Entropy and quasimorphisms

Let $S$ be a compact oriented surface. We construct homogeneous quasimorphisms on $Diff(S, area)$, on $Diff_0(S, area)$ and on $Ham(S)$ generalizing the constructions of Gambaudo-Ghys and Polterovich. We prove that there are infinitely many linearly independent homogeneous quasimorphisms on $Diff(S, area)$, on $Diff_0(S, area)$ and on $Ham(S)$ whose absolute values bound from below the topological entropy. In case when $S$ has a positive genus, the quasimorphisms we construct on $Ham(S)$ are $C^0$-continuous. We define a bi-invariant metric on these groups, called the entropy metric, and show that it is unbounded. In particular, we reprove the fact that the autonomous metric on $Ham(S)$ is unbounded.


INTRODUCTION
Let M be a smooth compact manifold with some fixed Riemannian metric. where the base of the log is two. It turns out that h( f ) does not depend on the choice of Riemannian metric, see [6,14]. It is notoriously difficult to compute topological entropy of a given diffeomorphism; even to detect whether entropy of a given diffeomorphism is non-zero is a difficult task in most cases.
We consider the case when M is a compact connected oriented surface S endowed with an area form. Let Diff(S, area) and Diff 0 (S, area) be groups, where the first group is the group of area-preserving diffeomorphisms of S, and the second group is the group of area-preserving diffeomorphisms of S isotopic to the identity. If S has a boundary, then we assume that diffeomorphisms are identity in some fixed neighborhood of the boundary.
In the first part of the paper we revise and extend the construction of quasimorphisms on Diff 0 (S, area) given by Gambaudo-Ghys [17] and Polterovich [23], see also [7,8]. The main advantage of our approach is that it allows us to treat all surfaces in an unified way and to show there are infinitely many linearly independent homogeneous quasimorphisms on Diff(S, area) whose restrictions on Diff 0 (S, area) are linearly independent.
In the second part of our work we show that there are infinitely many linearly independent homogeneous quasimorphisms on Diff(S, area) and on Diff 0 (S, area) whose absolute values bound from below the topological entropy. The same holds for the group Ham(S) of Hamiltonian diffeomorphisms of S. More precisely, we apply the construction described in Section 2 to quasimorphisms on mapping class groups constructed by Bestvina and Fujiwara in [3]. We prove that these quasimorphisms are Lipschitz with respect to the topological entropy. Our work is inspired by the paper of Gambaudo and Pecou [16], who constructed a dynamical cocycle on the group Diff(D 2 , area) that bounds from below the topological entropy.
Recall that a function ψ from a group G to the reals is called a quasimorphism if there exists D such that |ψ(a) − ψ(ab) + ψ(b)| < D for all a, b ∈ G. Minimal such D is called the defect of ψ and denoted by D ψ . A quasimorphism ψ is homogeneous if ψ(a n ) = nψ(a) for all n ∈ Z and a ∈ G. A quasimorphism ψ can be homogenized by setting ψ(a) := lim p→∞ ψ(a p ) p .

The vector space of homogeneous quasimorphisms on G is denoted by Q(G).
For more information about quasimorphisms and their connections to different branches of mathematics, see [13]. Throughout the paper we assume that the surface S is always connected. Our main result is the following: THEOREM 1. Let S be a compact oriented Riemannian surface and let the group G = Diff(S, area) or G = Diff 0 (S, area) or G = Ham(S). Then there exists an infinite dimensional subspace of Q(G) such that every Ψ in this subspace is Lipschitz with respect to the topological entropy, i.e., there exists a positive constant C Ψ , which depends only on Ψ, such that for every f ∈ G we have Let Ent(S) ⊂ G be the set of entropy-zero diffeomorphisms. This set is conjugation invariant and generates G, see Lemma 5.1. In other words, a diffeomorphism of S is a finite product of entropy-zero diffeomorphisms. One may ask for a minimal decomposition and this question leads to the concept of the entropy norm which we define by It is the word norm associated with the generating set Ent(S). This set is conjugation invariant and so is the entropy norm. The associated bi-invariant metric is denoted by d Ent . It follows from the work of Burago-Ivanov-Polterovich [12] and Tsuboi [26,27] that for many manifolds all conjugation invariant norms on Diff 0 (M ) are bounded. Hence the entropy norm is bounded in those cases.
We show that the situation is different for G. More precisely, as a corollary of our main result we obtain the following: THEOREM 2. Let S be a compact oriented Riemannian surface and let the group G be Diff 0 (S, area) or Diff(S, area) or Ham(S). Then the diameter of (G, d Ent ) is infinite. Moreover, in case when S is a closed disc, for each m ∈ N there exists a bi-Lipschitz injective homomorphism where Z m is endowed with the l 1 -metric.

REMARK.
There exists another conjugation invariant word norm on Ham(S), the autonomous norm. It is unbounded in the case when S is a compact oriented surface, see [8,9,10,11,17]. Theorem 2 together with the fact that every autonomous diffeomorphism of a surface has entropy zero, see [29], gives a new proof of unboundedness of the autonomous norm on Ham(S).

QUASIMORPHISMS ON DIFFEOMORPHISMS GROUPS OF SURFACES
Let S be a compact oriented surface endowed with an area form. In this section we define two linear maps G S,n : Q(MCG(S, n)) → Q(Diff(S, area)), Here MCG(S, n) is the mapping class group of S with n punctures and K (S, n) is a certain subgroup of MCG(S, n). If S is a disk, then the group K (S, n) is isomorphic to the braid group on n strands and G 0 S,n is the map defined by Gambaudo-Ghys [7,17]. If S is a closed surface of genus g ≥ 2, then K (S, 1) is isomorphic to the fundamental group of S and G 0 S,1 is the map defined by Polterovich [23].
The main difference in our approach is that we work with mapping class groups, instead of braid groups. Our definition of G 0 S,n is isotopy free, which allows us to define G 0 S,n in the case when π 1 (Diff 0 (S, area)) is non-trivial. It also allows us to extend the construction to the case of Diff(S, area).
The map G S,n is an extension of G 0 S,n in a sense that we have the following commutative diagram: The vertical maps are restrictions to subgroups. Many quasimorphisms we construct on Diff(S, area) restrict to non-trivial quasimorphisms on Diff 0 (S, area). In particular, they do not arise as a pull back of quasimorphisms on MCG(S) by the quotient map Diff(S, area) → MCG(S).
In the remaining part of this section we define maps G S,n , G 0 S,n and discuss their basic properties.

2.A. Configuration space.
Let D 2 be an open disc in the Euclidean plane, and let X n be the configuration space of n points in D 2 . We fix a tuple z = (z 1 , . . . , z n ) ∈ X n . Let ev z : It is shown in [16, Section 3.2 and Theorem 4], that there is a subset H n of X n with the following properties: it is a union of submanifolds of codimension 1, there exists a map h : X n \H n → Diff(D 2 ) which is a section of ev z , i.e., ev z • h is the identity on X n \H n .
Denote by Ω n = X n \H n , and let h be a section described as follows: let x = (x 1 , . . . , x n ) ∈ Ω n . Let P i be a geodesic segment connecting z i to x i . The fact that x ∉ H n guaranties, that x i and z i do not lie on P j for every i and j = i (for details see [16,Section 3.2]). By N (P i ) we denote the -neighborhood of P i . By the definition of H n , we can pick a small (x) ∈ R + such that x i , z i ∉ N (x) (P j ) for every i and j = i . The choice of (x) can be made such that the function (x) is C 1 -continuous on Ω n .
Let h x i ∈ Diff(D 2 ) be a map which pushes z i to x i along P i and is supported on By the definition, h x maps z i to x i . This is a C 1 -continuous section of ev z . The area form on S induces the volume form on X n (S), which is the configuration space of n-points in S. Spaces X n = X n (D 2 ) and Ω n are full measure subsets of X n (S). The group Diff(S, area) acts on X n (S) preserving the measure. Let MCG(S, n) = MCG(S, { j (z 1 ), . . . , j (z n )}), where {z i } are defined in Section 2.A. We define a cocycle γ S,n : Diff(S, area) × X n → MCG(S, n) by the formula To be fully correct, the map γ S,n ( f ) : X n → X n is not defined on X n , but on a full measure subset of X n that depends on f , namely, on the set Ω n, f = Ω n ∩ f −1 (Ω n ). It is easy to show that γ S,n is a cocycle, i.e., Consider the forgetful map F : MCG(S, n) → MCG(S) and denote K (S, n) = ker(F ). If f ∈ Diff 0 (S, area) then is homotopic to the identity in S, possibly by a homotopy which can move points j (z i ). Thus F (γ S,n ( f , x)) = 1 and γ S,n ( f , x) ∈ K (S, n). It follows that we can restrict γ S,n to Diff 0 (S, area) and obtain the cocycle: γ 0 S,n : Diff 0 (S, area) × X n → K (S, n). In the same way, γ S,n restricts to Ham(S).
2.B.1. Relation to braids. Let us recall a construction due to Gambaudo and Ghys [17,Section 5.2]. With an isotopy f t , t ∈ [0, 1] and a point x = (x 1 , . . . , x n ) ∈ X n (S), we associate a braid γ S,n ( f , x) ∈ B n (S) in the following way: we connect z with x using geodesic segments as in Section 2.A, then we connect x with f (x) by f t (x), t ∈ [0, 1] and at the end we connect f (x) again with z using geodesic segments.
Let us now describe the relation between γ S,n ( f , x) and γ S,n ( f , x). Recall the Birman map Push where B n (S) = π 1 (X n (S), z) is the braid group of S on n strings and the definition of Push is the following: let γ(t ), t ∈ [0, 1], be a loop in X n (S) based at z and ψ t ∈ Diff(S) an isotopy such that ψ t (z) = γ(t ). Then Push(

2.B.2. Finitely many mapping classes.
We say that a function γ defined on a probability space X has essentially finite image, if there exists a full measure subset of X on which γ has finite image.
has essentially finite image.
Before the proof we need some preparations. The following Proposition is an immediate consequence of the cocycle condition.
To prove Lemma 2.1, it is enough to prove it for a generating set of Diff(S, area). Let us consider the following three types of diffeomorphisms. Proof of Lemma 2.1. First we consider the case when S is a closed oriented surface. It follows from Lemma 2.2 that it is enough to prove the statement for Morse autonomous diffeomorphisms, Hamiltonian pushes and area-preserving Dehn twists. Let f be a Morse autonomous diffeomorphism. There exists a full measure subset X 0 n (S) < X n (S) where the set of braids associated to f is finite, see [8, Section 2.C]. It means that the set {γ S,n ( f , x) | x ∈ X 0 n (S)} is finite. The same analysis as in [8, Section 2.C] is applied to Hamiltonian pushes.
In the case of area-preserving Dehn twists we proceed as follows. Let f be a Dehn twist supported in annulus A ⊂ S. We can assume that z i ∉ A for all i = 1, . . . , n. Thus [ f ] ∈ MCG(S, n) and Since h x and h f (x) are isotopic to the identity, this implies that . Now the same analysis as in [8,Section 2.C] shows that there are finitely many braids of the form δ S,n ( f , x), and the proof follows for closed S.
Assume that S has a boundary. In this case we embed S into a closed surface S such that i : MCG(S, n) → MCG(S, n) is an embedding. For example, one can cap each boundary component of S with a torus with one boundary component. We extend the area form from S to S. It is possible to define the geodesic segments P i for S such that they agree with the geodesic segments defined for S (see Section 2.A). Now for x ∈ S and f ∈ Diff(S, area) we have that i • γ S,n ( f , x) = γ S,n (f , x), wheref is the extension of f to Diff(S, area) by the identity.
2.C. Definition of G S,n and G 0 S,n . Let ψ ∈ Q(MCG(S, n)) and let f be a diffeomorphism in Diff(S, area). By Lemma 2.1 the function In the last equality we used the fact that g preserves the measure, thus In a similar way one shows that G S,n (ψ)( f g ) ≥ G S,n ( f )+G S,n (g )−Area(S)D ψ .
The map G 0 S,n is defined in the same way as G S,n , except that now instead of γ S,n we use γ 0 S,n . In this situation we obtain a linear map G 0 S,n : Q(K (S, n)) → Q(Diff 0 (S, area)). It is also defined from Q(K (S, n)) to Q(Ham(S)).
2.D. Embedding Theorem. The proof of the following theorem is a variation of the proof of Ishida [19], see also [8,10]. We present it for the reader convenience. THEOREM 2.5. G S,n and G 0 S,n are injective. Proof. We give a proof in the case of G S,n . The argument for the injectivity of G 0 S,n goes along the same lines. Let ψ ∈ Q(MCG(S, n)) and let γ in MCG(S, n) such that ψ(γ) = 0. Dehn twists generate MCG(S, n), thus we can express γ as a product of Dehn twists along some finite set of simple loops C . We assume that z i does not lie on any loop in C . Let N be a small tubular neighborhood of loops in C such that z i ∉ N . We choose f such that [ f ] = γ and f is supported in N . The idea of the proof is to show that we can choose N in a way that G S,n (ψ)( f ) = 0. In what follows we will split X n into two pieces: one which has a small volume, and one on which f is the identity. This allows us to control the value of the integral.
Step 1. By definition f is the identity on D 2 \N . Let X N be a set of tuples in X n that have at least one coordinate in N . Then the volume of X N goes to zero when the area of N goes to zero. Let γ = γ S,n and D f be a full measure subset of X n on which γ( f ) has finite image. Let For all x in certain full measure subset of X N we have: Step 2. Let U = D 2 \N and consider U n ⊂ X n . Note that X n = U n ∪ X N and f acts identically on U n . The set U is open and has finitely many connected components. Denote them by U 1 , . . . ,U k . Let x ∈ U n . By c j (x) we denote the number of coordinates of x which belong to U j . The following two claims show that ψ (γ( f , x)) depends only on the numbers {c j (x)} k j =1 . Claim 1. Let x, y ∈ U n . Assume that x i = y σ(i ) for some permutation σ ∈ Sym n and i = 1, . . . , n . Then γ( f , x) and γ( f , y) are conjugated in MCG(S, n).

Proof. Consider a map h −1
x h y . This map permutes the points Claim 2. Let x, y ∈ U n such that x i , y i belong to the same connected component of U for i = 1, . . . , n. Then γ( f , x) and γ( f , y) are conjugated in MCG(S, n).
Proof. Let g be a diffeomorphism of S such that g (x i ) = y i and g is supported on U . In particular g can be taken to be a map that pushes x i towards y i and x i travels all the time in the same connected component of U . We consider the mapping class [h −1 The maps g and f have disjoint supports, hence they commute. Then The set U n splits into finitely many connected components of the form U s = U s(1) × · · · ×U s(n) , where s : {1, . . . , n} → {1, . . . , k}. Let C = (c 1 , . . . , c k ) be a partition such that c 1 +. . .+c k = n. Consider a component U s = U s(1) ×· · ·×U s(n) of U n for which c j = #s −1 ( j ). We say that U s is associated to C . The function ψ(γ( f )) is constant on connected components associated to C , and on each component it has the same value. Denote it by ψ (γ( f ,C )). Let L(C ) be the number of connected components associated to C . Every connected component associated to C has the same volume vol(C ) = area(U 1 ) c 1 · · · area(U k ) c k . Recall If we treat area(U i ) as a free variable, this integral is a homogeneous polynomial in k variables of degree n. Denote this polynomial by P . The coefficient of the monomial vol(C 0 ) is ψ(γ( f ,C 0 ))L(C 0 ) and is non-zero. The area(U i ) depends on the neighborhood N . If we start shrinking N such that area(N ) converges to zero, then area(U i ) converges to some number V i . We can assume that P (V 1 , . . . ,V k ) = 0. Indeed, if it is not the case, we can modify a little the loops C which were chosen to express [ f ] in terms of Dehn twists. Then the values of V i change freely, except that we have a constraint V 1 +· · ·+V k = area(S). It is easy to see, that a homogeneous polynomial P is non-trivial on every affine non-linear subspace of codimension one, so we can arrange V i such that P (V 1 , . . . ,V k ) = 0. Now we shrink N , so area(N ) → 0 and area(U i ) → V i . We have that A N → 0 and B N → P (V 1 , . . . ,V k ) = 0. Since G S,n (ψ)( f ) = A N + B N , then for some N we have G S,n (ψ)( f ) = 0.

CURVE COMPLEX
Let S be a connected oriented surface (possibly with boundary and punctures). A simple closed curve is called essential if it is not isotopic to a boundary curve, not isotopic to a curve going around exactly one puncture, and it is not isotopic to a point.
The curve complex C (S) of S was first defined by Harvey [18]. This simplicial complex is defined as follows: for vertices we take isotopy classes of essential simple closed curves in S. A collection of k + 1 vertices {α i } k i =1 form a k-simplex whenever this collection can be realized by pairwise disjoint closed curves in S. A celebrated result of Masur-Minsky states that C (S) is hyperbolic [22]. We write d C (S) for the induced combinatorial path-metric on C (S) which assigns unit length to each edge of C (S).
The intersection number ι S (α, β) between two simple closed curves α, β on S is defined to be the minimal number of geometric intersections between α and β where α is isotopic to α and β is isotopic to β. Recall that a surface S of genus g with k boundary components and n punctures is called non-sporadic if 3g + n + k − 4 > 0. Proof of the following lemma may be found in [25]. Proof. An analogous statement is proved in [1], c.f. [21,Lemma 4.2]. The difference is that there one works with homotopy classes of curves on the compact surface S and not on the punctured surface.

LEMMA 3.1. Let S be a non-sporadic surface. Then for all simple closed curves
We construct a specific metric on S such that we are able to use an argument from [1]. Then, by comparing metrics, the statement of the lemma holds for any Riemannian metric on S. Let D i be a small disc centered at p i and let S o = S (D 1 ∪ . . . ∪ D n ). We fix a hyperbolic metric on S o such that all boundary loops ∂D i are totally geodesic and have the same length . The induced length is denoted by l S o .
Let S r be a 2-dimensional round sphere of radius r and let B be a ball in S r of perimeter . We consider S r,o = S r B . By p we denote the point in S r which is antipodal to the center of the ball B . Let x and y be two different points in ∂B and let b ⊂ ∂B be an embedded arc that connects x to y. If the radius r of S r is big compared to , then the arc b has the following property. Let γ be an arc in S r,o p which connects x to y. Assume that b and γ are homotopic in S r,o {p} relatively to {x, y}. Then l (γ) ≥ l (b), where l is a Riemannian length with respect to the round metric on S r . Now we construct a metric on S. We start with the surface S o . To each boundary component ∂D i we glue a copy of S r,o along the boundary. We obtain a surface homeomorphic to S. Note that in each copy of S r,o there is one antipodal point p. These antipodal points naturally correspond to points {p i } n i =1 . On S we consider the path length l S induced by the hyperbolic length l S o on S o and round metrics on copies of S r,o .
Let α be an essential simple closed curve in S = S {p 1 , . . . , p n }. Since S o is a deformation retract of S , α is homotopic to a simple closed curve that is contained in the hyperbolic surface S o . Let γ α be the unique hyperbolic geodesic contained in S o which is homotopic to α in S .
Claim. The loop γ α has the minimal length among all simple loops homotopic to α in S .
Proof. Let γ be a simple loop homotopic to α in S . Assume, that γ is not contained in S o . Then there exists a boundary loop of S o , say ∂D i , which intersects γ in at least two points. Let x and y be two distinct points in γ ∩ ∂D i and let a be the arc contained in γ which connects x to y and is disjoint from the interior of S o . Since a is an embedded arc, it is homotopic in S r,o {p i } relative to {x, y} to one of the arcs in ∂D i whose end points are x and y. Denote this arc by b (see Figure 3.1). By construction, l S (b) ≤ l S (a). Hence if we substitute a by b, we obtain a new loop γ , which is homotopic to γ in S and l S (γ ) ≤ l S (γ). Repeating this procedure we find a loop γ such that γ ⊂ S o and l S (γ ) ≤ l S (γ). Then l S (γ α ) ≤ l S (γ ) and the claim follows.
Let α and β be essential simple loops in S . We prove that there is a constant C such that ι S (α, β) ≤ C l S (α)l S (β).
It follows from the claim that it is enough to prove that Let us repeat the argument from [1]. We can assume that γ α = γ β , otherwise this inequality is trivial. Let r 1 be a positive number which is less than the injectivity radius of the exponential map of the surface S o equipped with the hyperbolic length l S o . The geodesic γ α may be covered by fewer than l S (γ α ) r 1 + 1 geodesic arcs, each of which is contained in a geodesic disc. The same holds for γ β . Note that if an arc is close to a boundary of S o such a disk may contain part of the boundary of S o , but this does not affect the argument. Now a small arc of γ α intersects a small arc of γ β in at most one point. Thus we have Since the length l S of every essential simple closed curve in S is greater or equal than r 1 , we get Now let g be any Riemannian metric on S. It is easy to see, that the lengths l induced on S by g and l S are comparable. Thus, there exists C such that for every loop α we have l S (α) < C l (α). This finishes the proof of the lemma.

MAPPING CLASS GROUPS
Mapping class group MCG(S) of an oriented surface S is defined to be the group of orientation preserving diffeomorphisms of S which fix the boundary pointwise modulo diffeomorphisms which are isotopic to the identity. Since an element in MCG(S) takes homotopy classes of disjoint essential simple closed curves to homotopy classes of disjoint essential simple closed curves, MCG(S) acts by isometries on the curve complex (C (S), d C (S) ).
Let [ f ] ∈ MCG(S) and α an essential simple closed curve in S. Recall that the Translation length is independent of the choice of α and vanishes on all periodic and reducible elements of MCG(S).
for every essential simple closed curve α ⊂ S .
Proof. Let α ⊂ S be an essential simple closed curve. We have Note that by the definition of C (S ) we have d C (S ) (α, β) ≤ 1 if and only if ι S (α, β) = 0. We take a constant C 1 := max{C , 1}, where C is a constant from Lemma 3.2. Now Lemma 3.1 together with Lemma 3.2 gives us the following inequality: Combining the last two inequalities, we obtain where B = 2(log (C 1 l (α)) + 1).

4.A. Bestvina-Fujiwara quasimorphisms.
Here we describe a construction of quasimorphisms on mapping class groups due to Bestvina and Fujiwara [3]. Let S be an oriented surface and let ω be a finite oriented path in C (S). By |ω| we denote the length of ω. Let σ be a finite path. We set |σ| ω = {the maximal number of non − overlapping copies of ω in σ}.
Let α, β be two vertices in C (S) and let W be an integer such that 0 < W < |ω|.
where σ ranges over all paths from α to β.
Let α ∈ C (S). We define ψ ω : MCG(S) → R by Bestvina and Fujiwara proved that ψ ω is a quasimorphism [3]. The induced homogeneous quasimorphism is denoted by ψ ω . We denote by Q BF (MCG(S)) the space of homogeneous quasimorphisms on MCG(S) which is spanned by Bestvina-Fujiwara quasimorphisms. In [3] it is proved that Q BF (MCG(S)) is infinite dimensional whenever S is a non-sporadic surface. Let i * : Q(MCG(S, n)) → Q(K (S, n)) be a homomorphism induced by the inclusion map i : K (S, n) → MCG(S, n).

COROLLARY 4.2. Let S be a closed oriented surface and n ∈ N such that S with n punctures is non-sporadic. Then
G S,n (Q BF (MCG(S, n))) < Q (Diff(S, area)) and G 0 S,n • i * (Q BF (MCG(S, n))) < Q(Diff 0 (S, area)) are infinite dimensional.
Proof. The maps G S,n and G 0 S,n are injective by Theorem 2.5. Since the space Q BF (MCG(S, n)) is infinite dimensional [3], it follows that G S,n (Q BF (MCG(S, n))) is infinite dimensional.
It is left to prove that i * (Q BF (MCG(S, n))) is infinite dimensional. Since K (S, n) is an infinite normal subgroup of MCG(S, n), it is non-reducible by a theorem of Ivanov [20,Corollary 7.13]. In order to prove that the space i * (Q BF (MCG(S, n))) is infinite dimensional it is enough to show that K (S, n) is not virtually abelian, see [3,Theorem 12]. There are three cases.

Case 1.
The surface S is a sphere and n > 3. In this case the mapping class group of S is trivial and thus the group K (S, n) is nothing but MCG(S, n) which is not virtually abelian.

Case 2.
The surface S is a torus and n > 1. It follows from the Birman sequence for torus [4] that the group K (S, n) maps onto B n (S) modulo center, where B n (S) is the torus braid group on n strings. Let P n (S) be the pure torus braid group on n strings. By removing n − 2 strings we get an epimorphism from P n (S) to P 2 (S) which is isomorphic to Z 2 ×F 2 . It follows that B n (S) modulo center is not virtually abelian, and so is K (S, n).

Case 3.
The surface S is hyperbolic. It follows from the Birman exact sequence [4,5] that the group K (S, n) is isomorphic to B n (S) which is the braid group of S on n strings. Let P n (S) be the pure braid group of S on n strings. By removing n − 1 strings we get an epimorphism from P n (S) to P 1 (S) which contains F 2 . It follows that B n (S) is not virtually abelian, and so is K (S, n). Proof. From the definition of Q BF (MCG(S)), for each ψ in Q BF (MCG(S)) there exist k ∈ N, a 1 , . . . , a k ∈ R and ω 1 , . . . , ω k finite oriented paths in C (S) such that The limit exists, since the function p → c ω,W (α, f p (α)) is expressed as the difference of two sub-additive functions, namely, p → d C (S) (α, f p (α)) and p → inf σ∈P (α, f p (α)) (|σ| − W |σ| ω ), where P (α, β) is the set of all paths from α to β. By the definition of ψ ω i we have It follows that

5.
A. Proof of Theorem 1. Assume that S is a closed surface. We discuss the case of a surface with boundary at the end. Equip S with a metric as in Proposition 4.1. Denote G = G S,n , γ = γ S,n and assume that n is such that S with n punctures is non-sporadic. At the end of the proof we discuss the case when G = G 0 S,n . We pick an essential simple closed curve α in the punctured surface S = S {z 1 , . . . , z n } (see Section 2.A). Let ψ ∈ Q BF (MCG(S, n)) and f ∈ Diff(S, area). Then where the second inequality is by Lemma 4.3, the third inequality is by Proposition 4.1. Let x ∈ Ω n (see Section 2.A) and let U x be an open set such that x ∈ U x and the closure of U x is in Ω n . It follows from the Poincaré recurrence theorem that for almost all x, after passing to a subsequence, we can assume that f p (x) ∈ U x . Due to C 1 -continuity of the function h −1 x on Ω n , there exists a constant K x such that sup where · 1 is the C 1 -norm. Thus for almost every x ∈ X n we get This yields We apply Yomdin result [28, Theorem 1.4] and get that for almost every x ∈ X n Combining last two inequalities we get Since this inequality applies to any ψ ∈ Q BF (MCG(S, n)), then by Corollary 4.2 the space of quasimorphisms which are Lipschitz with respect to the entropy is infinite dimensional.
In the case when G = G 0 S,n the proof is the same. The fact that the space of quasimorphisms on Diff 0 (S, area) bounding entropy from below is infinite dimensional again follows from Corollary 4.2.
Let us discuss the case of Ham(S). It is a simple group which is isomorphic to the commutator subgroup of Diff 0 (S, area) [2]. Since every quasimorphism in Q BF (MCG(S, n)) vanishes on reducible elements, the space i * •Q BF (MCG(S, n)) contains no non-trivial homomorphisms to the reals. In addition, for every ψ ∈ i * •Q BF (MCG(S, n)) the map G 0 S,n (ψ) is not a homomorphism. Note that the the kernel of the restriction homomorphism Q(Diff 0 (S, area)) → Q(Ham(S)) is the space Hom(Diff 0 (S, area), R). Therefore the space Q(Diff 0 (S, area)) Hom(Diff 0 (S, area), R) is isomorphic to a subspace of Q (Ham(S)). It follows that the map is injective. Now the proof is identical to the case of Diff 0 (S, area).
At last let us comment on the case when S has a boundary. In this case we embed S into a closed surface S. Then each one of the groups Diff(S, area), Diff 0 (S, area) and Ham(S) embed in the usual way into the groups Diff(S, area), Diff 0 (S, area) and Ham(S) respectively. It follows from the proof of the embedding theorem that G S,n (ψ) is non-trivial on Diff(S, area) provided ψ is non-trivial. Similarly, G 0 S,n (i * • ψ) is non-trivial on Diff 0 (S, area) and on Ham(S) provided i * • ψ is non-trivial. 5.B. Proof of Theorem 2. We start with the following: Proof. Case 1. G = Ham(S). Denote by D the set of embedded discs in S of area less than or equal to half of area(S). Then by fragmentation lemma [2] for every f ∈ G there exists a finite collection of discs {D i } k i =1 in D and diffeomorphisms {h i } k i =1 such that each h i is supported in D i and f = h 1 • · · · • h k . Each h i ∈ Diff(D i , area) is generated by autonomous diffeomorphisms [9]. An autonomous diffeomorphism is a flow of a vector field, and as such has zero entropy, see [29], and the proof follows. Case 2. G = Diff 0 (S, area). There is a surjective homomorphism Flux from G to H 1 (S,R) Γ , where Γ is the flux group of Flux. The kernel of Flux is Ham(S) [2]. Take f ∈ G. Then there exists an autonomous (and therefore of zero entropy) diffeomorphism h of S such that Flux( f • h) = 0. Hence f • h ∈ Ham(S) which is generated by entropy-zero diffeomorphisms and so is G. Case 3. G = Diff(S, area). Mapping class group of S is isomorphic to Diff(S,area) Diff 0 (S,area) . It is generated by Dehn twists. Recall that every Dehn twist has a representative of zero entropy and the group Diff 0 (S, area) is generated by entropy-zero diffeomorphisms. Hence the group G is generated by entropy-zero diffeomorphisms. Proof. Let g ∈ G. Let s 1 , . . . , s k ∈ S such that g = s 1 • · · · • s k and g S = k, where · S is the induced word norm on G. Then since ψ vanishes on S we have Take h ∈ G such that ψ(h) = 0. Then for every n ∈ N we have It follows from Theorem 1 that there are infinitely many linearly independent homogeneous quasimorphisms on Diff(S, area), on Diff 0 (S, area) and on Ham(S) which vanish on the set of entropy-zero diffeomorphisms. By Lemma 5. Let r < 1 m . Denote by D r the disc in the plane of radius r centered at zero. Denote G = G D 2 ,n and G r = G D r ,n . The inclusion D r ⊂ D 2 induces an isomorphism K (D r , n) K (D 2 , n). Note that K (D 2 , n) = MCG(D 2 , n) which is isomorphic via Birman isomorphism to the Artin braid group B n . From now on for x ∈ X n and f ∈ Diff(D 2 , area)) we regard a mapping class γ( f , x) as an element of B n . We have G : Q(B n ) → Q (Diff(D 2 , area)), G r : Q(B n ) → Q (Diff(D r , area)).
We extend every diffeomorphism in Diff(D r , area) by identity on D 2 and get an injective homomorphism i r : Diff(D r , area) → Diff(D 2 , area). where D Ψ i ,n is the defect of the quasimorphism Ψ i ,n . We denote by D m := max i D Ψ i ,n and obtain the following inequality: Denote by M J := max i f i Ent . Now we have the following inequality: Last two inequalities conclude the proof of the theorem.
6. FINAL REMARKS 1. Let S be a compact oriented surface of positive genus, and let n be such that S with n punctures is non-sporadic. Since each quasimorphism in Q BF (MCG(S, n)) vanishes on reducible elements, the induced quasimorphism on Ham(S) vanishes on diffeomorphisms supported in a disc. Hence every quasimorphism that lies in the image of G 0 S,n is C 0 -continuous, see [15,Theorem 1.7]. 2. Let S be a compact oriented surface. One can easily show that there is an infinite family of egg-beater Hamiltonian diffeomorphisms { f i } ∞ i =1 of S (for definition see [24]), and a family of linearly independent quasimorphisms Ψ i ∈ Q(Diff(S, area)) which are Lipschitz with respect to the topological entropy such that Ψ i ( f i ) = 0. This implies that each f i has a positive topological entropy. 3. Since every autonomous diffeomorphism of a surface has zero entropy, entropy norm is bounded from above by the autonomous norm. It would be interesting to know whether these norms are equivalent. Note that the existence of a homogeneous quasimorphism on Ham(S) which does not vanish on the set of entropy-zero diffeomorphisms, but vanishes on every autonomous diffeomorphism, would imply that these norms are not equivalent.