Hut02] for thedetails.We can define the promised k grading. The definitions of the relativeChern class and relative self-intersection extend verbatim to relative homol-ogy classes A ∈ H ( Y ϕ , α, dγ ), once a trivialization τ over the simple orbitsin α and a trivialization τ over γ has been chosen. With the precedingunderstood, we now define k ( α, Z ) := c τ,τ ( Z ) , + Q τ,τ ( Z ) + CZ Iτ ( α ) . To simplify the notation, we will denote k ( α, Z ) by I ( Z ) below. We now use the twisted PFH to define various invariants. We begin bysummarizing for the reader what will be done in this section.To set the stage for what is coming, it is helpful to recall what was donein [CGHS20, Sec. 3.4]. There, we defined spectral invariants c d,k ( H ) for H ∈ H where H := { H ∈ C ∞ ( S × S ) : ϕ tH ( p − ) = p − , H ( t, p − ) = 0 , ∀ t ∈ [0 , , − < rot( { ϕ tH } , p − ) < } , where rot( { ϕ tH } , p − ) is the rotation number of the isotopy { ϕ tH } t ∈ [0 , at p − . It was shown in addition that these invariants depend only on the time1-map. Spectral invariants for compactly supported disc maps were thendefined by identifying the disc with the northern hemisphere.Our goal now is to define spectral invariants for all H ∈ C ∞ ( S × S )and to find invariants that depend only on ϕ ∈ Ham( S , ω ), rather than ona choice of generating Hamiltonian. Here is how we do this. First we extendthe procedure in [CGHS20] from H ∈ H to arbitrary H to get invariants c d,k , defined when k and d have the same parity. These c d,k extend the c d,k from our previous work: that is, if H ∈ H ⊂ C ∞ ( S × S ), then thedefinition of c d,k ( H ) here agrees with that in [CGHS20]. Similarly to ourprevious work, we can then define c d := c d, − d . This choice of k = − d isnot quite canonical, see Remark 3.4, but is convenient and suffices for ourpurposes: what is crucial is that c d (0) = 0.As alluded to in the introduction, these c d are in general not invariantsof the time 1-map, and so are not well-suited on their own for provingour main theorems. However, we can use the c d to form new invariants.16irst, we show that the c d for even d are invariants when we restrict tomean-normalized Hamiltonians; similarly, the homogenizations µ d , ζ d arealso invariants restricted to mean-normalized Hamiltonians. None of theseinvariants are C continuous, so we use a linear combination of the c d for d even to define another sequence η d .Thus, to summarize for the ease of the reader, the main product of thissection are invariants c d and η d defined for d even, and µ d , ζ d defined for all d , together with proofs of their properties that we will need. The µ d and ζ d are related by the formula (25). The µ d are used to prove Theorem 1.4,while the η d are used to prove Theorem 1.7; the c d are used to construct the µ d and the η d . We begin by introducing PFH spectral invariants c d,k ( H ) for Hamiltonians H ∈ C ∞ ( S × S ). This requires first recalling a construction of Hutchingsfor assigning a spectral invariant to every nonzero twisted PFH class. A Hamiltonian H ∈ C ∞ ( S × S ) is called d -nondegenerate if its time-1flow ϕ = ϕ H is d -nondegenerate. We now explain how to define PFH spectralinvariants for d -nondegenerate Hamiltonians by extending the definition in[CGHS20] in a natural way.We begin by explaining the aforementioned construction of Hutchingsfor assigning a spectral invariant to a nonzero twisted PFH class. A twistedPFH generator has an action defined by A ( α, Z ) = (cid:90) Z ω ϕ . The differential decreases the action, see for example [CGHS20, Sec 3.3], sothe action induces a filtration on the twisted PFH chain complex: we can de-fine (cid:94)
P F C L to be the subcomplex generated by twisted PFH generators withaction no more than L . Denote the homology of this complex by (cid:94) P F H L .For any nonzero class σ ∈ (cid:94) P F H ( S , ϕ, d ), we can now define c σ ( ϕ, γ, τ ) tobe the smallest L such that σ is in the image of the inclusion induced map (cid:94) P F H L → (cid:94) P F H.
We can think of this as the minimum action required to represent σ .The number c σ ( ϕ, γ, τ ) depends on the choice of reference cycle γ andtrivialization τ over γ ; we will now define the PFH spectral invariants associated to a d -nondegenerate Hamiltonian H by using the Hamiltonianflow to fix a natural reference cycle. 17o make this precise, let H be a d -nondegenerate Hamiltonian and write ϕ = ϕ H . Consider the trivializationΨ H : S × S → Y ϕ ( t, x ) (cid:55)→ (cid:0) ( ϕ tH ) − ( x ) , t (cid:1) . (12)Define γ H = Ψ H ( S × { p − } ). This is trivialized by the pushforward τ H of an S -invariant trivialization over p − . We will now use the twisted PFH chaincomplex for Y ϕ , with respect to the reference cycle γ H , to define the spectralinvariants.Assume first that H vanishes at p for all time. For each d ∈ N , we define c d,k ( H ) := c σ ( ϕ H , γ H , τ H ) , d ≡ k mod 2 , where σ is the unique nonzero class in (cid:94) P F H k ( S , ϕ, d ). We emphasize that,even fixing the Hamiltonian diffeomorphism, this can and will depend on H ,since the trivialized reference cycle γ H does. We note that for such an H , c d,k ( H ) = A ( α, Z ) , (13)for some twisted PFH generator ( α, Z ) . Indeed, as explained in [CGHS20,Sec. 3.3] this follows from the fact that the subset {A ( α, Z ) : ( α, Z ) ∈ (cid:94) P F C ( ϕ, d ) } ⊂ R is discrete, as under our nondegeneracy assumption thereare only finitely many orbit sets of degree d .Finally, for arbitrary H we reduce to the case of H vanishing at p bydemanding that the Shift property , stated in Proposition 3.2 below, holds.This says that c d,k ( H + h ) = c d,k ( H ) + d (cid:90) h ( t ) dt, (14)when h : S → R is any function.In principle, c d,k ( H ) could depend on the choice of admissible J , but wewill see by the Monotonicity property below that it does not. We now prove that the PFH spectral invariants have the following key prop-erties and extend to all, possibly degenerate, Hamiltonians.
Theorem 3.1.
The PFH spectral invariant c d,k ( H ) admits a unique exten-sion to all H ∈ C ∞ ( S × S ) such that the extended spectral invariant c d,k : C ∞ ( S × S ) → R satisfies the following properties. . Continuity: For any H, G ∈ C ∞ ( S × S ) , we have d (cid:90) S min( H t − G t ) dt ≤ c d,k ( H ) − c d,k ( G ) (cid:54) d (cid:90) S max( H t − G t ) dt.
2. Spectrality: c d,k ( H ) ∈ Spec d ( H ) . Before giving the proof, we note that the second item of the theoremimplies that if
H, G vanish at p − , then | c d,k ( H ) − c d,k ( G ) | ≤ d (cid:107) H − G (cid:107) , ∞ , (15)which is an alternative variant of the Hofer continuity property. Proof.
The proof proceeds along similar lines as [CG-H-S, Thm. 3.6].
Step 1: Reducing to the d -nondegenerate case. We now assume that thetheorem has been proved for d -nondegenerate H , and explain how this im-plies the result for all H . Given any H , take any sequence of d -nondegenerate H i which C converges to H , and define c d,k ( H ) = lim i →∞ c d,k ( H i ) . (16)This limit exists, and does not depend on the choice of approximating H i ,due to the Continuity property with H = H i and G = H j . The same in-equality implies that the extension from d -nondegenerate H is unique asclaimed; the Continuity and Shift properties for d -nondegenerate H implythese properties for all H . Spectrality for d -nondegenerate H implies Spec-trality for all H by Arzela-Ascoli. Step 2: Reducing to Hamiltonians that vanish at p − . It remains to prove Continuity and Spectrality in the nondegenerate case.We now show that by using the Shift property (14), it suffices to provethese properties for Hamiltonians vanishing at p − . We begin with Continu-ity. Consider arbitrary H, G . Then, we can write H = ˜ H + h, G = ˜ G + g, (17)where h and g are defined as the restriction of H, G to p − , and ˜ H, ˜ G vanishon p − . Then, by the Shift property, c d,k ( H ) − c d,k ( G ) = c d,k ( ˜ H ) − c d,k ( ˜ G ) + d (cid:90) S ( h ( t ) − g ( t )) dt. Thus, if Continuity holds for ˜ H and ˜ G , then we have c d,k ( H ) − c d,k ( G ) ≤ d (cid:90) S max( ˜ H t − ˜ G t ) dt + d (cid:90) S ( h ( t ) − g ( t )) dt. h, g only depend on t , we havemax( ˜ H t − ˜ G t ) = max( H t − G t ) + g ( t ) − h ( t ) . Combining this equality with the previous inequality proves the right-most inequality required for Continuity. Similarly, if Continuity holds for ˜ H and ˜ G , then we have c d,k ( H ) − c d,k ( G ) ≥ d (cid:90) S min( ˜ H t − ˜ G t ) dt + d (cid:90) S ( h ( t ) − g ( t )) dt, and we know thatmin( ˜ H t − ˜ G t ) = min( H t − G t ) + g ( t ) − h ( t ) , hence the leftmost inequality required for Continuity to hold.Similarly, if Spectrality holds for ˜ H in (17), then it holds for H by theShift property, because the addition of h does not change the set of criticalpoints of A H , hence by (8), Spec d ( H ) = Spec d ( ˜ H ) + d (cid:82) S h ( t ) dt. Thus, we can assume H and G vanish at p − . Step 3. Continuity when H and G vanish at p − .Under (12), the stable Hamiltonian structure ( dt, ω ϕ ) is of the form( dt, ω + dH ∧ dt ) , and R = ∂ t + X H . The natural symplectic form onthe symplectization X = R × Y ϕ under (12) is ω H = ds ∧ dt + ω + dH ∧ dt, where s is the coordinate on R . We henceforth identify Y ϕ with S × S using (12), we implicitly identify orbit sets on Y ϕ with the correspondingorbit sets on S × S , and we identify the trivialized reference cycle ( γ H , τ H )with the S -invariant trivialized cycle γ over p − .Given H and G , we pick a function β , which is 0 for sufficiently small s , 1 for s sufficiently large, and satisfies 1 + β (cid:48) ( H − G ) >
0, we define K = G + β ( s )( H − G ) and we consider the form ω X = ds ∧ dt + ω + d ( Kdt ) , which is symplectic and agrees with ω H for sufficiently positive s and ω G forsufficiently negative s .The general theory of (twisted) PFH cobordism maps, as developed byChen [Che18], guarantees a chain map between the twisted PFH chain com-plexes for H and G , counting ECH index zero J X -holomorphic buildingsfrom ( α, Z ) to ( β, Z (cid:48) ), and inducing an isomorphism, where J X is a fibra-tion compatible almost complex structure on X , in the sense that it preservesthe vertical tangent bundle and its ω X -orthogonal complement.20o, given d (cid:62) k ∈ Z of the same parity, let ( α , Z )+ . . . +( α m , Z m )be a cycle in (cid:94) P F C ( ϕ H , d ) representing σ d,k with c σ d,k ( ϕ H ) = A ( α , Z ) ≥ . . . ≥ A ( α m , Z m )and let ( β, Z (cid:48) ) be a generator in (cid:94) P F C ( ϕ G , d ) with maximal action amongthe support of Ψ H,G (( α , Z ) + . . . + ( α m , Z m )) . Thus, we have a J X -holomorphic building C from some ( α i , Z i ), which wewill denote by ( α, Z ), to ( β, Z (cid:48) ). Since, just as in [CGHS20], our argumentonly involves action and index considerations, we can assume that C consistsof a single level, and we know that Z (cid:48) + [ C ] = Z, as elements of H ( S × S , α, dγ ). Hence, as I ([ C ]) = 0, we must have I ( Z ) = I ( Z (cid:48) ) = k , so that c d,k ( ϕ H ) − c d,k ( ϕ G ) ≥ A ( α, Z ) − A ( β, Z (cid:48) ) . (18)We now claim the identity A ( α, Z ) − A ( β, Z (cid:48) ) = (cid:90) C ω + dK ∧ dt + K (cid:48) ds ∧ dt, (19)where K (cid:48) denotes the derivative with respect to s , and for the rest of thissection dK denotes the derivative in the S direction.The proof of this is just as in [CGHS20, Lem. 3.8]. Indeed, as in theproof of [CGHS20, Lem. 3.8], we have A ( α, Z ) = (cid:90) Z ω + d ( Hdt ) , A ( β, Z (cid:48) ) = (cid:90) Z (cid:48) ω + d ( Gdt ) , and (cid:90) C ω = (cid:90) Z ω − (cid:90) Z (cid:48) ω. Moreover, (cid:82) C d ( Kdt ) = (cid:82) Z d ( Hdt ) − (cid:82) Z (cid:48) d ( Gdt ), since
H, G vanish on γ . So,putting this all together, we have A ( α, Z ) − A ( β, Z (cid:48) ) = (cid:90) C ω + d ( Kdt ) , hence (19).Moreover, we have (cid:82) C ω + dK ∧ dt ≥
0, since as in the proof of [CGHS20,Lem. 3.8], the form ω + dK ∧ dt is pointwise nonnegative along C , and soin fact we obtain A ( α, Z ) − A ( β, Z (cid:48) ) ≥ (cid:90) C K (cid:48) ds ∧ dt, (20)21he argument in [CGHS20, Lem. 3.8] also shows that ds ∧ dt is pointwisenonnegative on C .Now we have (cid:90) C K (cid:48) ds ∧ dt = (cid:90) C β (cid:48) ( s )( H − G ) ds ∧ dt ≥ (cid:90) C β (cid:48) ( s ) min( H t − G t ) ds ∧ dt, since ds ∧ dt is pointwise nonnegative along C . We can evaluate the right-most integral in the above equation by projecting to the ( s, t ) plane; thisprojection has degree d , and (cid:82) β (cid:48) = 1, so the above inequality in combinationwith (20) and (18) give the leftmost inequality required for Continuity.To prove the other inequality, we switch the role of H and G in the aboveargument, and again combine the corresponding versions of (18) and (20)to get c d,k ( ϕ G ) − c d,k ( ϕ H ) ≥ (cid:90) C β (cid:48) ( s )( G − H ) ds ∧ dt, hence c d,k ( ϕ H ) − c d,k ( ϕ G ) ≤ (cid:90) C β (cid:48) ( s )( H − G ) ds ∧ dt ≤ (cid:90) C β (cid:48) ( s ) max( H t − G t ) ds ∧ dt, where in the rightmost inequality we have used the fact that ds ∧ dt ispointwise nonnegative. We then project to the ( s, t ) plane as above toobtain the rightmost inequality required for Continuity. Step 4. Spectrality when H vanishes at p − . Since H vanishes at p − , we know by (13) that any c d,k ( H ) = A ( α, Z )for some twisted PFH generator ( α, Z ).Recall that A ( α, Z ) is the action of some relative homology class. Wefirst construct a particular homology class Z α from a periodic orbit α , andshow that the action of this class lies in the action spectrum. More precisely,let x be a q periodic point of ϕ = ϕ H , and pick a capping disk u for theorbit γ ( t ) = ( ϕ tH ( x )) t ∈ [0 ,q ] , such that u (0 ,
0) = p − . Equip the disc with polarcoordinates ( θ, ρ ) with θ ∈ R /q Z , ρ ≤
1, and then consider the homologyclass Z α represented by R /q Z × [0 , → S × S , ( θ, ρ ) → ( θ mod 1 , u ( θ, ρ )) . We now compute A ( β, Z α ) = (cid:90) Z α ( ω + dH ∧ dt ) = (cid:90) Z α ( ω + d ( Hdt ))= (cid:90) u ∗ ω + (cid:90) ∂Z α Hdt = (cid:90) u ∗ ω + (cid:90) q H t ( γ ( t )) dt = A H ( γ, u ) ∈ Spec q ( H ) , H vanishesat p − .Now, given an arbitrary ( α, Z ), write α = { ( α i , q i ) } . We can write Z = (cid:80) Z α i + y [ S ] . Then A ( α, Z ) = y + (cid:88) A ( α i , Z α i ) . The right hand side of the above formula is an element of Spec d ( H ), sincewe can for example absorb the y into the capping of any particular orbit.Hence, c d,k ( H ) ∈ Spec d ( H ).We now collect some additional useful properties of the c d,k . Proposition 3.2.
The spectral invariant c d,k : C ∞ ( S × S ) → R satisfies:1. Normalization: c d,k (0) = 0 for − d ≤ k ≤ d
2. Monotonicity: Suppose that H (cid:54) G . Then, c d,k ( H ) (cid:54) c d,k ( G ) .
3. Shift: Let h : S → R be a function of time. Then, c d,k ( H + h ) = c d,k ( H ) + d (cid:90) S h ( t ) dt.
4. Symplectic invariance: c d,k ( H ◦ ψ ) = c d,k ( H ) for any ψ ∈ Ham( S , ω ) .5. Homotopy invariance: If H , H are mean-normalized and homotopic,then c d,k ( H ) = c d,k ( H )
6. Support-control: If the support of H is contained in a topological disc D with Area( D ) < d +1 , and − d ≤ k ≤ d , then | c d,k ( H ) | (cid:54) d Area( D ) .Proof. Normalization follows from our previous work [CGHS20, Thm. 3.6],since as mentioned previously the c d,k extend the spectral invariants wedefined there. The Shift property is immediate from the definition. TheMonotonicity property follows formally from Continuity: indeed, by Conti-nuity we have c d,k ( H ) − c d,k ( G ) ≤ d (cid:90) S max( H t − G t ) dt, and so if H ≤ G then the integrand in the above inequality is nonpositive,so that we obtain Monotonicity. 23o prove Symplectic invariance, let ψ t be a Hamiltonian isotopy such that ψ = Id , ψ = ψ . It is sufficient to show that the function t (cid:55)→ c d,k ( H ◦ ψ t ) isconstant. To see this, recall from Section 2.2 that Spec d ( H ◦ ψ t ) = Spec d ( H )and so the function t (cid:55)→ c d,k ( H ◦ ψ t ), which is continuous by the Continuityproperty of Theorem 3.1, takes values in the measure-zero set Spec d ( H ) andso it must be constant.The proof of Homotopy invariance is analogous. Let H s , ≤ s ≤
1, be asmooth path of mean-normalized Hamiltonians connecting H to H . Notethat, by the Homotopy invariance of the action spectrum from Section 2.2,we have Spec d ( H s ) = Spec d ( H ) for all d ∈ N and s ∈ [0 , s (cid:55)→ c d,k ( H s ) is constant because it takes values in themeasure zero set Spec d ( H ). We conclude that c d,k ( H ) = c d,k ( H ).It therefore remains to prove Support-control. The proof will rely on thefollowing lemma. We will say that a set U is d -displaced by a map Ψ ifthe sets U, Ψ( U ) , . . . , Ψ d ( U ) are all disjoint. Lemma 3.3.
Let F be a Hamiltonian and let B be an open topological discwhich is d -displaced by ϕ F . Then, for any Hamiltonian G which is supportedin B , we have c d,k ( G F ) = c d,k ( F ) . A similar lemma was established in [CGHS20, Lemma 4.4] but only formaps supported in the northern hemisphere. The argument presented hereis essentially the same and so we will be rather brief.
Proof of Lemma 3.3.
Let ( K s ) [0 , be a smooth one parameter family ofHamiltonians such that for any s ∈ [0 , K s is ϕ sG ϕ F and such that the isotopy of K s consists in following first the isotopy gen-erated by F and then that generated by sG ( st, x ). More precisely, we maytake K s = G s (cid:5) F, where G s ( x, t ) := sG ( x, st ). It generates the isotopy ϕ tK s = (cid:40) ϕ ρ (2 t ) F , if t ∈ [0 , ] ,ϕ sρ (2 t − G ϕ F , if t ∈ [ , . Then, for all s ∈ [0 , d ( K s ) = Spec d ( F ): the argument for this isexactly the same as the argument in [CGHS20, Lem. 4.4] and so we willomit it. This implies that for any ( d, k ) the continuous map s (cid:55)→ c d,k ( K s )take values in Spec d ( F ); the fact that this map is continuous is a consequenceof Hofer continuity of c d,k , see (15). Since this set is totally discontinuous,we deduce that these functions are all constant, and so c d,k ( K ) = c d,k ( K ).To finish the proof it is sufficient to show that c d,k ( K ) = c d,k ( F ) and c d,k ( K ) = c d,k ( G F ). To see this, note that the Hamiltonians flows To orient a reader who reads [CGHS20, Lem. 4.4], note that the argument thererefers to the spectrum of the time 1 maps of K s and F rather than to the Hamiltoniansthemselves; this is because in that proof, the Hamiltonians are all assumed zero on thesouthern Hemisphere so we can refer to the spectrum in terms of the time 1-map; howeverthe argument for that Lemma extends to the case here with no changes. tK and ϕ tF are homotopic rel. endpoints and, moreover, (cid:82) S (cid:82) S K ω dt = (cid:82) S (cid:82) S F ω dt . It then follows from the Homotopy invariance and Shift prop-erties that c d,k ( K ) = c d,k ( F ). The same reasoning implies that c d,k ( K ) = c d,k ( G F ); indeed, the flows of these two Hamiltonians are homotopic rel.endpoints and, moreover, (cid:82) S (cid:82) S K ω dt = (cid:82) S (cid:82) S G F ω dt .We will now use Lemma 3.3 to establish the Support-Control inequality.
Proof of the Support-control inequality.
Fix d >
0. Let H be a Hamiltoniansupported in a disc D of area smaller than d +1 . This area condition impliesthat we can find a Hamiltonian F such that the disc D is d -displaced by ϕ F .Furthermore, for any ε >
0, we may assume that (cid:107) F (cid:107) , ∞ (cid:54) Area( D ) + ε .To see this, note that we can find an area preserving diffeomorphism ψ such that ψ ( D ) is sandwiched between two meridians (that is, curves with θ = constant) of S which enclose a region of area Area(D) + ε . Sup-pose that ε is so small that Area(D) + ε < d +1 . Then, consider theHamiltonian K = Area(D)+ ε z whose time-1 map ϕ K is the horizontal ro-tation of angle Area(D) + ε , which d -displaces ψ ( D ). Then, we may set F = K ◦ ψ , whose time-1 map, ψ − ϕ K ψ , d-displaces the disc D . Clearly, (cid:107) F (cid:107) , ∞ = Area(D) + ε .By Lemma 3.3, we have c d,k ( H F ) = c d,k ( F ). Using this, and Equation15, we obtain | c d,k ( H ) − c d,k ( F ) | = | c d,k ( H ) − c d,k ( H F ) | (cid:54) d (cid:107) H − H F (cid:107) , ∞ = d (cid:107) F (cid:107) , ∞ . Hence, we have | c d,k ( H ) | (cid:54) | c d,k ( F ) | + d (cid:107) F (cid:107) , ∞ (cid:54) d (cid:107) F (cid:107) , ∞ = 2 d Area(D) + 2 ε. This completes the proof of support-control inequality.We have now completed the proof of Proposition 3.2.
The goal of this section is to introduce PFH spectral invariants, and otherrelated invariants, for Hamiltonian diffeomorphisms. c d We begin by noting that we can now define the PFH spectral invariants onthe universal cover (cid:93)
Ham( S , ω ) as follows. Given ˜ ϕ ∈ (cid:93) Ham( S , ω ), let H bea mean-normalized Hamiltonian such that the Hamiltonian path { ϕ tH } , ≤ t ≤
1, is a representative for ˜ ϕ . Define c d,k ( ˜ ϕ ) := c d,k ( H ) . (21)25he mapping c d,k : (cid:93) Ham( S , ω ) → R is well-defined as a consequence of the Homotopy invariance property inProposition 3.2. Note that for any (not necessarily normalized) Hamiltonian H , the Shift property yields c d,k ( ˜ ϕ ) = c d,k ( H ) − d (cid:90) S (cid:90) S H t ω dt, (22)where ˜ ϕ is the lift of ϕ given by the isotopy ( ϕ tH ) t ∈ [0 , .However, to prove our main theorems, we will want invariants ofHam( S , ω ) rather than (cid:93) Ham( S , ω ).To produce such invariants, we start by showing that, as mentionedabove, it turns out that we can use the c d,k to define invariants that areindependent of the choice of mean normalized Hamiltonian. To get startedwith this, let ˜ ϕ ∈ (cid:93) Ham( S , ω ) . Define c d ( ˜ ϕ ) := c d, − d ( ˜ ϕ ) . Next, we will prove that, for even d , the map c d : (cid:93) Ham( S , ω ) → R descends to Ham( S , ω ). In other words, we will show that there is a well-defined map c d : Ham( S , ω ) → R . This is the content of Proposition 3.5.
Remark 3.4.
The c d as defined here are not canonical. We could equallywell define c d ( ϕ ) := c d,k ( ϕ H )for any − d ≤ k ≤ d with the same party as d . What is important forthe applications in our paper is to choose a k such that the c d,k satisfy theNormalization property. It is also instructive to note that because additionof a sphere induces a canonical bijection of the twisted PFH chain complex,we have c d,k +2 d +2 ( H ) = c d,k ( H ) + 1 . (23)In particular, any c d,k (cid:48) is equivalent to a c d,k with − d ≤ k ≤ d , up to additionof spheres.To summarize, then, there are essentially d +1 possible spectral invariantscorresponding to degree d , and we have made a non-canonical choice of oneof them moving forward, with the main goal of simplifying the notation. (cid:74) For future use, we also define c d ( H ) := c d, − d ( H ) , for any H ∈ C ∞ ( S × S ). 26 roposition 3.5. For any positive even integer d and any even integer k , the invariant c d,k : (cid:93) Ham( S , ω ) → R descends to Ham( S , ω ) . In otherwords, it does not depend on the choice of mean-normalized H . In particular, we obtain a well-defined invariant c d : Ham( S , ω ) → R forany positive even integer d . Proof.
Let H be any Hamiltonian and K = ( z + 1) . Note that the Hamil-tonian K vanishes at p − and its time 1 flow is rotation by 2 π about the z -axis. We will show below that for any positive integer d and integer k ofthe same parity as d , c d,k ( H (cid:5) K ) = c d,k ( H ) + d . (24)This implies Proposition 3.5, by the following argument. Let H and H bemean-normalized Hamiltonians generating the same time 1-map. We canassume that H and H are not homotopic, or else the proposition holds byProposition 3.2, item 5. Then, H (cid:5) ( K − ) and H are homotopic, and H (cid:5) ( K − ) is mean-normalized, hence c d,k ( H (cid:5) ( K − )) = c d,k ( H ) . On the other hand, by the Shift property c d,k ( H (cid:5) ( K − )) = c d,k ( H (cid:5) K ) − d , so that the Proposition follows from (24).It remains to prove (24). To prove this, we first note that c d,k ( H (cid:5)
0) = c d,k ( H ) . Indeed, H (cid:5) H are homotopic, with the same mean. Thus, it sufficesto show that c d,k ( H (cid:5) K ) = c d,k ( H (cid:5)
0) + d . To prove this, by the Shift property we can assume that H vanishes at p − .Then, H (cid:5) K and H (cid:5) ϕ , and the same referencecycle γ ⊂ Y ϕ . The only difference between them is the trivialization of V over γ ; more precisely, if τ (cid:48) denotes the trivialization over γ induced by H (cid:5) K and τ denotes the trivialization induced by H (cid:5)
0, then we have τ (cid:48) = τ − d + d by [Hut02, Eq. 6,Lem. 2.5.b], we have c d,k ( H (cid:5) K ) = c d,k + d + d ( H (cid:5)
0) = c d,k ( H (cid:5)
0) + d .2.2 Homogenized invariants We introduced the homogenizations µ d and ζ d in the introduction (see Equa-tions (2) and (3). The next proposition states that they are well-defined. Proposition 3.6.
There are well defined maps µ d : Ham( S , ω ) → R and ζ d : C ∞ ( S ) → R given by µ d ( ϕ ) = lim sup n →∞ c d ( ˜ ϕ n ) n and ζ d ( H ) = lim sup n →∞ c d ( nH ) n , for any ϕ ∈ Ham( S , ω ) and H ∈ C ∞ ( S ) . Remark 3.7.
One can more generally define ζ d ( H ) := lim sup n →∞ c d ( H n ) n for any (non necessarily autonomous) Hamiltonian H ∈ C ∞ ( S × S ). How-ever, we choose to restrict ζ d to C ∞ ( S ) in analogy with [EP06], where asimilar map ζ was defined and was proved to satisfy the properties of asymplectic quasi-state. It would be interesting to see if our ζ d also has theseproperties. (cid:74) Proof.
The fact that both of the above lim sup exist follows directly fromthe Continuity property of c d := c d, − d in Theorem 3.1. This shows that ζ d is well defined on C ∞ ( S ) and that µ d is well-defined on the universal cover (cid:93) Ham( S , ω ). It remains to show that µ d descends to Ham( S , ω ).To see this, let ϕ ∈ Ham( S , ω ). Let ˜ ϕ, ˜ ϕ (cid:48) be two lifts of ϕ to (cid:93) Ham( S , ω ).Recall that π (Ham( S , ω ) has only two elements and that the non-trivial el-ement is represented by the isotopy { ϕ tR } , where R ( θ, z ) = z is the Hamilto-nian which generates a full 2 π rotation around the z -axis. As a consequence,for any n ∈ N , since ˜ ϕ n and ˜ ϕ (cid:48) n are both lifts of the same diffeomorphism ϕ n , we have either ˜ ϕ n = ˜ ϕ (cid:48) n or ˜ ϕ n = ˜ ϕ (cid:48) n ˜ ϕ R . In both cases, the Continuityproperty of Theorem 3.1 gives an upper bound which does not depend on n : (cid:12)(cid:12) c d ( ˜ ϕ n ) − c d ( ˜ ϕ (cid:48) n ) (cid:12)(cid:12) (cid:54) d (cid:107) R (cid:107) , ∞ . It then follows that lim sup n →∞ c d ( ˜ ϕ n ) n = lim sup n →∞ c d ( ˜ ϕ (cid:48) n ) n , which proves that µ d descends to Ham( S , ω ).We next state some of the properties of µ d which will be used in ourarguments. Proposition 3.8.
The invariant µ d : Ham( S , ω ) → R satisfies the follow-ing properties:1. Normalization: µ d (Id) = 0 . . Hofer continuity: For all ϕ, ψ we have | µ d ( ϕ ) − µ d ( ψ ) | (cid:54) d d H ( ϕ, ψ ) .
3. Calabi property: Suppose that supp( ϕ ) is contained in a topologicaldisc D . If Area( D ) < d +1 , then d µ d ( ϕ ) = − Cal( ϕ ) , where Cal : Ham c ( D, ω ) → R denotes the Calabi invariant.4. Relationship with ζ d : For any H ∈ C ∞ ( S ) , µ d ( ϕ tH ) = ζ d ( tH ) − t d (cid:90) S Hω, (25) for all t ∈ R .Proof. The first item follows immediately from the definition of µ d combinedwith the fact that c d (0) = 0.To prove the second item, let ϕ, ψ ∈ Ham( S , ω ) and H, K be mean-normalized Hamiltonians with ϕ H = ϕ and ϕ K = ψ . We also denote by˜ ϕ, ˜ ψ ∈ (cid:93) Ham( S , ω ) the lifts of ϕ, ψ respectively given by H, K . Then, bydefinition d | c d ( ˜ ϕ n ) − c d ( ˜ ψ n ) | = d | c d ( H (cid:5) n ) − c d ( K (cid:5) n ) | , for any n >
0. Here H (cid:5) n denotes the n -fold join H (cid:5) · · · (cid:5) H . By theContinuity property of c d , we have d | c d ( H n ) − c d ( K n ) | (cid:54) (cid:107) H (cid:5) n − K (cid:5) n (cid:107) , ∞ = n (cid:107) H − K (cid:107) , ∞ . Note that this last equality follows from (9). From this inequality, we deduce | µ d ( ϕ ) − µ d ( ψ ) | (cid:54) d (cid:107) H − K (cid:107) , ∞ . Since this holds for any choices of Hamiltonians
H, K , and since we canrestrict to mean-normalized Hamiltonians in computing the Hofer norm of ϕ − ψ , the Hofer continuity property follows.The Calabi property is a consequence of the Support-control propertyfrom Prop. 3.2. Indeed, given any Hamiltonian H with support in D , (22)yields c d ( ˜ ϕ ) = c d ( H ) − d (cid:90) S (cid:90) S H t ω dt, where ˜ ϕ is the lift of ϕ given by the isotopy ( ϕ tH ) t ∈ [0 , . The integral inthe second term in the right hand side above is nothing but − Cal( ϕ ), while29he first term is bounded from above by 2 d Area( D ), by Support-control.Applying this to ϕ n , for any n >
0, we get | n c d ( ˜ ϕ n ) + d Cal( ϕ n ) | = n | c d ( ˜ ϕ n ) + d Cal( ϕ n ) | = n | c d ( H n ) | (cid:54) n d Area( D ) . The Calabi property follows from this last inequality.As for the last item, it follows from the definitions of µ d and ζ d , and (22)that µ d ( ϕ tH ) = µ d ( ϕ tH ) = ζ d ( tH ) − d (cid:90) S (cid:90) S tH ω dt. η d Although we can use the invariants c d or µ d to get invariants of the time-1map, these invariants will not in general be C -continuous, as they requiremean normalizing the Hamiltonian. We obtain C -continuous invariants bydefining, for even d ∈ N , the numbers η d : Ham( S , ω ) → R ϕ (cid:55)→ c d ( ϕ ) − d c ( ϕ ) . (26)The fact that η d is well-defined is an immediate consequence of Proposi-tion 3.5. Observe that, by Proposition 3.5 and the Shift property of Propo-sition 3.2, we have η d ( ϕ ) = c d ( H ) − d c ( H ) , (27)where H is any Hamiltonian with time-1 flow ϕ .We now prove that η d satisfies various properties, most notably C –continuity. Proposition 3.9.
The invariant η d is well-defined and satisfies the followingproperties for all ϕ, ψ ∈ Ham( S , ω ) .1. Normalization: η d (Id) = 0 .2. Hofer continuity: | η d ( ϕ ) − η d ( ψ ) | (cid:54) d d H ( ϕ, ψ ) .
3. Support-control: If the support of ϕ ∈ Ham( S , ω ) , is contained in atopological disc D with Area( D ) < d +1 , then | η d ( ϕ ) | (cid:54) d Area( D ) .4. C continuity: The mapping η d : Ham( S , ω ) → R is continuous withrespect to the C topology on Ham( S , ω ) and, moreover, it extendscontinuously to Homeo ( S , ω ) . We recall in relation to the support-control inequality that the total areaof the sphere is assumed to be one. 30 roof of Proposition 3.9.
The first and third properties are immediate con-sequences of the same properties of the invariants c d,k ; see Theorem 3.1 andProposition 3.2. To prove the second, note first of all that if H and G areany Hamiltonians, then by the Continuity property of Theorem 3.1, we have( c d ( H ) − c d ( G )) − d c ( H ) − c ( G )) ≤ d || H − G || . Now let K be any Hamiltonian generating ϕ − ψ and let G generate ϕ . Then H := G K generates ψ and hence by the above inequality η d ( ψ ) − η d ( ϕ ) ≤ d || K || , η d ( ϕ ) − η d ( ψ ) ≤ d || K || , hence the Hofer continuity property, since K was arbitrary.We only have to establish the C -Continuity property. This takes upthe remainder of this subsection. Our proof will follow the lines (and usesome of the intermediate steps) of [CGHS20, Section 4], which establisheda similar result for the invariant c d restricted to maps supported in thenorthern hemisphere. We fix some degree d >
0. The result will follow fromthe next proposition.
Proposition 3.10.
For any h ∈ Homeo ( S , ω ) and ε > , there exists δ > such that for all f, g ∈ Ham( S , ω ) satisfying d C ( f, h ) < δ and d C ( g, Id) < δ , the inequality | η d ( gf ) − η d ( f ) | < ε holds. Let us temporarily assume this proposition and explain how it impliesthe C -Continuity property of η d . Let ( f i ) i ∈ N ∈ Ham( S , ω ) be a sequencewhich C converges to h ∈ Ham( S , ω ). We may write f i in the form g i h , with d C ( g i , Id) → i goes to ∞ . By the proposition we have | η d ( g i h ) − η d ( h ) | →
0, which proves the C -continuity. To prove extensionto Homeo ( S , ω ) let h ∈ Homeo ( S , ω ) and let ( f i ) i ∈ N ∈ Ham( S , ω ) be asequence which C -converges to h . Then, d C ( f i , f j ) = d C ( f i f − j , Id) be-comes arbitrarily small when i, j are large enough and so Proposition 3.10implies that | η d ( f i ) − η d ( f j ) | = | η d (( f i f − j ) f j ) − η d ( f j ) | becomes arbitrar-ily small for i, j large enough so that η d ( f i ) converges. Proposition 3.10also similarly implies that if ( f (cid:48) i ) i ∈ N is another sequence converging to h ,then | η d ( f i ) − η d ( f (cid:48) i ) | →
0, hence the limit does not depend on the choiceof limiting sequence. This allows us to consistently define η d ( h ) for any h ∈ Homeo ( S , ω ) by setting η d ( h ) := lim i →∞ η d ( f i ) , for any sequence f i which C -converges to h .We now prove Proposition 3.10. 31 roof of Proposition 3.10. We give the proof of this proposition in two steps.
Step A. Continuity in the non-finite order case.
We first assume that h is not of finite order in the group Homeo ( S , ω ).Then, there exists a point x ∈ S such that h d ! ( x ) (cid:54) = x . For such a point andfor any integers 0 (cid:54) p < q (cid:54) d , we have h q − p ( x ) (cid:54) = x . By composing with h p , we also have h q ( x ) (cid:54) = h p ( x ). Therefore, the points x, h ( x ) , . . . , h d ( x ) arepairwise distinct. Let B be a small ball centered at x , such that the closure B of B is d -displaced by h .Let ε >
0. We choose δ (cid:48) > f such that d C ( f, h ) < δ (cid:48) must also d -displace B .The next lemma says roughly that a C -small element of Ham( S , ω ) isHofer-close to being supported in B . Lemma 3.11.
Let B be any open topological disc. For all ε (cid:48) > , thereexists δ > , such that for all g ∈ Ham( S , ω ) with d C ( g, Id) < δ , there is φ ∈ Ham( S , ω ) supported in B such that d H ( φ, g ) (cid:54) ε (cid:48) . A similar result was proved in [CGHS20, Lemma 4.6] but only for maps g supported in the northern hemisphere. By conjugating by an appropriatearea preserving map, this particular case implies that Lemma 3.11 holds formaps g supported in any topological disk of area . In fact, the factor here is not essential to the proof of [CGHS20, Lemma 4.6]: the exact sameargument, which we omit for brevity, shows that it also holds for maps sup-ported in an embedded disk of any area The general case then immediatelyfollows from the next fragmentation lemma: indeed, given the lemma below,and given g , we can first fragment g into maps supported on embedded disksand then approximate each of these maps by maps supported in B . Lemma 3.12. [Sey13, Prop 3.1] There exists two open topological embeddeddiscs D , D which cover S , such that for any α > , there exists δ > such that for any g ∈ Ham( S , ω ) satisfying d C ( g, Id) < δ , there exists g , g ∈ Ham( S , ω ) , with supp( g i ) ⊂ D i and d C ( g i , Id) < α for i = 1 , ,such that g = g ◦ g . Having established Lemma 3.11, we can now continue the proof of Propo-sition 3.10. Let δ > ε (cid:48) = εd . Wemay assume without loss of generality that δ < δ (cid:48) . Then, by Lemma3.11, for any Hamiltonian diffeomorphism g with d C ( g, Id) < δ , thereexists φ ∈ Ham( S , ω ) supported in B satisfying d H ( φ, g ) (cid:54) εd . Now let To help the reader who reads the argument in [CGHS20, Lemma 4.6], we note thatthe only change is that the factors of 1 /
2, which come from the fact that the northernhemisphere has area 1 /
2, see the end of the second paragraph of the proof there, must bechanged to some number (cid:96) <
1. This change can be accommodated by choosing what arecalled N and m in the proof to be such that (cid:96)/N < area( B ) , (cid:96)/m < / (cid:96) N +1 m < (cid:15). With the preceding changes understood, the argument can then be repeated verbatim. , g ∈ Ham( S , ω ) be such that d C ( f, h ) < δ and d C ( g, Id) < δ . By Hofercontinuity of η d and Lemma 3.3, it follows that | η d ( gf ) − η d ( f ) | (cid:54) | η d ( gf ) − η d ( φf ) | + | η d ( φf ) − η d ( f ) | (cid:54) d d H ( gf, φf ) + 0 = d d H ( g, φ ) (cid:54) ε. This concludes the proof in the case where h is not of finite order. Step B. The non finite-order case.
We will now conclude the argument by reducing to the case where h is offinite order to the case where it is not. Let h be of finite order and let ε > ψ such that (cid:107) ψ (cid:107) < ε and hψ isnot of finite order Then, by Step A, there exists δ such that for any f (cid:48) , g (cid:48) satisfying d C ( f (cid:48) , hψ ) < δ and d C ( g (cid:48) , h ) < δ , we have | η d ( g (cid:48) f (cid:48) ) − η d ( f (cid:48) ) | < ε .Now take f, g as in the statement of the proposition. We now apply thetriangle inequality to obtain | η d ( gf ) − η d ( f ) | ≤ | η d ( gf ) − η d ( gf ψ ) | + | η d ( gf ψ ) − η d ( f ψ ) | + | η d ( f ψ ) − η d ( f ) | . By Hofer continuity and the above estimate on (cid:107) ψ (cid:107) , we have | η d ( gf ) − η d ( gf ψ ) | ≤ ε , | η d ( f ψ ) − η d ( f ) | ≤ ε . Thus, to finish the proof of the proposition, it remains to show that | η d ( gf ψ ) − η d ( f ψ ) | ≤ ε . This follows from the previous paragraph, since if d C ( f, h ) < δ, then d C ( f ψ, hψ ) < δ . Equipped with our new spectral invariants, we now prove Theorem 1.4. For the benefit of the reader, we briefly sketch why such a ψ exists. Since h has finiteorder, all x ∈ S are periodic and we let (cid:96) be the maximal period of a point. Then, theset of points of period (cid:96) is open, because it is { x ∈ S : h k ( x ) (cid:54) = x, ∀ k = 1 , . . . , (cid:96) − } . Itfollows that if we fix a point x of period (cid:96) , there exists an open set U containing x suchthat h (cid:96) | U = Id U and U, . . . , h (cid:96) − ( U ) are all disjoint. Now, let ψ be a C -small (henceHofer small) map supported in U which coincides with an irrational rotation around x ina smaller open subset V ⊂ U . Then, hψ does not have finite order. Indeed, for any n ∈ N , y ∈ V \ { x } , we have ( hψ ) n ( y ) = ψ n/(cid:96) ( y ) if (cid:96) divides n and ( hψ ) n ( y ) / ∈ U otherwise. Inboth cases, ( hψ ) n ( y ) (cid:54) = y . Thus, such ψ suits our needs. .1 Homogenization and monotone twists We begin by recalling the combinatorial model of Theorem 6.1 of [CGHS20],which gives an explicit formula for c d ( H ) where H : S → R is a monotonetwist, and which we will need below and for the proof of Theorem 1.7 aswell.Here and below we use the notations of [CGHS20, Section 5.2]. Tosummarize, recall that a lattice path P is the graph of a piecewise linearfunction Y : [0 , d ] → R ≥ , such that the vertices of P are at integer latticepoints; the number d is called the degree of the path and is assumed aninteger below. A lattice path is called concave if it never crosses below anyof its tangent lines. We can think of a lattice path as a collection of maximalline segments, called edges , joined end to end. We regard any edge as aninteger multiple m p,q of a primitive vector ( q, p ); these vectors are directedwith the convention that q ≥ . To any concave lattice path, we associate a number j ( P ) as follows. Weform the region R + bounded by the x -axis, the line x = d , and the part of P above the x -axis, we form the region R − bounded by the axes and the partof P below the x -axis, we define j + to be the number of lattice points in R + , not including lattice points on P , and we define j − to be the number oflattice points in R − , not including the lattice points on the x -axis. Finally,we define j ( P ) := j + − j − . See Figure 1. We define the combinatorialindex I ( P ) by I ( P ) := 2 j ( P ) − d.P d R + R − Figure 1: The lattice points which contribute to the count for j ( P ) arecircled. Here, j + ( P ) = 5, j − ( P ) = 5, d = 6, thus j ( P ) = 0 and I ( P ) = − H = h be a monotone twist. Assume that h ( −
1) = h (cid:48) ( −
1) =0 , h (cid:48)(cid:48) >
0, and h (cid:48) (1) is an integer. We call such a monotone twist nice . Wecall a lattice path P compatible with h if for every edge m p,q ( q, p ), thereexists some z p,q such that h (cid:48) ( z p,q ) = p/q. If P is a concave lattice path,34ompatible with a nice monotone twist, then we define the action A ( P ) bydefining A ( q, p ) := 12 ( p (1 − z p,q ) + qh ( z p,q )) , (28)and extending by linearity.We can now state the formula from [CGHS20, Thm. 6.1] for computingthe invariants c d,k . That is, for all degree d (cid:62) k , when H is a nice monotone twist we have c d,k ( H ) = max {A ( P ) : 2 j ( P ) − d = k } , (29)where the maximum is over concave lattice paths that are compatible with h .We are justified in invoking this formula because our nice monotone twistsare in H , and the c d,k defined here extend the definition from [CGHS20], seethe discussion at the beginning of Section 3. ζ d of monotone twists We now apply the combinatorial model from the previous section to computethe invariants ζ d in the case of monotone twists. In particular, we can nowgive the promised proof of Proposition 1.10. Proof of Proposition 1.10.
First note that ζ d is Lipschitz continuous withrespect to the uniform norm on C ∞ ( S ); this follows from Hofer continuity of c d . Thus, both sides of the equation in Proposition 1.10 are continuous withrespect to uniform norm. Since any monotone twist H can be approximateduniformly by nice monotone twists, we deduce that it is sufficient to proveProposition 1.10 for nice monotone twists. For the rest of the proof, wetherefore assume that H is a nice monotone twist.We note that the index I ( P ) = 2 j ( P ) − d from 4.1.1 is equivalently givenby the following formula I ( P ) = 2 A + y + w − e, (30)where y and w are respectively the minimal and maximal vertical coordinateof P , e is the number of edges in P and A is the (signed) area of the regionenclosed by P , the x -axis and the vertical line { d } × R . Indeed, by shiftingthe path if necessary, it suffices to prove this when y = 0, in which case itfollows from Pick’s formula that this corresponds to the definition given in4.1.1.Let us introduce some notation. For any i ∈ { , . . . , d } , we set a i = Y ( i ) − Y ( i − Y is the function [0 , d ] → R , such that P = graph( Y ) = { ( x, Y ( x )) : x ∈ [0 , d ] } . Then,2 A = 2 dy + a + (2 a + a ) + · · · + (2 a + · · · + 2 a d − + a d )= 2 dy + (2 d − a + (2 d − a + · · · + a d w = y + a + · · · + a d , the condition I ( P ) = − d becomes 2 dy + 2 y + 2 da + (2 d − a + · · · + 2 a d − e = − d. Therefore, under this condition, we may express y in terms of the a i as: y = − d +1 ( da + ( d − a + . . . a d + d − e ) = e − d d +1) + d (cid:88) i =1 ( − a i + id +1 a i ) . (31)Let us now turn our attention to the action. It is given by (28): we statehere a reformulated version with the a i , namely A ( P ) = y + d (cid:88) i =1 12 ( a i (1 − z i ) + h ( z i )) , where z i is the unique point such that h (cid:48) ( z i ) = a i . Using (31), we obtain A ( P ) = e − d d +1) + d (cid:88) i =1 ( id +1 a i − a i ) + d (cid:88) i =1 12 ( a i (1 − z i ) + h ( z i )) , = e − d d +1) + d (cid:88) i =1 (cid:16) h (cid:48) ( z i )( − id +1 − z i ) + h ( z i ) (cid:17) (32)Note that the term e − d d +1) belongs to ( − , P have horizontal displacement 1.By (29), c d = c d, − d is obtained by maximizing the value of A ( P ) over allpossible paths with I ( P ) = − d .To compute this maximum, consider the function F ( t , . . . , t d ) = d (cid:88) i =1 (cid:16) h (cid:48) ( t i )( − id +1 − t i ) + h ( t i ) (cid:17) , defined on the set E of tuples ( t , . . . , t d ) such that − (cid:54) t (cid:54) . . . (cid:54) t d (cid:54) A ( P ) = F ( z , . . . , z d ) + e − d d +1) . (33)We may compute the partial derivatives of F , ∂F∂t i = ( − id +1 − t i ) h (cid:48)(cid:48) ( t i ) , and we see that it is positive for t i < − id +1 and negative for t i > − id +1 . This implies that F attains its maximum at ( t , . . . , t n ) such that t i = − id +1 , for all i . Thus,max E F = d (cid:88) i =1 h ( − id +1 ) . (34)36ow it follows from the versions of (29), (33), (34) for nH , and the factthat e − d ≤
0, that for all n ,1 n c d ( nH ) (cid:54) d (cid:88) i =1 h ( − id +1 ) . (35)To proceed, let a n (cid:54) . . . (cid:54) a nd be sequences in n N which converge respec-tively to − d +1 , − d +1 , . . . , − d +1 . Set z ni such that h (cid:48) ( z ni ) = a ni .Then, F ( z n , . . . , z nd ) n →∞ −→ max E F = d (cid:88) i =1 h ( − id +1 ) . Moreover, we can construct a lattice path, for nH , such that (33) holds withthe z i = z ni , and e = d . We therefore deduce n c d ( nH ) (cid:62) F ( z n , . . . , z nd )and we conclude in combination with (35) thatlim n →∞ n c d, − d ( nH ) = d (cid:88) i =1 h ( − id +1 ) , as desired. Remark 4.1. (i) It follows from the previous proposition that we havethe following Composition property for monotone twists: For any twomonotone twists φ, ψ we have µ d ( φψ ) = µ d ( φ ) + µ d ( ψ ) . (36)Indeed, it follows from Proposition 1.10 that ζ d ( H + H ) = ζ d ( H ) + ζ d ( H ), hence (36).(ii) For any monotone twist ϕ ∈ Ham( S , ω ), it can easily be shown that µ d ( ϕ ) = lim n →∞ c d ( ˜ ϕ n ) n , i.e. the lim sup in (2) is in fact a limit for such ϕ . (cid:74) We will now construct a certain family of Hamiltonian diffeomorphismswhich will be used to establish Theorem 1.4. Let U ⊂ S be a properopen set containing the North Pole p + and let ι > i ∈ N , we denote D i := { ( z, θ ) ∈ S : 1 − d i < z ≤ } , where d i = 2 ι + i +1 .
37e choose ι large enough so that all the D i ’s are included in U . Notethat each D i is an embedded disc andArea( D i ) = 1 d i . The next lemma states the properties of our family of maps.
Lemma 4.2.
There exist autonomous Hamiltonians ( H i ) i ∈ N , such that H i is supported in D i and the following properties are satisfied for all t (cid:62) , i ∈ N :1. ϕ tH i is a monotone twist, for all t ∈ R ,2. Cal( ϕ tH i ) = t ,3. d H ( ϕ tH i , Id) (cid:54) t + 2 ,4. If j > i , then µ d i ( ϕ tH j ) = − t d i µ d i ( ϕ tH i ) > − t d i and if j < i , then µ d i ( ϕ tH j ) (cid:62) − t d i .6. ϕ tH i ϕ sH j = ϕ sH j ϕ tH i for all t, s ∈ R .Proof. Consider the functions f i : [ − , → R defined by f i ( z ) := (cid:40) z ∈ [ − , − d i ] ,d i (cid:16) z − (1 − d i ) (cid:17) , z ∈ [1 − d i , . These functions are non smooth but to ensure that our future Hamiltoniansare smooth, we approximate them by smooth functions h i satisfying thefollowing conditions:(i) h (cid:48) i ( z ) , h (cid:48)(cid:48) i ( z ) (cid:62) | f i ( z ) − h i ( z ) | (cid:54) d i for all z ∈ [ − , h i is contained in the interior of [1 − d i , (cid:82) − h i ( z ) dz = 2.Let H i ( z, θ ) = h i ( z ) and observe that H i is supported in D i and that ϕ tH i is a monotone twist for any t > ϕ tH i ) = t readily follows from Property (iii).To prove the third item, we will need the following lemma, whose proofwe postpone to the end of this section. The idea behind this lemma goesback to Sikorav, who implemented it in the case of R n in [Sik90]; see also[HZ94, Chap. 5 - 5.6]. 38 emma 4.3. Let H : S → R denote an autonomous Hamiltonian such thatthe support of H is contained in a disc D with the property that Area( D ) < N . Then, d H ( ϕ H , Id) ≤ N max( H ) + 2 . Since the support of H i is a disk of area less than d i , this lemma leadsto d H ( ϕ tH i , Id) (cid:54) d i max( tH i ) + 2 (cid:54) (cid:18) d i (cid:19) t + 2 (cid:54) t + 2 , which implies the third item.Item 4 is a consequence of the Calabi property from 3.8, because ϕ tH j issupported in D j and Area( D j ) < d i +1 for j > i . The last item of Lemma4.2 is also easy to check. Indeed, since the Hamiltonians H i are functions of z , they all Poisson commute, hence their flow commute.There remains to prove item 5. Since ζ d i ( tH j ) = µ d i ( ϕ tH j ) + d i (cid:90) S tH j ω = µ d i ( ϕ tH j ) + t d i , we just need to prove that ζ d i ( H i ) > ζ d i ( H j ) (cid:62) d i when i > j. As already mentioned, the above conditions (i) and (ii) ensure that ϕ tH j is a monotone twist for all t >
0. By Proposition 1.10, ζ d i ( H j ) = d i (cid:88) m =1 h j (cid:18) − md i + 1 (cid:19) . We can rewrite the above sum as ζ d ( H j ) = N (cid:88) k =0 h j (cid:18) − d i − k ) d i + 1 (cid:19) , where N is the largest integer such that − d i − N ) d i +1 is in the support of h j , that is − d i − N ) d i +1 > − d j . A simple computation reveals that N + 1 = d i d j . (37)Consider the (non-smooth) function f j ( z ). By the definition of h j , wehave h j (cid:62) f j − d j and so ζ d i ( H j ) (cid:62) (cid:34) N (cid:88) k =0 f j (cid:18) − d i − k ) d i + 1 (cid:19)(cid:35) − N + 1 d j . (38)The first term on the right hand side in the above equation may be com-puted explicitly. Indeed, f j ( z ) is linear in z and so the above is just an39rithmetic sum. First, note that − d i − k ) d i +1 = 1 − d i +1 ( k + 1), and so (cid:80) Nk =0 f j (cid:16) − d i − k ) d i +1 (cid:17) = (cid:80) Nk =0 f j (cid:16) − d i +1 ( k + 1) (cid:17) . Next, one can easilycheck that f j (cid:16) − d i +1 ( k + 1) (cid:17) = 2 d j − d j d i +1 ( k + 1). Thus, N (cid:88) k =0 f j (cid:18) − d i + 1 ( k + 1) (cid:19) = N (cid:88) k =0 (cid:34) d j − d j d i + 1 ( k + 1) (cid:35) = 2 d j ( N + 1) − d j d i + 1 ( N + 1)( N + 2)2= d i (cid:18) − d i + d j d i + 1 (cid:19) . (39)For i = j , (37), (38) and (39) yield ζ d i ( H i ) (cid:62) d i (cid:18) − d i d i + 1 − d i (cid:19) > , as desired, which implies the first part of item 5.For i > j , (37), (38) and (39) give ζ d i ( H i ) (cid:62) d i (cid:32) − d i + d j d i + 1 − d j (cid:33) (cid:62) d i (cid:18) − d i + d j d i − (cid:19) = 12 d i (cid:18) − d j d i (cid:19) (cid:62) d i . where we used d j (cid:54) for the second inequality and d j d i (cid:54) for the lastinequality. This concludes the proof of item 5.We end this section (and the proof of Lemma 4.2) with the proof ofLemma 4.3. Proof of Lemma 4.3.
Since Area( D ) < N , by Lemma 2.1, we can find ψ , . . . , ψ N ∈ Ham( S , ω ) such that1. ψ i ( D ) ∩ ψ j ( D ) = ∅ ,2. d H ( ψ i , Id) ≤ N .Define the Hamiltonians F i = N H ◦ ψ − i and note that ϕ F i = ψ i ϕ N H ψ − i .Let F = (cid:80) Ni =1 F i . Observe that, since F i is supported in ψ i ( D ), the F i ’shave disjoint supports, and so max( F ) = max( F i ) = N max( H ). Hence, d H ( ϕ F , Id) (cid:54) N max( H ) , d H ( ϕ H , ϕ F ) (cid:54)
2. This can beproved using Identities (5) and (6) as follows: d H ( ϕ H , ϕ F ) = d H (cid:32) N (cid:89) i =1 ϕ N H , N (cid:89) i =1 ψ i ϕ N H ψ − i (cid:33) (cid:54) N (cid:88) i =1 d H ( ϕ N H , ψ i ϕ N H ψ − i ) (cid:54) N (cid:88) i =1 d H ( ψ i , Id) (cid:54) . We are now ready to present a proof of Theorem 1.4. First note that withoutloss of generality we may assume that the open set U contains the Northpole p + . This allows us to use the constructions of the preceding section.We begin by proving the first part of the theorem, regarding the quasi-flat rank. Proof of Theorem 1.4(a).
Let R n + := { ( t , . . . , t n ) : t i ≥ } . We equip R n + with the distance induced by the sup norm, that is we define the distancebetween ( t , . . . , t n ) , ( s , . . . , s n ) ∈ R n + to be (cid:107) ( t , . . . , t n ) − ( s , . . . , s n ) (cid:107) ∞ = max {| t i − s i | : i = 1 , . . . , n } . The mapping Φ : R n + → Ham( S , ω )( t , . . . , t n ) (cid:55)→ ϕ t H ◦ . . . ◦ ϕ t n H n ◦ ϕ − ( t + ··· + t n ) H n +1 , takes values Ham U ( S , ω ). Moreover, since the H i all have the same integralover S , the mapping Φ takes values in the kernel of the Calabi homomor-phism.We will show that there exists an invertible n × n matrix A such that Φsatisfies the following inequality. (cid:107) A ( t − s ) (cid:107) ∞ (cid:54) d H (Φ( t ) , Φ( s )) (cid:54) n (2 (cid:107) t − s (cid:107) ∞ + 1 + n ) , (40)where t , s stand for ( t , . . . , t n ) and ( s , . . . , s n ), respectively. As a conse-quence, 1 (cid:107) A − (cid:107) op (cid:107) t − s (cid:107) ∞ (cid:54) d H (Φ( t ) , Φ( s )) (cid:54) n (cid:107) t − s (cid:107) ∞ + 2 n + 2 , where (cid:107) A − (cid:107) op is the operator norm of A − , as a linear map of the normedspace ( R n , (cid:107) · (cid:107) ∞ ). 41he above clearly implies that Φ is a quasi-isometric embedding of( R n + , (cid:107) · (cid:107) ∞ ) into (Ham( S , ω ) , d H ). This establishes Theorem 1.4 because( R n , (cid:107) · (cid:107) ∞ ) quasi-isometrically embeds into ( R n + , (cid:107) · (cid:107) ∞ ); an explicit formulafor such a quasi-isometric embedding is given by L : R n → R n + ( x , . . . , x n ) (cid:55)→ ( L ( x ) , . . . , L ( x n )) , where L : R → R is defined as L ( x ) := (cid:40) (0 , − x ) , x ≤ , ( x, , x ≥ . For a proof of the fact that L is a quasi-isometric embedding see [SZ18, Lem.8.12].We now turn our attention to (40), beginning with the following proofof the inequality on its right-hand side: d H (Φ( t ) , Φ( s )) = (cid:107) Φ( t ) ◦ Φ( s ) − (cid:107) = (cid:107) ϕ t − s H . . . ϕ t n − s n H n ϕ − (cid:80) i ( t i − s i ) H n +1 (cid:107) (cid:54) (cid:107) ϕ − (cid:80) i ( t i − s i ) H n +1 (cid:107) + n (cid:88) i =1 (cid:107) ϕ t i − s i H i (cid:107) (cid:54) n (cid:88) i =1 (4 | t i − s i | + 2) + 2 (cid:54) n (cid:18) (cid:107) t − s (cid:107) ∞ + 1 + 1 n (cid:19) . Above, the second equality on the first line is a consequence of the lastitem in Lemma 4.2, the first inequality on the second line follows from trian-gle inequality and the second inequality in the second line is a consequenceof Lemma 4.2, item 3. This proves the right hand side of (40).It remains to prove the left hand side. Let us consider the following twofamilies of monotone twists. α ( t ) = ϕ t H . . . ϕ t n H n , β ( t ) = ϕ ( t + ··· + t n ) H n +1 . By definition Φ( t ) = α ( t ) β ( t ) − and since monotone twists commute d H (Φ( t ) , Φ( s )) = (cid:107) β ( t ) α ( t ) − α ( s ) β ( s ) − (cid:107) = (cid:107) ( α ( t ) β ( s )) − α ( s ) β ( t ) (cid:107) = d H ( α ( t ) β ( s ) , α ( s ) β ( t )) . Now, combining the previous equality with the second item of Proposition3.8 gives max i =1 ,...,n (cid:12)(cid:12)(cid:12)(cid:12) µ d i ( α ( t ) β ( s )) d i − µ d i ( α ( s ) β ( t )) d i (cid:12)(cid:12)(cid:12)(cid:12) ≤ d H (Φ( t ) , Φ( s )) . (41)42y the fourth item of Proposition 3.8 and (36), we can write, µ d ( α ( t ) β ( s )) − µ d ( α ( s ) β ( t )) = n (cid:88) j =1 (cid:16) µ d ( ϕ H j ) − µ d ( ϕ H n +1 ) (cid:17) ( t j − s j ) , for any d . It follows from the above that the left hand side in (41) coincideswith the quantity (cid:107) A ( t − s ) (cid:107) ∞ , where A is the matrix whose ij entry (for 1 (cid:54) i, j (cid:54) n ) is A ij = µ d i ( ϕ H j ) − µ d i ( ϕ H n +1 ) d i . The fourth item in Lemma 4.2 tells us that µ d i ( ϕ H n +1 ) = µ d i ( ϕ H j ) = − d i , for j > i . It follows that A ij = 0 for j > i , i.e. the matrix A islower triangular. From the fifth item of the same lemma, we deduce thatthe diagonal entries of A are non-zero. Hence, A is invertible which proves(40). We have completed the proof of Theorem 1.4(a). In this section, we prove the remainder of Theorem 1.4, i.e. that the kernel ofthe Calabi Homomorphism defined on Ham U ( S , ω ) is not coarsely proper.Recall from the introduction that a metric space ( X, d ) is said to be coarselyproper if there exists R > X, d ) canbe covered by finitely many balls of radius R . Proof of Theorem 1.4(b).
For any fixed r > X r := { ϕ − rH ϕ rH i : i (cid:62) } , where the H i are the Hamiltonians provided by Lemma 4.2. Claim 4.4.
The set X r is r separated, i.e. for i (cid:54) = j we have r ≤ d H ( ϕ − rH ϕ rH i , ϕ − rH ϕ rH j ) . The above claim implies that the set X r , which is bounded by Lemma4.2, cannot be covered by finitely many balls of radius r . Since this holdsfor every value of r , and since X r is included in ker(Cal), we conclude thatker(Cal) is not coarsely proper, hence the Theorem.It remains to prove Claim 4.4. 43 roof of Claim 4.4. Suppose that i < j , pick k ∈ N such that 2 i < k < j and consider d k as in Lemma 4.2. By the Hofer continuity property of µ d k ,from Proposition 3.8, we have1 d k | µ d k ( ϕ rH i ) − µ d k ( ϕ rH j ) | (cid:54) d H ( ϕ rH i , ϕ rH j ) = d H ( ϕ − rH ϕ rH i , ϕ − rH ϕ rH j ) . By Lemma 4.2, we have µ d k ( ϕ rH j ) = − r d k and µ d k ( ϕ rH i ) (cid:62) − r d k . Thus, d k | µ d k ( ϕ rH i ) − µ d k ( ϕ rH j ) | (cid:62) r which completes the proof.We have now proved Theorem 1.4(b). Homeo ( S , ω ) We conclude by proving Theorem 1.7.
To prove non-simplicity of Homeo ( S , ω ), we explicitly construct a propernormal subgroup which we call the group of finite energy homeomor-phisms and denote by FHomeo( S , ω ). We introduced these homeomor-phisms in [CGHS20] where we proved that they form a proper normal sub-group of the compactly supported area-preserving homeomorphisms of thedisc. Here, we will give a slight variant of the definition in [CGHS20] whichis more natural from the point of view of Hofer’s geometry. Definition 5.1.
We say ϕ ∈ Homeo ( S , ω ) is a finite-energy homeomor-phism if there exists a sequence of Hamiltonian diffeomorphisms { ϕ i } i ∈ N which is bounded with respect to Hofer’s distance and which converges uni-formly to ϕ . We denote by FHomeo( S , ω ) the set of all finite-energy home-omorphisms. Theorem 1.7 follows immediately from the following result, which willoccupy the remainder of this section.
Theorem 5.2.
FHomeo( S , ω ) is a proper normal subgroup of Homeo ( S , ω ) . We prove the above using arguments similar to those given in [CGHS20].Here is a brief outline. As we shall see, it is not hard to show thatFHomeo( S , ω ) forms a normal subgroup of Homeo ( S , ω ); the main chal-lenge is proving that it is proper.To do this, we use the invariant η d : Ham( S , ω ) → R . We showed abovethat this is continuous with respect to the C topology on Ham( S , ω ) and,moreover, it extends continuously to Homeo ( S , ω ); see Proposition 3.9.44 straightforward argument shows that for any ϕ ∈ FHomeo( S , ω ) thereexists a constant C , depending on ϕ , such that for all (even) d we have η d ( ϕ ) d (cid:54) C. (42)We will then prove Theorem 5.2 by showing that certain so-called infi-nite twist homeomorphisms ψ ∈ Homeo ( S , ω ) satisfy the following;lim d →∞ η d ( ψ ) d = ∞ . (43)This violates (42). This last step requires estimating η d ( ψ ) for which werely on the combinatorial model from Section 4.1.1.We end this section by highlighting the differences between our proof,in this article, of non-simplicity of Homeo ( S , ω ) and the proof of non-simplicity of Homeo c ( D , ω ) given in [CGHS20]. In both articles we usePFH spectral invariants c d : C ∞ ( S × S ) → R . Given an arbitrary Hamil-tonian H , the value of c d ( H ) depends on H and so c d does not yield a well-defined invariant of Hamiltonian diffeomorphisms. However, in [CGHS20]we overcome this problem by restricting the domain of c d to a certain class ofHamiltonians which is suitable for the purposes of that article; see [CGHS20,Sec. 3.4]. In the current article, we do not have the possibility of restrictingthe domain of c d . Instead, we work with η d which is well-defined for allHamiltonian diffeomorphisms of the sphere as proved in Section 3.2.Another difference between the two proofs is the manner in which weshow properness of FHomeo. In both articles this is achieved by exhibitingarea-preserving homeomorphisms ψ satisfying Equation (43). The proof ofthis given in [CGHS20] involves verifying for certain smooth twist maps aconjecture of Hutchings, concerning recovering the Calabi invariant fromthe asymptotics of PFH spectral invariants, whereas our proof here, whichis shorter, uses the forthcoming Claim 5.4. The proof of this claim, however,relies on the combinatorial model for PFH developed in [CGHS20, Sec. 5].We should remark that part of the motivation for the somewhat longerargument in [CGHS20] was that Hutchings’ conjecture is of independentinterest, hence useful to verify for twist maps. We now carry out the above outline. We begin by describing the infinitetwist homeomorphisms ψ .Denote by p + ∈ S the North Pole of the sphere, i.e. the point whose z -coordinate is 1, in the cylindrical coordinate system introduced in Section2.1. We say a function F : S \{ p + } → R is an infinite twist Hamiltonian if it is of the form F ( z, θ ) = 12 f ( z ) , (44)45here f : [ − , → R is a smooth function such that f (cid:48) ≥ , f (cid:48)(cid:48) ≥ d →∞ d f (cid:18) − d + 1 (cid:19) = ∞ . (45)Observe that F defines a smooth Hamiltonian on S \ { p + } whose flow isgiven by ϕ tF ( θ, z ) = ( θ + 2 πtf (cid:48) ( z ) , z ) . We extend the flow ϕ tF to S by defining ϕ tF ( p + ) = p + ; this yields anarea-preserving flow on S which is non-smooth at the point p + . We say ψ ∈ Homeo ( S , ω ) is an infinite twist homeomorphism if it is of theform ψ := ϕ F (46)for some F . We will call ψ an adapted infinite twist if the corresponding f satisfies the following technical hypothesis: f (cid:48) (cid:18) − d + 1 (cid:19) ∈ ( d + 1) N , for all d ≥ Proof of Theorem 5.2.
We begin by noting that the argument in [CGHS20,Prop. 2.1], repeated verbatim, shows that FHomeo( S , ω ) forms a normalsubgroup of Homeo ( S , ω ). It remains to show that it is proper. Step 1. Linear growth in
FHomeo . We first show that for any ϕ ∈ FHomeo( S , ω ) the linear growth con-dition (42) holds. This is an immediate consequence of the properties inProposition 3.9. Indeed, let ϕ ∈ FHomeo( S , ω ) and choose a sequence ϕ i C −−→ ϕ that is uniformly bounded with respect to Hofer’s distance. Sincethe ϕ i are bounded and η d (Id) = 0, the Hofer continuity property ensures abound of the form η d ( ϕ i ) ≤ d · C for some uniform constant C ; then, by C continuity, the same bound holds for ϕ . Step 2. Superlinear growth of some infinite twists.
It remains to prove that FHomeo is proper. The structure of the remain-der of our argument will now be to prove properness, assuming the technicalClaim 5.3 below which makes use of the adapted condition, and then provethe Claim. From now until the end of the paper, we therefore assume that F is an adapted infinite twist Hamiltonian whose support is contained inthe interior of the disc { ( z, θ ) : z ≥ } which is of area . Imposing this as-sumption enables us to apply the following promised technical claim. Recallbelow that the η d are defined only for even d .46 laim 5.3. Fix d ≥ , define z := 1 − d +1 . Let H be a smooth monotonetwist Hamiltonian supported in a disc of area at most / . Assume that p := h (cid:48) ( z ) ∈ ( d + 1) N . Then η d ( ϕ H ) ≥ H ( z ) − d . We defer the proof for the moment. Assuming it, we can produce superlinear growth of the η d as follows. Claim 5.4. η d ( ϕ F ) ≥ f (1 − d +1 ) − d , for d ≥ .Proof of Claim 5.4. For every i ∈ N , let F i : S → R be a sequence ofsmooth Hamiltonians of the form F i ( z, θ ) = 12 f i ( z ) , where f i : [ − , → R is a smooth function such that f (cid:48) i ≥ , f (cid:48)(cid:48) i ≥ f i ( z ) = f ( z ) for z ∈ [ − , − i ].Observe that ϕ F i C −−→ ϕ F , because ( ϕ F i ) − ◦ ϕ F is supported in thedisc { ( z, θ ) : z ≥ − i } . Hence, by the C continuity of η d established inProposition 3.9, we have η d ( ϕ F ) = lim i →∞ η d ( ϕ F i ) . for every d . Applying Claim 5.3 to F i , for i sufficiently large with respect to d , yields η d ( ϕ F i ) ≥ F i (cid:16) − d +1 (cid:17) − d = f i (cid:16) − d +1 (cid:17) − d = f (cid:16) − d +1 (cid:17) − d , for d >
3. Hence, the claim.It follows immediately from the previous claim that ψ := ϕ F satisfiesEquation (43) which, as explained in Step 1, implies that an adapted infinitetwist ϕ F is not a finite-energy homeomorphism.To complete the proof of Theorem 5.2, it therefore remains to proveClaim 5.3. Proof of Claim 5.3.
Recall, from Equation (27), that η d ( ϕ ) = c d ( H ) − d c ( H ). Hence, to prove the Claim, it is sufficient to show that the fol-lowing two inequalities hold: c ( H ) (cid:54) . (47) c d ( H ) ≥ H (cid:18) − d + 1 (cid:19) . (48)47o prove (47), we invoke the Support-control inequality of Proposition3.2, which gives c ( H ) ≤ · · , since the area of the support of H isbounded by .Next, we prove (48). By the Continuity property of c d from Theorem3.1, we may perform a small perturbation of h , near z = 1, and assume that h (cid:48) (1) ∈ N , in other words that our twist is nice. This allows us to applyTheorem 6.1 of [CGHS20] whose statement we recalled in Section 4.1.1.Recall the notation z = 1 − d +1 . By Theorem 6.1 of [CGHS20] we have c d ( H ) ≥ A ( P ) , for any degree d lattice path P of combinatorial index I ( P ) = − d ; seeSection 4.1.1.Recall the notation p := h (cid:48) ( z ), which is by assumption an integer. Byassumption, there exists an integer a > p = a ( d + 1). Take P tobe the lattice path obtained by joining the lattice points (0 , − a ), ( d − , − a )and ( d, p − a ). This is a concave lattice path made of two edges. It satisfies A ( P ) = p − z ) + 12 h ( z ) − a,I ( P ) = 2 j ( P ) − d = 2(( p − a ) − da ) − d = − d. Hence, c d ( H ) ≥ p − z ) + 12 h ( z ) − a = p (cid:18) − z − d + 1 (cid:19) + 12 h ( z )= 12 h ( z ) = H (cid:18) − d + 1 (cid:19) . We have completed the proof of Theorem 5.2.
We close by briefly discussing the large scale geometry of FHomeo.It is possible to define Hofer’s distance for area-preserving homeomor-phisms as follows. Given ϕ ∈ FHomeo( S , ω ), we define its Hofer distancefrom the identity by ˜ d H ( ϕ, Id) := lim inf d H ( ϕ i , Id) , (49)where the infimum is taken over all sequences { ϕ i } ⊂ Ham( S , ω ) whichconverge uniformly to ϕ . Define ˜ d H ( ϕ, ψ ) := ˜ d H ( ϕ − ψ, Id).We leave it to the reader to check that this defines a bi-invariant distanceon FHomeo( S , ω ). 48t is a natural question to try to better understand this space. Forexample, one could ask if FHomeo has infinite quasi-flat rank. We stronglysuspect that the answer is, in fact, positive as our tools are robust withrespect to the C topology and so one can adapt the proof of Theorem 1.4to prove that FHomeo does have infinite quasi-flat rank. Similarly, it canbe shown that FHomeo is not coarsely proper.One could define ˜ d H ( ϕ, Id), via (49), for arbitrary ϕ ∈ Homeo ( S , ω ).If ϕ is not a finite energy homeomorphism, i.e. if ϕ / ∈ FHomeo( S , ω ),then we get ˜ d H ( ϕ, Id) = ∞ . Hence, we may view homeomorphisms which are not finite-energy as thosewhich are infinitely far from diffeomorphisms, in Hofer’s distance. This is thepoint of view expressed in Le Roux’s article [LR10, Question 1]. Theorem5.2 tells us that such homeomorphisms do exist.A question which arises immediately as a consequence of our definitionof ˜ d H is whether ˜ d H ( ϕ, ψ ) coincides with the usual Hofer distance d H ( ϕ, ψ )when ϕ, ψ ∈ Ham( S , ω ). We do not know the answer to this question.Note that this is equivalent to asking if the (usual) Hofer distance is lowersemi-continuous with respect to the C topology; this was posed as an openquestion by Le Roux in [LR10]. References [AGKK +
19] D. Alvarez-Gavela, V. Kaminker, A. Kislev, K. Kliakhandler,A. Pavlichenko, L. Rigolli, D. Rosen, O. Shabtai, B. Stevenson,and J. Zhang. Embeddings of free groups into asymptotic conesof Hamiltonian diffeomorphisms.
J. Topol. Anal. , 11(2):467–498, 2019.[Ban78] Augustin Banyaga. Sur la structure du groupe desdiff´eomorphismes qui pr´eservent une forme symplectique.
Comment. Math. Helv. , 53(2):174–227, 1978.[BEH +
03] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, andE. Zehnder. Compactness results in symplectic field theory.
Geom. Topol. , 7:799–888, 2003.[BS17] Michael Brandenbursky and Egor Shelukhin. The L p -diameterof the group of area-preserving diffeomorphisms of S . Geom.Topol. , 21(6):3785–3810, 2017.[Cal70] Eugenio Calabi. On the group of automorphisms of a symplec-tic manifold.
Problems in analysis (Lectures at the Sympos. inhonor of Salomon Bochner, Princeton Univ., Princeton, N.J.,1969) , pages 1–26, 1970.49CdlH16] Yves Cornulier and Pierre de la Harpe.
Metric geometry oflocally compact groups , volume 25 of
EMS Tracts in Mathe-matics . European Mathematical Society (EMS), Z¨urich, 2016.[CGHS20] Dan Cristofaro-Gardiner, Vincent Humili`ere, and Sobhan Sey-faddini. Proof of the simplicity conjecture. arXiv:2001.01792 ,2020.[Che18] Guanheng Chen. Cobordism maps on PFH induced by lef-schetz fibration over higher genus base. arXiv:1709.04270 ,2018.[CM05] Kai Cieliebak and Klaus Mohnke. Compactness for puncturedholomorphic curves.
J. Symplectic Geom. , 3(4):589–654, 2005.Conference on Symplectic Topology.[EP03] Michael Entov and Leonid Polterovich. Calabi quasimorphismand quantum homology.
Int. Math. Res. Not. , (30):1635–1676,2003.[EP06] Michael Entov and Leonid Polterovich. Quasi-states and sym-plectic intersections.
Comment. Math. Helv. , 81(1):75–99,2006.[EP09] Michael Entov and Leonid Polterovich. Rigid subsets of sym-plectic manifolds.
Compos. Math. , 145(3):773–826, 2009.[EPP12] Michael Entov, Leonid Polterovich, and Pierre Py. On continu-ity of quasimorphisms for symplectic maps. In
Perspectives inanalysis, geometry, and topology , volume 296 of
Progr. Math. ,pages 169–197. Birkh¨auser/Springer, New York, 2012. Withan appendix by Michael Khanevsky.[Eps70] D. B. A. Epstein. The simplicity of certain groups of homeo-morphisms.
Compositio Math. , 22:165–173, 1970.[Fat80] Albert Fathi. Structure of the group of homeomorphisms pre-serving a good measure on a compact manifold.
Ann. Sci.´Ecole Norm. Sup. (4) , 13(1):45–93, 1980.[Hig54] Graham Higman. On infinite simple permutation groups.
Publ.Math. Debrecen , 3:221–226 (1955), 1954.[Hof90] Helmut Hofer. On the topological properties of symplecticmaps.
Proc. Roy. Soc. Edinburgh Sect. A , 115(1-2):25–38, 1990.[HS05] Michael Hutchings and Michael Sullivan. The periodic Floerhomology of a Dehn twist.
Algebr. Geom. Topol. , 5:301–354,2005. 50HT07] Michael Hutchings and Clifford Henry Taubes. Gluing pseu-doholomorphic curves along branched covered cylinders. I.
J.Symplectic Geom. , 5(1):43–137, 2007.[HT09a] Michael Hutchings and Clifford Henry Taubes. Gluing pseu-doholomorphic curves along branched covered cylinders. II.
J.Symplectic Geom. , 7(1):29–133, 2009.[HT09b] Michael Hutchings and Clifford Henry Taubes. The Weinsteinconjecture for stable Hamiltonian structures.
Geom. Topol. ,13(2):901–941, 2009.[Hum12] Vincent Humili`ere. Hofer’s distance on diameters and theMaslov index.
Int. Math. Res. Not. IMRN , (15):3415–3433,2012.[Hut02] Michael Hutchings. An index inequality for embedded pseu-doholomorphic curves in symplectizations.
J. Eur. Math. Soc.(JEMS) , 4(4):313–361, 2002.[Hut14] Michael Hutchings. Lecture notes on embedded contact homol-ogy. In
Contact and symplectic topology , volume 26 of
BolyaiSoc. Math. Stud. , pages 389–484. J´anos Bolyai Math. Soc., Bu-dapest, 2014.[HZ94] Helmut Hofer and Eduard Zehnder.
Symplectic invariants andHamiltonian dynamics . Birkh¨auser Advanced Texts: BaslerLehrb¨ucher. [Birkh¨auser Advanced Texts: Basel Textbooks].Birkh¨auser Verlag, Basel, 1994.[Kha09] Michael Khanevsky. Hofer’s metric on the space of diameters.
J. Topol. Anal. , 1(4):407–416, 2009.[Kha16] Michael Khanevsky. Hamiltonian commutators with largeHofer norm.
J. Symplectic Geom. , 14(4):1175–1188, 2016.[KS18] Asaf Kislev and Egor Shelukhin. Bounds on spectral normsand barcodes. arXiv:1810.09865 , 2018.[LM95a] Fran¸cois Lalonde and Dusa McDuff. The geometry of symplec-tic energy.
Ann. of Math. (2) , 141(2):349–371, 1995.[LM95b] Fran¸cois Lalonde and Dusa McDuff. Hofer’s L ∞ -geometry:energy and stability of Hamiltonian flows. I, II. Invent. Math. ,122(1):1–33, 35–69, 1995.[LR10] Fr´ed´eric Le Roux. Six questions, a proposition and two pic-tures on Hofer distance for Hamiltonian diffeomorphisms on51urfaces. In
Symplectic topology and measure preserving dy-namical systems , volume 512 of
Contemp. Math. , pages 33–40.Amer. Math. Soc., Providence, RI, 2010.[LT12] Yi-Jen Lee and Clifford Henry Taubes. Periodic Floer ho-mology and Seiberg-Witten-Floer cohomology.
J. SymplecticGeom. , 10(1):81–164, 2012.[McD10] Dusa McDuff. Monodromy in Hamiltonian Floer theory.
Com-ment. Math. Helv. , 85(1):95–133, 2010.[MS17] Dusa McDuff and Dietmar Salamon.
Introduction to symplec-tic topology . Oxford Graduate Texts in Mathematics. OxfordUniversity Press, Oxford, third edition, 2017.[Oh05] Yong-Geun. Oh. Construction of spectral invariants of hamilto-nian paths on closed symplectic manifolds.
The breadth of sym-plectic and Poisson geometry. Progr. Math. , Birkhauser,Boston , pages 525–570, 2005.[Ost03] Yaron Ostrover. A comparison of Hofer’s metrics on Hamilto-nian diffeomorphisms and Lagrangian submanifolds.
Commun.Contemp. Math. , 5(5):803–811, 2003.[Pol93] Leonid Polterovich. Symplectic displacement energy for La-grangian submanifolds.
Ergodic Theory Dynam. Systems ,13(2):357–367, 1993.[Pol98] Leonid Polterovich. Hofer’s diameter and Lagrangian intersec-tions.
Internat. Math. Res. Notices , (4):217–223, 1998.[Pol01] Leonid Polterovich.
The geometry of the group of symplec-tic diffeomorphisms . Lectures in Mathematics ETH Z¨urich.Birkh¨auser Verlag, Basel, 2001.[PR14] Leonid Polterovich and Daniel Rosen.
Function theory on sym-plectic manifolds , volume 34 of
CRM Monograph Series . Amer-ican Mathematical Society, Providence, RI, 2014.[PS] Leonid Polterovich and Egor Shelukhin. Flats in the Hofer ge-ometry on the Hamiltonian group of the two-sphere. Preprint,to appear.[PS16] Leonid Polterovich and Egor Shelukhin. Autonomous Hamilto-nian flows, Hofer’s geometry and persistence modules.
SelectaMath. (N.S.) , 22(1):227–296, 2016.[Py08] Pierre Py. Quelques plats pour la m´etrique de Hofer.
J. ReineAngew. Math. , 620:185–193, 2008.52Sch00] Matthias Schwarz. On the action spectrum for closed symplec-tically aspherical manifolds.
Pacific J. Math. , 193(2):419–461,2000.[Sey13] Sobhan Seyfaddini. C -limits of Hamiltonian paths and theOh-Schwarz spectral invariants. Int. Math. Res. Not. IMRN ,(21):4920–4960, 2013.[Sey14] Sobhan Seyfaddini. Unboundedness of the Lagrangian Hoferdistance in the Euclidean ball.
Electron. Res. Announc. Math.Sci. , 21:1–7, 2014.[Sik90] Jean-Claude Sikorav. Syst`emes hamiltoniens et topologie sym-plectique. Dipartimento di Matematica dell’ Universita di Pisa,ETS, EDITRICE PISA, 1990.[SZ18] Vukaˇsin Stojisavljevi´c and Jun Zhang. Persistence modules,symplectic Banach-Mazur distance and Riemannian metrics. arXiv:1810.11151 , 2018.[Ush13] Michael Usher. Hofer’s metrics and boundary depth.
Ann. Sci.´Ec. Norm. Sup´er. (4) , 46(1):57–128 (2013), 2013.[Wen] Chris Wendl. Lectures on symplectic field theory. arXiv:1612.01009v2 . Dan Cristofaro-GardinerMathematics DepartmentUniversity of California, Santa Cruz1156 High Street, Santa Cruz, California, USASchool of MathematicsInstitute for Advanced Study1 Einstein Drive, Princeton, NJ, USA e-mail : [email protected] Humili`ereCMLS, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128Palaiseau Cedex, France. e-mail: [email protected] SeyfaddiniSorbonne Universit´e, Universit´e de Paris, CNRS, Institut de Math´ematiques deJussieu-Paris Rive Gauche, F-75005 Paris, France. e-mail: [email protected]@imj-prg.fr