DYNAMICS OF THE TEICHMÜLLER FLOW ON COMPACT INVARIANT SETS

A BSTRACT . Let S be an oriented surface of genus g ≥ 0 with m ≥ 0 punctures and 3 g − 3 + m ≥ 2. For a compact subset K of the moduli space of area-one holomorphic quadratic differentials for S , let δ ( K ) be the asymptotic growth rate of the number of periodic orbits for the Teichmüller ﬂow Φ t which are contained in K . We relate δ ( K ) to the topological entropy of the restriction of Φ t to K . Moreover, we show that sup K δ ( K ) = 6 g − 6 + 2 m .


Introduction
An oriented surface S is called of finite type if S is a closed surface of genus g ≥ 0 from which m ≥ 0 points, so-called punctures, have been deleted. We assume that S is nonexceptional, i.e. that 3g − 3 + m ≥ 2. This means that S is not a sphere with at most four punctures or a torus with at most one puncture. Since the Euler characteristic of S is negative, the Teichmüller space T (S) of S is the quotient of the space of all complete hyperbolic metrics on S of finite volume under the action of the group of diffeomorphisms of S which are isotopic to the identity.
The fibre bundle Q 1 (S) over T (S) of all holomorphic quadratic differentials of area one can naturally be viewed as the unit cotangent bundle of T (S) for the Teichmüller metric. The Teichmüller geodesic flow Φ t on Q 1 (S) commutes with the action of the mapping class group Mod(S) of all isotopy classes of orientation preserving self-homeomorphisms of S. Thus this flow descends to a flow on the quotient Q 1 (S)/Mod(S), again denoted by Φ t . This quotient is a non-compact orbifold.
In his seminal paper [V86], Veech showed that the asymptotic growth rate of the number of periodic orbits of the Teichmüller flow Φ t on Q 1 (S)/Mod(S) is at least h = 6g − 6 + 2m (we use here a normalization for the Teichmüller flow which differs from the one used by Veech). Recently Eskin and Mirzakhani [EM08] obtained a sharp counting result: They show that as r → ∞, the number of periodic orbits for Φ t of period at most r is asymptotic to e hr /hr. An earlier partial result for the Teichmüller flow on the space of abelian differentials is due to Bufetov [Bu09].
In this note we are interested in the dynamics of the restriction of the Teichmüller flow to a compact invariant set. For the formulation of our first result, a continuous flow Φ t on a compact metric space (X, d) is called expansive if there is a constant δ > 0 with the following property. Let x ∈ X and let s : R → R be any continuous function with s(0) = 0 and d(Φ t (x), Φ s(t) (x)) < δ for all t. If y ∈ X is such that d(Φ t (x), Φ s(t) (y)) < δ for all t then y = Φ τ (x) for some τ ∈ R [HK95]. Note that this definition of expansiveness does not depend on the choice of the metric d defining the topology on X so it makes sense to talk about an expansive flow on a compact metrizable space.
Let Γ < Mod(S) be a torsion free normal subgroup of Mod(S) of finite index. For example, the subgroup of all elements which act trivially on H 1 (S, Z/3Z) has this property. DefineQ(S) = Q 1 (S)/Γ. We show Theorem 1. The restriction of the Teichmüller flow to every compact invariant subset K ofQ(S) is expansive.
Periodic orbits of expansive flows on compact spaces are separated, and their asymptotic growth rate can be related to the topological entropy of the flow.
For a compact Φ t -invariant subset K ofQ(S) let h top (K) be the topological entropy of the restriction of the Teichmüller flow to K. For any subset U ofQ(S) (or of Q 1 (S)/Mod(S)) and for a number r > 0 define n U (r) (or n ∩ U (r)) to be the cardinality of the set of all periodic orbits for Φ t of period at most r which are entirely contained in U (or which intersect U ). Clearly n ∩ U (r) ≥ n U (r) for all r. We show It is not hard to see that Theorem 1 and Theorem 2 are equally valid for compact invariant subsets of Q 1 (S)/Mod(S). However we did not find an argument which avoids using some differential geometric properties for the action of the mapping class group on Teichmüller space which is not in the spirit of this paper, so we omit a proof.
By the variational principle, the topological entropy of a flow Φ t on a compact space K equals the supremum of the metric entropies of all Φ t -invariant Borel probability measures on K. Bufetov and Gurevich [BG07] showed that the supremum of the topological entropies of the restriction of the Teichmüller flow to the moduli space of abelian differentials is just the metric entropy of the invariant probability measure in the Lebesgue measure class, moreover this Lebesgue measure is the unique measure of maximal entropy.
The following counting result implies that the entropy h of the Φ t -invariant Lebesgue measure on Q 1 (S)/Mod(S) equals the supremum of the topological entropies of the restrictions of the Teichmüller flow to compact invariant sets.
The first part of Theorem 3 is immediate from the results of Eskin and Mirzakhani, however the proof given here is very short and easy. The lower bound for the growth of periodic orbits which remain in a fixed compact set is the technically most involved part of this work.
The main tool we use for the proofs of the above resuls is the curve graph C(S) of S and the relation between its geometry and the geometry of Teichmüller space. In Section 2 we introduce the curve graph, and we summarize some results from [H10] in the form used in the later sections. In Section 3 we investigate the Teichmüller flow Φ t onQ(S) and we show Theorem 1. In Section 4 we use the results from Section 3 to establish a version of the Anosov closing lemma for the restriction of the Teichmüller flow to compact invariant subsets ofQ(S) and show Theorem 2. The proof of Theorem 3 is contained in Section 5.

The curve graph and its boundary
Let S be an oriented surface of genus g ≥ 0 with m ≥ 0 punctures and 3g−3+m ≥ 2. The curve graph C(S) of S is the graph whose vertices are the free homotopy classes of essential simple closed curves on S, i.e. simple closed curves which are neither contractible nor freely homotopic into a puncture. Two such curves are joined by an edge if and only if they can be realized disjointly. Since 3g − 3 + m ≥ 2 by assumption, C(S) is connected (see [MM99]). However, the curve graph is locally infinite. In the sequel we often do not distinguish between a simple closed curve on S and its free homotopy class. Also, if we write α ∈ C(S) then we always mean that α is an essential simple closed curve, i.e. α is a vertex in the curve graph C(S).
Providing each edge in C(S) with the standard euclidean metric of diameter 1 equips the curve graph with the structure of a geodesic metric space. Since C(S) is not locally finite, this metric space (C(S), d) is not locally compact. Masur and Minsky [MM99] showed that nevertheless its geometry can be understood quite explicitly. Namely, C(S) is hyperbolic of infinite diameter. The mapping class group Mod(S) naturally acts on C(S) as a group of simplicial isometries. A mapping class ϕ ∈ Mod(S) is pseudo-Anosov if the cyclic subgroup of Mod(S) generated by ϕ acts on the curve graph C(S) with unbounded orbits.
A geodesic lamination for a complete hyperbolic structure on S of finite volume is a compact subset of S which is foliated into simple geodesics. A geodesic lamination λ on S is called minimal if each of its half-leaves is dense in λ. Thus a simple closed geodesic is a minimal geodesic lamination. A minimal geodesic lamination with more than one leaf has uncountably many leaves and is called minimal arational. A geodesic lamination λ is said to fill up S if every simple closed geodesic on S intersects λ transversely. This is equivalent to stating that the complementary components of λ are all topological discs or once punctured topological discs.
A measured geodesic lamination is a geodesic lamination λ together with a translation invariant transverse measure. Such a measure assigns a positive weight to each compact arc in S which intersects λ nontrivially and transversely and whose endpoints are contained in complementary regions of λ. The geodesic lamination λ is called the support of the measured geodesic lamination; it consists of a disjoint union of minimal components. Vice versa, every minimal geodesic lamination is the support of a measured geodesic lamination.
The space ML of measured geodesic laminations on S can be equipped with the weak * -topology. Its projectivization PML is called the space of projective measured geodesic laminations and is homeomorphic to the sphere S 6g−7+2m . There is a continuous symmetric pairing ι : ML × ML → (0, ∞), the so-called intersection form, which satisfies ι(aξ, bη) = abι(ξ, η) for all a, b ≥ 0 and all ξ, η ∈ ML. By the Hubbard Masur theorem (see [Hu06]), for every x ∈ T (S) the space PML of projective measured geodesic laminations can naturally be identified with the projectivized cotangent space of T (S) at x. Moreover, a quadratic differential q ∈ Q 1 (S) can be viewed as a pair (λ, ν) ∈ ML × ML with i(λ, ν) = 1 and the additional property that i(λ, ζ) + i(ν, ζ) > 0 for all ζ ∈ ML.
Since C(S) is a hyperbolic geodesic metric space, it admits a Gromov boundary ∂C(S) which is a (non-compact) metrizable topological space equipped with an action of Mod(S) by homeomorphisms (see [BH99] for the definition of the Gromov boundary of a hyperbolic geodesic metric space and for references). Following Klarreich [Kl99] (see also [H06]), this boundary can naturally be identified with the space of all (unmeasured) minimal geodesic laminations which fill up S, equipped with the topology which is induced from the weak * -topology on PML via the measure forgetting map. Now let F ML ⊂ PML be the Mod(S)-invariant Borel subset of all projective measured geodesic laminations whose support is minimal and fills up S. The discussion in the previous paragraph shows that there is a continuous Mod(S)-invariant surjection which associates to a projective measured geodesic lamination in F ML its support.
Since the curve graph is a hyperbolic geodesic metric space, for every c ∈ C(S) there is a visual metric δ c of uniformly bounded diameter on the Gromov boundary ∂C(S) of C(S) (we refer to Chapter III.H of [BH99] for details of this construction and for references). These distances are related to the intrinsic geometry of C(S) as follows.
For a point c ∈ C(S), the Gromov product at c associates to points x, y ∈ C(S) the value The Gromov product can be extended to a Gromov product ( | ) c for pairs of distinct points in ∂C(S) by defining where the supremum is taken over all sequences (x i ) and (y j ) in C(S) such that ξ = lim x i and ζ = lim y j . There are numbers β > 0, ν ∈ (0, 1) such that δ c ≤ e βd(c,a) δ a for all c, a ∈ C(S).
The distances δ c are equivariant with respect to the action of Mod(S) on C(S) and on ∂C(S). For c ∈ C(S), ξ ∈ ∂C(S) and r > 0 denote by D c (ξ, r) ⊂ ∂C(S) the ball of radius r about ξ with respect to the distance function δ c .
We will need a more precise quantitative relation between the distance functions δ c (c ∈ C(S)). Even though this property is well known, we did not find an explicit reference in the literature and we include a sketch of a proof.
For a formulation, for a number m > 1, an m-quasi-geodesic in a metric space Proof. Since C(S) is a hyperbolic geodesic metric space, there is a constant p > 1 depending on the hyperbolicity constant such that every point c ∈ C(S) can be connected to every point ξ ∈ ∂C(S) by a p-quasi-geodesic (for the particular case of the curve graph see [Kl99,H06,H10]). Similarly, any two points ξ = ζ ∈ ∂C(S) can be joined by a p-quasi-geodesic.
Let m ≥ p. By hyperbolicity, there is a number r(m) > 0 and for every m-quasigeodesic triangle T in C(S) with vertices c ∈ C(S), ξ = ζ ∈ ∂C(S) there is a point u ∈ C(S) whose distance to each of the sides of T is at most r(m). The (non-unique) point u is called a center of T . We claim that there is a number χ(m) > 0 only depending on m and the hyperbolicity constant such that Namely, let γ 1 , γ 2 : [0, ∞) → C(S) be m-quasi-geodesic rays which connect c to ξ, ζ. There is a universal constant b > 0 only depending on the hyperbolicity constant for C(S) such that (Remark 3.17.5 in Chapter III.H of [BH99]).
Now let γ : [0, ∞) → C(S) be any m-quasi-geodesic ray with endpoint γ(∞) = ξ ∈ ∂C(S), let ζ = ξ ∈ ∂C(S) and let T be an m-quasi-geodesic triangle with side γ and vertices γ(0), ξ, ζ. If u ∈ C(S) is a center for T and if σ ≥ 0 is such that d(u, γ(σ)) ≤ r(m), then for every s ∈ [0, σ] the distance between u and a center for any m-quasi-geodesic triangle with vertices γ(s), ξ, ζ is bounded from above by a constant only depending on m and the hyperbolicity constant for C(S). In particular, by the above discussion and the properties of an m-quasi-geodesic, there are constants a(m) > 0, b(m) > 0 such that From this the lemma follows.
By a result of Bers, there is a constant χ 0 = χ 0 (S) > 0 such that for every complete hyperbolic metric x on S of finite volume there is a pants decomposition for S consisting of simple closed geodesics of x-length at most χ 0 . Define a map by associating to a marked hyperbolic metric x ∈ T (S) a simple closed curve Υ T (x) whose x-length ℓ x (Υ T (x)) does not exceed χ 0 . There are choices involved in the definition of Υ T (x), but for any two such choices and any x the distance between the images of x is uniformly bounded.
Denote by d T the distance on T (S) defined by the Teichmüller metric. Lemma 2.2 of [H10] shows that there is a number L > 1 such that Choose a smooth function σ : [0, ∞) → [0, 1] with σ[0, χ 0 ] ≡ 1 and σ[2χ 0 , ∞) ≡ 0. For every x ∈ T (S) we obtain a finite Borel measure µ x on C(S) by defining where δ β denotes the Dirac mass at β. The total mass of µ x is bounded from above and below by a universal positive constant, and the diameter of the support of µ x in C(S) is uniformly bounded as well. The measures µ x are equivariant with respect to the action of the mapping class group on T (S) and C(S), and they depend continuously on x ∈ T (S) in the weak * -topology. This means that for every bounded function f : C(S) → R the function x → f dµ x is continuous.
Clearly the metrics δ x are equivariant with respect to the action of Mod(S) on T (S) and ∂C(S). Moreover, there is a constant κ > 0 such that for all x, y ∈ T (S) (see p.230 and p.231 of [H09]).
The main theorem of [H10] relates the geometry of Teichmüller space to the geometry of the curve graph via the map Υ T . For its formulation, denote for ǫ > 0 by T (S) ǫ the set of all hyperbolic metrics whose systole (i.e. the shortest length of a closed geodesic) is at least ǫ. For sufficiently small ǫ the set T (S) ǫ is connected, and the mapping class group acts properly and cocompactly on T (S) ǫ .
(1) For every L > 1 there is a number ǫ = ǫ(L) > 0 with the following property. Let J ⊂ R be a closed connected set of length at least 1/ǫ and let γ : J → T (S) be an L-quasi-geodesic. If Υ T • γ is an L-quasigeodesic in C(S) then there is a Teichmüller geodesic ξ : J ′ → T (S) ǫ such that the Hausdorff distance between γ(J) and ξ(J ′ ) is at most 1/ǫ.
(2) For every ǫ > 0 there is a number L(ǫ) > 1 with the following property.
Let J ⊂ R be a closed connected set and let γ : J → T (S) be a 1/ǫ-quasigeodesic. If there is a Teichmüller geodesic arc ξ : J ′ → T (S) ǫ such that the Hausdorff distance between γ(J) and ξ(J ′ ) is at most 1/ǫ then Υ T • γ is an L(ǫ)-quasi-geodesic in C(S).
Let Q 1 (S) be the bundle of area one quadratic differentials over Teichmüller space T (S) for S. The mapping class group Mod(S) acts properly discontinuously on Q 1 (S) as a group of bundle automorphisms. This action commutes with the action of the Teichmüller geodesic flow Φ t .
An area one quadratic differential q ∈ Q 1 (S) is determined by a pair (q v , q h ) of measured geodesic laminations, the vertical and the horizontal measured geodesic lamination of q, respectively. For every t ∈ R the pair (e t q v , e −t q h ) corresponds to the quadratic differential Φ t q. The strong stable manifold of q is defined as the set of all quadratic differentials of area one whose vertical measured geodesic lamination coincides precisely with the vertical measured geodesic lamination of q. The strong unstable manifold is the set of all quadratic differentials of area one whose horizontal measured geodesic lamination coincides precisely with the horizontal measured geodesic lamination of q. Define moreover the stable manifold W s (q) of q and the unstable manifold W u (q) of q by W s (q) = ∪ t∈R Φ t W ss (q) and W u (q) = ∪ t∈R Φ t W su (q). As q varies through Q 1 (S), the manifolds W ss (q), W su (q), W s (q), W u (q) define continuous foliations of Q 1 (S) which are called the strong stable, the strong unstable, the stable and the unstable foliation. These foliations are invariant under the action of Mod(S) and under the action of the Teichmüller geodesic flow Φ t .
For every q ∈ Q 1 (S) the map which associates to a quadratic differential its vertical measured geodesic lamination restricts to a homeomorphism of the unstable manifold W u (q) containing q onto an open dense subset of ML. The space ML admits a natural Mod(S)-invariant measure in the Lebesgue measure class which lifts to a locally finite measure on W u (q) in the Lebesgue measure class. The induced family of conditional measuresλ q (q ∈ Q 1 (S)) on strong unstable manifolds transform under the Teichmüller flow by dλ Φ t q • Φ t = e ht dλ q where as before, h = 6g − 6 + 2m. The measuresλ q are equivariant under the action of the mapping class group. Moreover, they are conditional measures for a Mod(S)-invariant locally finite measureλ on Q 1 (S) in the Lebesgue measure class. The measureλ is the lift of a finite measure λ on Q 1 (S)/Mod(S) which is Φ t -invariant and mixing. The measure λ gives full measure to the set of all quadratic differentials whose horizontal and vertical measured geodesic laminations are uniquely ergodic and fill up S (see [M82, V86] for details). Moreover, by the Poincaré recurrence theorem, λ-almost every q ∈ Q 1 (S)/Mod(S) is recurrent, i.e. it is contained in the ω-limit set of its own orbit under the Teichmüller flow.
be the map which associates to a quadratic differential its vertical projective measured geodesic lamination. For every q ∈ Q 1 (S) the restriction of the projection π to W su (q) is a homeomorphism of W su (q) onto the open subset of PML of all projective measured geodesic laminations µ which together with π(−q) jointly fill up S, i.e. are such that for every measured geodesic lamination η ∈ ML we have i(µ, η) + i(π(−q), η) = 0 (note that this makes sense even though the intersection form i is defined on ML rather than on PML). The measure class of the pushforward under π of the measureλ q on W su (q) does not depend on q and defines a Mod(S)-invariant ergodic measure class on PML.

Compact invariant sets are expansive
Let again Q 1 (S) be the bundle of area one quadratic differentials over Teichmüller space T (S) of an oriented surface of genus g ≥ 0 with m ≥ 0 punctures and where 3g − 3 + m ≥ 2. The mapping class group Mod(S) acts on Q 1 (S), but this action is not free and the quotient space Q 1 (S)/Mod(S) is a non-compact orbifold rather than a manifold.
To overcome this (mainly technical) difficulty we choose a torsion free normal subgroup Γ of Mod(S) of finite index. For example, the group of all elements which act trivially on H 1 (S, Z/3Z) has this property. DefineQ(S) = Q 1 (S)/Γ and let be the canonical projection. Since the action of Γ on Q 1 (S) is free, the map Π is a covering. The Teichmüller flow Φ t acts onQ(S).

The goal of this section is to show
Theorem 3.1. The restriction of the Teichmüller flow to every compact invariant subset K ofQ(S) is expansive.
We begin with the construction of a convenient metric on Q 1 (S) andQ(S) inducing the usual topology. To this end, call a distance d on a space X a length metric if the distance between any two points is the infimum of the lengths of all paths connecting these points. In the sequel denote by the canonical projection.
Lemma 3.2. There is a complete Mod(S)-invariant length metric d on Q 1 (S) with the following properties.
(1) The metric d induces the usual topology.
(2) The canonical projection P : is equipped with the Teichmüller metric.
(3) Every orbit of the Teichmüller flow with its natural parametrization is a minimal geodesic for d parametrized by arc length.
Proof. By the Hubbard Masur theorem (see [Hu06] for a presentation of this celebrated and by now classical result), the restriction of the canonical projection P : Q 1 (S) → T (S) to every unstable (or stable) manifold in Q 1 (S) is a homeomorphism onto T (S). Thus the Teichmüller metric on T (S) lifts to a length metric on the leaves of the stable and of the unstable foliation.
Call a path ρ : [0, 1] → Q 1 (S) admissible if there is a finite partition 0 = t 0 < · · · < t k = 1 such that the restriction of ρ to each interval [t i−1 , t i ] is entirely contained in a stable or in an unstable manifold. For each such admissible path ρ we can define its length to be the sum of the lengths with respect to the lifts of the Teichmüller metric of the subsegments of ρ entirely contained in a stable or an unstable manifold. For q 0 , q 1 ∈ Q 1 (S) define d(q 0 , q 1 ) to be the infimum of the lengths of all admissible paths connecting q 0 to q 1 . Then d is a (a priori non-finite) distance function on Q 1 (S) which satisfies the second requirement in the lemma. The third property holds true since the projection to T (S) of an orbit of the Teichmüller flow is a Teichmüller geodesic of the same length and hence realizes the distance between its endpoints. By the second property for d and the definition, the d-length of every path in Q 1 (S) which is entirely contained in a stable or an unstable manifold coincides with the length of its projection to T (S). As a consequence, the metric d is a length metric.
We are left with showing that d induces the usual topology on Q 1 (S). For this let q ∈ Q 1 (S) and let ǫ > 0. We have to show that the ǫ-ball about q for the distance d contains a neighborhood of q in Q 1 (S). For this denote for z ∈ Q 1 (S) and r > 0 by B i (q, r) the open r-ball about q in W i (q) with respect to the lift of the Teichmüller metric (i = s, u).
is an open neighborhood of z in W u (z) whose closure is compact and depends continuously on z in the Hausdorff topology for compact subsets of Q 1 (S) by the Hubbard Masur theorem.
is an open neighborhood of q in Q 1 (S). Moreover by construction, U is contained in the ǫ-ball about q with respect to the distance function d. This completes the proof of the lemma.
The distance d on Q 1 (S) constructed in Lemma 3.2 induces a metric onQ(S), again denote by d, via where as before, Π : Q 1 (S) →Q(S) is the canonical projection. In the sequel we always use these distance functions on Q 1 (S) andQ(S) without further mentioning.
With these preparations, we can prove Theorem 3.1.
Proof of Theorem 3.1. Let K ⊂Q(S) be a compact Φ t -invariant subset and let K ⊂ Q 1 (S) be the preimage of K under the canonical projection Π : Q 1 (S) → Q(S). Let as before P : Q 1 (S) → T (S) be the canonical projection. By the second part of Theorem 2.2, there is a constant p > 0 only depending on K such that for Since by inequality (9) the map Υ T is coarsely Lipschitz, this shows that lifts of orbits of Φ t |K which are contained in the same unstable manifold diverge linearly in forward direction. We therefore just have to relate distances in the curve graph to distances in Q 1 (S) for the metric d in a quantitative way. The remainder of the argument establishes such a control.
This flip is equivariant with respect to the action of the mapping class group and hence it descends to a continuous involution ofQ(S) which we denote again by F . Recall that the Gromov boundary ∂C(S) of C(S) can be identified with the set of all (unmeasured) minimal geodesic laminations on S which fill up S. Let again π : Q 1 (S) → PML be the canonical projection. If q ∈K ∪ F (K) then the support F (πq) of the vertical measured geodesic lamination of q is uniquely ergodic, which means that F (πq) admits a unique transverse measure up to scale, and F (πq) is minimal and fills up S [M82]. Moreover, the p-quasi-geodesic t → Υ T (P Φ t q) converges in C(S) ∪ ∂C(S) to F (πq) (the latter statement follows from the explicit identification of ∂C(S) with the set of minimal geodesic laminations which fill up S established in [Kl99,H06], see also [H10]).
Write A = π(K ∪ F (K)). Then A is a Γ-invariant Borel subset of PML. By the consideration in the previous paragraph, the restriction to A of the map F : F ML → ∂C(S) introduced in (1) of Section 2 is a Γ-equivariant continuous injection which associates to a projective measured geodesic lamination contained in A its support. In other words, we can identify the set A with a subset of ∂C(S).
Recall from equation (11) the definition of the distances δ x (x ∈ T (S)) on the Gromov boundary ∂C(S) of C(S). Since the map For simplicity of notation, we denote this distance again by δ P q . The topology on A defined by this distance is just the subspace topology of A as a subset of PML (this is the result of [Kl99]).
For q ∈K ∪ F (K) denote by D q (π(q), r) ⊂ A the ball of radius r in A about π(q) with respect to this distance. By continuity of the projection π and the map F A and by the relation (12) between the distances δ x and δ y for x, y ∈ T (S), the ball D q (π(q), r) depends continuously on q in the following sense. For every q ∈K ∪ F (K), every r > 0 and every ǫ ∈ (0, r) there is a neighborhood U of q iñ K ∪ F (K) such that for every u ∈ U we have By inequalities (12) and (17) and by Lemma 2.1, there are numbers α > 0, a > 1, b > 0 such that for every q ∈K and for all t > 0 we have For q ∈ Q 1 (S) and β > 0 denote by B(q, β) the ball of radius β about q in Q 1 (S) with respect to the length metric d defined in Lemma 3.2. Since the projection π is continuous, for every q ∈K ∪ F (K) there is a number ǫ(q) > 0 such that π(B(q, ǫ(q)) ∩ (K ∪ FK)) ⊂ D q (π(q), α) where α > 0 is as in the inequalities (19). By the continuity properties (18) of the balls D u (π(u), r) for u ∈K ∪ F (K) and r > 0, by invariance under the action of the group Γ < Mod(S) and cocompactness we can find a universal number β 0 > 0 such that Denote again by d the distance onQ(S) induced in equation (16) from the distance on Q 1 (S). Since K ∪ F (K) is compact and Π : Q 1 (S) →Q(S) is a covering, there is a number β < β 0 such that for every q ∈ K ∪ F (K) and every lift q of q to Q 1 (S) the ball B(q, β) inQ(S) of radius β about q is the homeomorphic image under Π of the ball B(q, β). Now the orbits of Φ t are geodesics for the distance d. Hence if x ∈ K, if s : R → R is a continuous function such that s(0) = 0 and d(Φ t q, Φ s(t) q) < β/2 for all t then by the choice of β we have |t − s(t)| < β/2 for all t.
This implies that for this function s, if u ∈ K is such that d(Φ t q, Φ s(t) u) < β/2 for all t then d(Φ t u, Φ s(t) u) < β/2 (since |t − s(t)| < β/2 and orbits of the Teichmüller flow are geodesics) and hence d(Φ t u, Φ t q) < β for all t by the triangle inequality. In particular, for a liftq ∈ Q 1 (S) of q and a liftũ of u with d(q,ũ) < β we have d(Φ tq , Φ tũ ) < β for all t ∈ R.
Let W s loc (q) be the connected component containing q of the intersection of B(q, β) with the stable manifold W s (q) of q. We claim that u ∈ W s loc (q). For this assume otherwise. Let againq be a preimage of q in Q 1 (S) and letũ ∈ Q 1 (S) be the preimage of u with d(q,ũ) < β. If π(q) = π(ũ) then δ Pq (π(q), π(ũ)) > 0 and by continuity, our choice of α and the estimates (19), there is a number t > 0 such that δ P Φ tq (π(q), π(ũ)) = α. On the other hand, for every s ∈ [0, t] the distance in Q 1 (S) between Φ sq , Φ sũ is smaller than β which is a contradiction to the choice of β < β 0 and the relation (20).
In the same way we conclude that u is contained in the intersection of B(q, β) with the local unstable manifold of q. Now the intersection of the local stable manifold W s loc (q) with the local unstable manifold W u loc (q) is contained in the orbit of q under the Teichmüller flow Φ t which completes the proof of Theorem 3.1.
For a compact Φ t -invariant subset K ofQ(S) let h top (K) be the topological entropy of the restriction of Φ t to K. For r > 0 let moreover n K (r) be the number of periodic orbits of Φ t of period at most r which are contained in K. Since by Theorem 3.1 the restriction of the flow Φ t to K is expansive, by Proposition 3.2.14 of [HK95] we have

An Anosov closing lemma
The goal of this section is to establish a version of the Anosov closing lemma for the restriction of the Teichmüller flow to a compact invariant set K ⊂Q(S). The classical Anosov closing lemma roughly states that for a hyperbolic flow on a closed Riemannian manifold, a closed curve consisting of sufficiently long orbit segments which are connected at the endpoints by sufficiently short arcs is closely fellow-traveled by a periodic orbit.
We continue to use the assumptions and notations from Section 2 and Section 3. In particular, we always use the distances on Q 1 (S) andQ(S) defined in Lemma 3.2 and in equation (16). For a precise formulation of our version of an Anosov closing lemma for the Teichmüller flow, using the assumptions and notations from Section 2 and Section 3 we define.
(1) For every j ≤ k the 2ǫ-neighborhood of q j is contained in a contractible subset ofQ(S).
The pseudo-orbit is contained in a compact set K if for all i and all t ∈ [0, t i ] we have Φ t q i ∈ K. The pseudo-orbit is called closed if q 0 = q k .
An (n, ǫ)-pseudo-orbit q 0 , . . . , q k determines an essentially unique arc connecting q 0 to q k which we call a characteristic arc. Namely, by assumption, for each j < k the 2ǫ-neighborhood of q j+1 is contained in a contractible subset ofQ(S). Hence the homotopy class with fixed endpoints inQ(S) of an arc of length smaller than 2ǫ connecting Φ tj q j to q j+1 is unique. We define a characteristic arc of the (n, ǫ)pseudo-orbit to be an arc connecting q 0 to q k which is obtained by successively joining the endpoint of the orbit segment {Φ t q j | 0 ≤ t ≤ t j } to q j+1 with an arc of length smaller than 2ǫ which is parametrized on the unit interval (j = 0, . . . , k − 1). The points q i (1 ≤ i ≤ k) are called the breakpoints of the characteristic arc of the pseudo-orbit. The characteristic homotopy class of the pseudo-orbit is the homotopy class with fixed endpoints of a characteristic arc connecting q 0 to q k . Note that this is independent of the choice of a characteristic arc. If the pseudo-orbit is closed then it determines a closed characteristic curve and hence a free homotopy class of closed curves inQ(S) which we call the characteristic free homotopy class of the closed pseudo-orbit.
By abuse of notation, denote again by P :Q(S) → T (S)/Γ the canonical projection.
(1) There is a number α ≤ δ such that the α-neighborhoods of P q 0 , P q k in T (S)/Γ with respect to the projection of the Teichmüller metric are contractible and contain P q, P Φ τ q.
(2) There is a liftζ to Q 1 (S) of the orbit segment ζ and a liftγ to Q 1 (S) of a characteristic arc γ for the pseudo-orbit with the following properties. The distance between the endpoints of Pγ, Pζ is at most α, and the Hausdorff distance betweenγ andζ is at most δ.
A closed pseudo-orbit inQ(S) is δ-shadowed by a periodic orbit if in addition to the above requirements the orbit {Φ t q | t ∈ [0, τ ]} is closed.
Note that every point in the orbifold Q 1 (S)/Mod(S) admits a contractible neighborhood, so we can use the above definition is the same way for the Teichmüller flow on the orbifold Q 1 (S)/Mod(S).
Using Definition 4.1 and Definition 4.2 we can now formulate a version of the Anosov closing lemma for the Teichmüller flow which is the main result of this section.
For the proof of Theorem 4.3 we need the following technical preparation which is also used in Section 5. To this end, recall that a point q ∈ Q 1 (S)/Mod(S) is recurrent if q is contained in the ω-limit set of its own orbit under the Teichmüller flow. For some m > 1, an unparametrized m-quasi-geodesic in C(S) is an arc γ : J → C(S) with the property that there is an orientation preserving homeomorphism ϕ : I → J such that γ • ϕ : I → C(S) is an m-quasi-geodesic. We show (1) For every compact Φ t -invariant set K ⊂Q(S) there are numbers ǫ 0 = ǫ 0 (K) > 0, n 0 = n 0 (K) > 0, ℓ 0 = ℓ 0 (K) > 1 depending on K with the following property. Let q 0 , . . . , q k ∈Q(S) be an (n 0 , ǫ 0 )-pseudoorbit contained in K and letγ be a lift to Q 1 (S) of a characteristic arc connecting q 0 to q k . Then the arc t → Υ T (Pγ(t)) is an ℓ 0 -quasi-geodesic in C(S).
(2) There is a number m > 1 and for every recurrent point q ∈ Q 1 (S)/Mod(S) there are numbers ǫ 0 (q) > 0, n 0 (q) > 0 with the following properties. Let q 0 , . . . , q k be an (n 0 (q), ǫ 0 (q))-pseudo-orbit with d(q i , q) ≤ ǫ 0 (q) for all i and letγ be a lift to Q 1 (S) of a characteristic arc γ connecting q 0 to q k . Then the arc t → Υ T (Pγ(t)) is an unparametrized m-quasi-geodesic in C(S).
Let K ⊂Q(S) be a compact Φ t -invariant set and letK ⊂ Q 1 (S) be the preimage of K under the natural projection. By the second part of Theorem 2.2 there is a number p > 1 such that for every q ∈K the assignment t → Υ T (P Φ t q) (t ∈ R) is a p-quasi-geodesic.
For q ∈K we have F (π(q)) = R(t → Υ T (P Φ t q)) ∈ ∂C(S). Let κ > 0 be as in (12) of Section 2. By continuity, for every q ∈K there is a number ǫ(q) > 0 such that for every pointq ∈K which is contained in the 2ǫ(q)-neighborhood of q the δ P q -distance between F (π(q)) and F (π(q)) is smaller than α(p)/κ where α(p) > 0 is as in the first paragraph of this proof. By continuity, invariance under the action of the mapping class group on Q 1 (S) and ∂C(S) and cocompactness of the action of Γ onK, there is a number ǫ 0 ∈ (0, 1/2) which has this property for all q ∈K (compare the proof of Theorem 3.1 for a similar statement).
Let m(p) > 0 be as in the first paragraph of this proof. Letγ be the lift to Q 1 (S) of a characteristic arc of an (m(p), ǫ 0 )-pseudo-orbit q 0 , q 1 , . . . , q k contained in K ⊂Q(S). Thenγ is a composition of curvesγ i (i = 1, . . . , k) where the curvẽ γ i is a lift to Q 1 (S) of an orbit segment for the Teichmüller flow of length at least m(p) beginning at x i−1 and an arc of length at most 2ǫ 0 < 1 parametrized on [0, 1], with endpoint x i . By the choice of m(p), of ǫ 0 > 0 and by the construction of the curvesγ i , the curve Υ T (Pγ) is of the form described in the second paragraph of this proof and hence it is a β(p)-quasi-geodesic in C(S). This shows the first part of the lemma.
The second part of the lemma follows in the same way. By [MM99] there is a number ℓ > 0 such that for everyq ∈ Q 1 (S) the map t → Υ T (P Φ tq ) is an unparametrized ℓ-quasi-geodesic (this number ℓ > 0 does not coincide with the number p > 1 above, see also [H10]). Let m(ℓ) > 0, α(ℓ) > 0, p(ℓ) > 1 be as in the first paragraph of this proof. Let q ∈ Q 1 (S)/Mod(S) be a recurrent point and letq ∈ Q 1 (S) be a lift of q. Then the vertical measured geodesic lamination ofq is uniquely ergodic and fills up S [M82], and the unparametrized ℓ-quasi-geodesic t → Υ T (P Φ tq ) is of infinite diameter. By continuity and by Lemma 2.4 of [H10] there is a neighborhood V ofq in Q 1 (S) and a number T (q) > 0 such that In particular, if u ∈ V and if ρ u : [0, a) → [0, ∞) (a ∈ (0, ∞]) is a homeomorphism with the property that the map t → Υ T (P Φ ρu(t) u) is a parametrized ℓquasi-geodesic in C(S) then ρ u (m(ℓ)) ≤ T (q).
The subset U of Q 1 (S) of all points with uniquely ergodic vertical measured geodesic lamination which fills up S is dense [M82]. Each u ∈ U defines a point F π(u) ∈ ∂C(S) which is just the endpoint of the infinite unparametrized ℓ-quasigeodesic t → Υ T (P Φ t u). The map u ∈ U → F π(u) ∈ ∂C(S) is continuous. Thus we can find a number ǫ 0 (q) ∈ (0, 1/2) which is small enough that the 2ǫ 0 (q)neighborhood W ofq in Q 1 (S) is contained in V and that for every point u ∈ W ∩U the δ Pq -distance between F (π(q)) and F (π(u)) is smaller than α(ℓ)/κ where as before, κ > 0 is as in (12) in Section 2. The second part of the lemma holds true for the numbers m = m(ℓ) > 0, ǫ 0 (q) > 0, n 0 (q) = T (q) > 0.
Proof of Theorem 4.3. Let K ⊂Q(S) be any compact Φ t -invariant set and let K be the preimage of K in Q 1 (S) under the natural projection. Let ǫ 0 = ǫ 0 (K) ∈ (0, 1/2), n 0 = n 0 (K) > 0 be as in the first part of Lemma 4.4. Let q 0 , . . . , q k be an (n 0 , ǫ 0 )-pseudo-orbit for Φ t which is contained in K and let t 0 , . . . , t k−1 ∈ [n 0 , ∞) be as in the definition of a pseudo-orbit such that d(Φ ti q i , q i+1 ) ≤ ǫ 0 for i < k. Let γ be a characteristic arc of this pseudo-orbit which is parametrized on [0, in such a way that for each j the restriction of γ to [ i<j t i + j, i<j+1 t i + j] is a reparametrization of the orbit segment {Φ t q j | t ∈ [0, t j ]} by a translation. The points q 1 , . . . , q k−1 are the breakpoints of the characteristic arc.
Letγ be a lift of γ to Q 1 (S). By Lemma 4.4, the assignment t → Υ T (Pγ(t)) is an ℓ 0 -quasi-geodesic in C(S) for a number ℓ 0 > 1 only depending on K. The map t → Pγ(t) ∈ (T (S), d T ) is one-Lipschitz. Since by inequality (9) there is a number L > 0 such that d T (P q, P z) ≥ d(Υ T (P q), Υ T (P z))/L − L for all q, z ∈ Q 1 (S), we conclude that the curve t → Pγ(t) is a uniform quasi-geodesic in T (S). By the first part of Theorem 2.2, this implies that there is a Teichmüller geodesic whose Hausdorff distance to Pγ is bounded from above by a universal constant. As a consequence, the Hausdorff distance betweenγ and the tangent line of this geodesic is bounded from above by a universal constant b > 0.
A mapping class g ∈ Mod(S) is pseudo-Anosov if the cyclic subgroup of Mod(S) generated by g acts on the curve graph C(S) with unbounded orbits. In this case the conjugacy class of g can be represented by a closed orbit for the Teichmüller flow Φ t on Q 1 (S)/Mod(S), and it can be represented by a closed orbit for the Teichmüller flow onQ(S) if the conjugacy class of g is contained in the normal subgroup Γ of Mod(S). Assume now that the (n 0 , ǫ 0 )-pseudo-orbit q 0 , . . . , q k contained in K is closed. Letγ be a lift to Q 1 (S) of a closed characteristic arc γ for the pseudoorbit. By the first part of Lemma 4.4, the curve t → Υ T (Pγ(t)) is an infinite ℓ 0 -quasi-geodesic in C(S) which is invariant under an element g ∈ Γ < Mod(S) of the mapping class group. The mapping class g acts on this quasi-geodesic as a translation and hence it is pseudo-Anosov. As a consequence, there is a unique g-invariant Teichmüller geodesic in T (S) whose cotangent line in Q 1 (S) projects to a periodic orbit of Φ t inQ(S) which defines the free homotopy class of γ. In other words, there is a closed orbit for Φ t inQ(S) which is freely homotopic to γ.
By the first part of Theorem 2.2, applied to the biinfinite quasi-geodesic Pγ in T (S), this orbit is contained in a compact subset C 0 ⊃ K ofQ(S) not depending on the pseudo-orbit. Moreover, it b-shadows the pseudo-orbit for a number b > 0 only depending on K, n 0 , ǫ 0 . This shows the first part of Theorem 4.3.
Let C ⊂ C 0 be the Φ t -invariant subset of C 0 of all points whose Φ t -orbit is entirely contained in C 0 . The periodic orbit defined by the conjugacy class of the pseudo-Anosov element g is contained in C. LetC be the preimage of C in Q 1 (S). Then every lift to Q 1 (S) of a periodic orbit inQ(S) determined as above by a closed (n 0 , ǫ 0 )-pseudo-orbit contained in K is contained inC.
Let again π : Q 1 (S) → PML be the canonical projection. Write A = π(C ∪ F (C)) where F : Q 1 (S) → Q 1 (S) is the flip q → F (q) = −q. As in Section 3, let F A = F |A : A → ∂C(S) be the measure forgetting injection. For q ∈ C ∪ F (C) let δ P q be the distance on ∂C(S) defined in equation (11) and denote by D q (π(q), r) the ball of radius r about π(q) in A with respect to the distance (x, y) ∈ A × A → δ P q (F A x, F A y) ∈ [0, ∞) which we denote again by δ P q (compare the proof of Theorem 3.1).
By the second part of Theorem 2.2, applied to the projection into T (S) of the preimageC of the compact Φ t -invariant set C ⊂Q(S), by Lemma 2.1 and by inequality (12) of Section 2, there are numbers α 0 < 1/2, a > 1, b > 0 such that for every q ∈C and for all t > 0 we have Moreover, for every α < α 0 there is a number β = β(α) < 1 such that for every q ∈C we have A ∩ πB(q, β) ⊂ D q (π(q), α) where B(q, β) is the ball of radius β about q in Q 1 (S) (compare the proof of Theorem 3.1).
Note that the argument in the previous paragraph together with the estimate (23) also shows that for all t ∈ [0, T ] with the additional property thatγ(t) ∈C (namely this holds true for every t such thatγ(t) projects to an orbit segment defining the pseudo-orbit).
Let againγ be a biinfinite lift toQ 1 (S) of a closed characteristic curve γ for an (n, σ)-pseudo-orbit contained in K. The curve Υ T (Pγ) is a uniform quasi-geodesic in C(S) which is invariant under a pseudo-Anosov element g ∈ Γ < Mod(S) on ∂C(S). The conjugacy class of γ defines the free homotopy class of γ. The oriented cotangent line of the axis of g is contained inC. If z ∈C is a point in this cotangent line then π(z) ∈ A is a fixed point for the action of g on PML. An inductive application to longer and longer subsegments ofγ of the argument which lead to the estimate (27) shows that for every t ∈ R such thatγ(t) ⊂K ⊂C the fixed point π(z) ∈ A of g is contained in the ball Dγ (t) (π(γ(t)), 4aα). The same argument also shows that the fixed point π(−z) for the action of g is contained in D −γ(t) (π(−γ(t)), 4aα). The periodic orbit onQ(S) defined by g is contained in the compact Φ t -invariant subset C ⊃ K ofQ(S) determined above.
By the considerations in Theorem 3.1 and its proof, applied to the compact Φ tinvariant subset C ofQ(S), this means that for every δ > 0 there is a constant β > 0 only depending on K with the following property. Let q 0 , . . . , q k be a closed (n, β)-pseudo-orbit contained in K. Then there is a closed orbit for Φ t contained in C whose Hausdorff distance to a closed characteristic curve defined by the pseudoorbit is at most δ. From this Theorem 4.3 follows.
The Anosov closing lemma implies the existence of many periodic orbits near any non-wandering point of a compact Φ t -invariant subset K ofQ(S). However, as for compact invariant hyperbolic sets in the usual sense of smooth dynamical systems (see [HK95]), these periodic orbits are in general not contained in K. The next corollary is an immediate adaptation of Corollary 6.4.19 of [HK95] and shows that the periodic orbits can be chosen to be contained in an arbitrarily small neighborhood of K.
Corollary 4.5. Let K be a compact Φ t -invariant subset ofQ(S) and let U be an open neighborhood of K. Then every non-wandering point q ∈ K is an accumulation point of periodic points of Φ t whose orbits are entirely contained in U .
In the case of a topologically transitive compact invariant set K ⊂Q(S) we can say more.
Lemma 4.6. Let K be a compact Φ t -invariant topologically transitive subset of Q(S). Then for every σ > 0 there is a periodic orbit for Φ t whose Hausdorffdistance to K (as subsets ofQ(S)) is at most σ.
Proof. Let K ⊂Q(S) be a compact Φ t -invariant topologically transitive set and let σ > 0. Let n = n(K) > 0, ǫ 2 = ǫ 2 (K, σ/2) < σ/2 be as in Theorem 4.3. Since K is topologically transitive by assumption, there is some q ∈ K and there is some T > n such that d(q, Φ T q) < ǫ 2 and that moreover the Hausdorff distance between the set K and its subset B = {Φ t q | 0 ≤ t ≤ T } is at most σ/2. By Theorem 4.3, applied to the closed (n, ǫ 2 )-pseudo-orbit defined by the orbit segment {Φ t q | 0 ≤ t ≤ T }, there is a periodic orbit for Φ t whose Hausdorff distance to B is at most σ/2. This means that the Hausdorff distance between this orbit and the set K is at most σ and shows the lemma.
For a compact Φ t -invariant subset K ⊂Q(S) denote by h top (K) the topological entropy of the restriction of Φ t to K. For an arbitrary subset U ⊂Q(S) and a number r > 0 let n U (r) be the number of all periodic orbits of Φ t of period at most r which are contained in U . The following corollary is another fairly immediate consequence of Theorem 4.3. Together with Corollary 3.3 it shows Theorem 2 from the introduction. Proof. Let K ⊂Q(S) be a topologically transitive compact Φ t -invariant set and let U be an open neighborhood of K. Then there is a number β > 0 such that U contains the β-neighborhood of K.
Let δ < β be sufficiently small that the δ-neighborhood of every point in K is contained in a contractible subset ofQ(S). Let n = n(K) > 0, ǫ 2 = ǫ 2 (K, δ/8) < 1 be as in Theorem 4.3. Since the Teichmüller flow on K is topologically transitive by assumption, by compactness of K × K there is a number N > n with the following property. Let q, q ′ ∈ K; then there is some u ∈ K and some T ∈ [n, N ] with d(u, q ′ ) < ǫ 2 and d(Φ T u, q) < ǫ 2 .
A subset E of K is called (m, δ)-separated for some m ≥ 0 if for any two points q = u ∈ E we have Let m > n and let E m ⊂ K be any (m, δ)-separated set. Let q ∈ E m . By the choice of N > n there is some u ∈ K and some T ∈ [n, N ] such that d(u, Φ m q) < ǫ 2 and d(Φ T u, q) < ǫ 2 . By Theorem 4.3, the closed (n, ǫ 2 )-pseudo-orbit q, u, q is δ/8shadowed by a periodic orbit which defines the characteristic free homotopy class of the pseudo-orbit. Since periodic orbits for Φ t inQ(S) minimize the length in their free homotopy class, the length of the periodic orbit does not exceed m + N + 2ǫ 2 . Moreover, by the choice of δ this periodic orbit is contained in U . There is a point ζ(q) on the orbit with d(q, ζ(q)) ≤ δ/8. In other words, there is a map ζ which associates to every point q ∈ E m a point ζ(q) ∈ U whose orbit under Φ t is entirely contained in U and is periodic of period at most m + N + 2ǫ 2 .
Since the points in the set E m are (m, δ)-separated by assumption and the orbits of Φ t are geodesics parametrized by arc length, the orbit segments c(q) = ∪ t∈(−δ/8,δ/8) Φ t ζ(q) (q ∈ E m ) are pairwise disjoint. Thus for a fixed periodic orbit γ for Φ t of length at most m+N +2ǫ 2 there are at most 4(m+N +2)/δ distinct points q ∈ E m with ζ(q) ∈ γ. As a consequence, there are at least δ card(E m )/4(m+N +2) distinct periodic orbits of period at most m+N +2 in U . This shows that the asymptotic growth as m → ∞ of the maximal cardinality of an (m, δ)-separated subset of K does not exceed the asymptotic growth of the numbers n U (r) as r → ∞. The corollary is now an immediate consequence from the definition of the topological entropy of a continuous flow on a compact space (recall also from Theorem 3.1 that the Teichmüller flow on K is expansive and hence for all sufficiently small δ > 0 its topological entropy is just the asymptotic growth rate of maximal (m, δ)-separated sets as m → ∞).

Lower bounds for the number of periodic orbits
In this section we complete the proof of Theorem 3 from the introduction. For this we continue to use the assumptions and notations from Sections 2 and 3. In particular, we always denote by d T the Teichmüller metric on Teichmüller space T (S) for S.
We begin with establishing the first part of Theorem 3 which is immediate from the work of Eskin and Mirzakhani [EM08]. Since the proof is short and easy, we include it for completeness.
The Poincaré series with exponent α > 0 at a point x ∈ T (S) is defined to be the series The critical exponent of Mod(S) is the infimum of all numbers α > 0 such that the Poincaré series with exponent α converges. Note that this critical exponent does not depend on the choice of x. Athreya, Bufetov, Eskin and Mirzakhani [ABEM06] showed that the critical exponent of the Poincaré series equals h = 6g − 6 + 2m and that the Poincaré series diverges at the critical exponent.
For r > 0 and for a compact set K ⊂ Q 1 (S)/Mod(S) let n ∩ K (r) be the number of all periodic orbits for the Teichmüller flow of period at most r which intersect K. The next lemma is the first part of Theorem 1. Let D be the diameter of K 1 and let x ∈ K 1 be any point. Let g ∈ Mod(S) be a pseudo-Anosov element whose axis (i.e. the unique g-invariant Teichmüller geodesic on which g acts as a translation) projects to a closed geodesic γ in moduli space which intersectsK. Then there is a pointx ∈ K 1 which lies on the axis of a conjugate of g which we denote again by g for simplicity. By the properties of an axis, the length ℓ(γ) of the closed geodesic γ equals d T (x, gx). On the other hand, we have Since the critical exponent of the Poincaré series equals 6g − 6 + 2m, for every ǫ > 0 the Poincaré series converges at the exponent α = 6g − 6 + 2m + ǫ. Let c(α) > 0 be its value. Then for every r > 0, the cardinality of the set {g ∈ Mod(S) | d(x, gx) ≤ r} does not exceed c(α)e αr (note that the term in the Poincaré series corresponding to such an element of Mod(S) is not smaller than e −αr ). This shows that lim sup 1 r log N (r) ≤ 6g − 6 + 2m + ǫ. Since ǫ > 0 and the compact set K ⊂ M(S) were arbitrarily chosen, the lemma follows.
Proof. Let K ⊂ Q 1 (S)/Mod(S) be a compact Φ t -invariant topologically transitive set, let q ∈ K be a point whose orbit under Φ t is dense in K and letq ∈Q(S) be a preimage of q under the natural projection Θ :Q(S) → Q 1 (S)/Mod(S). Let K be the closure of the orbit ofq; thenK is a compact Φ t -invariant topologically transitive set. By equivariance of the Teichmüller flow under the projection Θ, this set is mapped by Θ onto K. Moreover, by Corollary 4.7, for every open relative compact neighborhood U ofK we have Now the projection Θ maps periodic orbits for Φ t in U of period at most r to periodic orbits for Φ t of period at most r which are contained in the relative compact set Θ(U ) ⊂ Q 1 (S)/Mod(S). If the periodic orbits γ 1 = γ 2 in U are mapped to the same periodic orbit in Θ(U ) then there is some element g from the factor group G = Mod(S)/Γ which maps γ 1 to γ 2 . Since G is finite, the number of distinct periodic orbits in U which are mapped to the single orbit in Θ(U ) is uniformly bounded. Therefore by Lemma 5.1 we have This shows the corollary.
Now we are ready for the proof of the second part of Theorem 2 from the introduction.
Proof. As in Section 2, let F ML ⊂ PML be the Mod(S)-invariant Borel subset of all projective measured geodesic laminations whose support is minimal and fills up S and let F : F ML → ∂C(S) be the continuous Mod(S)-equivariant surjection which associates to a projective measured geodesic lamination in F ML its support. Let π : Q 1 (S) → PML be the natural projection as defined in (13) and define Let λ be the Φ t -invariant probability measure on Q 1 (S)/Mod(S) in the Lebesgue measure class constructed in [M82, V86]. This measure is ergodic and mixing under the Teichmüller flow, with full support. In particular, the Φ t -orbit of λ-almost every point q ∈ Q 1 (S)/Mod(S) returns to every neighborhood of q for arbitrarily large times. The measure λ lifts to a Mod(S)-invariant Φ t -invariant Radon measureλ on Q 1 (S) of full support which gives full measure to the Mod(S)-invariant Borel set A [M82].
The Lebesgue measureλ on Q 1 (S) is absolutely continuous with respect to the strong unstable foliation. More precisely, for every q ∈ Q 1 (S) there is a natural conditional measureλ q forλ on the strong unstable manifold W su (q), and these conditional measures transform under the Teichmüller flow via dλ Φ t q • Φ t = e ht dλ q where h = 6g−6+2m as before. The image under the projection π of the measureλ q on W su (q) is a locally finite Borel measure λ q on the open dense subset of PML of all projective measured geodesic laminations which together with π(−q) jointly fill up S. The measures λ q are all absolutely continuous, and they depend continuously on q ∈ Q 1 (S) in the weak * -topology. Moreover, for each q the measure λ q gives full measure to the set F ML and hence it can be mapped via the surjection F to a measure on ∂C(S) which we denote again by λ q .
Recall from (11) and (12) the definition of the distances δ x (x ∈ T (S)) on ∂C(S) and their properties. For q ∈ A and χ > 0 define D(q, χ) ⊂ ∂C(S) to be the closed δ P q -ball of radius χ about F π(q) ∈ ∂C(S). Note that D(q, χ) contains the image under the map F of the ball B δP q (π(q), χ) used in Section 4. Note moreover that D(q, χ) depends on q and not only on its center F π(q) and the radius χ since for t = 0 the distances δ P q and δ P Φ t q do not coinicide, Let q 0 ∈ Q 1 (S)/Mod(S) be a typical point for the Lebesgue measure λ (so that the Birkhoff ergodic theorem holds true for q 0 ) and let q 1 be a lift of q 0 to Q 1 (S). Assume without loss of generality that P q 1 is not fixed by any element of Mod(S). This is possible since the set of points in T (S) which are stabilized by a non-trivial element of Mod(S) is closed and nowhere dense and since the Lebesgue measure is of full support.
Let m > 1 be as in the second part of Lemma 4.4. We may assume that the image under the map Υ T of every Teichmüller geodesic in T (S) is an unparametrized mquasi-geodesic in C(S). Since q 0 is a typical point for the Lebesgue measure, the unparametrized quasi-geodesic t → Υ T (P Φ t q 1 ) is of infinite diameter.
Let κ > 0 be as in inequality (12). By Lemma 2.1 and the inequality (12), there is a number α > 0 depending on m and there is a neighborhood V of q 1 in Q 1 (S) of diameter at most log 2/κ and a number T 0 > 0 such that Since q 0 is recurrent and hence the vertical measured geodesic lamination of q 1 is uniquely ergodic and fills up S, by the second part of Lemma 3.2 of [H09] there is a number χ ≤ α/4 such that Let ǫ 0 = ǫ 0 (q 0 ) > 0 be as in the second part of Lemma 4.4. We may assume that the ǫ 0 -neighborhood of q 0 is contained in a contractible subset of Q 1 (S)/Mod(S). By continuity, there is a compact neighborhood K ⊂ V of q 1 with the following properties.
Following [F69], a Borel covering relation for a Borel subset C of a topological space X is a family V of pairs (x, V ) where V ⊂ X is a Borel set, where x ∈ V and such that For χ > 0 and the neighborhood K ⊂ Q 1 (S) of q 1 as above define V q0,χ,K ={(F π(q), gD(q 1 , χ)) | (38) By Proposition 3.5 of [H09], via possibly decreasing the size of χ and K we may assume that the covering relation V q0,χ,K is a Vitali relation for the measure λ q1 on ∂C(S). In our context, this means that for every T > 0 there is a covering of λ q1 -almost all of D(q 1 , χ/4) by pairwise disjoint sets from the relation of the form where u ∈ W su (q 1 ) ∩ A, F π(u) ∈ D(q 1 , χ/4), g ∈ Mod(S) and where t ≥ T is such that Φ t u ∈ gK (we refer to Section 3 of [H09] for a detailed discussion).
Since the measures λ q and the distances δ P q on ∂C(S) depend continuously on u ∈ Q 1 (S), there is a number a ≤ λ q1 D(q 1 , χ/4) such that λ q D(u, χ) ∈ [a, a −1 ] for all q ∈ K, u ∈ K ∩ A. By the transition properties for the measures λ u and invariance under the action of the mapping class group, if g ∈ Mod(S), if u ∈ W su (q 1 ) and if t > 0 are such that Φ t u ∈ gK for some t > 0 then λ q1 (gD(q 1 , χ)) ∈ [ae −ht , e −ht /a] (compare the discussion in [H09]).
The total λ q1 -mass of the balls from the covering is at least λ q1 D(q 1 , χ/4) ≥ a. Therefore there is a number T > T (ǫ) + 2 such that the total volume of those balls V (g, t) from the covering which correspond to a parameter t ∈ [T − 2, T − 1] is at least e −ǫT e 2h /a. Now the λ q1 -volume of each such ball is at most e −h(T −2) /a and hence the number of these balls is at least e (h−ǫ)T .
Thus the sets g j g i D(q 1 , χ) (i, j = 1, . . . , k) are pairwise disjoint. Namely, the sets g j D(q 1 , χ) (j = 1, . . . , k) are pairwise disjoint, and for each j the sets g j g i D(q 1 , χ) ⊂ g j D(q 1 , χ) (i = 1, . . . , k) are pairwise disjoint as well. By induction, we conclude that for any two distinct words w 1 = g i1 · · · g i ℓ and w 2 = g j1 · · · g jm in the letters g 1 , . . . , g k , viewed as elements of Mod(S), the images of D(q 1 , χ) under w 1 , w 2 are either disjoint or properly contained in each other. This shows that the elements g 1 , . . . , g k generate a free semi-subgroup Λ of Mod(S).
Since T (ǫ) ≥ n 0 , each word w of length ℓ ≥ 1 in the letters g 1 , . . . , g k defines a closed (n 0 , ǫ 0 )-pseudo-orbit u 0 , . . . , u ℓ in Q 1 (S)/Mod(S) with d(u i , q 0 ) < ǫ 0 . This pseudo-orbit consists of the successive projections to Q 1 (S)/Mod(S) of flow lines {Φ t u | t ∈ [0, τ ]} where u ∈ V ∩ W su (q 1 ) ∩ A and τ ∈ [T − 2, T − 1] are such that F π(u) ∈ g j D(q 1 , χ/4) for some j ≤ k and Φ τ u ∈ g j K. Thus by the second part of Lemma 4.4, ifγ is a lift to Q 1 (S) of a characteristic arc of such a pseudo-orbit then Υ T (γ) is a biinfinite unparametrized m-quasi-geodesic in C(S) which is invariant under the element of Λ ⊂ Mod(S) defined by w. In particular, this element is pseudo-Anosov, and its conjugacy class defines the characteristic free homotopy class of the closed pseudo-orbit.
The length of the periodic orbit of Φ t determined by w does not exceed the length of a characteristic closed curve for the pseudo-orbit and hence its is not bigger than T ℓ. Moreover, since by the choice of n 0 for any s < t with the property thatγ(s),γ(t) project to distinct breakpoints of γ the distance between Υ T (γ(s)), Υ T (γ(t)) is at least 2c(m), it follows from Lemma 2.4 of [H10] that the unparametrized m-quasi-geodesic Υ T (γ) is in fact a parametrized p-quasi-geodesic for some p > m. Using once more the first part of Theorem 2.2, this implies that the axis of the element of Λ ⊂ Mod(S) defined by w passes through a fixed compact neighborhood B of P q 1 in T (S), and the projection of its unit tangent line to Q 1 (S)/Mod(S) is a periodic orbit for Φ t which is contained in a compact subset C 0 of Q 1 (S)/Mod(S) not depending on w. If we denote by C the closed subset of C 0 of all points z ∈ C 0 whose orbit under Φ t is entirely contained in C 0 then each of these orbits is contained in C.
The above argument does not immediately imply that the asymptotic growth rate of the number of periodic orbits in C is at least h − ǫ. Namely, periodic orbits of the Teichmüller flow on Q 1 (S)/Mod(S) correspond to conjugacy classes of pseudo-Anosov elements in Mod(S). Thus if we want to count periodic orbits for Φ t in Q 1 (S)/Mod(S) using the semi-subgroup Λ of Mod(S) constructed above, then we have to identify those elements of Λ which are conjugate in Mod(S).
For this recall that the axis of each element of the semi-subgroup Λ of Mod(S) passes through the fixed compact neighborhood B of P q 1 . Thus if γ, ζ is the axis of v, w ∈ Λ and if v, w are conjugate in Mod(S) then there is some b ∈ Mod(S) with w = b −1 vb and the following additional property. Let γ[0, τ ] be a fundamental domain for the action of v on γ and such that γ(0) ∈ B. Such a fundamental domain always exists, perhaps after a reparametrization of γ. Then there is some t ∈ [0, τ ] such that b −1 γ(t) ∈ B.
As a consequence, the number of all elements w ∈ Λ which are conjugate to a fixed element v ∈ Λ is bounded from above by the number of elements b ∈ Mod(S) with bB ∩ γ[0, τ ] = ∅. In particular, if D is the diameter of B then this number does not exceed the cardinality of the set (40) {b ∈ Mod(S) | d T (bP q 1 , γ[0, τ ]) ≤ D}.
However, this cardinality is bounded from above by a universal multiple of τ . Therefore there is a constant c > 0 such that for all sufficiently large r > 0 the number of periodic orbits of Φ t contained in C of length at most r is not smaller than e (h−ǫ)r /cr. This completes the proof of the proposition. Remarks: 1. Proposition 5.3 is equally valid, with identical proof, for the Teichmüller flow Φ t onQ(S). Together with Theorem 2 it implies that the metric entropy h of the unique Φ t -invariant Lebesgue measure onQ(S) in the Lebesgue measure class equals the supremum of the topological entropies of the restriction of Φ t to compact invariant subsets ofQ(S). In [BG07], this fact was established for the Teichmüller flow on the moduli space of abelian differentials using symbolic dynamics.
2. The abundance of orbits of the Teichmüller flow which entirely remain in some compact set (depending on the orbit) was earlier established by Kleinbock and Weiss [KW04]. They show that this set is of full Hausdorff dimension.