Hofer's length spectrum of symplectic surfaces

Following a question of F. Le Roux, we consider a system of invariants $l_A : H_1(M; \mathbb{Z})\to\mathbb{R}$ of a symplectic surface $M$. These invariants compute the minimal Hofer energy needed to translate a disk of area $A$ along a given homology class and can be seen as a symplectic analogue of the Riemannian length spectrum. When $M$ has genus zero we also construct Hofer- and $C_0$-continuous quasimorphisms $Ham(M) \to H_1(M;\mathbb{R})$ that compute trajectories of periodic non-displaceable disks.


Introduction and results
Let (M, ω) be an open symplectic surface of finite type, possibly with boundary. Pick an open disk D ⊂ M of Area(D) = D ω = A and a compactly supported Hamiltonian φ ∈ Ham(M ) such that φ(D) = D. We define the trajectory traj D (φ) ∈ π 1 (M ) in the following way. Pick a Hamiltonian isotopy φ t which connects the identity to φ. Roughly speaking, traj D (φ) is the free homotopy class of the trajectory of D under φ t . More formally, pick x ∈ D and denote by γ ⊂ D a curve which connects φ(x) with x. We catenate the trajectory of x under φ t with γ: It is not difficult to see that traj D (φ) does not depend on the choice of x, γ and φ t .
Since all disks in M of the same area are equivalent, l A depends only on the area A and not on a choice of particular D.
Clearly, l A satisfies the triangle inequality: let a, b be two based loops with the same base point, denote c = a * b. Then l A ([c]) ≤ l A ([a]) + l A ([b]). Using the argument from [LM], one shows that l A (α) ≥ A for all α = 0. Namely, given φ ∈ Ham(M ) with traj D (φ) = 0, the lift of φ to the universal cover displaces a lift of D, hence φ ≥ A by the energy-capacity inequality. Therefore l A is a nontrivial system of invariants which behaves as a symplectic analog of the length spectrum.
Question 5 from [Rou] addresses a computation of l A in the case when M is an annulus and D is nondisplaceable. Its solution appeared in [Kha1], this paper extends the results to general surfaces.
Remark. One may change the definition of l A and restrict attention to those φ that fix D pointwise. All estimates and results presented in this article remain true also for this setting, the same ideas apply after minor adjustments of the argument.
Date: October 23, 2018. The author was supported by NSF grant DMS-1105813 and also a Simons instructorship.
In this article we discuss a weaker version of this invariant where we consider trajectories in a class α ∈ H 1 (M ; Z). We prove the following: Theorem 1. given 0 < A < Area(M ) and α ∈ H 1 (M ; Z), the following estimates hold: • If A ≤ Area(M )/2, then l A (α) ≤ 2A.
• Otherwise (genus(M ) = 0 and A > Area(M )/2) l A : H 1 (M ; Z) → R is comparable to a homogeneous norm on H 1 (M ; Z). It is not important which norm we consider: H 1 (M ; R) is finitely generated, hence all homogeneous norms on H 1 (M ; Z) are equivalent.
Certain partial results are available also for the homotopical version l A : π 1 (M ) → R, see the discussion in Section 4.2.
We also show: . P A contains continuum of such quasimorphisms that are linearly independent.
These ρ A can be seen as one of many ways to generalize the rotation number to dimension two. The Lipschitz property |ρ A (φ)| ≤ c φ implies the lower bound for the third case of Theorem 1: Additional properties and applications of invariants ρ A are discussed in Section 4.4.
This paper is organized as follows. In order to show the upper bounds for l A as in Theorem 1 we describe explicit Hamiltonian isotopies that achieve the desired energy bounds. Details are provided in Section 2. In Section 3 we construct ρ A for Theorem 2 from the Calabi quasimorphism Ham(S 2 ) → R described in [EP]. (Note that the conditions on M in Theorem 2 are equivalent to the statement that M embeds to a sphere.) Section 4 discusses possible generalizations of l A and provides few applications of the quasimorphisms from Theorem 2.
Acknowledgements: The author would like to thank D. Calegari, V. Humiliere, F. Le Roux and F. Zapolsky for useful discussions and comments.

Upper bounds
In this section we show upper bounds for Theorem 1. We construct explicit Hamiltonian isotopies that translate a disk along a given homology class and remain within the prescribed energy bound.
2.1. Suppose A = Area D ≤ Area(M )/2. We assume first that A < Area(M )/2. Consider the following Hamiltonian in (R 2 , dx∧dy). We pick two disjoint disks D 1 and D 2 of area A in the plane, connect them by two narrow nonintersecting strips ('pipes') as described in Figure 1. Let H be an autonomous Hamiltonian which equals one in the area bounded by the two disks and the pipes, zero outside and linearly interpolated in between. We apply to H a C 0 -small smoothing near the singular points.
The flux of the Hamiltonian flow through a curve γ equals to the difference of values of H at the endpoints of γ. Hence if we choose γ to be a cut going to the inner region (where H = 1) from the outside, the flux through γ is one. This implies that the resulting flow is relatively slow inside the disks and accelerates in the pipes where such a cut γ can be very short. For appropriate choice of the smoothing of H the time-(A + ε) map of the flow of H will move D 1 to D 2 . (In fact, it is possible to arrange that the two disks will be swapped.) The Hofer norm is at most A+ε 0 (max H − min H)dt = A + ε. ε depends on the area of the pipes and the choice of smoothing of H and can be made arbitrarily small. Note that the energy cost of (A + ε) is optimal since every Hamiltonian that displaces D 1 has energy ≥ A by the energy-capacity inequality.
We augment this construction as in Figure 2: we choose the first pipe to go along an interval connecting D 1 with D 2 and let the second pipe lie in a small neighborhood of the two disks and the first pipe. This way the Hamiltonian H is supported in a small neighborhood of the two disks and an interval. This construction can be embedded to any surface given two disjoint copies of a disk and a simple path connecting boundary points of the disks. This allows us to translate a disk along a simple path.
Let D 1 , D 2 be two disjoint copies of D in M and let γ 1 be a simple path connecting D 1 to D 2 and γ 2 be a simple path connecting D 2 to D 1 . γ 1 and γ 2 may intersect one another but not the disks. We apply the construction above in order to move D 1 to D 2 along γ 1 and then move back to D 1 along γ 2 . The total energy cost of such process equals 2 Area(D) + 2ε and the result is a translation of the disk D 1 along a trajectory in class [γ 1 * γ 2 ]. (By abuse of notation we extend γ 1 , γ 2 inside the disks so that they have the same endpoints and catenation makes sense.)  The upper bound follows from the following topological lemma: Lemma 3. Let M be a surface and α ∈ π 1 (M ). Then there exist two simple curves γ 1 , γ 2 with the same endpoints such that α = [γ 1 * −γ 2 ]. (−γ 2 denotes γ 2 with reversed orientation.) Indeed, pick a class α ∈ π 1 (M ). By this lemma there exist two points p, q ∈ M and two simple curves γ 1 , γ 2 connecting them so that [γ 1 * γ 2 ] = α. We replace p and q by small disks D 1 , D 2 which intersect both curves only in small intervals near the endpoints. Pick a symplectic form ω on M so that both disks have area A and the total area is M ω = M ω. (M, ω ) is symplectomorphic to (M, ω) by Moser's argument. We construct a Hamiltonian in (M, ω ): translate D 1 along γ 1 to D 2 and then back along γ 2 . The resulting deformation φ satisfies at the cost of 2 Area(D) + 2ε, where ε is arbitrarily small.
Assume that A = Area(M )/2 and fix α = [γ 1 * γ 2 ] ∈ π 1 (M ). Pick a smaller disk D ⊂ D, suppose that Area(D) = A − . We apply the previous argument and construct φ ∈ Ham(M ) which translates D along α. Then we deform φ(D) back to D by another Hamiltonian isotopy f which fixes the smaller disk D. Note that the annulus φ(D \ D) is spread along γ 1 ∪ γ 2 and its shape does not depend on the choice of D. This implies that f can be chosen with f ≤ c where c depends on γ 1 , γ 2 but not on . Therefore The result follows by letting ε, → 0.
Remark. Note that we proved the upper bound for α ∈ π 1 (M ) which, in particular, implies the same bound for the homological spectrum.
Proof of lemma. Pick α ∈ π 1 (M ). Let γ be a loop representing α which has finite number of transverse intersections. We deform γ so that it can be cut into a pair of simple curves. Pick arbitrary point p ∈ γ which will be a starting point of γ 1 . We draw γ 1 by traversing γ starting from p. We prevent self-intersections of γ 1 as follows. Once we arrive to a self-crossing we stretch the previously drawn path an push it in a small neighborhood in front of the pen (see Figure 4). After passing several crossing we will have a number of strings pushed in this manner. During of this process we construct a simple path which is homotopic to the corresponding arc of γ relative the endpoints. We stop the process when we arrive to a point q ∈ γ such that the arc [q, p] ⊂ γ is simple. Set γ 2 = [q, p] ⊂ γ. Lemma follows since γ 1 is homotopic to [p, q] ⊂ γ relative the endpoints.
. We show first that one can translate a disk along a simple loop using energy Area(D) + ε. Then we consider separately the cases genus(M ) = 0 and genus(M ) > 0. Consider the following Hamiltonian in (R 2 , dx ∧ dy). We pick a disk D ⊂ R 2 of area A and connect two boundary points of the disk by a circular pipe. Let H be an autonomous Hamiltonian which equals one in the inner region, zero outside and linearly interpolated in between. We apply to H a C 0 -small smoothing near the singular points (see Figure 5).
The Hamiltonian flow of H is relatively slow inside the disk D and fast within the pipe. For an appropriate choice of a smoothing of H there exists ε > 0 such that D is a fixed set of the time-(A + ε) map of the flow of H. The energy is at most A + ε and ε depends on the area of the pipe and a choice of smoothing and can be made arbitrarily small.
We cut H off inside the circle so that it is supported in a small neighborhood of the disk and the pipe. Note that this cutoff does not affect the flow of H in the disk D and the pipe. This construction can be copied to any symplectic surface given a disk and a simple curve which connects boundary points of the disk and does not intersect it away from the endpoints. This implies the following result.
Lemma 4. Suppose that α = 0 is represented by a simple loop. Then l A (α) = A.
Proof. Let γ be a simple loop representing α. As in the previous subsection, we deform ω so that there exists a disk of area A whose boundary points are connected by an arc of γ and this arc does not intersect the disk except for the endpoints. An application of the Hamiltonian described above implies l A (α) ≤ A + ε. The lemma follows by letting ε → 0 and from the fact that l A (α) ≥ A.
Corollary 5. Let S ⊆ H 1 (M ; Z) be the set of classes represented by simple loops. It is easy to see that S generates H 1 (M ; Z). Denote by · S the norm on H 1 (M ; Z) given by word length with respect to S. The triangle inequality implies l A (α) ≤ A · α S . The same argument holds also for the homotopical length spectrum.
Given any surface M , H 1 (M ; Z) admits a basis B represented by simple loops. By the triangle inequality, l A (α) ≤ A · α B where · B is the word length norm with respect to B. · B is a homogeneous norm on H 1 (M ; Z). This shows the upper bound for the third case in Theorem 1. A more careful argument may provide better constants. For example, in the case of an annulus M the bound can be improved to (see [Kha1]).
Suppose that genus(M ) > 0. The upper bound for Theorem 1 follows from the lemma below and Corollary 5.
Lemma 6. Let M be a surface with positive genus. Then every homology class in H 1 (M ; Z) can be represented by a sum of two simple loops. Proof.
Step 1: where each summand is represented by a simple loop (see Figure 6). α β Step 2: Suppose M = S 1 × S 1 \ {p 1 , . . . , p k } is a punctured torus. Without loss of generality we assume that all punctures lie on a horizontal meridian. H 1 (M ; Z) admits a basis of k vertical meridians α 1 , . . . , α k and a horizontal meridian β (see Figure 7). Then every where each summand is represented by a simple loop.
Step 3: Suppose genus(M ) > 1. M can be decomposed into a connected sum of punctured tori We identify H 1 (T i ; Z) with their image in H 1 (M ; Z) by abuse of notation.
Pick α ∈ H 1 (M ; Z). There exist α i ∈ H 1 (T i ; Z) such that α = α i . By the previous step we choose homology classes β i , β i ∈ H 1 (T i ; Z) represented by simple loops such that Both β i and β i are represented by simple loops: consider a connected sum of the corresponding simple loops in each T i . The result is a simple loop in M since T i are disjoint.

Construction of quasimorphisms ρ
The quasimorphism r is homogeneous if it satisfies r(g m ) = mr(g) for all g ∈ G and m ∈ Z. Any homogeneous quasimorphism satisfies r(f g) = r(f ) + r(g) for commuting elements f, g and is invariant under conjugations.
We construct quasimorphisms ρ A described in Theorem 2. Recall that M is a surface of finite type of genus zero. M has finite area, so without loss of generality we rescale ω to get Area(M ) = 1. Such M is symplectomorphic to the interior of a k-times punctured disk U \ {p 1 , . . . , p k } where k is the rank of H 1 (M ; Z) and Area(U ) = 1.
is a homogeneous quasimorphism which satisfies all properties of Theorem 2. Continuum of linearly independent choices of ρ A implies that for a choice of ρ A .
Proof of Proposition 7 takes the rest of this section. It can be found in [Kha1], we give it below for the sake of completeness. We use notation A = S 1 × (0, 1) R/Z × (0, 1) equipped with the standard symplectic form ω = dθ ∧ dh, Area(A) = 1. Without loss of generality we assume that the generator α ∈ H 1 (A; Z) is represented by the positively oriented circle S 1 × {0.5}. Let D ⊂ A be a disk with Area(D) = A, denote L = ∂D.
Pick a function H : A → R with compact support such that H(θ, h) = h away from a small neighborhood of ∂A. Denote by Φ the time-1 map of the Hamiltonian flow generated by H. It is easy to see that Φ ∈ S with trajectory [traj D ( Φ)] = α. Note that S = n∈Z S n = n∈Z Φ n S 0 . We construct a quasimorphism ρ A .
Step I: it is sufficient to construct a homogeneous quasimorphism ρ : Ham(A) → R which satisfies: • ρ is Hofer-Lipschitz, Proof. ρ vanishes on Hamiltonians supported in a disk. It follows from the results of [EPP] that such ρ is continuous in the C 0 -topology. Any φ ∈ S n decomposes as φ = Φ n • s for some s ∈ S 0 . Hence (R denotes the defect of ρ). It follows that We set ρ A = ρ c . Suppose that φ ∈ Ham(A) has an invariant disk D of Area(D) ≥ A.
(1) We consider the case D = D. Then φ ∈ S n for some n ∈ Z and (2) The case Area(D) = A. Pick g ∈ Ham(A) such that g(D) = D. Then gφg −1 ∈ S hence (3) The case Area(D) > A. Pick a smaller disk D 1 ⊂ D with Area(D 1 ) = A. φ(D) = D, choose g ∈ Ham(D) such that g(φ(D 1 )) = D 1 . Then D 1 is a fixed disk of area A for gφ hence We claim that ρ A (gφ) = ρ A (φ). Indeed, the disk D is fixed for gφ, φ and their iterates. Moreover, (gφ) n differs from φ n by a Hamiltonian f n ∈ Ham(D). Hence by the homogenuity of ρ A is bounded by the defect of ρ A . Letting n → ∞ we show ρ A (φ) = ρ A (gφ).
Step II: in order to build ρ we use the Calabi quasimorphism on Ham(S 2 ) which was constructed by M. Entov and L. Polterovich in [EP]. We give a brief recollection of the relevant facts.
Let U be an open disk equipped with a symplectic form ω. Let F t : U → R, t ∈ [0, 1] be a time-dependent smooth function with compact support. We define As ω is exact on U , Cal descends to a homomorphism Cal U : Ham(U ) → R which is called the Calabi homomorphism. Clearly, for φ ∈ Ham(U ), |Cal U (φ)| ≤ Area(U ) · φ . Let S 2 be a sphere equipped with a symplectic form ω. Suppose Area(S 2 ) = 2A. For a smooth function F : S 2 → R the Reeb graph T F is defined as the set of connected components of level sets of F (for a more detailed definition we refer the reader to [EP]). For a generic Morse function F this set, equipped with the topology induced by the projection π F : S 2 → T F , is homeomorphic to a tree. We endow T F with a measure given by µ(X) = π −1 F (X) ω for any X ⊆ T F with measurable π −1 F (X). x ∈ T F is called a median of T F if the measure of each connected component of T F \ {x} does not exceed A. By [EP] a median exists and is unique. This construction can be extended to functions F such that F supp(F ) is Morse. [EP] describes construction of a homogeneous quasimorphism Cal S 2 : Ham(S 2 ) → R. It has the following properties: Cal S 2 is Hofer-Lipschitz (|Cal S 2 (φ)| ≤ 2A · φ ). In the case when φ ∈ Ham(S 2 ) is supported in a disk U which is displaceable in S 2 , Cal S 2 (φ) = Cal U (φ U ). Moreover, for φ ∈ Ham(S 2 ) generated by an autonomous function F : S 2 → R, Cal S 2 (φ) can be computed in the following way. Let x be the median of T F and X = π −1 F (x) be the corresponding subset of S 2 . Then Given a symplectic embedding j : A → S 2 into a sphere of area 2A, consider the pullback Cal j = j * (Cal S 2 ) : Ham(A) → R. Namely, given φ ∈ Ham(A), extend j * (φ) toφ ∈ Ham(S 2 ) by identity on the complement of j(A). Then Cal j (φ) = Cal S 2 (φ). Clearly, Cal j is a homogeneous quasimorphism. It has the following properties: • Cal j (φ) = Cal D (φ D ) for any φ supported in a disk D of area A. To see that note that the correspondingφ ∈ Ham(S 2 ) is supported in a displaceable disk j(D) in • for an autonomous φ generated by a compactly supported function H : A → R, where X ⊆ A is the level set component which is sent by j to the median set of j * (H) in S 2 .
Consider the embeddings j s : A → S 2 (0 ≤ s ≤ 2A − 1) into a sphere of area 2A that are given by gluing a disk of area s to S 1 × {0} and a disk of area (2A − 1 − s) to S 1 × {1}. This construction ensures that j s (L) = j s (∂D) bisects S 2 into two displaceable disks.
We claim that ρ satisfies conditions of Step I. Obviously, ρ is a homogeneous quasimorphism on Ham(A) which satisfies the Lipschitz property: Consider the function H which was used to define the Hamiltonian Φ described above. It is easy to see that the "median" level set X s of H which is relevant for the computation

This implies
It is left to show that ρ vanishes on S 0 . Denote by S 0 the subgroup of S 0 which fixes a neighborhood of L pointwise.
Lemma 8. Let q be a homogeneous quasimorphism which is Hofer-continuous and vanishes on S 0 . Then q vanishes on S 0 .

Proof.
Pick an open disk U ⊂ A \ L. Ham(U ) ⊂ S 0 , therefore q vanishes on Ham(U ). It follows from the results of [EPP] that q is continuous in the C 0 -topology.
φ P is Hamiltonian in A, but after the restriction to P we have justφ P = φ P P ∈ Symp c (P ). In the argument below we apply a sequence of deformations to φ P in order to get φ P whose restrictionφ P ∈ Ham(P ). All deformations involved in the process preserve the value ρ(φ P ). Finally, we show that ρ(φ P ) = 0 by explicit computation.
The mapping class group π 0 (Symp c (P )) is isomorphic to Z 3 and is generated by Dehn twists near the three boundary components. For the proof of this fact we refer the reader to [FM] where the authors show that π 0 (Dif f c (P )) Z 3 and is generated by Dehn twists. Note that φ, ψ ∈ Symp c (P ) are isotopic in Symp c if and only if they are isotopic in Dif f c . As Dehn twists belong to Symp c (P ), the statement for π 0 (Symp c (P )) follows.
Denote by T 1 , T 0 Dehn twists near S 1 × {1}, S 1 × {0} and by T L a Dehn twist in P near L = ∂D. There are Hamiltonians ψ L in S 0 with arbitrary small Hofer norm whose restriction to P realizes the Dehn twist T L . For example, consider a bump function supported near D which has small height but is very steep in an annulus near L.φ P is isotopic in Symp c (P ) to some T k1 As ψ L can be chosen to be arbitrarily small, by continuity of ρ it is enough to show the desired statement for the deformed φ. After the replacement k L vanishes, hence the modifiedφ P ∼ T k1 1 T k0 0 . Note thatφ P is induced by a Hamiltonian φ ∈ S. The definition of traj D (φ) implies that k 1 α = [traj D (φ)] = −k 0 α (recall that α is the chosen generator of H 1 (A; Z)). The minus sign appears because opposite orientations of the boundary components result in the opposite directions of the corresponding Dehn twists. Moreover, as φ ∈ S 0 , [traj D (φ)] = 0 hence k 1 = k 0 = 0. Therefore the restriction φ P belongs to the identity component of Symp c (P ).
Pick K : A → R supported in a small neighborhood of D such that K = 1 in a neighborhood of the closure D. Denote by χ t the time-t map generated by the Hamiltonian flow of K. χ t is supported in a disk in A, hence ρ(χ t ) = 0 for all t.
Consider the homomorphism i * : H 1 c (P ; R) → H 1 c (A; R) induced by inclusion i : P → A. Both χ t , φ P are Hamiltonian in A, hence their fluxes are zero in H 1 c (A; R). After restriction to P , f lux(χ t P ), f lux(φ P ) belong to one-dimensional subspace ker i * ⊂ H 1 c (P ; R). f lux(χ t P ) = 0, therefore one can find an appropriate t φ ∈ R such that φ P = χ t φ • φ P restricts toφ P = χ t φ P •φ P with zero flux in P . The results of [Ban] implyφ P ∈ Ham(P ).
Pick a compactly supported function F t : P × [0, 1] → R whose flow generatesφ P . Denote by U s the complement of the closed disk j s (D) in S 2 , it is a displaceable disk. j s, * (φ P ) ∈ Ham(S 2 ) and it is supported in U s , therefore is bounded by the defect of ρ. It follows that ρ is bounded on the subgroup S 0 . As ρ is homogeneous, it vanishes there.
Remark. Choice of different parameters s 1 , s 2 in (1) gives rise to different quasimorphisms ρ A = ρ A,s1,s2 . If we fix s 1 and let s 2 vary, we get a linearly independent family. We show that by an argument from [BEP]: If we consider the set of all Hamiltonians generated by functions H = H(h), the values of the family {ρ A,s1,s2 } s2 applied to this set are linearly independent.

4.1.
Comparison with the Riemannian length spectrum. In the case of a Riemannian manifold (M, g) the marked length spectrum l : π 1 (M ) → R contains a lot of information regarding the metric g. For example, in a closed negatively curved surface (M, g), g is completely determined by l ( [Ota]). The symplectic analogue looks much less rewarding. The only case when l A : H 1 (M ; Z) → R is not bounded is the case of a punctured disk. Therefore Hofer's length spectrum is able to detect genus zero surfaces of bounded area. It may happen that more accurate estimates of l A provide additional information regarding the topology of M .
On the other side, the symplectic setting is less rich than the Riemannian one. The equivalence class of (M, ω) is determined by the genus, rk(H 1 (M ; Z)) and Area(M ). In the case of finite area ω is just a scaling factor, hence asymptotic behavior behavior of l A depends only on the topology of M . Clearly, l A is able to extract some of the topological information of M . This result may be interpreted as the fact that the geometry of Ham(M ) sees some of the topology of M . 4.2. Homotopical spectrum. It would be interesting to provide similar estimates for l A : π 1 (M ) → R. When we consider the asymptotical behavior of l A , it is more convenient to discuss the stabilized versionl n .
The homological version ofl A is a norm on H 1 when 2A > Area(M ) and genus(M ) = 0. In all other casesl A ≡ 0.
If we replace H 1 by π 1 , the argument from the previous sections gives the following partial information: • Suppose that 2A ≤ Area(M ). Thenl A ≡ 0. This is true since the argument of Section 2.1 applies for π 1 and not just H 1 .
The author does not know estimates for most of the remaining cases. For example, let 2A > Area(M ) and genus(M )geq2. For a primitive self intersecting class α the number of simple curves needed to represent nα is not bounded as n → ∞ (see [Cal]), so the argument for the upper bound does not give a lot. On the other hand, we do not know any tools that may give a nontrivial lower bound. There are several constructions of quasimorphisms on positive genus surfaces, but none of them is known to respect Hofer's metric. When genus(M ) = 0 and α belongs to the commutator subgroup [π 1 (M ), π 1 (M )], the argument used in this paper fails as translation by α cannot be realized by an autonomous flow. It may happen that Entov-Polterovich quasimorphisms used in our argument still imply a nontrivial lower bound. Unfortunately, the author was not able to compute them on appropriate non-autonomous examples.
D. Calegari observed that if one takes the word length norm with respect to the generating set of simple loops, then A times its stabilized version A · S satisfies all properties ofl A described above (we assume 2A > Area(M )). Indeed, the discussion in Section 2 impliesl A ≤ A · S and we do not know any example where the inequality is strict.

Higher dimensions.
Most arguments and constructions used in this paper completely fail when one considers balls or tori in higher dimension manifolds. For example, upper bounds for l A (α) were induced by decomposition of α into simple loops. In dimension ≥ 4 any loop can be perturbed into a simple one. However, we cannot translate a ball through a pipe containing such simple loop because of the 'symplectic camel' phenomenon.
4.4. The quasimorphism ρ A . Theorem 2 describes quasimorphisms ρ A : Ham(M ) → H 1 (M ; R) for a k times punctured disk M . ρ A can be seen as a generalization of the rotation number ρ : Homeo + (S 1 ) → R. Indeed, ρ satisfies the following: it is Lipschitz with respect to the C 0 -norm on Homeo + (S 1 ). If f ∈ Homeo + (S 1 ) has a fixed point x, then ρ(f ) computes the degree of the trajectory of x under f . In fact, ρ is determined by the second property, while there is a continuous family if linearly independent homogeneous quasimorphisms satisfying same properties as ρ A .
Let f t be a Hamiltonian isotopy and D a disk of area A or greater. The computation in Section 3 shows that ρ A vanishes on those isotopies f t that fix ∂D for all t. This implies that the value ρ A (f 1 ) is determined up to a bounded defect by the trajectory f t (∂D). That is, for any g t such that g t (∂D) = f t (∂D), ρ A (g 1 ) − ρ A (f 1 ) is bounded. Therefore ρ A can be thought as an invariant of Hamiltonian isotopies of ∂D.
In particular, ρ A nearly ignores those dynamical attributes that do not affect a large enough disk in M , it sees only 'global' features that deform all large disks. In this sense it is different from most other generalizations of rotation number to dimension two.
A possible application is the ability to detect non-existence of large fixed or periodic disks. Namely, if we pick two such quasimorphisms ρ A , ρ B and ρ A (f ) = ρ B (f ), then f has neither fixed nor periodic disks of area ≥ max{A, B}. More than that, if one wishes to perturb f in order to have such a periodic disk, ρ A (f ) − ρ B (f ) allows to estimate a lower bound on Hofer's energy of such perturbation. However, current techniques allow to compute the value of ρ A in very limited number of scenarios (like the case of autonomous Hamiltonians, locally supported and etc.) which limits practical use of this observation.
Another application is as follows. Pick a non-displaceable disk D in M and two linearly independent quasimorphisms ρ A , ρ B where A, B ≤ Area D. Denote by L the space of Lagrangians Hamiltonian isotopic to ∂D. Suppose that f t is a Hamiltonian isotopy. A simple computation shows that τ = ρ A − ρ B depends (up to a bounded defect) just on f 1 (∂D) ∈ L. This way τ gives rise to an invariant on L which is Lipschitz with respect to the induced Hofer's norm. τ is not bounded since it is a nonzero homogeneous quasimorphism, therefore by Lipschitz property the space L is not bounded in Hofer's norm as well (a similar argument with more details can be found in [Kha2]).