Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization

For a closed connected manifold N, we construct a family of functions on the Hamiltonian group G of the cotangent bundle T^*N, and a family of functions on the space of smooth functions with compact support on T^*N. These satisfy properties analogous to those of partial quasi-morphisms and quasi-states of Entov and Polterovich. The families are parametrized by the first real cohomology of N. In the case N=T^n the family of functions on G coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of G, to Aubry-Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity.

1 Introduction and results

Overview
Fix a closed connected manifold N of dimension n. The cotangent bundle T * N has a natural symplectic structure. We let G be the Hamiltonian group with compact support of T * N . We construct two families of functions, µ a : G → R, and ζ a : C ∞ c (T * N ) → R, where a ∈ H 1 (N ; R). These functions possess properties analogous to those of partial quasi-morphisms and partial quasi-states of Entov and Polterovich [EP2]. 1) The precise properties are listed in theorems 1.3, 1.8 below.
In the case N = T n , the family µ a is equivalent to Viterbo's symplectic homogenization. The symplectic homogenization is an operator (see [Vi2]) Identify H 1 (T n ; R) = R n . Then we have Theorem 1.1. Let N = T n . Then H(p) equals the value of µ p on the time-1 map of H, for any H ∈ C ∞ c ([0, 1] × T * T n ) and any p ∈ R n .
The properties of µ a , ζ a lead to various applications. Briefly, these include lower bounds on the fragmentation norm on G relative to displaceable subsets, symplectic invariance of Mather's alpha function, Hofer and spectral geometry of G, restrictions on the Poisson brackets and symplectic rigidity of subsets of T * N . Most of these applications have appeared in the literature in some form or another; we indicate the connection to the existing results where appropriate. We would like to point out, however, that here we present a unified approach which provides transparent and elementary proofs of all of the above results, together with new ones.
The main technical tool in the construction of µ a and ζ a is the spectral invariants in Lagrangian Floer homology, which themselves are functions on G. The µ a are obtained from the spectral invariants via homogenization, and ζ a are obtained from µ a by pulling them back via the exponential map.
Spectral invariants have been used for some time now to prove interesting and deep results in symplectic topology; to list but a few references: [Vi1], [Oh1], [EP1]. Our contribution to their theory in this paper is twofold. Firstly, we prove a sharp triangle inequality for them, which implies, in particular, that the invariants descend to the Hamiltonian group. Secondly, we prove an inequality relating spectral invariants coming from Lagrangian and Hamiltonian Floer homology. This allows us to prove a vanishing property for the µ a and ζ a .
Floer-homological spectral invariants have become standard objects in symplectic topology, and in our opinion the fact that symplectic homogenization is expressible with their help, makes the latter fit nicely into the general theory.
The rest of the paper is organized as follows. The remainder of this section is devoted to precise formulations of the properties of the µ a and ζ a and their applications. In section 2 we present the construction and properties of Lagrangian and Hamiltonian spectral invariants on G. Section 3 contains proofs of the results formulated in subsections 1.2 and 1.3. The reader interested in proofs of the applications can go directly to section 3, after reviewing subsection 2.5. With rare exceptions, the proofs presented in section 3 rely only on the properties of the spectral invariants appearing there.

Preliminaries and notations
The symplectic form on T * N is ω = dλ = dp ∧ dq, where λ = p dq is the Liouville form. The zero section of a cotangent bundle T * Q is denoted by Q, unless a confusion may arise, in which case we employ the more explicit notation O Q .
We implicitly fix an auxiliary Riemannian metric on N and other closed manifolds appearing below, and the lengths of cotangent vectors are measured relative to this metric.
A time-dependent Hamiltonian, that is a smooth function H: [0, 1] × T * N → R, is either denoted by H or by explicitly pointing out the time-dependence, H t . This symbol also means the function H(t, ·) ∈ C ∞ (T * N ). The time-t map of the flow of H is denoted by φ t H and the time-1 map by φ H . The collection of time-1 maps of all the Hamiltonians with compact support is the Hamiltonian group G of T * N .
For an open subset U ⊂ T * N we let G U ⊂ G be the subgroup generated by Hamiltonians with compact support in U . We let T * <r N = {(q, p) | p < r}, for r > 0. An interesting subgroup of G consists of all the Hamiltonian diffeomorphisms fixing the zero section N as a set. It is denoted by G 0 .
Proposition 1.2. There is a natural action homomorphism A: G 0 → R.
See subsection 2.1.2 for a precise formulation and a proof. For now let us just note that if H is a time-dependent Hamiltonian which equals c ∈ R when restricted to the zero section, then A(φ H ) = c.
A subset S ⊂ T * N is called displaceable if there is φ ∈ G with S ∩ φ(S) = ∅. We say that S is dominated by an open subset U if there is φ ∈ G such that S ⊂ φ(U ). In subsection 2.2 we introduce the spectral norm Γ: G → R. The spectral displacement energy of a displaceable subset S is by definition e(S) = inf{Γ(ψ) | ψ(S) ∩ S = ∅}. The spectral displacement energy of a family S = {S i } i of subsets is e(S) = sup i e(S i ).
We also introduce the fragmentation norm. This is defined as follows. If V is an open covering of T * N , then Banyaga's fragmentation lemma [Ban] states that any φ ∈ G can be represented as a finite product φ = i φ i where every φ i belongs to G Ui for some U i ∈ V. The fragmentation norm of φ relative to the covering V is the minimal number of such factors. We will need the following version of the fragmentation norm. Let U be an arbitrary family of open subsets and consider the open covering V consisting of all open subsets V for which there is ψ ∈ G with ψ(V ) ∈ U. We let φ U be the fragmentation norm of φ relative to the covering V.
We define φ t H := φ t−k H φ k H for t ∈ [k, k + 1], where k ∈ Z; here φ k H := (φ H ) k . Whenever H is defined for all t ∈ R and is 1-periodic in t, the time-t flow of H equals φ t H .

Properties of µ a and ζ a
The following theorem lists the properties of µ a . Recall the notion of the spectral displacement energy e of a family of subsets introduced above. Theorem 1.3. Let N be a closed connected manifold. For every a ∈ H 1 (N ; R) there is a function µ a : G → R with the following properties: (i) µ a (φ k ) = kµ a (φ) for k ≥ 0 an integer; (ii) µ a is conjugation-invariant; (iii) if φ, ψ ∈ G are generated by the Hamiltonians H, K, then in particular µ a is Lipschitz with respect to the Hofer metric; (iv) the restriction of µ a to G U vanishes for any displaceable U ; (v) for any collection U of open subsets with e(U) < ∞ we have |µ a (φψ) − µ a (ψ)| ≤ e(U) φ U ; (vi) the restriction of µ 0 to G 0 coincides with the action homomorphism A; (vii) if φ ∈ G is generated by a Hamiltonian whose restriction to the graph of a closed 1-form in the class a is ≥ c (respectively, ≤ c, = c), where c is some number, then (viii) for commuting φ, ψ we have µ a (φψ) ≤ µ a (φ) + µ a (ψ); (ix) for fixed φ ∈ G the function H 1 (N ; R) → R, a → µ a (φ), is Lipschitz, the Lipschitz constant being given by a semi-norm.
We may call these µ a partial quasi-morphisms in the sense of Entov-Polterovich.
Remark 1.4. Combining this theorem with theorem 1.1 we see that now we have a definition of the symplectic homogenization (as an operator G → C c (H 1 (N ; R)) for any base. In fact, the proof of convergence in [Vi2] relies on more assumptions than that of the existence of µ a . 2) This means that theorem 1.3 gives an alternative definition of symplectic homogenization. The properties of µ a , listed in this theorem, give properties of symplectic homogenization; in particular, the Lipschitz property of H mentioned ibid. is a consequence of point (ix) of our theorem.
In the original work [Vi2] the author constructs the symplectic homogenization as a limit in certain variational metric of a sequence of flows when one passes to coverings of arbitrary large degrees. More precisely, consider the conformal symplectic covering r k : T * T n → T * T n , r k (q, p) = (kq, p). Hamiltonian flows can be pulled back via this covering, namely if H is a time-dependent Hamiltonian generating φ, put H k (t, q, p) = H(kt, kq, p) and let φ k be the time-1 map of H k . Then the symplectic homogenization of φ is a continuous Hamiltonian which only depends on p, which generates in a certain precise sense the limit in the aforementioned metric of the sequence φ k . At least philosophically, it follows from the equivalence of the symplectic homogenization and our functionals µ p , p ∈ R n , that the latter are invariant under this passage to coverings. More precisely, we have the following claim.
This can be extracted as a byproduct of the proof of theorem 1.1. Remark 1.6. As pointed out by L. Polterovich, this result makes our construction fit nicely in the philosophy of homogenization, which in particular manifests itself in such objects as the Gromov-Federer stable norm, and on the other hand shows that the classical notion of homogenization, in this sense, is a particular case of a Floer-homological construction applicable to general (not necessarily convex) Hamiltonian dynamical systems.
For applications it is important to extend the definition of µ a to more general diffeomorphisms. In subsection 2.1.5 we show how to define µ a (φ H ) in case H is a Hamiltonian with complete flow. For time-dependent Hamiltonians H t , H ′ t on symplectic manifolds Z, Z ′ , respectively, we define the direct sum H ⊕ H ′ to be the time-dependent Hamiltonian on Z × Z ′ given by (H ⊕ H ′ )(t, z, z ′ ) = H(t, z) + H ′ (t, z ′ ). An easy observation is that whenever H, H ′ have complete flows, so does their sum H ⊕ H ′ . With this observation at hand we formulate Proposition 1.7. Assume that N = N 1 × N 2 , and that µ (i) ai , i = 1, 2 are the corresponding functionals given by theorem 1.3, extended to the set of complete flows, where a i ∈ H 1 (N i ; R). Let H (i) be a time-dependent Hamiltonian on T * N i for i = 1, 2, and assume both have complete flows. Then for a = (a 1 , We let ζ a : C ∞ c (T * N ) → R be defined as ζ a (H) = µ a (φ H ). The following theorem lists the properties of ζ a .
Theorem 1.8. (i) ζ a (λF ) = λζ a (F ) for λ ≥ 0 a real number; (ii) ζ a is invariant under the natural action of G on C ∞ c (T * N ); 2) It relies, as far as we understand, on the existence of certain capacity bounds, see also subsection 1.4 below.
(iv) ζ a (F ) = 0 for F with displaceable support; (v) for displaceable U , any F ∈ C ∞ c (T * N ) and any G with support dominated by U , we have in particular, if F, G commute and the support of G is displaceable then ζ a (F + G) = ζ a (F ) + ζ a (G) = ζ a (F ); (vi) if F ≥ c (respectively, ≤ c) when restricted to the graph of a closed 1-form in the class a, then ζ a (F ) ≥ c (respectively, ≤ c); Similarly to µ a , ζ a can be defined on autonomous Hamiltonians with complete flow. For these we have the following product formula, which follows from the one formulated in proposition 1.7: Proposition 1.9. Assume that N = N 1 × N 2 and that ζ (i)

Applications
In particular, if φ is generated by a Hamiltonian whose restriction to L is at least c in absolute value, where L is a Lagrangian submanifold Hamiltonian isotopic to the zero section, and c is a number, then φ U ≥ c e(U) .
For the second claim it suffices to note that for such φ we have |µ 0 (φ)| ≥ c.
Similar results are proved in [EP1], [La2]. The difference is in the class of manifolds under consideration (closed manifolds in the first reference and certain types of open convex manifolds in the second, including the unit disk cotangent bundle of a torus) and, in the case of the first reference, that there the Hamiltonian diffeomorphism φ is itself required to have displaceable support. This has to do with the fact that the Calabi quasi-morphism used there coincides with the Calabi invariant on displaceable subsets while our µ a (and Lanzat's functionals) vanish on displaceable subsets.

Connection with Mather's alpha function
Aubry-Mather theory, among other things, associates a function on H 1 (N ; R) to a Tonelli Hamiltonian H, the so-called alpha function α H : H 1 (N ; R) → R. A Hamiltonian H: [0, 1] × T * N → R is called Tonelli if it is fiberwise strictly convex and superlinear, and has complete flow. We refer the reader to [Mat] for a more detailed exposition. The functions µ a appearing in theorem 1.3 can be correctly defined on Hamiltonians having complete flow. This is done in subsection 2.1.5. We have Theorem 1.11. Let H be a time-periodic Tonelli Hamiltonian. Then for a ∈ H 1 (N ; R) One way to interpret this result is that now we have a way of defining the alpha function for an arbitrary Hamiltonian H with complete flow: α H (a) := µ a (φ H ). Theorem 1.11 first appeared in [Vi2] in the case N = T n .
An immediate consequence of this theorem is formulated in the following Proof. The extended functionals µ a are still invariant under conjugation by elements of G.
Since H • φ generates the diffeomorphism φ −1 φ H φ, the desired conclusion follows from this conjugation invariance.
The reader can find the proof of the symplectic invariance of the alpha function in [Ber] and the references therein, in the case of Tonelli Hamiltonians. It is also implicit in [PPS], in case H is autonomous. The advantage of our approach is that this invariance follows formally from the conjugation invariance of µ a , and it is applicable to any Hamiltonian with complete flow.

Hofer geometry and spectral norm on G
For φ ∈ G put where osc = max − min and the infimum is over all the compactly supported Hamiltonians whose time-1 map is φ. Also put It is a highly nontrivial fact that ρ is a metric on G, called the Hofer metric. It is biinvariant. 4) There is another norm on G, various variants of which were introduced by Viterbo, Schwarz, Oh, and in the present context, by Frauenfelder and Schlenk [FS]. Namely, there are two spectral invariants c ± : G → R and the spectral norm is defined to be 3) Since φ has compact support, H • φ automatically has complete flow. 4) The reader is referred to [Pol] for preliminaries on Hofer geometry. See subsection 2.2 for more details. Since this norm is conjugation-invariant [FS], it gives rise to another biinvariant metric on G, which we call the spectral metric, via It is known [FS] that Γ(φ, ψ) ≤ ρ(φ, ψ) .
Contrast this with [Py], where the author constructs, using the energy-capacity inequality, quasi-isometric embeddings of R k , k ≥ 1, into the Hamiltonian group of a symplectic manifold admitting a π 1 -injective Lagrangian embedding of a Riemannian manifold of non-positive sectional curvature.
We define the asymptotic Hofer norm As with the Hofer norm, we can introduce the asymptotic version We then have Proposition 1.14. Let φ ∈ G. Then homogenizing, we obtain Related results can be found in [PS], [Si2], [SV], [MZ]. There is also a connection between Aubry-Mather theory and Hofer geometry, as studied in [Si1]. We let H be the space of Hamiltonian functions on the closed unit disk cotangent bundle B ⊂ T * N which vanish at the boundary and which admit smooth extensions to the whole cotangent bundle which only depend on p and t outside the unit ball bundle. There is the associated notion of Hofer norm: This was proved in [Si1] for N = T n and in [ISM] for a class of Hamiltonians on the cotangent bundles over a general base, using different methods. Note that the minimum in the right-hand side only depends on H. Of course, since we have a definition of the alpha function for any Hamiltonian with complete flow, and the Hofer norm is defined for any compactly supported Hamiltonian, proposition 1.14 provides a more natural formulation of the relation between the Hofer norm and the alpha function, so we only include this result for completeness's sake and to illustrate the power of the methods developed here.

Poisson brackets and symplectic rigidity
We abbreviate ζ = ζ 0 . Property (v) of ζ implies the following restrictions on Poisson brackets.
Theorem 1.16. There are constants 5) C, C ′ > 0 such that the following holds. If {f i } K i=1 are smooth functions such that the support of each one of them is dominated by an element in a fixed collection U of displaceable subsets, and which satisfy Moreover, if there is a number k such that the number of supports of the f j intersecting at any point of T * N is at most k, then The proof is a verbatim repetition of the one in [EPZ] and will be omitted.
We now turn to non-displaceability. We refer the reader to [EP2], [EP3] for a treatment of the rigidity of subsets in closed symplectic manifolds.
Following [EP3], we make Definition 1.17. Call a compact subset X ⊂ T * N ζ-superheavy, or superheavy for brevity, if for any Remark 1.18. Since ζ is invariant under the action of G, so is the collection of superheavy subsets. Also we would like to point out that in this paper only superheavy (not heavy) subsets appear, since it is easy to construct examples of superheavy subsets but we could not find a heavy subset which is not superheavy.
Example 1.19. The zero section is superheavy by property (vi) in theorem 1.8.
Lemma 1.20. A subset X is superheavy if and only if for any f we have ζ(f ) ≤ max X f .

5)
It is true that C ≥ 9 8 and C ′ ≥ 1/2. Lemma 1.20 is proved in subsection 3.5. In fact, the original definition of a superheavy subset used this weaker characterization, which is more easily checked. Superheavy subsets are rigid in the sense that any two must intersect: Proposition 1.21. Let X, X ′ be two superheavy subsets; then X ∩ X ′ = ∅.
This implies in particular that superheavy subsets are non-displaceable. The proof is short and instructive, thus we include it here.
Proof. Assume the contrary and choose f, , and such that the supports of f, f ′ are disjoint. In particular this means that they Poisson commute. Then we have, by property (vii) of ζ Using the same argument, one can show that if X is superheavy and has a finite number of connected components, then only one of these connected components is superheavy.
The following proposition, proved in subsection 3.5 allows us to construct many examples of superheavy subsets.
Proposition 1.22. Let X be a compact subset such that T * N − X = U ∞ ∪ i U i is a finite disjoint union with U ∞ being the unbounded connected component (the union of the unbounded connected components in case dim N = 1). Assume that U ∞ is disjoint from the zero section and that each one of U i is displaceable. Then X is superheavy.
Example 1.23. The codimension 1 skeleton of a triangulation (or, more generally, a polygonal subdivision) of the closed unit disk cotangent bundle in T * N , considered as a manifold with boundary, satisfies the assumptions of the proposition and thus is superheavy.
Finally, in order to obtain yet more examples, we formulate the following result, also proved in subsection 3.5.
This implies the following . . , k be subsets as in proposition 1.22. Then

Connection with existing constructions and generalizations
Here we indicate connections to analogous constructions. This is an expository subsection, therefore no proofs are given.
Lanzat [La1], [La2] produces examples of open symplectic manifolds whose Hamiltonian group with compact support (or its universal cover) admits genuine (not partial) quasi-morphisms, and whose space of smooth functions with compact support, as a consequence, admits a symplectic quasi-state. He also shows how to construct a partial quasi-morphism and a partial symplectic quasi-state on a general (strongly semipositive) convex manifold. In particular his construction applies to cotangent bundles. The spectral invariant c + : G → R, introduced in subsection 2.2, satisfies the triangle inequality and so can be homogenized to yield a functional ν: G → R, whose pullback to C ∞ c (T * N ) is denoted by η. These ν, η, in fact, coincide with Lanzat's functionals for cotangent bundles, and enjoy properties analogous to those of µ a , ζ a . Moreover, owing to the comparison of Lagrangian and Hamiltonian spectral invariants (subsection 2.3), we can conclude that µ a ≤ ν and ζ a ≤ η for any a ∈ H 1 (N ; R). In particular, any ζ a -superheavy set is η-heavy and so sets described in proposition 1.22. This means that η can be used to prove non-displaceability of such subsets. However, since η(f ) = 0 for nonpositive functions f , the collection of η-superheavy sets is empty, and therefore the applications to symplectic rigidity end there. It is instructive to note here that in contrast, the collection of ζ a -superheavy subsets is not empty, and this allows for more flexible rigidity results.
We would like to point out that certain cotangent disk bundles, such as those of tori T n , admit symplectic embeddings into closed symplectic manifolds whose Hamiltonian group carries a genuine quasi-morphism, which can be pulled back to yield quasi-morphisms on the Hamiltonian group of these disk bundles. It is an intriguing question whether this pull-back coincides with the restriction of µ 0 . A partial result in this direction is presented in [MZ]. In particular, it is unclear whether the quasi-morphism on the disk cotangent bundle of a torus is invariant under coverings, like µ 0 (see proposition 1.5).
Next, we mention that the construction of Lagrangian spectral invariants on the Hamiltonian group can be performed for any symplectically aspherical Lagrangian in a completely analogous matter. One only needs to work with the space of paths with endpoints on the Lagrangian which represent a trivial element in the relative π 1 , and consistently introduce spanning half-disks. The rest of the theory comes though. It is appropriate to mention that a related approach, although in a different context, was pursued by Rémi Leclercq in [Lec]. He defined invariants of Lagrangian submanifolds instead of Hamiltonian diffeomorphisms, but in fact his construction allows for a generalization of the results presented here to the case of symplectically aspherical Lagrangians. We have not done this in detail, but it is likely that this would yield results analogous to those listed above, namely, applications to the fragmentation norm, to Hofer and spectral geometry on the Hamiltonian group of the symplectic manifold in which the chosen Lagrangian is contained, to restrictions on Poisson brackets, and to symplectic rigidity. 6) Lastly, we mention a conjecture due to Viterbo concerning a certain bound on Lagrangian spectral invariants, namely, it states that there is a constant κ such that if φ ∈ G is generated by a Hamiltonian whose support is contained in the unit disk bundle, then ℓ + (φ)−ℓ − (φ) ≤ κ (here ℓ ± are the Lagrangian spectral invariants introduced in section 2 below). If this conjecture is true, the triangle inequality and Poincaré duality will immediately imply that ℓ + and µ 0 are quasi-morphisms when restricted to the subgroup of G generated by Hamiltonians with support inside the unit disk bundle. This will have applications to second bounded cohomology of this subgroup, to asymptotics of the Hamilton-Jacobi equation, more restrictions on Poisson brackets, and more.
Acknowledgements. We would like to thank Frédéric Bourgeois, Michael Entov, Albert Fathi, Vincent Humilière, Joe Johns, Sergei Lanzat, Slava Matveyev, Maxim Maydanskiy, Marco Mazzucchelli, Dusa McDuff, Fabien Ngô, Andreas Ott, Sheila Sandon, and Matthias Schwarz for stimulating discussions, and Leonid Polterovich and Karl Friedrich Siburg for reading a preliminary version of the paper and making valuable suggestions.
AM is partially supported by the German National Academic Foundation. NV is partially supported by the ANR grant "Floer Power", ANR-08-BLAN-0291-03/04. FZ thanks the Max Planck Institute for Mathematics in the Sciences, Leipzig, where part of this work was carried out, for hospitality and an excellent research atmosphere. AM and FZ profited from their visit to the University of Chicago, and wish to thank Leonid Polterovich for the invitation. We would like to express our collective gratitude to the organizers of Edi-Fest at ETH Zürich, where the idea to write a joint paper was born.
And finally, we wish to acknowledge our intellectual debt to the fascinating paper [Vi2]. In fact, the present work grew out of an attempt to understand it. We are grateful to Claude Viterbo for explaining to us some of its more difficult parts.

Spectral invariants for Hamiltonian diffeomorphisms
In this section we present the construction and properties of Lagrangian and Hamiltonian spectral invariants on the group G. Subsection 2.1 contains the construction and properties of Lagrangian spectral invariants arising in Floer homology of the zero section N ⊂ T * N . Subsection 2.2 describes Hamiltonian spectral invariants, subsection 2.3 compares them to the Lagrangian invariants. In subsection 2.4 we briefly review Lagrangian spectral invariants coming from generating functions, and their comparison to the Floer-homological ones. Finally, subsection 2.5 summarizes the various properties of the spectral invariants.
Fix a closed connected manifold N . All homology is with Z 2 coefficients, and all moduli spaces are counted modulo 2. We identify N with the zero section in T * N via the embedding N → T * N .
All the material in this section is known and more or less standard, with the exception of the sharp triangle inequality for Lagrangian spectral invariants, proposition 2.4, its consequence, the independence of spectral invariants of isotopy, lemma 2.6, and the comparison of Lagrangian and Hamiltonian spectral invariants, proposition 2.14. The exposition is terse, but on the other hand it is extensive enough so as to provide sufficient background both for the sake of proof of the new results, and for the reader who is familiar with Floer homology, but not with spectral invariants.

Lagrangian spectral invariants from Floer homology
Here we define Lagrangian spectral invariants for Hamiltonian diffeomorphisms via Lagrangian Floer homology and prove some of their properties. The general reference we use is Oh's works [Oh1], [Oh2]. Whatever statements we make without proof or reference can be found there. We would like to point out that our sign conventions are different from those of Oh. The effect of this difference is that our invariants are "dual" to his. This is discussed in subsection 2.4.2.
The setup is as follows. Let H ∈ C ∞ c ([0, 1] × T * N ). We define the action functional A H on the space of paths Let M ⊂ N be a closed connected submanifold. Consider the path space We let the action spectrum of H relative to M be the set Let J: [0, 1] → End(T T * N ) be a path of almost complex structures, compatible with ω in the sense that ω(·, J t ·) is a path of Riemannian metrics on T * N . There is an induced L 2 -metric on Ω(M ), as follows: for ξ, η ∈ T γ Ω(M ) put ξ, η = 1 0 ω(ξ(t), J t η(t)) dt. The gradient of A H relative to this metric reads The corresponding negative gradient equation for u: For γ ± ∈ Crit(H : M ) we let M(γ − , γ + ) denote the set of solutions u of this equation such that u(±∞, ·) = γ ± ; this set admits a natural action of R by translation in the s variable, and

Generic Hamiltonian
For a generic choice of H the intersection φ H (N ) ∩ ν * M is transverse and so Crit(H : M ) is finite, and the various spaces CF are all finite-dimensional; we also refer to such a Hamiltonian as regular. If in addition J is chosen generically, then for any γ ± ∈ Crit(H : M ) the moduli spaces M(γ − , γ + ), M(γ − , γ + ) are finite-dimensional smooth manifolds; we also call such a J regular for H.
We have ∂ 2 = 0 and the corresponding Floer homology groups are HF * (H : M ). Since elements of M(γ − , γ + ) are negative gradient lines of the action functional, it decreases along any such element; therefore ∂ induces a differential on the subspace CF <a (ii) if H k is a sequence of regular Hamiltonians which tends to 0 in the C 1 -topology, then in particular the spectral invariants are Lipschitz with respect to the C 0 -norm.
We refer to property (iii) as the continuity of the spectral invariants.
Similarly, we can define spectral invariants associated to cohomology classes of M . To this end, consider the dual Floer complex CF * (H : M ) = Hom(CF * (H : M ), Z 2 ) ≡ (CF * (H : M )) * . The universal coefficient theorem implies that the cohomology of this cochain complex taken with the dual differential ∂ * is canonically isomorphic to the dual of its homology, that is to (H * (M )) * , which with coefficients in a field is the same as the singular cohomology H * (M ). The dual complex is similarly filtered by the action, that is, it increases along the differential. More precisely, we consider the subcomplex CF

Arbitrary Hamiltonian and the action homomorphism
If H is an arbitrary compactly supported Hamiltonian, it can be approximated by regular (that is, generic) Hamiltonians H k , in the C ∞ sense; it follows from the continuity of spectral invariants that ℓ(α, H k : M ) is a convergent sequence and that its limit only depends on H. Thus spectral invariants can be uniquely extended to the set of all Hamiltonians. It can be proved that these extended invariants satisfy the spectrality axiom (see, for instance [Oh3]; in our case it is even easier since one does not have to keep track of spanning disks), that is Of course, the extended invariants are also continuous in the sense of property (iii) above, and so they are Lipschitz with respect to the C 0 -norm.
Let us prove proposition 1.2 which states that there is a natural homomorphism on the subgroup G 0 ⊂ G which consists of Hamiltonian diffeomorpisms fixing the zero section as a set.
Let us first see that the above action does not depend on the choice of the point q. Indeed, . Then it is a smooth embedding and has as its image the set of critical points of A H . Since any function attains the same value on a connected submanifold which consists solely of critical points, we see that A H (γ x ) is independent of x. Thus the isotopy φ t H has as its spectrum only one point. Remark 2.7 shows that the action spectrum is independent of the isotopy representing a given element of G and therefore A is well-defined. It is a homomorphism because action is additive under concatenations.
As a consequence of spectrality, we have the following observation, which turns out to be crucial for many applications of Lagrangian spectral invariants: Lemma 2.1. The restriction of any spectral invariant ℓ(α, · : N ) to the group G 0 coincides with the action homomorphism. It follows that Proof. For the first assertion let H be a Hamiltonian generating an element φ ∈ G 0 . Spectrality implies that ℓ(α, H : N ) equals the action of an orbit of the flow of H. Proposition 1.2 shows that this action equals A(φ). This proves the first assertion, and in particular shows that ℓ(α, H : N ) only depends on φ.
Assume now that H| N = c. The zero section being Lagrangian, the flow of H preserves it, since H| N is constant. The action of any orbit equals c, thus the proof in this particular case is done. Now if H| N ≥ c, we can find another time-dependent Hamiltonian K with compact support which satisfies H ≥ K and K| N = c. The claim then follows from the particular case we just considered and the continuity of spectral invariants. The other inequality is proved similarly.
Remark 2.2. In what follows we will need from time to time to use Hamiltonians defined on [0, τ ] × T * N with τ different from 1. All the preceding constructions are modified in the obvious way, for example, the action functional is now defined on paths γ: ) dt − γ * λ, and so on. We will not mention this modification explicitly, and the context will always make clear the domain of definition of Hamiltonians, paths, and action functionals.

Poincaré duality
In this subsection M = N . Let H be regular, that is φ H (N ) intersects N transversely. By standard duality considerations (see [Sch], for example) we obtain Consider the Hamiltonian H defined by H(t, x) = −H(1 − t, x). It generates the isotopy obtained from the one generated by H by retracing it backward, that is In fact, one can prove, using an argument similar to (and actually, a little simpler than) that of [EP1], that the following more general version of Poicaré duality holds: or using homology only, We will not need this more general version, however.

Triangle inequality and independence of isotopy
If H, H ′ are smooth and H(1, ·) = H ′ (0, ·) with all the time derivatives, H♯H ′ is smooth as well.
The first result of this subsection reads is the intersection product in homology.
Remark 2.5. There is a procedure (see [Pol], for instance) which allows to replace any given time-dependent Hamiltonian with one which vanishes for values of time close to 0 and 1, which we call smoothing. This procedure leaves intact all the spectral invariants of the Hamiltonian. Also, the concatenation of any two smoothed Hamiltonians is again smooth. This works as follows.
(i) Consider H(t, x), a time-dependent Hamiltonian on T * N with compact support. Let f : [0, 1] → [0, 1] be a smooth function with f ′ ≥ 0 everywhere and f (0 . This is also a smooth Hamiltonian with compact support.
H . Thus there is a bijection between the sets of solutions of the corresponding Hamiltonian ODEs with boundary conditions on the zero section, given by Crit(H : ). This bijection preserves the corresponding actions: is a continuous family of smooth functions with f 0 = id [0,1] , f 1 = f and f τ (0) = 0, f τ (1) = 1, f ′ τ ≥ 0, then the action spectrum Spec(H fτ : N ) is independent of τ , and consequently, by spectrality, so is any spectral invariant.
(ii) Now let f satisfy the additional requirement that f (t) = 0 for t near 0 and f (t) = 1 for t near 1. Consider another time-dependent Hamiltonian K and another function g with the same properties as f . The concatenation H f ♯K g is then smooth, and its spectral invariants are independent of the functions f, g used for smoothing; moreover, if the concatenation H♯K is smooth, then H♯K and H f ♯K g have the same spectral invariants as well. If H is a regular Hamiltonian, then so is H f . If J is an almost complex structure regular for H, then J f = J(f ′ (·), ·) is for H f , with an obvious identification between the various moduli spaces relative to H, J and H f , J f .
Proof (of proposition 2.4). The above remark, together with the continuity of spectral invariants, shows that it suffices to prove the statement for H, H ′ regular and smoothed, that is, H = H ′ = 0 for times t near 0, 1. Let ε > 0. Consider the concatenation H ′′ 0 = H♯H ′ . It may not be regular any more, so we perturb it to a regular Hamiltonian H ′′ such that H ′′ − H ′′ 0 C0 < ε. Moreover, we choose an additional smooth function K: R × [0, 2] × T * N → R such that K(s, t, ·) = H(t, ·) for s ≤ 1 and t ∈ [0, 1], K(s, t, ·) = H ′ (t − 1, ·) for s ≤ 1 and t ∈ [1, 2], K(s, t, ·) = H ′′ (t, ·) for s ≥ 2 and all t and for s ∈ [1, 2] we have ∂K ∂s < ε for all t. Fix a t-dependent almost complex structure J, defined for t ∈ [0, 2], which coincides with the metric almost complex structure outside a compact. For γ, γ ′ , γ ′′ critical points of A H , A H ′ , A H ′′ respectively, we consider the moduli space M(γ, γ ′ ; γ ′′ ) of maps u: Υ → T * N , where Υ is the strip with a slit 7) appearing in [AS], with coordinates (s, t), where t ∈ [0, 2], satisfying This Υ is a Riemann surface with boundary which is conformally equivalent to a closed disk with three boundary punctures; we put on it the conformal coordinates coming from the identification of its interior with the domain R × (0, 2) − (−∞, 0] × {1} ⊂ R 2 = C. The conformal coordinate near the point (0, 1) is given by the square root.
Examining the boundary of the compactification of the 1-dimensional such moduli spaces, we see that this bilinear map is in fact a chain map, hence descends to homology, We claim that, under the natural identifications HF * = H * (N ), this map corresponds to the intersection product. Indeed, Oh proved that a different version of this Υ-product corresponds to the cup product in singular cohomology. In his version the Hamiltonian K on the strip with a slit vanishes for s near 0. It can be seen that if we use such a Hamiltonian in the definition of our moduli space, we will obtain the same map on homology. Indeed, one can define the corresponding moduli space of paths of solutions to the above equation where the Hamiltonian depends on the variable of the path, say K τ . Examining the boundary of the 1-dimensional such moduli spaces, one can see that counting the 0-dimensional moduli spaces amounts to a chain homotopy between the chain maps constructed from Hamiltonians K 0 and K 1 , which implies that they define the same map in homology. Thus it is immaterial whether to use our Hamiltonian K, "glued" from H, H ′ , H ′′ , or Oh's Hamiltonian which vanishes for s near 0. Now, Oh's sign conventions make his Floer homologies isomorphic to H * (N ) (see subsection 2.4.2). Passing to our sign conventions amounts to applying the Poincaré duality in each variable, which transforms the cup product on cohomology into the intersection product on homology. Now, a computation shows (compare with [AS]) that if u ∈ M(γ, γ ′ ; γ ′′ ), then where E(u) ≥ 0 is the energy of u. It follows that the above chain map restricts to a map on filtered subcomplexes: for any a, b, ε ′ such that a / ∈ Spec(H : N ), b / ∈ Spec(H ′ : N ), ε ′ > ε, and a + b + ε ′ / ∈ Spec(H ′′ : N ). This implies that ℓ(α ∩ β, H ′′ ) ≤ ℓ(α, H) + ℓ(β, H ′ ) + ε .
Since H ′′ was chosen ε-close to the concatenation H♯H ′ , passing to the limit as ε → 0, we obtain the desired triangle inequality As a consequence, we have Then the spectral invariants of H, H ′ coincide.
Proof. First, let G ∈ C ∞ c ([0, 1] × T * N ) be a Hamiltonian generating a loop, that is φ G = id. We claim that its spectral invariants all vanish. First, observe that we may replace G by a smoothed version, without altering the spectral invariants, and such that φ G is still the identity map. Note that ℓ(α, G) is, by spectrality, the action of a Hamiltonian arc γ ∈ Ω(N ). Since G generates a loop and is smoothed, this arc is in fact a smooth closed orbit. A standard computation shows (see [Sch]) that the actions A G (γ x ) are all the same, where γ x (t) = φ t G x.
It follows that they are all zero, because we can take x to be outside the support of G. Thus A G (γ) = A G (γ γ(0) ) = 0, as claimed.
Then we have Since H♯H ′ generates a loop, its spectral invariants vanish and we obtain ℓ(α, H) ≤ ℓ(α, H ′ ) , and the reverse inequality follows by exchanging H and H ′ . To prove that ℓ(α, H) = ℓ(α, H♯H ′ ♯H ′ ) , we proceed as follows. Since we smoothed H ′ , it is true that H It is easy to see that for all τ ∈ [0, 1] the Hamiltonians K τ , K τ are smooth. Now let H τ be the concatenation of H, then K τ running in time τ and then K τ running in time τ . An immediate computation shows that Spec(H τ : N ) is independent of τ and that H 0 = H and H 1 = H♯H ′ ♯H ′ . The assertion now follows from spectrality.
Remark 2.7. The fact that the spectrum Spec(H : M ) only depends on the time-1 map of H can be proved in a similar, though much more elementary, way, since no triangle inequality is needed. To wit, as we mentioned in the beginning of the proof of lemma 2.6, for a Hamiltonian generating a loop the action of any Hamiltonian orbit vanishes. Now let H, H ′ have the same time-1 map, and be smoothed, without loss of generality. Let z ∈ T * N and γ(t) = φ t H (z), γ ′ (t) = φ t H ′ (z), and let γ ′′ be the concatenation of γ and the reversal of γ ′ . Then since γ ′′ is an orbit of H♯H ′ , which generates a loop.

Hamiltonians with complete flow
Here we describe how to define the various spectral invariants ℓ(α, · : M ) for a Hamiltonian having complete flow. Proof. This follows from the fact that H can be continuously deformed into H ′ such that the action spectrum stays intact during the deformation. More precisely, let H τ = τ H ′ + (1 − τ )H. Then H τ is a smooth Hamiltonian whose flow sends U into V for all times and which coincides with H and H ′ when restricted to V . It follows that H τ has the same set of Hamiltonian orbits in Ω(M ) regardless of τ and those have actions independent of τ . The claim follows.
If H has complete flow, there is R > 0 such that φ t H (N ) ⊂ T * <R N for all t ∈ [0, 1]. Any two compactly supported cutoffs H ′ , H ′′ of H outside T * <R N satisfy the assumptions of the lemma and so have identical spectral invariants; we declare the common value ℓ(α, H ′ : M ) = ℓ(α, H ′′ : M ) to be the spectral invariant ℓ(α, H : M ). Note that these extended spectral invariants share the properties of the usual ones, that is, spectrality, continuity, the triangle inequality, independence of isotopy, and, what is also important in applications, the product formula below, for which, incidentally, Hamiltonians with complete flow provide natural subjects.
For future use, we formulate Remark 2.10. A word of warning is in order. It is not true that one can consistently define the Floer complex for a Hamiltonian with complete flow, since moduli spaces of Floer trajectories may fail to be compact without additional assumptions on the behavior of the Hamiltonian at infinity, such as quadratic growth or similar. It is also not true that the Floer complexes of two cutoffs are isomorphic. What is true, and this is what makes the whole theory work, is that the Floer complex of any cutoff is well-defined, and that the complexes of different cutoffs are related by canonical chain maps (continuation morphisms) which descend to level-preserving isomorphisms on homology.

The product formula
In this subsection we prove the product formula for spectral invariants, which turns out to be important for applications to symplectic rigidity. Recall the definition of the direct sum of two time-dependent Hamiltonians, subsection 1.2.
Theorem 2.11. Let H, H ′ be time-dependent Hamiltonians with complete flows on T * N , T * N ′ , respectively. Then, for any α ∈ H * (N ) − {0} and α ′ ∈ H * (N ′ ) − {0} we have Before passing to the proof, we need some preparations. By definition, a filtered graded chain complex is a quadruple V = (V, v, A, ∂), where v = (v 1 , . . . , v k ) is a graded finite set, V = Z 2 ⊗ v is the Z 2 -vector space spanned by v, A: v → R is a 1-1 function, called the action, and ∂: V → V is a differential, which lowers the grading by 1, and respects the action filtration, that is it preserves V <a := Z 2 ⊗ ( v ∩ {A < a}) ⊂ V for every a ∈ R. Following the usual procedure, one can define the spectral invariants of V relative to homology classes in H(V, ∂), which we denote by ℓ(α, V) for α ∈ H(V, ∂) − {0}. Given two filtered graded chain complexes V = (V, v, A, ∂) and V ′ = (V ′ , v ′ , A ′ , ∂ ′ ), one can form the product filtered graded chain The spectral invariants of filtered graded chain complexes satisfy the following product property: The proof, though elementary, is somewhat involved, and can be extracted from [EP3].
We can now pass to the proof of theorem 2.11.
Proof (of theorem 2.11). Given an arbitrary Hamiltonian with complete flow, we can always perturb it (say, in C ∞ topology) to a generic one, meaning that the Floer complex of any cutoff is a filtered graded chain complex in the sense of the discussion above. Moreover, given two such Hamiltonians, we can perturb both of them in such a way that both the perturbations and their direct sum are generic. Since spectral invariants are continuous with respect to C 0 norm, it suffices to restrict attention to Hamiltonians H, H ′ which are generic in this sense, and such that the sum H ⊕ H ′ is generic as well, which is what we choose to do. Let G, G ′ be cutoffs of H, H ′ . The direct sum G ⊕ G ′ has complete flow. We choose regular almost complex structures J, J ′ on T * N, T * N ′ , respectively, which coincide, outside a large compact, with the metric almost complex structures. We let R > 0 be large enough so that T * <R (N × N ′ ) contains the product T * <r N × T * <r ′ N ′ , where r is large enough so that T * <r N contains the images of all the critical points of A G , as well as the images of the Floer trajectories between pairs of critical points of A G of index difference 1, and similarly for r ′ , T * N ′ and G ′ . Let G ′′ be a cutoff of G ⊕ G ′ outside T * <R (N × N ′ ). Then it is also a cutoff of Moreover, G ′′ is generic by construction, and J ′′ := J ⊕J ′ is a regular almost complex structure. It then follows that the Floer complex of G ′′ relative to J ′′ is a filtered graded chain complex, which is the product of the Floer complexes of G, G ′ . Applying lemma 2.12, we see that This concludes the proof.

Hamiltonian spectral invariants
These were defined for weakly exact symplectic manifolds convex at infinity in [FS], and in the more general setting of semipositive symplectic manifolds convex at infinity in [La1]. We only present a sketch of the construction, referring the reader to the aforementioned sources for details.
Standard Floer homology cannot be correctly defined for compactly supported Hamiltonians because they are degenerate. To circumvent this difficulty, one considers Hamiltonians which have support in some fixed cotangent ball bundle and which have a certain prescribed behavior near the boundary.
These spectral invariants satisfy all the standard properties, including Lipschitz continuity, triangle inequality, and spectrality. This implies that if H is an arbitrary compactly supported Hamiltonian, and we C 0 -approximate it by non-degenerate Hamiltonians H k , k ∈ N, whose behavior for p ∈ [R − ε k , ∞) is prescribed by the function h as above, and ε k → 0, then the sequence c(α, H k ; h, R) is Cauchy and we declare its limit to be the spectral invariant c(α, H; h, R). It can be shown, by a standard but a little lengthy argument that this spectral invariant is independent of the choices, that is, h and R. Moreover, it actually only depends on φ H , and so we will use the notation c(α, φ H ) for it. We will only need the invariant c − (φ) = c(pt, φ) and its dual counterpart c + (φ) = −c − (φ −1 ).
In [FS] it is also shown that if U ⊂ T * N is displaceable by ψ ∈ G then for any φ ∈ G U it is true that where Γ(ψ) = c + (ψ) − c − (ψ) is the spectral norm of ψ.
Remark 2.13. We would like to point out that Hamiltonian spectral invariants, unlike Lagrangian ones, are not defined for Hamiltonians with complete flow. This has to do with the fact that Floer homology constructed from periodic orbits, may be ill-defined for such Hamiltonians. Moreover, there is no consistent way of cutting such Hamiltonians off, like in the Lagrangian case, in order to use the compactly supported theory.

Comparison of Lagrangian and Hamiltonian spectral invariants
Our goal in this subsection is the following proposition.
Proposition 2.14. Let φ ∈ G. Then 8) It is normalized so as to equal the Morse index of critical points of a C 2 -small Hamiltonian, considered as 1-periodic orbits.
This implies the following chain of inequalities: where the rightmost inequality follows by duality.
Proof. The proof is essentially contained in [Alb], the only point of difference being that there the theory is restricted to closed manifolds. This has to do with compactness of moduli spaces of perturbed pseudo-holomorphic curves. In our case, since the almost complex structure is assumed to coincide, outside a large compact, with the one coming from the auxiliary Riemannian metric, there are no additional compactness issues beyond the closed case, and in fact, since the form is exact, there is no bubbling off of spheres or disks, and the proofs are actually much simpler, and in particular the PSS morphisms constructed in the aforementioned paper are defined for all degrees and are isomorphisms. Therefore we only present a sketch of the argument, emphasizing the essential point of comparison of the spectral invariants.
Albers defines a map ι: CF * (H : N ) → CF * (H), as follows. First, one can assume that H t is time-independent near t = 0, 1, and that the Floer homology for it is defined as in subsection 2.2. Given a Hamiltonian arc γ and a periodic orbit x of H, consider the moduli space M(γ, x) consisting of solutions of the Floer equation defined on the Riemann surface Υ ′ , conformal to a closed disk with one boundary and one interior puncture, obtained from the above strip with a slit Υ through identifying the top and the bottom boundary components, that is, solutions to ∂u ∂s where the boundary puncture is asymptotic to γ and the interior puncture is asymptotic to x, while the boundary is mapped to the zero section. What makes this equation well-defined is the presence of global conformal coordinates (s, t) on Υ ′ . He shows that this moduli space is a smooth manifold of dimension m H:N (γ) − m H (x). It is compact in dimension 0, which follows from the usual convexity considerations. Let ι be the linear extension of He then shows that this is a chain map, and the canonical identifications HF * (H : N ) = H * (N ) and HF * (H) = H * (T * N ) intertwine it with the isomorphism H * (N ) → H * (T * N ) induced by the inclusion of the zero section into T * N . We have the sharp action-energy identity for an element u ∈ M(γ, x): Spec(H : N ). It follows that c − (H) ≤ ℓ − (H). Using the continuity of spectral invariants, we conclude that this inequality holds for all smooth Hamiltonians with compact support.
It follows from the previous subsection that if U is an open subset displaceable by ψ, then for any φ ∈ G U we have

Lagrangian spectral invariants from generating functions
We also need to use another definition of Lagrangian spectral invariants, namely those coming from generating functions, due to Viterbo. The reason is that we need both definitions in the proof theorem 1.1 which states that the symplectic homogenization is a particular case of our functionals µ a .

Definition
Let us recall briefly Viterbo's construction of Lagrangian spectral invariants using generating functions [Vi1]. A generating function quadratic at infinity, or gfqi for short, is a function We let E = E + ⊕ E − be the splitting into the positive and negative subspaces of B.
Consider the relative homology H * ({S < a}, {S < b}). It follows from the definition that for a large enough and b small enough this group is independent of a, b and is canonically isomorphic to H * (N )⊗ H * (E − , E − − 0) ≃ H * +d (N ), where d = dim E − and the last isomorphism ("Thom isomorphism") is given by tensoring with the generator of H d (E − , E − − 0) ≃ Z 2 . We denote this group by H * (S : N ). There is a natural inclusion morphism i b : H * ({S < b}) → H * (S : N ). To each α ∈ H * (N ) we associate the spectral invariant Similarly, if M ⊂ N is a closed submanifold, we can consider the restriction S| M×E as a gfqi with base M and define spectral invariants associated to classes in H * (M ).
These invariants are defined also for Lagrangian submanifolds of T * N , as follows. A regular gfqi gives rise to a Lagrangian immersion, see [Vi1]. A Lagrangian submanifold Hamiltonian isotopic to the zero section admits a gfqi unique up to gauge a transformation, stabilization, and the addition of a constant [Vi1], [The]. Except the addition of a constant, the elementary operations do not alter the spectral invariants. We can then say that the spectral invariants of this gfqi are attached to the Lagrangian submanifold in question, and they are all defined up to simultaneous addition of a constant.

Sign conventions
In what follows we will rely on certain results due to Milinković and Oh on the equality of spectral invariants coming from Floer homology and from generating functions [MO1], [MO2]. Since our sign conventions differ from theirs, it is necessary to relate the two conventions regarding spectral invariants.
Our sign convention follows the philosophy that the Floer theory of the action functional is a perturbation of the Morse theory of a function on a closed manifold, in particular the Hamiltonian enters the action functional with a positive sign.
Notation 2.15. In this subsection, as well as in the rest of the paper, we will denote objects defined with the sign conventions of Milinković and Oh, with the overline, with the exception of H, which we reserve for the "reversed" Hamiltonian. Namely, assume that H is a compactly supported time-dependent Hamiltonian on T * N . The action functional A H : Ω(M ) → R is defined as The symplectic form ω = −ω = −dλ = −dp ∧ dq. The Hamiltonian vector field X H is defined by the equation ω(X H , ·) = dH and so X H = X H . In particular, the flows in the two sign conventions coincide.

Relation between Floer-homological invariants and invariants from generating functions
In [MO1], [MO2] Milinković and Oh show that the Lagrangian invariants coming from Floer homology and those coming from generating functions coincide provided the generating function is suitably normalized. The normalization is as follows. Recall that if a generating function W defined on the total space of a submersion π: E → N generates a Lagrangian embedding L ⊂ T * N , then it induces a function, denoted W |L, on the image of the embedding via the formula W |L := W • (i W ) −1 : L → R, where i W : Σ W → T * N is the canonical map from the fiberwise critical locus Σ W of W to T * N . It follows from the fact that the differential d(W |L) coincides with λ| L and that two generating functions of L induce functions on it whose difference is constant. A particularly important case is the action functional A H , defined on the path space Ω, where the submersion is given by Ω → N , γ → γ(1). It generates φ H (N ).
We have the following lemma, whose proof can be extracted from [MO1], [MO2]: In order to see that our normalization condition implies the conclusion of Milinković and Oh, one has to translate it into the language of wavefronts, see [Oh1]. It suffices to note that when S is normalized as in the lemma, it is possible to find an interpolation between S and A H , in the sense of [MO1], [MO2], which has a constant wavefront, and this allows their argument to work.

Summary
Here we summarize for further reference the properties of spectral invariants proved above, together with some immediate consequences.
Theorem 2.17. Let N be a closed connected manifold. To each α ∈ H * (N ) − {0} we associate a function ℓ(α, ·): G → R such that: (vi) the restriction of ℓ(α, ·) to G 0 coincides with the action homomorphism A; in particular if H generates φ and H| N = c (respectively H| N ≥ c, H| N ≤ c) for some c ∈ R, then ℓ(α, φ) = c (respectively ℓ(α, φ) ≥ c, ℓ(α, φ) ≤ c); (vii) if U ⊂ T * N is an open subset and ψ ∈ G is such that ψ(U ) ∩ U = ∅, then Proof. With the exception of points (iv) and (viii), these statements are proved in the previous subsections. For point (iv) the triangle inequality implies, for example: because ℓ(α, id) = 0, the identity map being generated by the zero Hamiltonian. Point (xi) is a consequence of the triangle inequality. For instance, 3 Proofs

Main result
Proof (of theorem 1.3). We define µ 0 : G → R by The limit exists because the sequence {ℓ + (φ k )} k is subadditive. It is finite because of property (ii) in theorem 2.17. As an immediate consequence of this definition, we obtain point (i).
(ii) Point (viii) in theorem 2.17 implies that for any φ, ψ ∈ G we have Dividing by k and taking k → ∞ yields µ 0 (ψφψ −1 ) = µ 0 (φ). Let us define the functionals µ a . For a ∈ H 1 (N ; R) let α ∈ a and define the symplectomorphism T α : T * N → T * N by T α (q, p) = (q, p + α(q)). Put Since µ 0 is conjugation-invariant, it follows that if we replace α by a cohomologous form α ′ , then µ 0 (T −α φT α ) = µ 0 (T −α ′ φT α ′ ) and therefore µ a is unambiguously defined. Indeed, assume that α−α ′ = df for some f ∈ C ∞ (N ). Let B be a cotangent ball large enough to contain the support of a Hamiltonian generating T −α ′ φT α ′ . Let F be a compactly supported Hamiltonian obtained from π * f by cutting it off outside B. We then have (iii) Let us show the upper bound for µ 0 , for example; the rest follows similarly. We have In order to pass to µ 0 , we need to homogenize and to this end, to concatenate Hamiltonians. Let f be a smoothing function as in remark 2.5. Then For any ε > 0 there is such a smoothing function for which It follows that and passing to the limit k → ∞, and then letting ε → 0, we obtain (iv) If U is displaceable by ψ ∈ G, we have, by theorem 2.17: for any φ ∈ G U . Then For a ∈ H 1 (N ; R) and α ∈ a we have that T −α φT α ∈ G T−α(U) and that T −α (U ) is displaceable by T −α ψT α .
(v) Let us say for brevity that a diffeomorphism φ ∈ G is dominated by an open subset U if φ is generated by a Hamiltonian whose support is dominated by U . Let φ be dominated by one of the elements in U. The triangle inequality and duality for ℓ ± implies that for any Put φ j = ψ j φψ −j . Then φ j is also dominated by one of the elements in U. We have which implies, using induction and the above inequality, that Since all φ j are dominated by a displaceable subset with displacement energy ≤ e(U), it follows, using property (vii) in theorem 2.17, that and upon dividing by k and taking k → ∞ we get The claim follows by induction on φ U . (vi) Since the restriction of any spectral invariant to G 0 coincides with the action homomorphism, it follows by homogenization that so does the restriction of µ 0 .
(vii) It suffices to restrict the attention to the zero section and a = 0. From theorem 2.17, point (iv), we know that if, for example, H| N ≥ c, then Again, to prove the corresponding property for µ 0 , we need to concatenate. Consider the Hamiltonian K given by where f is a smoothing function. Of course, K is not compactly supported, but this is easily circumvented by cutting it off outside a large compact. The action spectrum of K is that of H f , shifted upward by the amount dt. This number can be made as small as we wish by suitably choosing f . It follows that for any ε > 0 there is a smoothing function f such that the spectral invariants of K differ from those of H f by not more than ε. We already know that the spectral invariants of H f and of H coincide. This discussion shows that But K equals c near t = 0, 1 and so can be concatenated with itself to yield a smooth function. It follows that and passing first to the limit k → ∞ and then taking ε → 0 we obtain µ 0 (φ) ≥ c .
(viii) Follows from the triangle inequality for ℓ + and the fact that where α ∈ a, β ∈ b. The right-hand side is bounded from above by For any 1-form χ on N we have where we use the notation where we identified T vert (q,p) T * N = T * q N and ·, · is the pairing between T * q N and T q N . It follows that where |dH|: H 1 (N ; R) → R is the semi-norm defined by This means that a → µ a (φ) is Lipschitz, the Lipschitz constant being replaced by the seminorm |dH|.
Remark 3.1. The functions µ a have been defined via µ 0 , which in turn is the homogenization of the spectral invariant ℓ + . An equivalent construction of the µ a can be achieved as follows. Fix α ∈ a and let L α be the graph of α and λ α = λ − π * α. Then L α is an exact Lagrangian submanifold of (T * N, λ α ), on which λ α vanishes, and so we can perform the constructions of section 2 in the same fashion, with the zero section being replaced by L α . It is easy to see that this construction leads, through the corresponding version of the spectral invariants ℓ +,α , to the same functions µ α .
Proposition 1.7 follows from the product formula for spectral invariants, theorem 2.11, homogenization, and shifting by T α for appropriate 1-forms α.
Proof (of theorem 1.8). Points (ii-iv), (vi), (vii) are immediate consequences of the relevant properties of µ a . Point (i) is proved by invoking the semi-homogeneity of µ a to obtain the desired identity first for natural, then rational, and finally, using continuity, for arbitrary λ ≥ 0. Point (v) is proved as in [EPZ], carefully keeping track of the constants.

Equivalence to symplectic homogenization on T n
Here we prove proposition 1.1 which states that our present construction is equivalent to the symplectic homogenization if N = T n .

Overview of the proof
Before giving the details, let us present an overview of the construction and an intuitively clear argument why the two constructions are equivalent. We use the notation T * T n to indicate that the symplectic form is the negative of the usual one.
One starts with a Hamiltonian H ∈ C ∞ c ([0, 1] × T * T n ) and its flow φ t . The graph of φ t is a Lagrangian submanifold Γ φ t ⊂ T * T n × T * T n , and it is the image of the diagonal ∆ ≡ ∆ T * T n ⊂ T * T n × T * T n under the Hamiltonian isotopy id ×φ t . There is a symplectic covering τ : T * ∆ → T * T n × T * T n which sends the zero section diffeomorphically onto ∆. The isotopy id ×φ t lifts to a unique Hamiltonian isotopy (which no longer has compact support) Φ t which maps the zero section to a Lagrangian submanifold L(t) = Φ t (O ∆ ). This Lagrangian submanifold maps diffeomorphically onto Γ φ t under the covering τ .
In this form it is almost obvious. The point is that the restriction S(k)| T n ×{0}×E generates a Lagrangian submanifold in T * T n , which is the image of φ k H (O T n ) under the involution (q, p) → (q, −p). Since the action functional corresponding to H ♯k generates the same Lagrangian, its spectral invariants will coincide with those of S(k) 0 if the two induce the same function on the Lagrangian. The bulk of the proof below is devoted to showing this fact. What allows to conclude is, roughly speaking, the fact that the action functional corresponding to the lifted isotopy Φ k , as well as the gfqi S(k), generate the same Lagrangian L(k), which is the lift to T * ∆ via τ of the graph Γ φ k . But these two are normalized to equal zero at points of ∆ = T * T n outside a large compact. It then follows that their spectral invariants, and in particular those of the reduced functionals S(k) 0 and A H ♯k , coincide. Note the multiple instances of the use of lemma 2.16.

Details
Let us describe the construction of the symplectic homogenization. Fix H ∈ C ∞ c ([0, 1] × T * T n ) and let φ t ≡ φ t H be the isotopy generated by H t , and φ = φ 1 . Also denote Φ t = id ×φ t : T * T n × T * T n → T * T n × T * T n . The isotopy Φ t is generated by the Hamiltonian . We have the following commutative diagram: Here we view explicitly T n = R n /Z n . The left and the right arrows are induced from the quotient maps 11) R n × R n → T n × T n and T * R n → T * T n . The top map is given by (q, p; Q, P ) → (Q, p; p − P, Q − q).

Consider the Hamiltonian
We can now extract spectral invariants from H t . This Hamiltonian is not compactly supported, but for finite t it suffices to cut it off outside a large enough ball, and consider the action functional corresponding to that function. By abuse of notation we denote this modified action functional also by A H . It has the same values on all Hamiltonian arcs starting at the zero section and following the flow Φ t as the original functional before the cutoff.
Again, the fact that H t has compact support implies that L differs from the zero section only inside a compact subset of T * ∆ and so we can compactify all the objects in sight to T * (T n × S n ) = T * (∆ ∪ T n × {∞}). We denote them by the same letters as their counterparts on T * ∆.
Viterbo gives a formula for a gfqi generating the Lagrangian L [Vi2]. The precise formula is irrelevant, since, as we mentioned in subsection 3.2.1, the spectral invariants of a gfqi for L are uniquely determined as soon as we normalize it to equal a quadratic form outside K × E, 11) Usually if there is a smooth map f : X → Y , there is no natural way of associating a smooth map between the corresponding cotangent bundles, however if this map is a local diffeomorphism, then we get the induced map f * : T * X → T * Y given by f * (α) = α • (dxf ) −1 for α ∈ T * x X, and it is symplectic: (f * ) * ω T * Y = ω T * X .
K ⊂ ∆ being a certain compact subset. We denote the gfqi of L, normalized in this fashion, by S: ∆ × E → R, until the end of this subsection. From the definition of H it is clear that the points of T n × {∞}, considered as constant curves, are Hamiltonian arcs with respect to H t , starting and ending at the zero section, and moreover that H t actually equals zero on an open neighborhood of T n ×{∞} inside T * (T n ×S n ), which means, in particular, that the action of a point in T n ×{∞}, considered as a Hamiltonian arc, is zero.
Recall that both A H and S generate L, and so induce functions on L ⊂ T * (T n × S n ) (see subsection 2.4) and that these functions differ by a constant. Now, it follows from the previous paragraphs that the values of both these functions at a point of T n ×{∞} is zero, which implies that they functions coincide. In particular, if γ: [0, 1] → T * (T n × S n ) is a Hamiltonian arc relative to H t , starting at the zero section, and z = γ(1) ∈ L, then Symplectic homogenization is defined in terms of the spectral invariants of the functions S p : T n × E → R, p ∈ R n , where S: T * T n × E → R is the generating function of L = Φ 1 (O ∆ ) described above, and S p (q; ξ) := S(q, p; ξ). It turns out that S p generates the Lagrangian submanifold of T * T n given by This is a simple computation which can be checked using the above commutative diagram. Since we want to prove proposition 3.3, we restrict ourselves to the case p = 0, so that S 0 generates the following Lagrangian: where for a Lagrangian Y ⊂ T * X we denote by Y the flipped Lagrangian, that is, the image of Y by the involution (q, p) → (q, −p).
The same Lagrangian submanifold is generated by the action functional A H . We want to show that the spectral invariants of H coincide with those of S 0 , namely ℓ ± (H) = ℓ ± (S 0 ). First, we have Postponing the proof of the lemma for now, let us show how it allows to conclude. Since both A H and S generate L 0 , it follows that both A H = −A H and −S 0 generate L 0 = φ(O T n ), and that (−A H )|L 0 = (−S 0 )|L 0 , which yields (see subsections 2.4.2, 2.4.3) By duality considerations (see [Vi1] for example), Note that the whole construction up to this point can be performed with φ replaced by φ k and therefore where, as before, S(k) is the gfqi of the Lagrangian Φ k (O ∆ ) ⊂ T * ∆. The discussion after the formulation of proposition 3.3 shows that this suffices to prove the proposition and therefore we are done. It only remains to prove lemma 3.4.
Proof. Since both A H and S 0 generate the same Lagrangian L 0 , it suffices to show their equality at one point of L 0 . Choose a point z ∈ L 0 ∩ O T n . It exists by Lagrangian intersection theory. Let γ be the Hamiltonian arc ending at z, relative to the flow φ t H , that is γ(t) = φ t H (γ(0)) and γ(1) = z. Denote (q, 0) = γ(0) ∈ T * T n . In coordinates, γ(t) = (Q t , P t ). Note that the curve t → Q t ∈ T n has lifts to R n , and for any such lift, say, δ(t), the difference δ(t) − δ(0) is independent of the lift. We denote this difference by Q t − q ∈ R n .
Consider the following arc γ: [0, 1] → T * ∆: Let us compute the action of this arc relative to the Hamiltonian H t , that is, where λ ∆ is the Liouville form on T * ∆. We have in the first integral: The second integral equals In total we get −A H ( γ) = A H (γ) .
Denoting z = γ(1), we have as asserted. The first of these equalities follows from the fact that L 0 is obtained from L by symplectic reduction (which is just a reformulation of the fact that L 0 is generated by the gfqi S 0 which itself is the restriction of S to the zero section O T n ⊂ T * T n = ∆), and that under this reduction z is mapped to z.

Alpha function
Proof (of theorem 1.11). It suffices to show the equality for a = 0. We have the following expression for the alpha function at zero (this is implicit in [Mat]; see for example the proof of the proposition on page 178): where k ∈ N and A k L (γ) = k 0 L(t, γ(t),γ(t)) dt , L: R × T N → R being the time-periodic Lagrangian function associated to H by the Fenchel duality. We claim that the infimum in the right-hand side equals −ℓ + (φ k H ). Assuming this claim for a moment, we obtain as required.
To prove the claim, consider the functional A k L : P k → R, where P k = {γ: [0, k] → N }. The evaluation map π k : P k → N , γ → γ(k), is a submersion, therefore one can consider A k L as a generating function. It generates the Lagrangian submanifold φ k H (N ). The above infimum is in fact a minimum, therefore a critical value of A k L , and as such, it is the action of a Hamiltonian arc running from the zero section back to itself. We would like to show that this critical value is a spectral invariant of H. First, it is possible to find a genuine finite-dimensional generating function S k for φ k H (N ) whose associated quadratic form is positive-definite; 12) in this case min S k = ℓ − (S k ) .
Since any two generating functions for the same Lagrangian submanifold induce functions on it whose difference is constant, by normalizing S k we can assume that its critical values coincide with those of A k L . Thus min A k L = min S k = ℓ − (S k ) . Our sign conventions imply that the Hamiltonian action functional is the negative of the Lagrangian one when evaluated at a critical point. Therefore S k |φ k H (N ) = A k L |φ k H (N ) = −A k H |φ k H (N ), and it follows that the spectral invariants of S k coincide with those of −A k H . Therefore, by duality, min A k L = ℓ − (S k ) = −ℓ + (φ k H ) , as claimed.
12) A proof of this fact can be found in the latest version of [Vi2], appendix D.
The argument for (i) is an elaboration of this trick. Fix a non-singular closed 1-form α and let a = [α] ∈ H 1 (N ; R). Let H: T * N → R be smooth, such that the restriction of H to the graph of tα equals t for t ∈ [0, 1]. This is possible because α has no zeros. Now define a map C ∞ c (0, 1) → C ∞ c (T * N ) by f → H f := f • H. This is a linear map. Define ι: C ∞ c (0, 1) → G by ι(f ) ≡ φ f := φ H f . This map is a group homomorphism. We have max H f = max f and same for min and osc. Consequently ρ(ι(f ), ι(g)) ≤ osc(H f − H g ) = osc(f − g) .
On the other hand, if F, G are time-dependent Hamiltonians generating φ f , φ g , respectively, then by construction, since f • H equals f (t) on the graph of tα, and similarly for g. It follows that and similarly min(f − g) ≥ 1 0 min(F t − G t ) dt .
The proof for the spectral metric is analogous.
Next we prove proposition 1.14.
Proof. If H is a Hamiltonian generating φ, then for any a ∈ H 1 (N ; R) we have It follows that and the assertion about the (asymptotic) Hofer metric follows. For the spectral metric we have the comparison inequality c − (φ) ≤ ℓ + (φ) ≤ c + (φ) .
The triangle inequality for c ± implies that the sequence {c + (φ k )} k≥1 is subadditive, the sequence {c − (φ k )} k≥1 is superadditive, which means The spectral invariants c ± are invariant under the symplectomorphisms T α (see the proof of theorem 1.3 for their definition), therefore as claimed. Finally, note that the spectral norm satisfies Γ(φ) ≤ ρ(φ).
We have (see, for example [SV]): next, for any a ∈ H 1 (N ; R) such that a < 1, it is true that α Kε (a) = α H (a) .

Symplectic rigidity
Proof (of lemma 1.20). Assume that X is superheavy and for smooth f let c = max X f . It is possible, for any ε > 0, to find g ∈ C ∞ c (T * N ) such that g| X = c and g − f C 0 ≤ ε. Then ζ(f ) − ζ(g) ≤ max(f − g) ≤ ε; on the other hand ζ(g) = c, thus ζ(f ) ≤ c + ε. Now take ε → 0.
Conversely, if f | X = c, then ζ(f ) ≤ c. On the other hand where we used propertiy (vii) of theorem 1.8. Therefore ζ(f ) = c.

13)
Here · is the Gromov-Federer stable norm; see [PPS] for more information.
Proof (of proposition 1.22). We need to show that if f ∈ C ∞ c (T * N ) satisfies f | X = c then ζ(f ) = c. By the C 0 continuity of ζ it suffices to show this for all f which equal c on an open neighborhood of X. Therefore let f be such a function. Let X = X ∪ i U i = T * N − U ∞ and let f be defined as follows: it coincides with f on U ∞ and equals c on X. Then f is smooth and f | N = c since X ⊃ N . It follows that ζ( f ) = c. On the other hand, if we define the function f i by f i | U c i = 0 and f i | Ui = c − f for each i, it follows that f i is a smooth function with compact support inside U i , which is displaceable, that all the f i commute with each other and with f , and that f = f + i f i . This implies, together with the properties of ζ, that ζ(f ) = ζ( f ) = c.
Proof (of theorem 1.24). Put X = X 1 × X 2 . It is enough to show that for any f such that f | X = c, we have ζ(f ) ≤ c. Due to the Lipschitz continuity of ζ, it suffices to show the above for any function f which equals c on a neighborhood of X. So choose such a function f and let U be the neighborhood. Let U i ⊃ X i be neighborhoods of X i such that U 1 × U 2 ⊂ U . Let S i be a closed cotangent disk bundle in T * N i which contains the image under the projection T * (N 1 × N 2 ) → T * N i of the support of f . Finally, let M > 0 be a real number which satisfies min(2M, M + c/2) ≥ max f . Consider functions f i ∈ C ∞ c (T * N i ) such that f i | Xi = c/2, f i | Ui−Xi ≥ c/2, f i | Si−U c i = M , and f i | S c i ≥ 0. As a case-by-case verification shows, f 1 ⊕ f 2 ≥ f on S, and moreover f 1 ⊕ f 2 is positive on a small neighborhood V of S 1 × S 2 . Moreover, the flow of f 1 ⊕ f 2 keeps the zero section inside S 1 × S 2 . Let g be a cutoff of f 1 ⊕ f 2 outside V . Then g is a compactly supported function verifying g ≥ f , and in addition the Hamiltonian flow of g keeps the zero section inside S 1 × S 2 .
The definition of ζ for Hamiltonians with complete flow shows that if h has complete flow and h ′ is a cutoff outside a compact which contains the image of the zero section under the flow of h, then ζ(h) = ζ(h ′ ) (lemma 2.9). The properties of ζ (theorem 1.8) then show that ζ(f ) ≤ ζ(g) = ζ 1 (f 1 ) + ζ 2 (f 2 ) = c .