Lagrangian configurations and Hamiltonian maps
aa r X i v : . [ m a t h . S G ] F e b LAGRANGIAN CONFIGURATIONSAND HAMILTONIAN MAPS
LEONID POLTEROVICH AND EGOR SHELUKHIN
Abstract.
We study configurations of disjoint Lagrangian submanifolds incertain low-dimensional symplectic manifolds from the perspective of the ge-ometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamil-tonian group of the two-sphere equipped with Hofer’s metric, prove constraintson Lagrangian packing, find instances of Lagrangian Poincar´e recurrence, andpresent a new hierarchy of normal subgroups of area-preserving homeomor-phisms of the two-sphere. The technology involves Lagrangian spectral invari-ants with Hamiltonian term in symmetric product orbifolds.
Contents
1. Introduction and main results 21.1. Overview 21.2. Flats in Hofer’s geometry 31.3. Lagrangian packing 51.4. Lagrangian Poincar´e recurrence 51.5. Area-preserving homeomorphisms of S
62. Lagrangian estimators 73. Hofer’s geometry: proofs and further results 133.1. Proof of Theorem A 133.2. Proof of Theorem B 153.3. Flats in the kernel of Calabi 163.4. The asymptotic Hofer norm 173.5. Stabilization 184. Lagrangian packing via Hofer’s geometry 195. Lagrangian Poincar´e recurrence: proof 196. C -continuity and non-simplicity 20
7. Lagrangian spectral invariants and estimators 237.1. Lagrangian Floer homology with bounding cochains and bulkdeformation 237.2. Lagrangian spectral invariants 267.3. Orbifold setting 288. Further directions 398.1. Other configurations 398.2. Next destination: asymptotic cone of Ham( S ) 408.3. Comparison with periodic Floer Homology? 41Acknowledgements 41References 411. Introduction and main results
Overview.
A symplectic structure ω on an even-dimensional manifold M n can be viewed as a far reaching generalization of the two-dimensional area on sur-faces: by definition, ω is a closed differential 2-form whose top wedge power ω n is avolume form. The group Symp( M, ω ) of all symmetries of a symplectic manifold,i.e. of diffeomorphisms preserving the symplectic structure, contains a remarkablesubgroup of
Hamiltonian diffeomorphisms
Ham(
M, ω ). When M is closed and itsfirst cohomology vanishes, Ham coincides with the identity component of Symp.At the same time, in classical mechanics, where M models the phase space, Hamarises as the group of all admissible motions. This group became a central objectof interest in symplectic topology in the past three decades. In spite of that, somevery basic questions about the algebra, geometry and topology of Ham( M, ω ) arefar from being understood even when M is a surface.Let us briefly outline the contents of the paper. First, we obtain new results onHofer’s bi-invariant geometry of Ham( M, ω ) in the case where M = S is thetwo-dimensional sphere. We establish, roughly speaking, that Ham( S ) containsflats of arbitrary dimension, thus solving a question open since 2006. A similarresult was obtained simultaneously and independently, by using a different tech-nique, in a recent paper by Cristofaro-Gardiner, Humili`ere and Seyfaddini [13].Furthermore, our method yields an infinite hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere, all of which contain the normalsubgroup of homeomorphisms of finite energy discovered in [13]. Additionally, wefind a new constraint on rotationally symmetric homeomorphisms of finite energy. AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 3
Second, we extend a number of elementary two-dimensional phenomena takingplace on S to the stabilized space S × S , where the area of the second factor is“much smaller” than that of the first one. These phenomena include constraintson packing by circles, which correspond to packings by two-dimensional tori af-ter stabilization, and a version of Poincar´e recurrence theorem for sets of zeromeasure in the context of Hamiltonian diffeomorphisms. Furthermore, we provea stabilized version of our results on Hofer’s geometry presented above.To illustrate the stabilization paradigm, an elementary area count shows that onecannot fit into the sphere of unit area k pair-wise disjoint Hamiltonian images ofa circle L bounding a disc of area > /k . We shall show that the same is true for L × equator in the stabilized space. Here the area/volume control miserably fails:a two-dimensional torus does not bound any volume in a four-manifold!The first instance of such a stabilization was discovered in a recent paper by Makand Smith [37], who studied constraints on the displaceability of collections ofcurves on S . Recall, that a set X ⊂ M is called displaceable if there exists aHamiltonian diffeomorphism φ with φ ( X ) ∩ X = ∅ . This notion, introduced byHofer in 1990, gives rise to a natural “small scale” on symplectic manifolds. Makand Smith noticed that certain “small sets” on M become more rigid when onelooks at them in the symmetric product, a symplectic orbifold ( M ) k / Sym k whereSym k stands for the permutation group. On the technical side, our main observa-tion is that a powerful Floer-theoretical tool, Lagrangian spectral invariants withHamiltonian term, as developed by Leclercq-Zapolsky and Fukaya-Oh-Ohta-Ono,extends to Lagrangian tori in symmetric product orbifolds and can be applied tothe study of the above-mentioned questions on Hamiltonian maps. With this lan-guage, our paper provides a toolbox for measurements of large energy symplecticeffects on small geometric scales by using Floer theory in symmetric products.Let us note that the idea of looking at configuration spaces of points on a sur-face, the two-sphere in particular, in order to construct invariants of Hamiltoniandiffeomorphisms goes back to Gambaudo and Ghys [23]. Technically, it turnsout that for this paper it is beneficial to work with symmetric products, whichare certain compactifications of unordered configuration spaces, and instead ofthe sphere to look at a certain stabilization of it to a four-manifold. Thus ourapproach can be considered as a “symplectization” of the one by Gambaudo andGhys. Furthermore, the central objects of interest to us are certain collectionsof pair-wise disjoint Lagrangian submanifolds which in some sense govern Hamil-tonian dynamics. These are the Lagrangian configurations appearing in the titleof the paper.1.2. Flats in Hofer’s geometry.
In [29] Hofer has introduced a remarkablebi-invariant metric on the group Ham(
M, ω ) of Hamiltonian diffeomorphisms ofa symplectic manifold (
M, ω ) . It can intuitively be thought of as the minimal L ∞ , energy required to generate a given Hamiltonian diffeomorphism. For a LEONID POLTEROVICH AND EGOR SHELUKHIN time-dependent Hamiltonian H in C ∞ ([0 , × M, R ) we denote by { φ tH } t ∈ [0 , theHamiltonian isotopy generated by the vector field X tH given by ω ( X tH , · ) = − dH t ( · ) ,H t ( · ) = H ( t, · ) for all t ∈ [0 , . We say that H has zero mean if R H t ω n = 0for all t ∈ [0 , . When the symplectic manifold is closed, the Hofer distance of φ ∈ Ham(
M, ω ) to the identity is defined as d Hofer ( φ, id) = inf φ H = φ Z max M | H ( t, − ) | dt, where the infimum is taken over all zero mean Hamiltonians generating φ . It isextended to arbitary pairs of diffeomorphisms by the bi-invariance, d Hofer ( ψφ, ψφ ′ ) = d Hofer ( φψ, φ ′ ψ ) = d Hofer ( φ, φ ′ ) , for all φ, φ ′ , ψ ∈ Ham(
M, ω ) . The non-degeneracy of d Hofer was studied in many further publications such as[45, 52] and was proven to hold for arbitrary symplectic manifolds in [32]. Letus mention also that Hofer’s metric naturally lifts to a (pseudo)-metric on theuniversal cover ] Ham(
M, ω ), where the question about its non-degeneracy is stillopen in full generality. We refer to [47] for a detailed introduction to the Hofermetric and many of its aspects and properties.The main question related to the Hofer metric, Problem 20 in [40], is whether itsdiameter is infinite for all symplectic manifolds, and when it is, which unboundedgroups can be quasi-isometrically embedded into (Ham(
M, ω ) , d Hofer ) . Theorem A.
The additive group G = C ∞ c ( I ) of compactly supported smoothfunctions on an open interval I with the C distance embeds isometrically into Ham( S ) endowed with d Hofer . This settles a question of the first author and Kapovich from 2006, cf. Problem21 [40]. A similar result was obtained simultaneously and independently in [13]by completely different methods based on periodic Floer homology.For closed surfaces of higher genus, flats of arbitrary dimension were found in[50]. The proof is based on the fact that such a surface admits an incompressibleannulus foliated by non-displaceable closed curves. Symplectic rigidity of thesecurves yields the result. The lack of such an annulus in the case of S requiresa new tool, Lagrangian estimators , which we develop by using Lagrangian Floertheory in orbifolds, see Sections 2 and 7.In fact, our method of proof yields the following statement about Hofer’s geometryin dimension four. Throughout the paper S ( b ) stands for the sphere equippedwith the standard area form normalized in such a way that the total area equals b . We abbreviate S = S (1). AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 5
For a >
0, consider a symplectic manifold M a = S × S (2 a ). The naturalmonomorphism ι : Ham( S ) → Ham( M a ) , φ φ × id , satisfies d Hofer ( ι ( φ ) , ι ( ψ )) ≤ d Hofer ( φ, ψ ). Theorem B.
Assume that a > is small enough. The isometric monomorphism Φ : G → Ham( S ) from Theorem A can be chosen in such a way that Ψ = ι ◦ Φ : ( G , d C ) ֒ → (Ham( M a ) , d Hofer ) is a bi-Lipschitz embedding. Furthermore, the monomorphism of the universalcovers e Ψ : G → ] Ham( M a ) covering Ψ is an isometric embedding. Let us mention that the interval I in Theorem B is in general slightly smallerthan the one in Theorem A.1.3. Lagrangian packing.
Let K r be a simple closed curve on the sphere S = S (1) bounding a disc of area 1 /k > r > / ( k + 1), k ∈ N , k ≥
2. Let S ⊂ S (2 a ) be the equator. Note that the Lagrangian torus Λ r = K r × S can beHamiltonianly k -packed into M a , that is, there are k Hamiltonian diffeomorphisms φ = id , φ , . . . , φ k of M a such that { φ j (Λ r ) } ≤ j ≤ k are pairwise disjoint. Indeed,such a packing exists for K r ⊂ S . Theorem C (Lagrangian packing) . One cannot pack M a = S × S (2 a ) by k + 1 Hamiltonian images of Λ r . We present an argument, based on asymptotic Hofer’s geometry, in Section 4.1.4.
Lagrangian Poincar´e recurrence.
Using the Lagrangian packing obstruc-tions, we are able to make the following progress on the well-known LagrangianPoincar´e recurrence conjecture in Hamiltonian dynamics [25]. It is a version ofthe classical Poincar´e recurrence theorem, but in the setting of Lagrangian sub-manifolds instead of sets of positive measure. Note that except for the case ofsurfaces, this is a purely symplectic question, since Lagrangian submanifolds donot bound and they have zero measure.Let Λ r ⊂ M a be a Lagrangian torus as in Section 1.3. For an arbitrary Hamilton-ian diffeomorphism φ ∈ Ham( M a ), consider the recurrence set R φ := { n ∈ N : φ n Λ r ∩ Λ r = ∅} . Theorem D (Lagrangian recurrence) . The lower density of R φ is at least /k . We remark that while in [24] a similar statement was proven for specific rigidHamiltonian diffeomorphisms of complex projective spaces (pseudo-rotations) andarbitrary Lagrangians, we provide the first non-trivial higher-dimensional exam-ples of Lagrangian submanifolds satisfying the recurrence property for all
Hamil-tonian diffeomorphisms.
LEONID POLTEROVICH AND EGOR SHELUKHIN
Area-preserving homeomorphisms of S . It has been established byCristofaro-Gardiner, Humili`ere and Seyfaddini [13] that the group G S of sym-plectic homeomorphisms of the sphere possesses a non-trivial normal subgroup G FS of homeomorphisms of finite energy. These are the homeomorphisms φ forwhich there exists a sequence of Hamiltonian diffeomorphisms ψ j and a constant C > d C ( ψ j , φ ) → d Hofer ( ψ j , id) ≤ C. This is achieved byshowing that radially symmetric Hamiltonian homeomorphisms of bounded en-ergy satisfying a certain monotone twist condition must, in a precise sense, havefinite Calabi invariant. We extend this result as follows.Let h : [ − / , / → R be a smooth function that vanishes on [ − / , . Consider φ ∈ G S generated by H = h ◦ z : it is the C -limit of φ i = φ H i ∈ Ham c ( D ) ⊂ Ham( S ) for H i = h i ◦ z for h i approximations to h constant near 1 / h on [ − / , / − /i ] . Theorem E. If φ ∈ G FS then the primitive of h is a bounded function. The proof is based on a combination of [13, Lemma 3.11], an interesting soft resultrelating C -smallness, supports, and the Hofer metric, with our Lagrangian spec-tral estimators. It turns out that suitable linear combination of these invariantsare continuous with respect to the C topology on Ham( S ) and extend to thegroup G S . In fact, our invariants give rise to an infinite series of pair-wise distinctnormal subgroups of G S containing G FS , see Section 6 for a precise formulation. Remark . We note that a suitable generalized limit construction (e.g., the Banachlimit or the limit with respect to a non-principal ultrafilter) yields the existenceof many homomorphisms φ C ( φ ) ∈ R on the group of φ as in Theorem E thatcoincide with the Calabi invariant of φ ∈ Ham c ( D ) if H = h ◦ z extends smoothlyto S . Organization of the paper:
In Section 2 we introduce and list the propertiesof Lagrangian estimators, our main technical tool. The detailed construction ofthe estimators via Lagrangian Floer theory in symmetric products, as well as theproof of their properties is presented in Section 7.Section 3 deals with flats in Hofer geometry. We prove Theorems A and B,and present a result on infinite-dimensional flats in subgroups of Hamiltoniandiffeomorphisms of the disc with vanishing Calabi invariant. Additionally, wederive an estimate on the asymptotic Hofer’s norm which is used in the proof ofTheorem C on Lagrangian packing. This proof can be found in Section 4.In Section 5 we deduce Theorem D on Lagrangian recurrence from the packingconstraint by a combinatorial argument.
AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 7
In Section 6 we prove the C -continuity of certain linear combinations of ourinvariants and present applications to normal subgroups of the group of area-preserving homeomorphisms of S .We conclude the paper with a discussion of further directions in Section 8.2. Lagrangian estimators
In this section we introduce three slightly different flavors of
Lagrangian estima-tors , the main technical tool of the present paper. These are functionals with anumber of remarkable properties defined on the space of time-dependent Hamilto-nians (spectral estimators) , on the group of Hamiltonian diffeomorphisms (groupestimators) , and on the Lie algebra of functions on a symplectic manifold (algebraestimators) . They somewhat resemble, but are different from, partial symplecticquasi-morphisms and quasi-states introduced in [15], respectively (see Remark 4below).We rely on the theory of Lagrangian spectral invariants [22, 34, 35] in the settingof bulk-deformed Lagrangian Floer homology [18] for symplectic orbifolds [10]and crucially its recent investigation [37] in the context of Lagrangian links insymplectic four-manifolds. The output of our construction is a new invariant of aHamiltonian diffeomorphism of S that instead of being supported on a single non-displaceable Lagrangian circle is supported on a non-displaceable configuration ofpair-wise disjoint and (in general) individually displaceable Lagrangian circles.We start with a couple of preliminary notions and notations. Let z : S → [ − / , /
2] be the moment map for the standard S = R / Z -action on S . Itis instructive to think that S = S (1) is the round sphere of radius 1 / R equipped with the standard area form divided by π , and z is simply the verticalEuclidean coordinate.Let k ≥ < C < B be two positive rationalnumbers such that 2 B + ( k − C = 1.Denote by L k,B ⊂ S be the configuration of k disjoint circles given by L k,B = S ≤ j Moreover, Cal defines a homomorphism Cal : ] Ham c ( U ) → R . Finally, we call an open set U displaceable from a subset V if there exists aHamiltonian diffeomorphism θ such that θ ( U ) ∩ V = ∅ . Now we are ready to formulate the main result of the present section. Theorem F (Lagrangian spectral estimators) . For each k, B, a as above, with B, a rational, there exists a map c k,B : C ∞ ([0 , × M a , R ) → R satisfying the following properties:1. (Hofer-Lipschitz) For each G, H ∈ C ∞ ([0 , × M a , R ) , | c k,B ( G ) − c k,B ( H ) | ≤ Z max | G t − H t | dt. 2. (Monotonicity) If G, H ∈ C ∞ ([0 , × M a , R ) satisfy G ≤ H as functions,then c k,B ( G ) ≤ c k,B ( H ) . 3. (Normalization) For each H ∈ C ∞ ([0 , × M a , R ) and b ∈ C ∞ ([0 , , R ) ,c k,B ( H + b ) = c k,B ( H ) + Z b ( t ) dt. 4. (Lagrangian control) If ( H t ) | L jk,B ≡ c j ( t ) ∈ R for all ≤ j < k then c k,B ( H ) = 1 k X ≤ j 5. (Independence of Hamiltonian) For H ∈ C ∞ ([0 , × M a , R ) with zeromean, the value c k,B ( H ) = c k,B ( φ H ) depends only on the class φ H = [ { φ tH } ] ∈ ] Ham( M a ) in the universal cover of Ham( M a ) generated by H. 6. (Subadditivity) For all φ, ψ ∈ ] Ham( M a ) ,c k,B ( φψ ) ≤ c k,B ( φ ) + c k,B ( ψ ) . AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 9 7. (Calabi property) If a Hamiltonian H ∈ C ∞ ([0 , × M a , R ) is supportedin an open set of the form [0 , × U , where U ⊂ M a is disjoint from L k,B , then c k,B ( φ H ) = − M a ) Cal( { φ tH } ) . 8. (Controlled additivity) If ψ = φ H ∈ ] Ham( M a ) for a Hamiltonian H ∈ C ∞ ([0 , × M a , R ) supported in [0 , × U for an open set U ⊂ M a disjointfrom L k,B , then for all φ ∈ ] Ham( M a ) c k,B ( φψ ) = c k,B ( φ ) + c k,B ( ψ ) . In turn, this implies by homogenization that the following holds. Theorem G (Lagrangian group estimators) . For each k, B, a as above, with B, a rational, there exists a map µ k,B : C ∞ ([0 , × M a , R ) → R satisfying the following properties:1. (Hofer-Lipschitz) For each G, H ∈ C ∞ ([0 , × M a , R ) , | µ k,B ( G ) − µ k,B ( H ) | ≤ Z max | G t − H t | dt. 2. (Monotonicity) If G, H ∈ C ∞ ([0 , × M a , R ) satisfy G ≤ H as functions,then µ k,B ( G ) ≤ µ k,B ( H ) . 3. (Normalization) For each H ∈ C ∞ ([0 , × M a , R ) and b ∈ C ∞ ([0 , , R ) ,µ k,B ( H + b ) = µ k,B ( H ) + Z b ( t ) dt. 4. (Lagrangian control) If ( H t ) | L jk,B ≡ c j ( t ) ∈ R for all ≤ j < k then µ k,B ( H ) = 1 k X ≤ j 5. (Independence of Hamiltonian) For H ∈ C ∞ ([0 , × M a , R ) with zeromean, the value µ k,B ( H ) = µ k,B ( φ H ) depends only on the class φ H = [ { φ tH } ] ∈ ] Ham( M a ) in the universal cover of Ham( M a ) generated by H. 6. (Conjugation invariance) For all φ, ψ ∈ ] Ham( M a ) we have µ k,B ( ψφψ − ) = µ k,B ( φ ) . 7. (Positive homogeneity) For all φ ∈ ] Ham( M a ) and m ∈ Z ≥ µ k,B ( φ m ) = m · µ k,B ( φ ) . 8. (Commutative subadditivity) If φ, ψ ∈ ] Ham( M a ) commute, φψ = ψφ, then µ k,B ( φψ ) ≤ µ k,B ( φ ) + µ k,B ( ψ ) . 9. (Calabi property) If a Hamiltonian H ∈ C ∞ ([0 , × M a , R ) is supportedin an open set of the form [0 , × U , where U ⊂ M a is displaceable from L k,B , then µ k,B ( φ H ) = − M a ) Cal( { φ tH } ) . The proof is given in Section 7.3.4 below. Remark . The rationality of B, a is necessary for the Lagragian control propertyand the Calabi property. In our applications to Hofer’s geometry this will not leadto a loss generality because Q is dense in R . Note also that ψφψ − in Property6. depends only on φ and the image of ψ under the natural map ] Ham( M a ) → Ham( M a ) . We note that if ψ = id and φψ = ψφ in ] Ham( M a ) then(2) µ k,B ( φψ ) = µ k,B ( φ ) . Indeed commutative subadditivity implies that − µ ( ψ ) ≤ µ ( φψ ) − µ ( φ ) ≤ µ ( ψ ) , while positive homogeneity yields µ ( ψ ) = 0 . Moreover, note that by the Hofer-Lipschitz property one can naturally extend µ k,B to a map C ([0 , × M, R ) → R satisfying a directly analogous list of properties. Remark . While it is not directly pertinent to our applications in this paper,it would be interesting to explicitly calculate the restriction of c k,B and µ k,B to π (Ham( M a )) . It will be determined by the valuation of a suitable Seidel rep-resentation evaluated on Gromov’s loop of infinite order in π (Ham( M a )) andrespectively its homogenization.It turns out to be useful to consider the restriction of µ k,B to the space C ∞ ( M a , R )of Hamiltonians which do not depend on time. We recall that the Poisson bracketof two functions F, G ∈ C ∞ ( M a , R ) is defined as { F, G } = dF ( X G ) . The followinglist of properties is a direct consequence of those in Theorem G: note that quasi-additivity and vanishing follow from commutative subadditivity and the Calabiproperty of µ k,B . AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 11 Theorem H (Lagrangian algebra estimators) . The map C ∞ ( M a , R ) → C ∞ ([0 , × M a , R ) , F H ( t, x ) = F ( x ) , induces a map ζ k,B : C ∞ ( M a , R ) → R ζ k,B ( F ) = µ k,B ( H ) which satisfies the following properties:1. ( C -Lipschitz) For each G, H ∈ C ∞ ( M a , R ) , | ζ k,B ( G ) − ζ k,B ( H ) | ≤ | G − H | C . 2. (Monotonicity) If G, H ∈ C ∞ ( M a , R ) satisfy G ≤ H as functions, then ζ k,B ( G ) ≤ ζ k,B ( H ) . 3. (Normalization) ζ k,B (1) = 1 . 4. (Lagrangian control) If H | L jk,B ≡ c j ∈ R for all ≤ j < k then ζ k,B ( H ) = 1 k X ≤ j 5. (Invariance) For all ψ ∈ Ham( M a ) and H ∈ C ∞ ( M a , R ) we have ζ k,B ( H ◦ ψ − ) = ζ k,B ( H ) . 6. (Positive homogeneity) For all H ∈ C ∞ ( M a , R ) and t ∈ R ≥ ,ζ k,B ( t · H ) = t · ζ k,B ( H ) . 7. (Quasi-additivity and vanishing) If F, G ∈ C ∞ ( M a , R ) Poisson-commute, { F, G } = 0 , then µ k,B ( F + G ) ≤ µ k,B ( F ) + µ k,B ( G ) and if in addition G is supported in U displaceable from L k,B , then µ k,B ( F + G ) = µ k,B ( F ) + µ k,B ( G ) = µ k,B ( F ) . Finally, we consider the restriction of the invariants µ k,B , ζ k,B to Hamiltonians on S by means of the stabilization by the zero Hamiltonian. Theorem I. The map C ∞ ([0 , × S ) → C ∞ ([0 , × M a ) , F H = F ⊕ , thatis, H ( t, x, y ) = F ( t, x ) induces maps c k,B : C ∞ ([0 , × S , R ) → R ,µ k,B : C ∞ ([0 , × S , R ) → R ,ζ k,B : C ∞ ( S , R ) → R , by means of c k,B ( F ) = c k,B ( H ) , µ k,B ( F ) = µ k,B ( H ) , ζ k,B ( F ) = ζ k,B ( H ) . These maps satisfy the corresponding lists of properties as in Theorems F,G,Hwith M a replaced by S , L jk,B by L ,jk,B , and L k,B by L k,B everywhere.In addition, c k,B and µ k,B satisfy the following stronger independence of theHamiltonian property: c k,B ( H ) and µ k,B ( H ) for a mean-normalized Hamilton-ian H ∈ C ∞ ([0 , × S , R ) depend only on φ = φ H ∈ Ham( S ) . As a function c k,B : Ham( S ) → R it satisfies the subadditivity, Calabi, and controlled additivity properties. As afunction µ k,B : Ham( S ) → R it satisfies the conjugation invariance, positive homogeneity, commutative subad-ditivity, and the Calabi properties. In particular, | ζ k,B ( H ) | ≤ d Hofer ( φ H , id) for all H ∈ C ∞ ( S , R ) . Proof. The proof of all statements is immediate, except for stronger independenceof the Hamiltonian. To this end, we observe that by a classical result of Smale π (Ham( S )) ∼ = Z / Z . Let ψ be its generator. Since π (Ham( S )) lies in the centerof ] Ham( S ) , the result for µ k,B follows from (2). For c k,B we have c k,B ( ψ ) = 0by Lagrangian control, since ψ is generated by the mean-zero Hamiltonian F = z : S → [ − / , / . Hence by subadditivity, for all φ ∈ ] Ham( S ) we have c k,B ( φψ ) ≤ c k,B ( φ ) . Replacing φ by φψ and using ψ = id , we obtain the inequality c k,B ( φ ) ≤ c k,B ( φψ ) in the reverse direction. (cid:3) Remark . We observe that the maps ζ k,B , and hence also ζ k,B , are not a partialsymplectic quasi-states for k > . Indeed, ζ k,B equals 1 /k for the cut-off of theindicator function of a small neighbourhood of L , k,B , a circle of our configurationhaving the smallest area. But this circle is displaceable, a contradiction with thevanishing axiom for quasi-states. However, it is not hard to see that the choicesinvolved in the Floer data defining ζ , / can be chosen in such a way that itcoincides with the symplectic quasi-state ζ on S (which is in fact unique: see[48, Exercise 5.4.29]). AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 13 Hofer’s geometry: proofs and further results Here we apply the techniques of Lagrangian estimators to the proof of our mainapplications to Hofer’s geometry. Note that for proofs of Theorems A and B weneed the simplest Lagrangian configurations consisting of two circles.3.1. Proof of Theorem A. Step 1: Construction. We start with a moreexplicit formulation of the theorem. Consider a symmetric interval I = ( − b, b )for b < / . Let F I ⊂ C ∞ c ( I )be the space of even compactly supported smooth functions on I. In other words F I consists of functions h ∈ C ∞ c ( I ) satisfying h ( x ) = h ( − x ) for all x ∈ I. Endow F I with the C norm | h | C = max I | h | , which induces the distance function d C ( h , h ) = | h − h | C . Observe that thegroup G = C ∞ c ((0 , b )) with the C distance naturally embeds into F I by evenextension.Consider the standard symplectic sphere ( S , ω ) of total area 1 . It admits a Hamil-tonian S -action whose zero-mean normalized moment map z : S → R has image[ − / , / . We define the following embeddingΦ : F I → Ham( S ) , keeping in mind that the Hofer metric works with zero-mean Hamiltonians. Wefirst build an embedding of F I to the space I of even functions of integral zero in C ∞ ([ − / , / . To a function h ∈ F I we associate h ∈ I defined by h | ( − b,b ) = h, h | (1 / − b, / ( x ) = − h (1 / − x ) , h | [ − / , − / b ) ( x ) = − h ( − / − x ) , extendedby zero to [ − / , / . Now, for h ∈ F I we consider the mean-zero HamiltonianΓ( h ) ∈ C ∞ ( S , R ) given by Γ( h ) = h ◦ z, and let Φ( h ) = φ h ) be the time-one map of Γ( h ) . It is immediate by construction that this mapΦ : F I → Ham( S ) is a homomorphism and for all h , h ∈ F (3) d Hofer (Φ( h ) , Φ( h )) = d Hofer (Φ( h − h ) , id) ≤ | h − h | C . We claim that the monomorphism of groupsΦ : ( F I , d C ) ֒ → (Ham( S ) , d Hofer )is an isometric embedding. Step 2: Proof of the claim. Note that by (3), the main step in the proof ofthe claim is the inequality(4) d Hofer (Φ( h ) , id) ≥ | h | C ∀ h ∈ F I . As | ( − h ) | C = | h | C and d Hofer (Φ( − h ) , id) = d Hofer (Φ( h ) − , id) = d Hofer (Φ( h ) , id)for all h ∈ F I , it is sufficient to prove (4) under the assumption that | h | C = h ( x ) > . Note that in this case h ( x ) = h ( − x ) , and hence either x = 0 , or wecan assume that x ∈ (0 , b ) . Consider B = 1 / − x . For x ∈ [0 , b ) we consider ζ i = ζ ,B i , where B i = 1 / − x i , and x i ∈ (0 , b ) is a sequence of rational numbersconverging to x . Here we fix a rational parameter 0 < a < / − b for defining ζ ,B i as in Theorem I. By continuity of h we now have h ( x i ) i →∞ −−−→ h ( x ) = | h | C . Now by the Lagrangian control property of ζ i we have ζ i (Γ( h )) = h ( x i ) , and by the Hofer-Lipschitz and independence of Hamiltonian properties d Hofer (Φ( h )) ≥ ζ i (Γ( h ))for all i. Therefore taking limits as i → ∞ we obtain d Hofer (Φ( h )) ≥ | h | C as required. This finishes the proof. (cid:3) Remark . We note that the interval I = ( − / , / 6) in the claim of Step 1is the best possible using the method we describe in this paper. Indeed, thedisconnected Lagrangian given by z − ( {± (1 / δ ) } ) yields a Lagrangian in thesymmetric square that is displaceable from itself. (See Section 7 for a descriptionof the framework.). Moreover, Sikorav’s trick in Section 4 below shows that thisembedding is not isometric for b > / . Remark . Note also that for any odd function h ( z ) ∈ C ∞ ( S , R ) , that is h ( − z ) = − h ( z ) , φ h is conjugate to its inverse. Indeed for the involution R ∈ Ham( S ) givenby R ( x, y, z ) = ( x, − y, − z ) we have h ◦ R − = − h so ( φ h ) − = φ − h = φ h ◦ R − = R ◦ φ h ◦ R − . Therefore, d Hofer ( φ h , id) ≤ C = 2 d Hofer ( R, id) . Hence all such oddautonomous Hamiltonians generate one-parametric subgroups in the ball of Hoferof radius C around the identity. From this perspective, it is natural that ourconstruction is based on even functions. When x = 0 , (4) follows from a result of the first author [46]. See [14, 16] and [35] foralternative proofs. AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 15 Proof of Theorem B. Let I = ( − b, b ) for b < / < a < / − b. Arguing as in the proof of Theorem A we get that the monomorphism e Ψ : F I → ] Ham( M a ) covering Ψ is an isometric embedding.In order to pass from the universal cover to the group itself, we shall restrictourselves to the subspace F I ⊂ F I consisting of functions h with h (0) = 0. Notethat the image of G = C ∞ c ((0 , b )) in F I lies in F I .Let us show that the monomorphism of groupsΨ = ι ◦ Φ : ( F I , d C ) ֒ → (Ham( M a ) , d Hofer )is a bi-Lipschitz embedding. It suffices to show that there exists K ≥ h ∈ F I (5) d Hofer (id , Ψ( h )) ≥ K | h | C . Suppose without loss of generality that || h || C = h ( r ) > r ∈ (0 , b ).Consider the invariant µ ,B : ] Ham( M a ) → R provided by Theorem G for1 / > B = 1 / − r ≥ / − b . In order to proceed further, we have to understand the effect of the fundamen-tal group π (Ham( M )). It is a finitely generated abelian group. In fact by [1,Theorem 1.1] we have π (Ham( M )) ∼ = Z ⊕ Z / Z ⊕ Z / Z . The Z / Z terms appearfrom the natural maps π (Ham( S )) → π (Ham( M )) corresponding to the twocomponents of M = S × S . The Z term is the well-known Gromov loop [26],investigated in detail by Abreu and McDuff [1, 38]. Set T = Z / Z ⊕ Z / Z forthe torsion part of G = π (Ham( M )) , and let A = G / T ∼ = Z be its free part.Note that π (Ham( M )) ⊂ Z ( ] Ham( M )) is a central subgroup. As in the proof ofTheorem A it is easy to see that µ ,B vanishes on T and by commutative subaddi-tivity descends to ] Ham( M ) / T . However, the same is not clear for A . We proceeddifferently.Observe that by [43, Theorem 1.2] there exists a homogeneous Calabi quasi-morphism ρ : ] Ham( M ) → R that is 1-Lipschitz in the Hofer metric and re-stricts to a non-trivial homomorphism G → R . It vanishes on the torsion T sowe can consider it to be a homomorphism ρ : A → R . Choosing a generator g of A , we have ρ ( g k ) = k · ρ ( g ) for a positive constant ρ ( g ) . It is also knownthat ρ (Ψ( h )) = h (0) = 0 for h ∈ F I . This follows from ρ yielding a symplecticquasi-state on C ∞ ( M a , R ) and S × S ⊂ M a , where S ⊂ S is the equator, beinga stem (see [15, 16, 48]).Define the map ν r : ] Ham( M ) → R ,ν r ( e φ ) = ( µ ,B ( e φ ) , ρ ( e φ )) . Let e Ψ : F I → ] Ham( M ) be the homomorphism covering Ψ . Observe that ν r is 1-Lipschitz in Hofer’s metric, where R is endowed with the l ∞ norm.By the above-mentioned properties of ρ , and by Lagrangian control of µ ,B , wecan calculate for an arbitrary element e Ψ( h ) g k f covering Ψ( h ) , where f ∈ T , that ν r ( e Ψ( h ) g k f ) = ν r ( e Ψ( h ) g k ) = ( µ ,B ( e Ψ( h ) g k ) , ρ ( e Ψ( h ) g k )) = ( µ ,B ( e Ψ( h ) g k ) , kρ ( g )) . This implies that | ν r ( e Ψ( h ) g k f ) | ≥ max {| h | C − | k | d Hofer ( g, id) , | k | ρ ( g ) } ≥ C | h | C , for C = ρ ( g ) ρ ( g )+ d Hofer ( g, id) , the last step being an easy optimization in | k | . Therefore we have C | h | C ≤ | ν r ( e Ψ( h ) g k f ) | ≤ d Hofer ( e Ψ( h ) g k f, id) , and hence d Hofer (Ψ( h ) , id) ≥ K | h | C for K = 1 /C ≥ . This finishes the proof. (cid:3) Flats in the kernel of Calabi. Let I k = ( − / / ( k + 1) , − / /k ) . Considering µ k,B , with 1 /k > B > / ( k + 1) , we prove that the space of allfunctions G k ⊂ C ∞ c ( I k ) with zero mean admits an isometric embedding into(Ham( S ) , d Hofer ) . Recall that for a proper open subset U ⊂ S its symplecticform is exact, and the Calabi homomorphismCal U : Ham c ( U ) → R is defined as Cal U ( φ ) = Cal U ( { φ tH } ) for any H ∈ C ∞ c ([0 , × U, R ) with φ = φ H . Observe that by the natural constant extension we have the inclusion Ham c ( U ) → Ham( S ) . In particular, ker(Cal U ) can and shall be considered to be a subgroupof Ham( S ) . Starting from the following result we require configurations L k,B forall values of k. Theorem J. Let Φ k : ( G k , d C ) → (cid:0) Ham( S ) , d Hofer (cid:1) be given by Φ k ( h ) = φ H for H = k · h ◦ z. Then Φ k is an isometric embedding, whose image lies in ker(Cal D ) where D = D /k is the open cap of area /k around the south pole. Furthermore,for each proper open set U ⊂ S there is an isometric embedding of C ∞ c ( I ) of anopen interval I into ker(Cal U ) . AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 17 A similar result is proved in [13].The proof of this statement is almost identical to that of Theorem A, with theadditional observation that by the conjugation invariance of µ k,B we may supposethat the open set U contains D /k for k sufficiently large. Moreover, passingfrom smooth functions on an interval, say C ∞ c (( − + k +1 , − + k +1) + k )) tofunctions in G k can be easily carried out by odd extension about the midpoint υ k = − + k +1) + k of I k . Finally, analogously to Theorem B, we see that these isometric embeddings ex-tend to stabilizations on the level of universal covers and remain bi-Lipschitzembeddings on the level of groups. Theorem K. Let k ≥ , / ( k + 1) < B < /k and < a < k +1 k − B − k − . Themonomorphism of groups Ψ k = ι ◦ Φ k : ( G k , d C ) ֒ → (Ham( M a ) , d Hofer ) is a bi-Lipschitz embedding. In fact, there exists K ≥ such that for all h , h ∈ G k K | h − h | C ≤ d Hofer (Ψ k ( h ) , Ψ k ( h )) ≤ | h − h | C . At the same time, the monomorphism e Ψ k : G k → ] Ham( M a ) covering Ψ k is anisometric embedding.Question . In [21] a family of non-displaceable Lagrangian tori was constructedon S × S . It is possible to prove that these tori yield the existence of large flatsin Ham( S × S ) . These flats do not come by stabilization from S . Is it possibleto prove that they cannot be brought to be at a finite Hofer distance from a flatsupported in an arbitrarily small neighborhood of a symplectic divisor of the form { pt } × S ?3.4. The asymptotic Hofer norm. Let us fix B > C > B + ( k − C =1. Let(6) z j = − / B + ( j − C, j = 1 , . . . , k . Denote by σ B,C the measure k P kj =1 δ z j . By using the Lagrangian estimator asin Theorem G, we get that for every smooth function h = h ( z ) on S with thezero mean the asymptotic Hofer norm satsfies(7) d Hofer (id , φ h ) := lim t → + ∞ d Hofer (id , φ th ) t ≥ Z hdσ B,C . Let us emphasize that in this definition the flow φ th naturally lifts to the universalcover ] Ham and we consider Hofer’s metric there. Question . Is this estimate is sharp? As a test, we fix small δ > r ∈ (cid:18) k + 1 , k (cid:19) , and consider h r,δ to be a smoothing of the indicator function of[ − / r − δ, − / r + δ ]which we arrange to have zero mean by extending it to [ − / r − δ, − / r +5 δ ]as an odd function about the midpoint − / r +2 δ of the interval. Choose B = r ,put C = (1 − B ) / ( k − h r,δ has | h r,δ | C = 1 and equals 1 on thecircle K r := L , k,B of area r and is supported in its δ -neighbourhood. Since B > C ,we can apply inequality (7) with the measure σ B,C and get u ( r ) := lim inf δ → d Hofer (id , φ h r,δ ) ≥ k . At the same time, note that there exists a packing of S by k copies of the supportof h , so by Sikorav’s trick (which we recall in Section 4 below) u ( r ) ≤ /k . Thus u ( r ) = 1 /k , so the estimate is sharp.Presumably, progress in the direction outlined in Section 8.1 will yield efficientlower bounds on the asymptotic Hofer norm for more general autonomous Hamil-tonians.3.5. Stabilization. Consider now the stabilization of the Hamiltonian diffeomor-phism φ h r,δ constructed in Section 3.4. Fix a closed symplectic manifold ( P, Ω)and put Φ r,δ := φ h r,δ × id ∈ Ham( S × P ) . Put u P ( r ) = lim δ → d Hofer (id , Φ r,δ ) . Question . Is u P ( r ) > P, Ω) is a 2-sphere of area 2 a with B + a > C . Infact, we have in this case u P ( r ) = 1 /k as above.Now we discuss another version of the stabilization. Let S ⊂ P be a closedLagrangian submanifold, and let f W be a smoothing of the characteristic functionof a small Weinstein neighbourhood W of S . Consider the Hamiltonian h r,δ f W .We denote by u S ( r ) the lower limit of the corresponding asymptotic Hofer’s normwhen r → W shrinks to S . Question . Under which assumptions on S , one has u S ( r ) > AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 19 For instance, when P is S of area 2 a with B > C + a and S is the equator,Hamiltonians h r,δ f W , are concentrated near Λ r := K r × S , and the same argumentbased on Theorem G and Sikorav’s trick yields(8) u K ( r ) = 1 /k . Lagrangian packing via Hofer’s geometry Here we use identity (8) for the asymptotic Hofer’s norm in order to deduceTheorem C. Proof of Theorem C: Assume on the contrary that there exists a packing of M a by k + 1 Hamiltonian copies of Λ r . Then Sikorav’s trick [30, 51] would yield u K ( r ) ≤ / ( k + 1), in contradiction with (8).For the reader’s convenience we now recall this trick in detail. Set l = k + 1 , andsuppose that M a can be packed by l copies of Λ r . Suppose that φ = id , . . . , φ l ∈ Ham( M a ) yield an l -packing of Λ r in M a : { φ j (Λ r ) } ≤ j ≤ l are pairwise disjoint.Consider the stabilized function f = h r,δ · f W , where δ is sufficiently small, and W is sufficiently thin, so that f j = f ◦ φ − j have disjoint supports for all 1 ≤ j ≤ l. We claim that d Hofer ( φ f , id) ≤ /l, as calculated in ] Ham( M a ) , which implies ourclaim.Indeed by the conjugation invariance of Hofer’s pseudo-metric on ] Ham( M, ω ) ,d Hofer ( φ ltf , φ tf · ... · φ tf l ) ≤ C, where C ≤ d Hofer ( φ , id) + . . . + d Hofer ( φ l , id)) is uniform in t. However φ tf · ... · φ tf l = φ tF ,F = f + . . . + f l . As | F | C = 1 we obtain that d Hofer ( φ f ) = l d Hofer ( φ F ) ≤ /l. (cid:3) Question . Is it possible to prove that for a larger than B − C one can pack k + 1 Hamiltonian images of Λ or more? The recent methods of Hind and Kerman[28] might help produce packings of this kind.5. Lagrangian Poincar´e recurrence: proof Proof of Theorem D: Denote by K the complete graph with vertices Z ≥ . Edge-color K as follows: the edge ij is blue if φ i Λ ∩ φ j Λ = ∅ , and it is red otherwise.By Corollary C, this coloring does not possess any red complete subgraph with k + 1 vertices. Fix a maximal red complete subgraph, say, Q .Since Q is maximal, each vertex outside Q is connected to some vertex in Q bya blue edge. Put N = mk , and consider the graph B N with vertices { , . . . , N } connected only by the blue edges. The positive integer m will play the role of the large parameter in the proof. In particular, we assume that the maximal element q of Q is ≤ mk .The number of vertices in B N \ Q is at least ( m − k + 1. Denote by d themaximal degree of a vertex from Q in B N . Then, by counting outcoming blueedges from Q we get ( m − k + 1 ≤ dk which yields d ≥ m . It follows that somevertex p ∈ Q has at least m blue outcoming edges in B N .Note now that the coloring is invariant under positive translations. It follows that0 has at least m − q outcoming blue edges (we can lose at most q edges as roughlyspeaking the corresponding vertices will become negative after the shift by − p ),yielding | R φ ∩ [0 , mk ] | ≥ m − q . We conclude the proof by noticing that ( m − q ) /mk → /k as m → + ∞ . (cid:3) We remark that the proof above actually gives the slightly stronger statement,that if A ⊂ N is a subset of cardinality | A | = N then | R φ ∩ A | /N ≥ /k + O (1 /N ) . Furthermore, we observe that the same estimates work in general whenever asubset Λ ⊂ X of a set X cannot be ( k + 1)-packed into X by powers of a giveninvertible map φ : X → X. For instance this is the case when ( X, ν ) is a measurespace of total measure 1 , Λ is a subset of positive measure, k = ⌊ /ν (Λ) ⌋ , and φ is any invertible measure-preserving transformation. In this setting a strongerstatement follows from the Ergodic Theorem [5, Theorem 1.2]: there exists asequence i m of density ≥ /ν (Λ) such that ν (Λ ∩ φ − i Λ ∩ . . . ∩ φ − i m Λ) > m. However, in our Lagrangian situation the same method is not applicable:indeed our Lagrangian Λ is a measure-zero subset which does not bound a positive-measure subset. For a different example in symplectic topology, we could consider X to be a symplectic manifold, φ ∈ Symp( X ) a symplectomorphism, Λ an openball of a given capacity, and k its packing number. That this sometimes givessharper bounds than simply the volume constraint is one of the paradigms ofmodern symplectic topology, initiated in [6, 26, 39].6. C -continuity and non-simplicity We observe that the collection of Lagrangian spectral estimators c k,B containssufficient data to provide new C -continuous invariants on Ham( S ) that extendto the group of area-preserving homeomorphisms G S of the two-sphere. Fol-lowing the strategy of [12, 13] this is shown to yield alternative proofs of thenon-simplicity of the group of compactly supported area-preserving homeomor-phisms of the two-disk, known as the “simplicity conjecture” [40, Problem 42] byway of proving the “infinite twist conjecture” [40, Problem 43], as well as that of G S . We refer to the above references for the original proofs of these conjecturesby using periodic Floer homology, as well as for ample further information about AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 21 the non-simplicity questions and their historical context. We shall also rely onthe following result, [13, Lemma 3.11], proven by soft fragmentation methods. Lemma 12. Let U ⊂ S be a disk. Then for all ǫ > there exists δ > suchthat if φ ∈ Ham( S ) satisfies d C ( φ, id) < δ then there exists ψ supported in U such that d Hofer ( φ, ψ ) < ǫ. This allows us to prove the following C -continuity result. Let G S = Ham( S ) C and G D = Ham c ( D ) C denote the groups of area-preserving homeomorphismsof S and of D (with compact support). Theorem L. The map τ k,k ′ ,B,B ′ : Ham( S ) → R ,τ k,k ′ ,B,B ′ = c k,B − c k ′ ,B ′ is -Lipschitz in Hofer’s metric, is C -continuous, and extends to G S . Proof. Let U be the open disk of area min { B, B ′ } disjoint from L k,k ′ ,B,B ′ = L k,B ∪ L k ′ ,B ′ . Then by the Calabi property we obtain that τ = τ k,k ′ ,B,B ′ satisfies τ ( ψ ) = 0 forall ψ supported in U. (Indeed, in this case it is easy to find a Hamiltonian H supported in U that generates ψ. ) For ǫ > δ > θ ∈ Ham( S ) define the C -neighborhood U θ,δ = { θφ | d C ( φ, id) < δ } . Then for each θφ ∈ U θ,δ | τ ( θφ ) − τ ( θψ ) | ≤ ǫ for ψ supported in U provided by Lemma 12. However, in view of the controlledadditivity property and the above vanishing property of τ, we have τ ( θψ ) = τ ( θ ) . Hence | τ ( θφ ) − τ ( θ ) | ≤ ǫ, which proves the C -continuity of τ. Note that it proves more: in fact τ is uni-formly continuous with respect to the uniform structure on Ham( S ) given by theneighborhoods V δ = { ( θ, ψ ) | d C ( θ − ψ, id) < δ } , for δ > , of the diagonal inHam( S ) × Ham( S ) . Therefore τ extends to G S . (cid:3) It is convenient to consider L , / = S ⊂ S to be the standard equator. Let τ k,B = τ k, ,B, / , where c , / is the Lagrangian spectral invariant of S ⊂ S . ByTheorem L, τ k,B extends to G FS . Proof of Theorem E: Recall that we are given a smooth function h : [ − / , / → R vanishing on [ − / , . Consider φ ∈ G D ⊂ G S generated by H = h ◦ z . As-sume that a homeomorphism φ , generated by the Hamiltonian H , lies in G FS ,i.e., φ is the C -limit of Hamiltonian diffeomorphisms of Hofer’s norm ≤ C . Then | τ k,B ( φ ) | ≤ C for all k . Observe now that by Lagrangian control and normaliza-tion properties, for every rational B ∈ (0 , / , (9) Z / − B h ( s ) ds = lim k → + ∞ τ k,B ( φ ) ≤ C . Applying the same argument also to ( − h ) yields that h has bounded primitive. (cid:3) Question . Formula (9) shows that one can reconstruct the Calabi invariantof rotationally-symmetric Hamiltonian diffeomorphisms by using invariants τ k,B .Does such a reconstruction exist for more general symplectomorphisms?We remark that with minor modifications the same argument works for τ k,k ′ ,B,B ′ with k ′ , B ′ fixed. However, using a special property of τ k, ,B, / we prove thefollowing (compare [33]). Call two functions ρ , ρ : (0 , / → R > equivalent ifthere exist constants C , C , D , D ≥ ρ ≤ C ρ + D , ρ ≤ C ρ + D . Let ρ : (0 , / → R > be a smooth function constant near 1 / 2. Define the subset G ρ ⊂ G S as follows: a homeomorphism φ ∈ G ρ if and only if for every B ∈ (0 , / 2) there exist a positive integer κ ( B, φ ), and positive constants C ( φ ) , D ( φ )such that(10) max( τ k,B ( φ ) , τ k,B ( φ − )) ≤ C ( φ ) · ρ ( B ) + D ( φ ) ∀ k ≥ κ ( B, φ ) . Theorem M. The subset G ρ is a nontrivial normal subgroup of G S containing G FS . The subgroups corresponding to non-equivalent ρ are different.Proof. Note that τ k,B ( φψ ) ≤ τ k,B ( φ ) + τ k,B ( ψ ) + δ for a constant δ > c k,B and the quasi-morphism property of c , / (see [31, 35]) | c , / ( ψθ ) − c , / ( ψ ) − c , / ( θ ) | ≤ δ ∀ ψ, θ ∈ Ham( S ) . Hence G ρ being a subgroup is immediate. It being normal follows immediatelyfrom (12) below. That G ρ contains G FS follows immediately from the proof ofTheorem E.Consider now a Hamiltonian function H = h ◦ z as above generating a diffeomor-phism φ ∈ G ρ . Note that by (9) and (10), the primitive f of h satisfies(11) f (1 / − B ) ≤ C ( φ ) · ρ ( B ) + D ( φ ) . Furthermore, putting h ( s ) = − ρ ′ (1 / − s ) for s ∈ (0 , / 2) and h ( s ) = 0 for s ≤ B > k large enough τ k,B ( φ ) ≈ ρ ( B ) − ρ (1 / τ k,B ( φ − ) ≈ − ρ ( B ) + ρ (1 / φ ∈ G ρ . Denotethis Hamiltonian diffeomorphism by φ ρ , and observe that ρ ( s ) = f (1 / − s ), where f is a primitive of h .If G ρ = G ρ , we have that φ ρ ∈ G ρ and φ ρ ∈ G ρ . By (11), ρ and ρ areequivalent. This proves the last statement of the theorem. AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 23 In particular, G ρ = G S . This completes the proof. (cid:3) Remark . It is easy to see by construction that direct analogues of TheoremsE and M hold for G D and G F D . Remark . Here we explain that a direct analogue of Theorem L should applyto the four-manifolds V a = D × S (2 a ) and W a = D × D (2 a ) for 0 < a < min { ( k +1) B − k − , ( k ′ +1) B ′ − k ′ − } when k, k ′ ≥ < a < ( k +1) B − k − when k ′ = 1 . Indeed, let us discuss the case of W a . It is not hard to prove an analogue of Lemma12 with S replaced by W a , Ham( S ) replaced by G = Ham c ( W a ) , and U replacedby U × D (2 a ) . Indeed, as was described in [17, Section 5.3], by Gromov’s result[26] that Symp c ( W ) = Ham c ( W ) for each topological ball W ⊂ R , fragmentationresults for Hamiltonian maps apply in dimension four. Specifically, the argumentproving [33, Lemma 4.4, Proposition 4.2] extends to our case and the proofs ofanalogues of [36, Lemma 4.7] and hence of [12, Lemma 4.6], [13, Lemma 3.11] gothrough.Furthermore it is easy to see that in this case c k,B , c k ′ ,B ′ are defined on G = e G bythe obvious map e G → ] Ham( M a ) . (Recall that in this case π ( G ) = { } by Gromov[26].) Finally, the argument proving Theorem L proceeds without change, exceptfor using Gromov’s technique again to show that ψ supported in U × D (2 a ) isgenerated by a Hamiltonian supported in [0 , × U × D (2 a ). Remark . Finally, it easy to see that for instance Theorem J about the existenceof flats in Ham( S ) extends naturally to the groups Hameo( S ) , Hameo c ( D ) withtheir respective Hofer’s metrics, and possibly further to G FS , G F D . Lagrangian spectral invariants and estimators Here we construct Lagrangian estimators described in Section 2 by using La-grangian spectral invariants in symmetric products, and prove Theorems F andG. Interestingly enough, a remarkable Toeplitz tridiagonal matrix, the A k Cartanmatrix, naturally appears in the course of our calculation of the critical points ofthe Landau-Ginzburg superpotential combined from smooth and orbifold terms.7.1. Lagrangian Floer homology with bounding cochains and bulk de-formation. We briefly discuss the general algebraic properties of the LagrangianFloer homology theory with weak bounding cochains and bulk. We refer to [37]for a slightly more detailed discussion, and to the original work [18, 19, 20, 21, 22]for all detailed definitions. We also remark that the Fukaya algebra of a Bohr-Sommerfeld Lagrangian submanifold in a rational symplectic manifold, that iswhen [ ω ] is contained in the image of H ( M ; Q ) inside H ( M ; R ) , was constructedin [8] by classical transversality techniques. Hence one could feasibly carry out allconstructions in this paper by the above techniques, when restricted to the ratio-nal setting, which is sufficient for our purposes. We also note that we establish our result as a rather formal consequence of the methods of [18, 22, 37], hence itapplies with whichever perturbation schemes these papers do.For a subgroup Γ ⊂ R , define the Novikov field with coefficients in the field K = C as Λ Γ = { X j a j T κ j | a j ∈ K , κ j ∈ Γ , κ j → + ∞} . This field possesses a non-Archimedean valuation ν : Λ Γ → R ∪ { + ∞} given by ν (0) = + ∞ , and ν ( X a j T κ j ) = min { κ j | a j = 0 } . For now we may assume that Γ = R , but later it will be important to choosea smaller subgroup. We often omit the subscript Γ , and write Λ for Λ Γ . SetΛ = ν − ([0 , + ∞ )) ⊂ Λ to be the subring of elements of non-negative valuation,and Λ + = ν − ((0 , + ∞ )) ⊂ Λ the ideal of elements of positive valuation.Given a closed connected oriented spin Lagrangian submanifold L ⊂ M, andan ω -tame almost complex structure on M, considering the moduli spaces of J -holomorphic disks with boundary on L and with boundary and interior punctures,and suitable virtual perturbations required to regularize the problem, as well assuitable homological perturbation techniques, yields the following maps. First,considering only k + 1 boundary punctures, for k ≥ , we have the maps m k : H ∗ ( L, Λ) ⊗ k → H ∗ ( L, Λ) . These maps satisfy the relations of a curved filtered A ∞ algebra. Furthermore,these maps decompose as m k = P m k,β T ω ( β ) , where the sum runs over the relativehomology classes β ∈ H ( M, L ; Z ) of disks in M with boundary on L. Moreover,considering also l interior punctures yields maps q l,k : H ∗ ( M, Λ) ⊗ l ⊗ H ∗ ( L, Λ) ⊗ k → H ∗ ( L, Λ) . Given a “bulk” class b ∈ H ∗ ( M ; Λ + ) we can deform the m k operations to: m b k ( x ⊗ . . . ⊗ x k ) = X r ≥ q r,k ( b ⊗ r ⊗ x ⊗ . . . ⊗ x k ) . The operations m b k for k ≥ A ∞ algebra, and decompose into a sum m b k = P m b k,β T ω ( β ) as above, over the relativehomology classes β ∈ H ( M, L ; Z ) of disks in M with boundary on L. Furthermore, given a “cochain” class b = b + b + for b ∈ H ( L, C ) and b + ∈ H ( L, Λ + ) , we define the b -twisted A ∞ operations with bulk b ∈ H ∗ ( M ; Λ + ) asfollows: first set m b ,b k,β = e h b ,∂β i m b k,β . AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 25 These operations again satisfy the curved filtered A ∞ equations. Now, to furtherdeform by b + ∈ H ( L, Λ + ) , we set m b ,bk,β ( x ⊗ . . . ⊗ x k ) = X l ≥ X m b ,b k + l,β ( b + ⊗ . . . ⊗ b + ⊗ x ⊗ b + . . . ⊗ b + ⊗ x k , b + ⊗ . . . ⊗ b + ) , where the interior sum runs over all possible positions of the k symbols x , . . . , x k appearing in this order, and l symbols b + , and finally we set m b ,bk = X β T ω ( β ) m b ,bk,β . The operations m b ,bk also satisfy the curved filtered A ∞ equations. Moreover, if b − b ′ ∈ H ( L, π √− Z ) , then m b ,bk = m b ,b ′ k , b = b + b + , b ′ = b ′ + b + . Given b ∈ H ∗ ( M ; Λ + ) , we say that b = b + b + ∈ H ( L ; Λ ) is a weak boundingcochain for { m b k } if there exists a constant c ∈ Λ + such that X k m b ,b k ( b + ⊗ . . . ⊗ b + ) = c · L , where 1 L ∈ H ( L ; Λ ) is the cohomological unit. We call c = W b ( b ) the potentialfunction of the weak bounding cochain b. Note that it depends only on the classof b in H ( L ; Λ ) /H ( L ; 2 π √− Z ) , which is well-defined for L without torsion in H ( L ; Z ) , for example for tori. We shall henceforth consider bounding cochainsas elements of H ( L ; Λ ) /H ( L ; 2 π √− Z ) . Now we have the following result regarding Lagrangian tori. Theorem N ([19, Theorem 4.10], [20, Theorem 3.16]) . If L ∼ = T n is a Lagrangiantorus and all elements of H ( L ; Λ ) /H ( L ; 2 π √− Z ) are weak bounding cochainsfor L, then if b is a critical point of the potential function W b : H ( L ; Λ ) /H ( L ; 2 π √− Z ) → Λ + with H ( L ; Λ ) /H ( L ; 2 π √− Z ) ∼ = (Λ / π √− Z ) n identified with Λ \ Λ + by theexponential function, then m b ,b = 0 and hence the ( b , b ) -deformed Floer coho-mology of L is isomorphic to H ∗ ( L ; Λ) . Moreover, this implies that L is non-displaceable by Hamiltonian isotopies in M. Finally we note that H ∗ ( L ; Λ) ∼ = H ∗ ( L ; C ) ⊗ C Λ possesses a non-Archimedeanfiltration function A determined by requiring that the basis E = E ⊗ , for E =( e , . . . , e B ) a basis of H ∗ ( L ; C ) be orthonormal in the sense that A ( e j ⊗ 1) = 0for all 1 ≤ j ≤ B, A (0) = −∞ , and for all ( λ , . . . , λ B ) ∈ Λ B , A ( X λ j e j ⊗ 1) = max { A ( e j ⊗ − ν ( λ j ) } . It is important to note that the A ∞ algebra { m b ,bk } from Theorem N has unit1 L ∈ H ∗ ( L ; Λ) of non-Archimedean filtration level A (1 L ) = 0 . Lagrangian spectral invariants. In this section we discuss Lagrangianspectral invariants. While essentially going back to Viterbo [52], they were de-fined in Lagrangian Floer homology by a number of authors in varying degreesof generality, starting with Oh [42], Leclercq [34], and Monzner-Vichery-Zapolsky[41]. However, the two main contributions that are relevant to our goals are thepaper of Leclercq-Zapolsky [35] in the context of monotone Lagrangians, and ofFukaya-Oh-Ohta-Ono [22, Definition 17.15] in the context of Floer homology withbounding cochains and bulk deformation.7.2.1. Gapped submonoids and their associated subgroups. First we fomulate thespaces of possible values of our spectral invariants. Following [22], consider ele-ments b ∈ H ∗ ( M ; Λ ) , b ∈ H ( L ; Λ ) . We say that they are gapped if they can bewritten as b = X g ∈ G ( b ) b g T g , b g ∈ H ∗ ( M ; C ) b = X g ∈ G ( b ) b g T g , b g ∈ H ( L ; C )where G ( b ) , G ( b ) are discrete submonoids of R ≥ . In practice all relevant elements b , b will be gapped, so we assume that they are for the rest of this section.Given an ω -tame almost complex structure J on M define the submonoid G ( L, ω, J )to be generated by the areas ω ([ u ]) of all J -holomorphic disks u with boundaryon L. It is gapped by Gromov compactness. Note that in the orbifold settingbelow, one includes both areas of smooth holomorphic disks and those of orbifoldholomorphic disks. Definition 17. Let G ( L, b , b ) ⊂ R ≥ be the discrete submonoid generated by theunion G ( b ) ∪ G ( b ) ∪ G ( L, ω, J ) . Furthermore let Γ( b ) , Γ( b ) , Γ( L, ω, J ) , Γ( L, b , b )be the subgroups of R generated by the monoids G ( b ) , G ( b ) , G ( L, ω, J ) , G ( L, b , b )respectively. We call ( L, b , b ) rational if Γ( L, b , b ) is a discrete subgroup of R . Given a Hamiltonian H ∈ C ∞ ([0 , × M, R ) we consider the chords x : [0 , → M with x (0) , x (1) ∈ L, satisfying ˙ x ( t ) = X tH ( x ( t )) for all t ∈ [0 , . Furthermore, werestrict attention to only those chords that are contractible relative to L. SetSpec( H, L ) to be the set of all actions of pairs ( x, x ) each consisting of a chord x and its contraction x to L, called a capping, x : D → M, x | ∂ D ∩{ Im( z ) ≥ } = x,x ( ∂ D ∩ { Im( z ) ≤ } ) ⊂ L. The action of ( x, x ) is given by A H,L ( x, x ) = Z H ( t, x ( t )) − Z x ω. Definition 18. Define the ( b , b )-deformed spectrum Spec( H, L, b , b ) of H, L bySpec( H, L, b , b ) = Spec( H, L ) + Γ( L, b , b ) . AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 27 We recall that for two subgroups A, B of R , A + B is the subgroup of R given by A + B = { a + b | a ∈ A, b ∈ B } . Filtered Floer complex and spectral invariants. Consider a Hamiltonian H and L ⊂ M a Lagrangian as above. Let ( b , b ) be a weak bounding cochainwith bulk deformation. Assuming that ( H, L ) is non-degenerate, that is φ H ( L )intersects L transversely, and { J t } t ∈ [0 , a time-dependent ω -compatible almostcomplex structure on M, following [18, 22] one constructs a filtered bimodule CF ( H, L, b , b ; Λ ) over ( H ∗ ( L ; Λ + ) , { m b ,bk } ) , which is also a finite rank Λ -module,where Λ = Λ Γ for Γ = Γ( L, b , b ) . We set CF ( H, L, b , b ) = CF ( H, L, b , b ; Λ ) ⊗ Λ Λ . The module CF ( H, L, b , b )comes with an non-Archimedean filtration function A H,L whose values are con-tained in Spec( H, L, b , b ) ∪ {−∞} . Indeed, as a Λ-module CF ( H, L, b , b ) is givenby the completion with respect to the action functional of the vector space gen-erated by pairs ( x, x ) of Hamiltonian H -chords from L to L contractible relativeto L, where we identify between ( x, x ) , ( x, x ′ ) if v = x ′ x − : ( D , ∂ D ) → ( M, L ) , defined by gluing suitably reparametrized x ′ and x − ( z ) = x ( z ) along their com-mon boundary chord, satisfies h [ ω ] , [ v ] i = 0 . In this case the filtration A is de-fined by declaring any Λ-basis [( x i , x i )] , for { x i } the finite set of contractible H -chords from L to L an orthogonal Λ-basis of CF ( H, L, b , b ) . Finally, the ho-mology HF ( H, L, b , b ) of CF ( H, L, b , b ) is naturally isomorphic to self-Floer ho-mology HF ( L, b , b ) = HF (( L, b , b ) , ( L, b , b )) of L deformed by ( b , b ) . We writeΦ H : HF ( L, b , b ) → HF ( H, L, b , b ) for this isomorphism. In the setting of Theo-rem N there is a natural isomorphism between HF ( H, L, b , b ) and H ∗ ( L, Λ) , whichin particular explains why L is not Hamiltonianly displaceable: if φ H ( L ) ∩ L = ∅ then CF ( H, L, b , b ) = 0 and hence one must have HF ( H, L, b , b ) = 0 . For a ∈ R the subspace CF ( H, L, b , b ) a generated over Λ by all [( x, ¯ x )] satisfy-ing A H,L ( x, ¯ x ) < a forms a subcomplex of C ( H, L ) . We denote its homology by HF ( H, L, b , b ) a . It comes with a natural map HF ( H, L, b , b ) a → HF ( H, L, b , b ) . Given a class z ∈ HF ( L, b , b ) , we define its spectral invariant with respect to H as in [22, Definition 17.15] by c ( L, b , b ; z, H ) = inf { a ∈ R | Φ H ( z ) ∈ im( HF ( H, L, b , b ) a → HF ( H, L, b , b )) } . In the sequel we will primarily work with z = 1 L , the unit of the algebra on HF ( L, b , b ) induced by the m b ,b operation. Following the arguments of [22, The-orem 7.2], [35, Theorem 35] together with the argument in [48, Remark 4.3.2] forrational spectrality, it is straightforward to show the following properties of thespectral invariant c ( L, b , b ; z, H ):(1) non-degenerate spectrality: for each z ∈ HF ( L, b , b ) \{ } , and Hamiltonian H such that H, L is non-degenerate, c ( L, b , b ; z, H ) ∈ Spec( H, L, b , b ) . (2) rational spectrality: if Γ( H, b , b ) is rational, then for z ∈ HF ( L, b , b ) \ { } , the condition c ( L, b , b ; z, H ) ∈ Spec( H, L, b , b )holds for each Hamiltonian H, (3) non-Archimedean property: for each Hamiltonian H, c ( L, b , b ; − , H ) is anon-Archimedean filtration function on HF ( L, b , b ) as a module over theNovikov field Λ with its natural valuation.(4) Hofer-Lipschitz: for each z ∈ HF ( L, b , b ) , | c ( L, b , b ; z, F ) − c ( L, b , b ; z, G ) | ≤ max { E + ( F − G ) , E − ( F − G ) } where for a Hamiltonian H, we set E + ( H ) = R max M ( H t ) dt and E − ( H ) = E + ( − H ) . Note that E ± ( H ) ≤ R max M | H t | dt. (5) Lagrangian control: if Γ( H, b , b ) is rational and ( H t ) | L = c ( t ) ∈ R for all t ∈ [0 , 1] then setting c + ( H ) = c ( L, b , b ; 1 L , H ) we have c + ( H ) = Z c ( t ) dt hence for all H ∈ C ∞ ([0 , × M, R ) , R min L H t dt ≤ c + ( H ) ≤ R max L H t dt. (6) monotonicity: if two Hamiltonians F, G satisfy F t ≤ G t for all t ∈ [0 , , then c ( L, b , b ; z, F ) ≤ c ( L, b , b ; z, G ) for each z ∈ HF ( L, b , b )(7) homotopy invariance: for H t of mean-zero for all t ∈ [0 , , c ( L, b , b ; z, H )depends only on the class e φ H ∈ ] Ham( M, ω ) of the Hamiltonian isotopy { φ tH } t ∈ [0 , , (8) triangle inquequality: for each z, w ∈ HF ( L, b , b ) , and Hamiltonians F, G,c ( L, b , b ; m b ,b ( z, w ) , F G ) ≤ c ( L, b , b ; z, F ) + c ( L, b , b ; w, G ) . Orbifold setting. We shall consider only a very simple kind of a symplecticorbifold X. It is called a global quotient symplectic orbifold, and consists of thedata of a closed symplectic manifold f M and an effective symplectic action of afinite group G on it.In fact, we shall only consider f M = M k = M × . . . × M for a symplectic manifold M and the symmetric group G = Sym k acting on f M by permutations of thecoordinates. The corresponding global quotient orbifold is called the symmetricpower X = Sym k ( M ) of M. The inertia orbifold IX of a global quotient orbifold X is itself a global quotientorbifold given by the action of G on the disjoint union ⊔ g ∈ G f M g , where for g ∈ G, f M g is the fixed point submanifold of g, and f ∈ G acts by f M g → f M fgf − , x f x. See [2] for further details. AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 29 Orbifold Hofer metric. While on a symplectic orbifold one can define smoothfunctions, and hence Hamiltonian diffeomorphisms, in the case of a global quotientorbifold X, the data of a smooth Hamiltonian H ∈ C ∞ ([0 , × X, R ) is equiv-alent to the data of a G -invariant Hamiltonian e H ∈ C ∞ ([0 , × f M , R ) , that is H ( t, g · x ) = H ( t, x ) for all t ∈ [0 , , x ∈ f M , g ∈ G. In our particular case f M = M k and our Hamiltonians satisfy H ( t, x , . . . , x k ) = H ( t, x σ − (1) , . . . , x σ − ( k ) ) for all t ∈ [0 , , x , . . . , x k ∈ M, and σ ∈ Sym k . Given an orbifold Hamiltonian diffeo-morphism φ of X, which is equivalently a G -equivariant Hamiltonian diffeomor-phism e φ of f M that is generated by a G -invariant Hamiltonian H on f M , we defineits orbifold Hofer distance to the identity by d Hofer ( φ, id) = inf φ H = e φ Z max f M H ( t, − ) − min f M H ( t, − ) dt where the infinimum runs over all such G -invariant Hamiltonians generating e φ. Fi-nally we remark that orbifold Hamiltonian diffeomorphisms form a group Ham( X ) , it is isomorphic the the identity component Ham( f M ) G of the subgroup of Ham( f M )consisting of G -equivariant Hamiltonian diffeomorphisms, and d Hofer extends toa bi-invariant (non-degenerate) metric on Ham( X ) . We denote by ] Ham( X ) theuniversal cover at this group with basepoint at the identity. This is also a group.7.3.2. Orbifold Lagrangian Floer homology with bulk, and spectral invariants. Itwas explained in [10] and summarized in [37] that the above setup of LagrangianFloer homology with weak bounding cochain and bulk deformation b , b holds inthe setting of a smooth Lagrangian submanifold L in the regular locus of a closedeffective symplectic orbifold X (which for us will be a global quotient orbifold f M /G, and whose regular locus is f M /G where f M is the set of points with trivialstabilizer) with the additional feature that we may consider bulk deformationsby classes in the homology H ∗ ( IX ; Λ + ) of the inertia orbifold of X by countingholomorphic disks with possible orbifold singularities at the interior punctures.Furthermore, the above constructions of spectral invariants and arguments prov-ing their properties go through in this situation.7.3.3. Lagrangians in symmetric products and their spectral invariants. Consider S with the symplectic form ω of total area 1 . Under this normalization considerthe Lagrangian configuration L ,j ,B , j = 0 , 1, see (1) above. In [37] Mak and Smithshow, as translated to our normalizations, that if M = ( S × S , ω ⊕ aω ) , with 0 < a < B − B − C, then the Lagrangian link L ,B = L ,B × S in M is Hamiltonianly non-displaceable: for all φ ∈ Ham( X ) , φ ( L ,B ) ∩ L ,B = ∅ . The key observation of this paper is that in fact the proof of [37] gives strictlystronger information. We recall their approach. The two connected components of L ,B are L j ,B = L ,j ,B × S where S ⊂ S (2 a ) is the equator. Consider theproduct Lagrangian L ′ ,B = L ,B × L ,B ⊂ f M = M × M. Since L ′ ,B is disjoint from the diagonal ∆ M ⊂ M × M, it descends to a smoothLagrangian submanifold L ,B in the regular locus X reg of the symplectic globalquotient orbifold X = ( M × M ) / ( Z / Z ) , where Z / Z acts by exchanging thefactors. Similarly, in the situation of configurations for arbitrary values ( k, B ) ofparameters, one defines L ′ k,B = L k,B × . . . × L k − k,B ⊂ M k , and it descends to a Lagrangian L k,B in the regular locus of X reg the symplecticorbifold X = Sym k ( M ) = M k / Sym k , where the symmetric group Sym k acts on M k by permutations of the coordinates.In [37] the authors have proved for k = 2 , and we show in the next section thatthe same extends to arbitrary values of parameters ( k, B ) and a < ( k +1) B − k − thefollowing statement. Theorem O. There exists an integrable almost complex structure J M on M suchthat with respect to the complex structure J = Sym k ( J M ) on X = Sym k ( M ) , theLagrangian L k,B ⊂ X has a well-defined Fukaya algebra in the sense of [10, 18,22]. Moreover, this Fukaya algebra admits gapped orbifold bulk deformation b andweak bounding cochain b with the bulk-deformed Floer homology HF (( L k,B , b , b ) , ( L k,B , b , b )) ∼ = HF ( L k,B , b , b ) of ( L k,B , b , b ) with itself well-defined and isomorphic to H ∗ ( L k,B ; Λ) with co-efficients in the Novikov field Λ = Λ Γ , where Γ = Γ( L, b , b ) . Moreover, for B ∈ (1 / ( k + 1) , / , a ∈ (0 , ( k +1) B − k − ) rational, b , b can be chosen in such away that Γ is rational. Furthermore, HF ( L k,B , b , b ) is an associative unital alge-bra, and we denote its unit by L k,B . For H ∈ C ∞ ([0 , × X, R ) set σ k,B ( H ) = c ( L k,B , b , b ; 1 L k,B , H ) . Then σ k,B satisfies the properties of Lagrangian spectral invariants, includingrational spectrality, from Section 7.2:(1) spectrality: Γ( H, b , b ) being rational, the condition σ k,B ( H ) ∈ Spec( H, L, b , b )holds for each Hamiltonian H. (2) Hofer-Lipschitz: | σ k,B ( F ) − σ k,B ( G ) | ≤ max { E + ( F − G ) , E − ( F − G ) } . AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 31 (3) Lagrangian control: Γ( H, b , b ) being rational, if ( H t ) | L k,B = c ( t ) ∈ R forall t ∈ [0 , , then we have σ k,B ( H ) = Z c ( t ) dt, hence for each Hamiltonian H, R min L k,B H t dt ≤ σ k,B ( H ) ≤ R max L k,B H t dt. (4) monotonicity: if two Hamiltonians F, G satisfy F t ≤ G t for all t ∈ [0 , , then σ k,B ( F ) ≤ σ k,B ( G ) . (5) homotopy invariance: for H t of mean-zero for all t ∈ [0 , , σ k,B ( H ) de-pends only on the class φ H ∈ ] Ham( X, ω ) of the Hamiltonian isotopy { φ tH } t ∈ [0 , , (6) triangle inquequality: for each two Hamiltonians F, G,σ k,B ( F G ) ≤ σ k,B ( F ) + σ k,B ( G ) . Proof of Theorems F and G. We now prove Theorems F and G. For aHamiltonian F ∈ C ∞ ( M a , R ) let H ∈ C ∞ ( X, R ) be the Hamiltonian on X deter-mined by the Sym k -invariant Hamiltonian e H = F ⊕ . . . ⊕ F in C ∞ ([0 , × M k , R ) , that is e H ( t, x , . . . , x k ) = F ( t, x ) + . . . + F ( t, x k ) . Observe that if F is mean-zero,then so is H. Furthermore, if F H , F H , then F F H H andthe map F H induces a map ] Ham( M a ) → ] Ham( X ) . We set c k,B ( F ) = 1 k σ k,B ( H ) ,µ k,B ( F ) = lim m →∞ m c k,B ( F m ) . The limit exists by Fekete’s lemma by the subadditivity property of σ k,B . TheHofer-Lipschitz property of c k,B and µ k,B follows from that of σ k,B by the subaddi-tivity of Hofer’s energy functional. The monotonicity, normalization, Lagrangiancontrol, and independence of Hamiltonian properties are immediate consequencesof those for σ k,B . Note the factor 1 /k in the definition of c k,B : it serves for instanceto obtain the Hofer-Lipschitz property with coefficient 1 , as E + ( H ) ≤ k E + ( F ) for F H. Similarly, if F t | L jk,B ≡ c j ( t ) , then H t | L k,B ≡ P ≤ j 1] and hence c k,B ( F H ) = c k,B ( F ) . The proof is now concluded by the normalization axiomto pass to the mean-zero Hamiltonian G as in (13) instead of H. (cid:3) Remark . In fact, the controlled additivity property of c k,B holds under themore general assumption that( H t ) | L jk,B ≡ c j ( t ) ∈ R for all 0 ≤ j < k. However, since we do not use this stronger version, we chose toomit it for simplicity of exposition. AGRANGIAN CONFIGURATIONS AND HAMILTONIAN MAPS 33 Multiple level circles.