The growth rate of symplectic Floer homology

. The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is deﬁned to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies that symplectic zeta function has a positive radius of convergence. This also establishes a connection between Floer homology and geometry of 3-manifolds.


Introduction
The main theme of this paper is to study for a symplectomorphism φ : M → M in a given mapping class g of a compact surface M , the asymptotic invariant F ∞ (g), introduced in [11], which is defined to be the growth rate of the sequence dim HF * (φ n ) of the total dimensions of symplectic Floer homologies of the iterates of φ. We prove a conjecture from [11] which suggests that the asymptotic invariant coincides with asymptotic Nielsen number and with the largest dilatation of the pseudo-Anosov components of g and its logarithm coincides with topological entropy. This establishes a connection between Floer homology and geometry of 3-manifolds. The asymptotic invariant also provides the radius of convergence of the symplectic zeta function F g (t) = F φ (t) = exp Theorem 1.2. Letφ be a perturbed standard form map φ (as in [35,19,4]) in a reducible mapping class g of a compact surface of genus greater than or equal to 2, and let λ be the largest dilatation of the pseudo-Anosov components (λ = 1 if there are no pseudo-Anosov components). Then where ψ is a canonical representative of mapping class g. Remark 1.3. The genus-one case follows from Pozniak's thesis [34]. [35,19,4]) in a reducible mapping class g of compact surface of genus greater than or equal to 2, and let λ be the largest dilatation of the pseudo-Anosov components (λ = 1 if there are no pseudo-Anosov components). Then the symplectic zeta function F g (t) = Fφ (t) has positive radius of convergence R = 1/λ.

Theorem 1.4. Letφ be a perturbed standard form map φ (as in
Although the exact evaluation of the asymptotic invariant would be desirable, in general, its estimation is a more realistic goal and as we shall show, one that is sufficient for some applications.
We suggested in [11] that the asymptotic invariant potentially may be important for the applications. A recent paper by Smith [38] gives an application of the asymptotic invariant to the important question of faithfulness of a representation of extended mapping class group via considerations motivated by homological mirror symmetry. Because c φ controls the index, or expected dimension, of moduli spaces of holomorphic curves under change of homology class and ω φ controls their energy under change of homology class, the monotonicity condition ensures that the energy is constant on index-one components of the moduli space, which implies compactness and, as a corollary, finite count in a differential of the Floer complex.
We recall the fundamental properties of Symp m (M, ω) from [35,19]. Let Diff + (M ) denotes the group of orientation-preserving diffeomorphisms of M .
Now let g be a mapping class of M , i.e., an isotopy class of Diff + (M ). Pick an area form ω and a representative φ ∈ Symp m (M, ω) of g. HF * (φ) is an invariant as φ is deformed through monotone symplectomorphisms. These imply that we have a symplectic Floer homology invariant HF * (g) canonically assigned to each mapping class g given by HF * (φ) for any monotone symplectomorphism φ. Note that HF * (g) is independent of the choice of an area form ω by Moser's isotopy theorem [31] and naturality of Floer homology.
We give now, following Cotton-Clay [4], a notion of weak monotonicity such that HF * (φ) is well defined and invariant among weakly monotone symplectomorphisms. Monotonicity implies weak monotonicity, and so HF * (g) = HF * (φ) for any weakly monotone φ in a mapping class g. The properties of weak monotone symplectomorphism of a surface play a crucial role in the computation of Floer homology for pseudo-Anosov and reducible mapping classes (see [4]). Fundamental properties of monotone symplectomorphisms listed above are also satisfied when replacing monotonicity with weak monotonicity.
is generated by tori T such that π| T : T → S 1 is a fibration with fiber S 1 , where the map π : T φ → S 1 is the projection. Throughout this paper, Symp wm (M, ω) denotes the set of weakly monotone symplectomorphisms.

Floer homology.
Let φ ∈ Symp(M, ω). There are two ways of constructing Floer homology detecting its fixed points, Fix(φ). Firstly, the graph of φ is a Lagrangian submanifold of M × M, (−ω) × ω and its fixed points correspond to the intersection points of graph(φ) with the diagonal Δ = {(x, x) ∈ M × M }. Thus we have the Floer homology of the Lagrangian intersection HF * (M × M, Δ, graph(φ)). This intersection is transversal if the fixed points of φ are nondegenerate, i.e., if 1 is not an eigenvalue of dφ(x), for x ∈ Fix(φ). The second approach was mentioned by Floer in [15] and presented with details by Dostoglou and Salamon in [6]. We follow here Seidel's approach [35] which, comparable with [6], uses a larger class of perturbations, but such that the perturbed action form is still cohomologous to the unperturbed one. As a consequence, the usual invariance of Floer homology under Hamiltonian isotopies is extended to the stronger property stated above. Let Vol. 12 (2012) The growth rate of symplectic Floer homology 9 7 now φ be monotone or weakly monotone. Firstly, we give the definition of HF * (φ) in the special case where all the fixed points of φ are nondegenerate, i.e., for all y ∈ Fix(φ), det(id −dφ y ) = 0, and then following Seidel's approach [35] we consider the general case when φ has degenerate fixed points.
Let Ω φ = {y ∈ C ∞ (R, M) | y(t) = φ(y(t+1))} be the twisted free loop space, which is also the space of sections of T φ → S 1 . The action form is the closed one-form α φ on Ω φ defined by The tangent bundle of any symplectic manifold admits an almost complex structure J : dt defines a metric on the loop space Ω φ . So the critical points of α ω are the constant paths in Ω φ and hence the fixed points of φ. The negative gradient lines of α ω with respect to the metric above are solutions of the partial differential equations with boundary conditions These are exactly Gromov's pseudoholomorphic curves [20]. For y ± ∈ Fix(φ), let M(y − , y + ; J, φ) denote the space of smooth maps u : R 2 → M which satisfy conditions (2). Now with every u ∈ M(y − , y + ; J, φ) we associate a Fredholm operator D u which linearizes (2) in suitable Sobolev spaces. The index of this operator is given by the so-called Maslov index μ(u), which satisfies μ(u) = deg(y + ) − deg(y − ) mod 2, where (−1) deg y = sign(det(id −dφ y )). We have no bubbling, since for surface π 2 (M ) = 0. For a generic J, every u ∈ M(y − , y + ; J, φ) is regular, meaning that D u is onto. Hence, by the implicit function theorem, M k (y − , y + ; J, φ) is a smooth kdimensional manifold and is the subset of those u ∈ M(y − , y + ; J, φ) with μ(u) = k ∈ Z. Translation of the s-variable defines a free R-action on one-dimensional manifold M 1 (y − , y + ; J, φ) and hence the quotient is a discrete set of points. The energy of a map u : R 2 → M is given by E(u) = R 1 0 ω ∂ t u(s, t), J t ∂ t u(s, t) dt ds for all y ∈ Fix(φ). Seidel [35] and Cotton-Clay [4] have proved that if φ is monotone or weakly monotone, then the energy is constant on each M k (y − , y + ; J, φ). Since all fixed points of φ are nondegenerate, the set Fix(φ) is a finite set and the Z 2 -vector space admits a Z 2 -grading with (−1) deg y = sign(det(id −dφ y )), for all y ∈ Fix(φ). By Gromov compactness, the boundedness of the energy E(u) for monotone or weakly monotone φ implies that the zero-dimensional J FPTA quotients M 1 (y − , y + , J, φ)/R are compact and thus actually finite sets. Denoting by n(y − , y + ) the number of points mod 2 in each of them, one defines a differential ∂ J : CF * (φ) → CF * +1 (φ) by ∂ J y − = y+ n(y − , y + )y + . Due to gluing theorem, this Floer boundary operator satisfies ∂ J • ∂ J = 0. For gluing theorem to hold one needs again the boundedness of the energy E(u). It follows that (CF * (φ), ∂ J ) is a chain complex and its homology is by definition the Floer homology of φ denoted HF * (φ). It is independent of J and is an invariant of φ. If φ has degenerate fixed points, one needs to perturb equations (2) in order to define the Floer homology.

Nielsen classes and Reidemeister trace
Before discussing the results of the paper, we briefly describe the few basic notions of Nielsen fixed point theory which will be used. We assume X to be a connected, compact simplicial complex and f : X → X to be a continuous map. Let p :X → X be the universal cover of X andf :X →X a lifting of f ; i.e., p Let f : X → X be given, and let a specific liftingf :X →X be chosen as a reference. Let Γ be the group of covering translations ofX over X. Then every lifting of f can be written uniquely as α •f , with α ∈ Γ. So elements of Γ serve as coordinates of liftings with respect to the referencef . Now for every α ∈ Γ the compositionf • α is a lifting of f , so there is a unique α ∈ Γ such that α •f =f • α. This correspondence α → α is determined by the referencef , and it is obviously a homomorphism. The endomorphism f * : Γ → Γ determined by the liftingf of f is defined byf * (α) •f =f • α. It is well known that Γ ∼ = π 1 (X). We shall identify π = π 1 (X, x 0 ) and Γ in the usual way.
We have seen that α ∈ π can be considered as the coordinate of the lifting α •f . We can tell the conjugacy of two liftings from their coordinates: So we have the Reidemeister bijection: Lifting classes of f are in 1-1 correspondence withf * -conjugacy classes in group π, the lifting class [α •f ] corresponds to thef * -conjugacy class of α.

Vol. 12 (2012)
The growth rate of symplectic Floer homology 99 By an abuse of language, we say that the fixed point class p(Fix α •f ), which is labeled with the lifting class [α •f ], corresponds to thef * -conjugacy class of α. Thus thef * -conjugacy classes in π serve as coordinates for the fixed point classes of f , once a reference liftingf is chosen.

Reidemeister trace.
The results of this section are well known (see [24,10,14]). We shall use this results later to estimate the radius of convergence of the symplectic zeta function. The fundamental group π = π 1 (X, x 0 ) splits intof * -conjugacy classes. Let π f denote the set off * -conjugacy classes, and Zπ f denote the Abelian group freely generated by π f . We will use the bracket notation a → [a] for both projections π → π f and Zπ → Zπ f . Let x be a fixed point of f . Take a path c from x 0 to x. Thef * -conjugacy class in π of the loop c · (f • c) −1 , which is evidently independent of the choice of c, is called the coordinate of x. Two fixed points are in the same fixed point class F if and only if they have the same coordinates. Thisf * -conjugacy class is thus called the coordinate of the fixed point class F and denoted by cd π (F, f ) (compare with the description in Section 2.2). The generalized Lefschetz number or the Reidemeister trace [24] is defined as where the summation is over all essential fixed point classes F of f . The Nielsen number N (f ) is the number of nonzero terms in L π (f ), and the indices of the essential fixed point classes appear as the coefficients in L π (f ). This invariant used to be called the Reidemeister trace because it can be computed as an alternating sum of traces on the chain level as follows [24].
Assume that X is a finite cell complex and f : X → X is a cellular map. A cellular decomposition e d j of X lifts to a π-invariant cellular structure on the universal coveringX. Choose an arbitrary liftẽ d j for each e d j . They constitute a free Zπ-basis for the cellular chain complex ofX. The liftf of f is also a cellular map. In every dimension d, the cellular chain mapf gives rise to a Zπ-matrixF d with respect to the above basis, i.e., Now we describe an alternative approach to the Reidemeister trace formula proposed by Jiang [24]. This approach is useful when we study the periodic points of f , i.e., the fixed points of the iterates of f .
The mapping torus Then the map f : X → X is the return map of the semiflow φ.
A point x ∈ X and a positive number τ > 0 determine the orbit curve φ (x,τ ) := φ t (x) 0≤t≤τ in T f . Take the base point x 0 of X as the base point J FPTA of T f . It is known that the fundamental group H := π 1 (T f , x 0 ) is obtained from π by adding a new generator z and adding the relations z −1 gz =f * (g) for all g ∈ π = π 1 (X, x 0 ). Let H c denote the set of conjugacy classes in H. Let ZH be the integral group ring of H, and let ZH c be the free Abelian group with basis H c . We again use the bracket notation a → [a] for both projections H → H c and ZH → ZH c . If F n is a fixed point class of f n , then f (F n ) is also a fixed point class of f n and ind(f (F n ), f n ) = ind(F n , f n ). Thus f acts as an index-preserving permutation among fixed point classes of f n . By definition, an n-orbit class O n of f is the union of elements of an orbit of this action. In other words, two points x, x ∈ Fix(f n ) are said to be in the same n-orbit class of f if and only if some f i (x) and some f j (x ) are in the same fixed point class of f n . The set Fix(f n ) splits into a disjoint union of n-orbits classes. Point x is a fixed point of f n or a periodic point of period n if and only if the orbit curve φ (x,n) is a closed curve. The free homotopy class of the closed curve φ (x,n) will be called the H-coordinate of point x, It follows that periodic points x of period n and x of period n have the same H-coordinate if and only if n = n and x, x belong to the same n-orbit class of f . Thus it is possible to equivalently define x, x ∈ Fix(f n ) to be in the same n-orbit class if and only if they have the same H-coordinate. Jiang [24] has considered the generalized Lefschetz number with respect to H: and he has proved the following trace formula: whereF d is Zπ-matrices defined above and zF d is regarded as a ZH-matrix.

Twisted Lefschetz numbers and twisted Lefschetz zeta function.
Let R be a commutative ring with unity. Let GL n (R) be the group of invertible n × n matrices in R, and let M n×n (R) be the algebra of n × n matrices in R. Suppose a representation ρ : H → GL n (R) is given. It extends to a representation ρ : ZH → M n×n (R). Following Jiang [24], we define the ρ-twisted Lefschetz number where h ρ is the ρ-image of h ∈ ZH. It has the trace formula (see [24]) where for a ZH-matrix A, its ρ-image A ρ means the block matrix obtained from A by replacing each element a ij with n × n R-matrix a ρ ij . Twisted

Vol. 12 (2012)
The growth rate of symplectic Floer homology 101 Lefschetz zeta function is defined as formal power series: It is in the multiplicative subgroup 1 + tR [[t]] of the formal power series ring R [[t]]. The trace formula for the twisted Lefschetz numbers implies that L f ρ (t) is a rational function in R given by the formula where E stands for suitable identity matrices. Twisted Lefschetz zeta function enjoys the same invariance properties as that of L H (f n ).

Computation of symplectic Floer homology
In this section we describe known results from [4,19,11,12] about computation of symplectic Floer homology for different mapping classes.

Thurston's classification theorem and standard form maps.
We recall firstly Thurston classification theorem for homeomorphisms of surface M of genus greater than or equal to 2.   (1) or (2).
The map f above is called a Thurston canonical representative of φ. A key observation is that if f is canonical representative, so are all iterates of f .
Thurston's classification theorem for homeomorphisms of surface implies that every mapping class of M is precisely one of the following: periodic, pseudo-Anosov or reducible.
In this section, we review standard form maps as discussed in [19,4]. These are special representatives of mapping classes adopted to the symplectic geometry. For the identity mapping class, a standard form map is a small perturbation of the identity map by the Hamiltonian flow associated with a Morse function for which the boundary components are locally minima and maxima. Every fixed point is in the same Nielsen class. This Nielsen class has an index given by the Euler characteristic of the surface. For nonidentity periodic mapping classes, a standard form map is an isometry with respect J FPTA to a hyperbolic structure on the surface with geodesic boundary. Every fixed point is in a separate Nielsen class and each of the Nielsen classes, for which there is a fixed point, has index +1. For a pseudo-Anosov mapping class, a standard form map is a symplectic smoothing (see [4]) of the singularities and boundary components of the canonical singular representative. Each singularity has a number p ≥ 3 of prongs and each boundary component has a number p ≥ 1 of prongs. If a singularity or boundary component is (setwise) fixed, it has some fractional rotation number modulo p (see [4]). There is a separate Nielsen class for every smooth fixed point, which is of index +1 or −1; for every fixed singularity, which when symplectically smoothed gives p − 1 fixed points all of index −1 if the rotation number is zero modulo p or one fixed point of index +1 otherwise; and for every fixed boundary component with rotation number zero modulo p, which when symplectically smoothed gives p fixed points all of index −1 (see [4]).
From this discussion, we see that for nonidentity periodic and pseudo-Anosov mapping classes, the standard form map is such that all fixed points are nondegenerate of index +1 or −1 and, for every Nielsen class F , the number of fixed points in F is |ind(F )|. We now turn to reducible maps and the identity map. By Thurston's classification (see [40,7]; see also [19,Definition 8] and [4, Definition 4.6]), in a reducible mapping class g, there is a (not necessarily smooth) map φ which satisfies the following. maps S to itself, the map φ | S is area preserving and is either isotopic to the identity, periodic or pseudo-Anosov. In these cases, we require the map to be in a standard form as above.

Vol. 12 (2012)
The growth rate of symplectic Floer homology 103 Thurston's classification theorem for homeomorphisms of surface implies that every mapping class of M is precisely one of the following: periodic, pseudo-Anosov or reducible. [19,12]). If φ is a nontrivial, orientation-preserving, standard form periodic diffeomorphism of a compact connected surface M of Euler characteristic χ(M ) ≤ 0, then φ is a monotone symplectomorphism with respect to some φ-invariant area form and

Theorem 2.3 (See
where L(φ) and N (φ) denote the Lefschetz and the Nielsen number of φ, respectively.

Algebraically finite mapping classes.
A mapping class of M is called algebraically finite if it does not have any pseudo-Anosov components in the sense of Thurston's theory of surface diffeomorphism. The term algebraically finite goes back to J. Nielsen.
In [19] the diffeomorphisms of finite type were defined. These are reducible maps in standard form which are special representatives of algebraically finite mapping classes adopted to the symplectic geometry.
By M id we denote the union of the components of M \ int(U ), where φ restricts to the identity.
The monotonicity of diffeomorphisms of finite type was investigated in details by Gautschi in [19]. Let φ be a diffeomorphism of finite type and let be as in (1). Then φ is the product of (multiple) Dehn twists along U . Moreover, two parallel Dehn twists have the same sign. We say that φ has uniform twists if φ is the product of only positive, or only negative, Dehn twists. Furthermore, we denote by the smallest positive integer such that φ restricts to the identity on M \ U .
If ω is an area form on M which is the standard form dq∧dp with respect to the (q, p)-coordinates on U , then ω := i=1 (φ i ) * ω is standard on U and φinvariant, i.e., φ ∈ Symp(M, ω). To prove that ω can be chosen such that φ ∈ Symp m (M, ω), Gautschi distinguished two cases: uniform and nonuniform twists. In the first case he proved the following stronger statement. [19]). If φ has uniform twists and ω is a φ-invariant area form, then φ ∈ Symp m (M, ω).

Lemma 2.4 (See
In the nonuniform case, monotonicity does not hold for arbitrary φinvariant area forms. [19]). If φ does not have uniform twists, then there exists a φ-invariant area form ω such that φ ∈ Symp m (M, ω). Moreover, ω can be chosen such that it is the standard form dq ∧ dp on U . J FPTA Theorem 2.6 (See [19]). Let φ be a diffeomorphism of finite type, then φ is monotone with respect to some φ-invariant area form and

Lemma 2.5 (See
Here, L denotes the Lefschetz number.

Pseudo-Anosov mapping classes.
For a pseudo-Anosov mapping class, a standard form map is a symplectic smoothing of the singularities and boundary components of the canonical singular representative. Full description of the symplectic smoothing is given by Cotton-Clay in [4]. [4], see also [12]). If φ is any symplectomorphism with nondegenerate fixed points in a given pseudo-Anosov mapping class g, then φ is weakly monotone, HF * (φ) is well defined and

Theorem 2.7 (See
where ψ is the singular canonical pseudo-Anosov representative of g.

Reducible mapping classes. Recently, Cotton-Clay [4] calculated Seidel's symplectic Floer homology for reducible mapping classes. This result completes all previous computations.
In the case of reducible mapping classes, an energy estimate forbids holomorphic discs from crossing reducing curves except when a pseudo-Anosov component meets an identity component (with no twisting). Let us introduce some notation following [4]. Recall the notation of M id for the collection of fixed components as well as the three types of boundary: (1) ∂ + M id , ∂ − M id denote the collection of components of ∂M id on which we have joined up with a positive (resp., negative) twist; (2) the collection of components of ∂M id which meet a pseudo-Anosov component will be denoted ∂ p M id . Additionally, let M 1 be the collection of periodic components and let M 2 be the collection of pseudo-Anosov components with punctures (i.e., before any perturbation) instead of boundary components wherever there is a boundary component that meets a fixed component. We further subdivide M id . Let M a be the collection of fixed components which do not meet any pseudo-Anosov components. Let M b,p be the collection of fixed components which meet one pseudo-Anosov component at a boundary with p prongs. In this case, we assign the boundary components to ∂ + M id (this is an arbitrary choice). Let M o b,p be the collection of the M b,p with each component punctured once. Let M c,q be the collection of fixed components which meet at least two pseudo-Anosov components such that the total number of prongs over all the boundaries is q. In this case, we assign at least one boundary component to ∂ + M id and at least one to ∂ − M id (and beyond that, it does not matter). [4]). Ifφ is a perturbed standard form map φ (as in [35,19,4]) in a reducible mapping class g with choices of the signs of components Vol. 12 (2012) The growth rate of symplectic Floer homology 105 of ∂ p M id . Thenφ is weakly monotone, HF * (φ) is well defined and dim HF * (g) = dim HF * (φ)

Theorem 2.8 (See
where L(φ|M 1 ) is the Lefschetz number ofφ|M 1 , the L(φ|M 1 ) summand is all in even degree, the other two summands (with p − 1 and q) are all in odd degree, and HF * (φ|M 2 ) denotes the Floer homology forφ on the pseudo-Anosov components M 2 .
Remark 2.9. The first summand and the L(φ|M 1 ) are as in Gautschi's theorem (Theorem 2.6); see [19]. The last summand comes from the pseudo-Anosov components and is calculated via Theorem 2.7. The sums over p and q arise in the same manner as the first summand.

Topological entropy and Nielsen numbers
The most widely used measure for the complexity of a dynamical system is the topological entropy. For the convenience of the reader, we include its definition. Let f : X → X be a self-map of a compact metric space. For given > 0 and n ∈ N, a subset E ⊂ X is said to be (n, )-separated under f if for each pair x = y in E there is 0 ≤ i < n such that d(f i (x), f i (y)) > . Let A basic relation between topological entropy h(f ) and Nielsen numbers was found by Ivanov [22]. We present here a very short proof of Ivanov's inequality by Jiang [24]. [22]).

Lemma 3.1 (See
Proof. Let δ be such that every loop in X of diameter less than 2δ is contractible. Let > 0 be a smaller number such that d(f (x), f(y)) < δ whenever d(x, y) < 2 . Let E n ⊂ X be a set consisting of one point from each essential fixed point class of f n . Thus |E n | = N (f n ). By the definition of h(f ), it suffices to show that E n is (n, )-separated. Suppose it is not so. Then there would be two points This inequality is remarkable in that it does not require smoothness of the map and it provides a common lower bound for the topological entropy of all maps in a homotopy class.

Asymptotic invariant
Let Γ = π 0 (Diff + (M )) be the mapping class group of a closed connected oriented surface M of genus greater than or equal to 2. Pick an everywhere positive two-form ω on M . An isotopy theorem of Moser [31] says that each mapping class of g ∈ Γ, i.e., an isotopy class of Diff + (M ), admits representatives which preserve ω. Due to Seidel [35] and Cotton-Clay [4] we can pick a monotone (weakly monotone) representative φ ∈ Symp m (M, ω) (or φ ∈ Symp wm (M, ω)) of g such that HF * (φ) is an invariant as φ is deformed through monotone (weakly monotone) symplectomorphisms. These imply that we have a symplectic Floer homology invariant HF * (g) canonically assigned to each mapping class g given by HF * (φ) for any monotone (weakly monotone) symplectomorphism φ.
Note that HF * (g) is independent of the choice of an area form ω by Moser's theorem and naturality of Floer homology.
The growth rate of a sequence a n of complex numbers is defined by Growth(a n ) := max 1, lim sup n→∞ |a n | 1/n which could be infinity. Note that Growth(a n ) ≥ 1 even if all a n = 0. When Growth(a n ) > 1, we say that the sequence a n grows exponentially.

Vol. 12 (2012)
The growth rate of symplectic Floer homology 107 In [11] we have introduced the asymptotic invariant F ∞ (g) assigned to a mapping class g ∈ Mod M = π 0 (Diff + (M )) via the growth rate of the sequence {a n = dim HF * (φ n )} for a monotone (or weakly monotone) representative φ ∈ Symp m (M, ω) of g:

Example 3.2.
If φ is a nontrivial orientation-preserving standard form periodic diffeomorphism of a compact connected surface M of Euler characteristic χ(M ) < 0, then the periodicity of the sequence dim HF * (φ n ) implies that for the corresponding mapping class g, the asymptotic invariant F ∞ (g) := Growth(dim HF * (φ n )) = 1.

Example 3.3.
Let φ be a monotone diffeomorphism of finite type of a compact connected surface M of Euler characteristic χ(M ) < 0 and g a corresponding algebraically finite mapping class. Then the total dimension of HF * (φ n ) grows at most linearly (see [4,38,12]). Taking the growth rate in n, we get that the asymptotic invariant For any set S, let ZS denote the free Abelian group with the specified basis S. The norm in ZS is defined by when the s i in S are all different. For a ZH-matrix A = (a ij ), define its norm by A := i,j a ij . Then we have the inequalities AB ≤ A | · B when A, B can be multiplied, and tr A ≤ A when A is a square matrix. For a matrix A = (a ij ) in ZS, its matrix of norms is defined to be the matrix A norm := ( a ij ) which is a matrix of nonnegative integers. In what follows, the set S will be π, H or H c . We denote by s(A) the spectral radius of A, s(A) = lim n n A n , which coincides with the largest module of an eigenvalue of A. We define the asymptotic absolute Lefschetz number [24] to be the growth rate L ∞ (f ) = Growth( L π (f n ) ).
We also define the asymptotic Nielsen number [22] to be the growth rate N ∞ (f ) = Growth(N (f n )).
All these asymptotic numbers are homotopy type invariants. J FPTA Lemma 3.5. If φ is any symplectomorphism with nondegenerate fixed points in a given pseudo-Anosov mapping class g, then where ψ is a singular canonical pseudo-Anosov representative of g.
Proof. It is known that for a pseudo-Anosov map ψ, fixed points are topologically separated, i.e., each essential fixed point class of ψ consists of a single fixed point (see [40,22,10]). Then the generalized Lefschetz number or the Reidemeister trace [24] is where the summation is over all essential fixed point classes F of ψ, i.e, over all fixed points of ψ. So, the result follows from Theorem 2.7 and the definition of the norm L π (ψ) .
Remark 3.6. Lemma 3.5 provides, via Reidemeister trace formula, a new combinatorial formula to compute dim HF * (g) comparable to the train-track combinatorial formula of Cotton-Clay in [4]. [7,22,24]). Let f be a pseudo-Anosov homeomorphism with dilatation λ > 1 of surface M of genus greater than or equal to 2.

Theorem 3.7 (See
Then [24]). Suppose f is a canonical representative of a homeomorphism of surface M of genus greater than or equal to 2 and λ is the largest dilatation of the pseudo-Anosov components (λ = 1 if there are no pseudo-Anosov components).

Theorem 3.8 (See
Then Theorem 3.9. If φ is any symplectomorphism with nondegenerate fixed points in a given pseudo-Anosov mapping class g with dilatation λ > 1 of surface M of genus greater than or equal to 2. Then where ψ is a canonical singular pseudo-Anosov representative of g. Proof. By Lemma 3.5 we have that dim HF * (φ n ) = L π (ψ n ) for every n. So, the result follows from Theorem 3.7.
Theorem 3.10. Letφ be a perturbed standard form map φ (as in [35,19,4]) in a reducible mapping class g of compact surface of genus greater than or equal to 2, and let λ be the largest dilatation of the pseudo-Anosov components (λ = 1 if there are no pseudo-Anosov components). Then where ψ is a canonical representative of the mapping class g.

Vol. 12 (2012)
The growth rate of symplectic Floer homology 109 Proof. It follows from Theorem 2.8 that for every n, We need to investigate only the growth of the last summand in this formula because the rest part in the formula grows at most linearly [4,38]. We have dim HF * ((φ) n |M 2 ) = j dim HF * ((φ j ) n |M 2 ), where the sum is taken over different pseudo-Anosov components of φ n |M 2 . It follows from Theorem 3.9 that dim HF * ((φ j ) n |M 2 ) grows as λ n j , where λ j is the dilatation of the pseudo-Anosov componentφ j |M 2 .
Taking growth rate in n, we get

Corollary 3.11. The asymptotic invariant F ∞ (g) > 1 if and only if ψ has a pseudo-Anosov component.
Although the exact evaluation of the asymptotic invariant F ∞ (g) would be desirable, its estimation is a more realistic goal. We carry out such estimation using notations and results from Sections 2.2.1 and 2.2.2. Everything below in this section goes through via twisted Lefschetz zeta function. Proposition 3.12. Suppose ρ : H → U (n) is a unitary representation and ψ is a canonical representative of a reducible mapping class g of compact surface of genus greater than or equal to 2. Let w be a zero or a pole of the rational function L ψ ρ (t) ∈ C(t). Then Proof. We know from complex analysis and definition of the twisted Lefschetz zeta function that Growth(L ρ (ψ n )) is the reciprocal of the radius of convergence of the function log(L ψ ρ (t)), hence On the other hand, according to Section 2.2.1, H-coordinates of n-orbit classes are in the form [z n ]g with g ∈ π. So we can assume L H (ψ n ) = i k i [z n g i ], where the [z n g i ] are different conjugacy classes in H. Since the trace of a unitary matrix is bounded by its dimension, we get that L ρ (ψ n ) ∈ C J FPTA are bounded by Hence Growth(L ρ (ψ n )) ≤ L ∞ (ψ). From Theorem 3.10 it follows that F ∞ (g) = L ∞ (ψ). So we get the estimation from below 1/|w| ≤ F ∞ (g). The initial data of our lower estimation is the knowledge of the ZH-matricesF d provided by a cellular map, which enables us to compute the twisted Lefschetz zeta function. There is also a way to derive an upper bound from the same data. We have Hence Remark 3.13. A practical difficulty in the use of L ρ (ψ n ) and twisted Lefschetz zeta function L ψ ρ (t) for estimation is to find a useful representation ρ. Following the approach of Jiang in [24, Section 1.7], we can weaken the assumption on ρ in Proposition 3.12. There are many examples in [24,Chapter 4] which illustrate the method of estimation above for surface homeomorphisms.

Symplectic Floer homology and geometry of 3-manifolds
A three-dimensional manifold M 3 is called a graph manifold if there is a system of mutually disjoint two-dimensional tori T i in M 3 such that the closure of each component of M 3 cut along union of tori T i is a product of surface and S 1 . The growth rate of symplectic Floer homology 111 structure of finite volume, then asymptotic invariant F ∞ (g) > 1. If asymptotic invariant F ∞ (g) > 1, then φ has an infinite set of periodic points with pairwise different periods.
Proof. Kobayashi [25] has proved that the mapping torus T φ is a graph manifold if and only if the Thurston canonical representative for φ does not contain pseudo-Anosov components. So, Example 3.3 and Corollary 3.11 imply the first statement of the theorem. Thurston has proved [41,39] that Int (T φ ) admits a hyperbolic structure of finite volume if and only if φ is isotopic to pseudo-Anosov homeomorphism. This proves the second statement of the theorem. It is known [25] that a pseudo-Anosov homeomorphism has infinitely many periodic points whose periods are mutually distinct. This proves the last statement of the theorem.
Recall that the set of isotopy classes of orientation-preserving home- Arnoux-Yoccoz [1] and Ivanov [23] proved that spec (Mod(M )) = spec F (Mod(M )) is a closed discrete subset of R. It follows that spec F (Mod(M )) has, for each M , a least element, which we shall denote by F (M ). We can think of F (M ) as the systole of M(M ).
From a result of Penner [32] it follows that there exist constants 0 < c 0 < c 1 so that for all closed surfaces M with χ(M ) < 0, one has The proof of the lower bound comes from a spectral estimate for Perron-Frobenius matrices, with c 0 > log(2)/6 (see [32,29]). As such, this lower bound is valid for all surfaces M with χ(M ) < 0, including punctured surfaces. The upper bound is proved by constructing pseudo-Anosov homeomorphisms φ g : M g → M g on each closed surface of genus g ≥ 2 so that λ(φ g ) ≤ e c1/(2g−2) . J FPTA The best known upper bound for {F (M g )|χ(M g )|} follows from Hironaka and Kin [18] and from Minakawa [30], and it is 2 log(2+ √ 3). The situation for punctured surfaces is more mysterious.
For a pseudo-Anosov homeomorphism φ, let τ W P (φ) denote the translation length of φ, thought of as an isometry of Teich(M ) with the Weil-Petersson metric. Brock [3] has proved that the volume of the mapping torus T φ and τ W P (φ) satisfy a bi-Lipschitz relation, and in particular Moreover, there is a relation between the Weil-Petersson translation length and the Teichmüller translation length τ Teich (φ) = log(λ(φ)) (see [26]), which implies Then Theorem 3.9 implies the following estimation: where g is a pseudo-Anosov mapping class of φ. However

Radius of convergence of the symplectic zeta function
In [11] we have introduced a symplectic zeta function assigned to a mapping class g via zeta function F φ (t) of a monotone (or weakly monotone) representative φ ∈ Symp m (M, ω) of g. Symplectomorphisms φ n are also monotone (weakly monotone) for all n > 0 (see [19,4]) so, symplectic zeta function F φ (t) is an invariant as φ is deformed through monotone (or weakly monotone) symplectomorphisms in g. These imply that we have a symplectic Floer homology invariant F g (t) canonically assigned to each mapping class g. A motivation for the definition of this zeta function was a connection [11,19] between Nielsen numbers and Floer homology and nice analytic properties of Nielsen zeta function [33,8,11,9,10,13,14]. We denote by R the radius of convergence of the symplectic zeta function F g (t) = F φ (t).
In this section we give exact algebraic lower estimation for the radius R using Reidemeister trace formula for generalized Lefschetz numbers from Section 2.2.1. The growth rate of symplectic Floer homology 113 R admits the following estimations: and Proof. It follows from Lemma 3.5 that dim HF * (φ n ) = L π (ψ n ) . By the homotopy type invariance of the right-hand side we can estimate it. We can suppose that ψ is a cell map of a finite cell complex. The norm L H (ψ n ) is the sum of absolute values of the indices of all the n-orbits classes O n . It equals L π (ψ n ) , the sum of absolute values of the indices of all fixed point classes of ψ n , because any two fixed point classes of ψ n contained in the same n-orbit class O n must have the same index. From this we have The radius of convergence R is given by the Caushy-Adamar formula Therefore, we have The inequalities dim HF * (φ n ) = L π (ψ n ) = L H (ψ n ) =  [35,19,4]) in a reducible mapping class g of compact surface of genus greater than or equal to 2, and let λ be the largest dilatation of the pseudo-Anosov components (λ = 1 if there are no pseudo-Anosov components). Then the symplectic zeta function F g (t) = Fφ(t) has positive radius of convergence R = 1 λ , where λ is the largest dilatation of the pseudo-Anosov components (λ = 1 if there are no pseudo-Anosov components).
Proof. The radius of convergence R is given by the Caushy-Adamar formula 1 R = lim sup n n dim HF * (φ n ) n = lim sup n n dim HF * (φ n ).
Example 3.17. Let X be a surface with boundary and let f : X → X be a map. Fadell and Husseini (see [24]) devised a method of computing the matrices of the lifted chain map for surface maps. Suppose {a 1 , . . . , a r } is a free basis for π 1 (X). Then X has the homotopy type of a bouquet B of r circles which can be decomposed into one 0-cell and r 1-cells corresponding to the a i , and f has the homotopy type of a cellular map g : B → B. By the homotopy type invariance of the invariants, we can replace f with g in computations. The homomorphismf * : π 1 (X) → π 1 (X) induced by f and g is determined by the images b i =f * (a i ), i = 1, . . . , r. The fundamental group π 1 (T f ) has a presentation π 1 (T f ) = a 1 , . . . , a r , z|a i z = zb i , i = 1, . . . , r . Let D = ∂b i ∂a j be the Jacobian in Fox calculus (see [24]). Then, as pointed out in [24], the matrices of the lifted chain mapg arẽ Now, we can find estimations for the radius R as above. This example deals with surfaces with boundary, this is not what most of the paper concerns.
Let μ(d), d ∈ N, be the Möbius function. The growth rate of symplectic Floer homology 115 surface M of Euler characteristic χ(M ) < 0. Then the symplectic zeta function F g (t) = F φ (t) is a radical of a rational function and where the product is taken over all divisors d of the period m, and P (d) is the integer We denote by L φ (t) the Weil zeta function where L(φ n ) is the Lefschetz number of φ n . [11]). If φ is a hyperbolic diffeomorphism of a two-dimensional torus T 2 , then the symplectic zeta function F g (t) = F φ (t) is a rational function and F g (t) = F φ (t) = (L φ (σ · t)) (−1) r , where r is equal to the number of λ i ∈ Spec (φ) such that |λ i | > 1, p is equal to the number of μ i ∈ Spec (φ) such that μ i < −1 and σ = (−1) p , hereφ is a lifting of φ to the universal cover.

Concluding remarks and questions
Remark 3.20. For a symplectic manifold X, the Floer-type entropy of g ∈ Symp (X)/Ham (X), a mapping class of φ, is defined in [38] as h F (g) = lim sup 1 n log rk HF (φ n ) = log F ∞ (g).
This is a kind of robust version of the periodic entropy, robust in the sense that it depends on a symplectic diffeomorphism only through its mapping class; by contrast topological and periodic entropy are typically very sensitive to perturbation. As we proved above, for area-preserving diffeomorphisms of a surface M , the Floer-type entropy coincides with the topological entropy of the canonical representative in the corresponding mapping class; moreover, h F (g) > 0 if and only if g has a pseudo-Anosov component.