Symbolic dynamics for the Teichmüller flow
Let be an oriented surface of genus with punctures and . The Teichmüller flow acts on the moduli space (or ) of area one holomorphic quadratic (or abelian) differentials preserving a collection of so-called strata. For each component of a stratum, we construct a subshift of finite type and Borel suspension which admits a finite-to-one semi-conjugacy into the Teichmüller flow on . This is used to show that the -invariant Lebesgue measure on is the unique measure of maximal entropy. If is the entropy of then for every there is a compact set such that the entropy of any -invariant probability measure with does not exceed . Moreover, the growth rate of periodic orbits in does not exceed . This implies that the number of periodic orbits for in of period at most is asymptotic to . Finally we give a unified and simplified proof of exponential mixing for the Lebesgue measure on strata.
Research partially supported by the Hausdorff Center Bonn
An oriented surface of finite type is a closed surface of genus from which points, so-called punctures, have been deleted. We assume that , i.e. that is not a sphere with at most punctures or a torus with at most puncture. We then call the surface nonexceptional. The Euler characteristic of is negative.
The Teichmüller space of is the quotient of the space of all complete finite area hyperbolic metrics on under the action of the group of diffeomorphisms of which are isotopic to the identity. The sphere bundle
of all holomorphic quadratic differentials of area one can naturally be identified with the unit cotangent bundle for the Teichmüller metric. If the surface has punctures, i.e. if , then we define a holomorphic quadratic differential on to be a meromorphic quadratic differential on the closed Riemann surface obtained from by filling in the punctures, with a simple pole at each of the punctures and no other poles.
The mapping class group of all isotopy classes of orientation preserving self-diffeomorphisms of naturally acts on . The quotient
is the moduli space of area one quadratic differentials. It can be partitioned into so-called strata. Namely, let be a sequence of positive integers with
The stratum defined by the -tuple is the moduli space of pairs where is a closed Riemann surface of genus and where is an area one meromorphic quadratic differential on with zeros of order and simple poles and which is not the square of a holomorphic one-form.
A stratum is a real hypersurface in a complex orbifold of complex dimension
Masur and Smillie [MS93] showed that the stratum is non-empty unless (and is a closed surface of genus 2). The strata need not be connected, however they have at most two connected components [L08]. The closure in of a component of a stratum is a union of components of strata where (note that we always fix the number of simple poles).
If the surface is closed, i.e. if , then we can also consider the bundle
of area one abelian differentials. It descends to the moduli space of holomorphic one-forms defining a singular euclidean metric of area one. Again this moduli space decomposes into a union of strata corresponding to the orders of the zeros of the differentials. Strata are in general not connected, but the number of their connected components is at most 3 [KZ03]. The stratum is a real hypersurface in a complex orbifold of dimension
The Teichmüller flow acts on (or ) preserving the strata. If is a component of a stratum of abelian differentials then Rauzy induction for interval exchange transformations can be used to construct a symbolic coding for the Teichmüller flow on . Such a coding consists in a suspension of a subshift of finite type by a positive roof function and a semi-conjugacy of an invariant Borel subset of the suspension flow into ([V82], see also [AGY06] for a discussion and references). Rauzy induction has been extended to strata of quadratic differentials by Boissy and Lanneau [BL09].
Our first goal is to construct a new coding for the Teichmüller flow on any component of a stratum.
Let be a component of a stratum of quadratic or abelian differentials. Then there is a subshift of finite type , a - invariant Borel set , a suspension of over given by a positive bounded continuous roof function on and a finite-to-one semi-conjugacy which induces a continuous bijection of the space of -invariant Borel probability measures on onto the space of -invariant Borel probability measures on .
Our construction is valid for strata of abelian differentials, but it is different from Rauzy induction. A dictionary between these two codings has yet to be established.
Let again be a component of a stratum in or . We use Theorem 1 to investigate the space of all -invariant Borel probability measures on equipped with the weak-topology. A specific example of such a measure in the Lebesgue measure class was constructed by Masur and Veech [M82, V86]. This measure is ergodic [M82, V86], and its entropy coincides with the complex dimension (or ) of the complex orbifold defining the stratum (note that we use a normalization for the Teichmüller flow which is different from the one used by Masur and Veech). In particular, the entropy of the Lebesgue measure on the open connected stratum equals .
Denote by the entropy of a measure . Define
A measure of maximal entropy for the component is a measure such that . A priori, such a measure need not exist. However, using Rauzy induction, Bufetov and Gurevich [BG07] showed that for components of strata of abelian differentials, the -invariant probability measure in the Lebesgue measure class is the unique measure of maximal entropy for the component. We use Theorem 1 and the work of Buzzi and Sarig [BS03] to extend this result to all components of strata of quadratic or abelian differentials, with a simpler proof.
For every component of a stratum in or , the -invariant Borel probability measure in the Lebesgue measure class is the unique measure of maximal entropy.
Components of strata are non-compact. Our next result shows that there is an entropy gap for measures supported in the complement of large compact subsets of .
For every there is a compact subset such that the entropy of every -invariant Borel probability measure which gives full mass to does not exceed .
We use related ideas to count periodic orbits of the Teichmüller flow in the component .
For every there exists a compact set and a number with the following property. The number of periodic orbits of contained in of period at most does not exceed .
A saddle connection of a quadratic differential is a geodesic segment for the singular euclidean metric defined by which connects two singular points and does not contain a singular point in its interior. A closed subset of a component of a stratum is compact if and only if there is a number so that the shortest length of a saddle connection of any point in is at least . Thus Theorem 4 is equivalent to the following
For every there exists a number and a number with the following property. The number of periodic orbits of contained in of period at most which consist of quadratic differentials with at least one saddle connection of length at most does not exceed .
For the open stratum in , Theorem 4 is due to Eskin and Mirzakhani [EM11]. The general case was independently shown Eskin, Mirzakhani and Rafi [EMR11]. They also obtain a more precise statement for the number of periodic orbits in the subset of of all points with short saddle connection.
Our method provides an upper bound for the number of periodic orbits outside a large compact subset of a stratum which can explicitly be calculated. The growth rate of the number of orbits which are entirely contained in the thin part of the stratum corresponding to a Riemann surface which contains at least one simple closed curve of small extremal length can be estimated in the same way. We leave the precise calculation to the forthcoming paper [H12] where we will show that all these upper bounds are sharp. For example, we find that for every component of a stratum, the growth rate of the number of periodic orbits with one saddle connection of length at most tends to as (with the only exception of some strata of small complexity). However, for every there are components of strata on a closed surface of genus for which the asymptotic growth rate of the number of periodic orbits in the thin part of moduli space equals .
As , the number of periodic orbits for of length at most which are contained in is asymptotic to .
The Lebesgue measure on components of strata is exponentially mixing for the Teichmüller flow.
The organization of the paper is as follows. In Section 2 we begin with establishing some properties of train tracks and geodesic laminations needed in the sequel. In Section 3 we associate to each component of a stratum a family of train tracks. This is used in Section 4 to construct for every connected component of a stratum a subshift of finite type . In Section 5 we define a bounded roof function on an invariant Borel subset of this subshift of finite type and show that there is a semi-conjugacy of the suspension of with roof function into the Teichmüller flow. This completes the proof of Theorem 1. In Section 6 we use the results of the earlier sections and the work of Buzzi and Sarig [BS03] to establish Theorem 2. The proof of Theorem 5 is contained in Section 7. This is then used in Section 8 to show Theorem 3. The proof of Theorem 4 is contained in Section 9.
Acknowledement: During the work on this paper, I benefitted from numerous discussions with more collegues than I can list here. Special thanks go to Artur Avila, Yves Benoist, Alex Eskin, Francois Ledrappier, Howard Masur, Curtis McMullen, Maryam Mirzakhani, Kasra Rafi and Omri Sarig. Part of the work was carried out during two visits of the MSRI in Berkeley (in fall 2007 and in fall 2011) and at the Hausdorff Institut in Bonn in spring 2010. I am very grateful to these two institutions for their hospitality.
2. Train tracks and geodesic laminations
In this section we summarize some constructions from [T79, PH92, H10c] which will be used throughout the paper. Furthermore, we introduce a class of train tracks which will be important in the later sections, and we discuss some of their properties.
2.1. Geodesic laminations
Let be an oriented surface of genus with punctures and where . A geodesic lamination for a complete hyperbolic structure on of finite volume is a compact subset of which is foliated into simple geodesics. A geodesic lamination is called minimal if each of its half-leaves is dense in . 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. Every geodesic lamination consists of a disjoint union of finitely many minimal components and a finite number of isolated leaves. Each of the isolated leaves of either is an isolated closed geodesic and hence a minimal component, or it spirals about one or two minimal components [CEG87].
A geodesic lamination on is said to fill up if its complementary regions are all topological discs or once punctured monogons. A maximal geodesic lamination is a geodesic lamination whose complementary regions are all ideal triangles or once punctured monogons.
A geodesic lamination is called large if fills up and if moreover can be approximated in the Hausdorff topology by simple closed geodesics. A maximal large geodesic lamination is called complete.
Since every minimal geodesic lamination can be approximated in the Hausdorff topology by simple closed geodesics [CEG87], a minimal geodesic lamination which fills up is large. However, there are large geodesic laminations with finitely many leaves.
The topological type of a large geodesic lamination is a tuple
such that the complementary regions of which are topological discs are -gons. Let
be the space of all large geodesic laminations of type equipped with the restriction of the Hausdorff topology for compact subsets of .
A measured geodesic lamination is a geodesic lamination together with a translation invariant transverse measure. Such a measure assigns a positive weight to each compact arc in with endpoints in the complementary regions of which intersects nontrivially and transversely. The geodesic lamination is called the support of the measured geodesic lamination; it consists of a disjoint union of minimal components. The space of all measured geodesic laminations on equipped with the weak-topology is homeomorphic to . Its projectivization is the space of all projective measured geodesic laminations.
The measured geodesic lamination fills up if its support fills up . This support is then necessarily connected and hence minimal, and for some tuple , it defines a point in the set . The projectivization of a measured geodesic lamination which fills up is also said to fill up . We call strongly uniquely ergodic if the support of fills up and admits a unique transverse measure up to scale.
There is a continuous symmetric pairing , the so-called intersection form, which extends the geometric intersection number between simple closed curves.
2.2. Train tracks
A train track on is an embedded 1-complex whose edges (called branches) are smooth arcs with well-defined tangent vectors at the endpoints. At any vertex (called a switch) the incident edges are mutually tangent. Through each switch there is a path of class which is embedded in and contains the switch in its interior. A simple closed curve component of contains a unique bivalent switch, and all other switches are at least trivalent. The complementary regions of the train track have negative Euler characteristic, which means that they are different from discs with or cusps at the boundary and different from annuli and once-punctured discs with no cusps at the boundary. We always identify train tracks which are isotopic. Throughout we use the book [PH92] as the main reference for train tracks.
A train track is called generic if all switches are at most trivalent. For each switch of a generic train track which is not contained in a simple closed curve component, there is a unique half-branch of which is incident on and which is large at . This means that every germ of an arc of class on which passes through also passes through the interior of . A half-branch which is not large is called small. A branch of is called large (or small) if each of its two half-branches is large (or small). A branch which is neither large nor small is called mixed.
Remark: As in [H09a], all train tracks are assumed to be generic. Unfortunately this leads to a small inconsistency of our terminology with the terminology found in the literature.
A trainpath on a train track is a -immersion such that for every the restriction of to is a homeomorphism onto a branch of . More generally, we call a -immersion a generalized trainpath. A trainpath is closed if and if the extension defined by and is a trainpath.
A generic train track is orientable if there is a consistent orientation of the branches of such that at any switch of , the orientation of the large half-branch incident on extends to the orientation of the two small half-branches incident on . If is a complementary polygon of an oriented train track then the number of sides of is even. In particular, a train track which contains a once punctured monogon component, i.e. a once punctured disc with one cusp at the boundary, is not orientable (see p.31 of [PH92] for a more detailed discussion).
A train track or a geodesic lamination is carried by a train track if there is a map of class which is homotopic to the identity and maps into in such a way that the restriction of the differential of to the tangent space of vanishes nowhere; note that this makes sense since a train track has a tangent line everywhere. We call the restriction of to a carrying map for . Write if the train track is carried by the train track . Then every geodesic lamination which is carried by is also carried by .
A train track fills up if its complementary components are topological discs or once punctured monogons. Note that such a train track is connected. Let be the number of those complementary components of which are topological discs. Each of these discs is an -gon for some . The topological type of is defined to be the ordered tuple where ; then . If is orientable then and is even for all . A train track of topological type is called maximal. The complementary components of a maximal train track are all trigons, i.e. topological discs with three cusps at the boundary, or once punctured monogons.
A transverse measure on a generic train track is a nonnegative weight function on the branches of satisfying the switch condition: for every trivalent switch of , the sum of the weights of the two small half-branches incident on equals the weight of the large half-branch. The space of all transverse measures on has the structure of a cone in a finite dimensional real vector space, and it is naturally homeomorphic to the space of all measured geodesic laminations whose support is carried by . The train track is called recurrent if it admits a transverse measure which is positive on every branch. We call such a transverse measure positive, and we write (see [PH92] for more details).
A subtrack of a train track is a subset of which is itself a train track. Then is obtained from by removing some of the branches, and we write . If is a small branch of which is incident on two distinct switches of then the graph obtained from by removing is a subtrack of . We then call a simple extension of . Note that formally to obtain the subtrack from we may have to delete the switches on which the branch is incident.
A simple extension of a recurrent non-orientable connected train track is recurrent. Moreover,
An orientable simple extension of a recurrent orientable connected train track is recurrent. Moreover,
If is a simple extension of a train track then can be obtained from by the removal of a small branch which is incident on two distinct switches . Then is an interior point of a branch of .
If is connected, non-orientable and recurrent then there is a trainpath which begins at , ends at and such that the half-branch is small at and that the half-branch is small at . Extend to a closed trainpath on which begins and ends at . This is possible since is non-orientable, connected and recurrent. There is a closed trainpath which can be obtained from by replacing the trainpath by the branch traveled through from to . The counting measure of on satisfies the switch condition and hence it defines a transverse measure on which is positive on . On the other hand, every transverse measure on defines a transverse measure on . Thus since is recurrent and since the sum of two transverse measures on is again a transverse measure, the train track is recurrent as well. Moreover, we have .
Let be the number of branches of . Label the branches of with the numbers so that the number is assigned to . Let be the standard basis of and define a linear map by for and where is the weight function on defined by the trainpath . The map is a surjection onto a linear subspace of of codimension one, moreover preserves the linear subspace of defined by the switch conditions for . In particular, the corank of in is at most one. However, is contained in the space of solutions of the switch conditions on and consequently the dimension of the space of transverse measures on is not smaller than the dimension of the space of transverse measures on minus one.
Together with the first paragraph of this proof, we conclude that . This completes the proof of the first part of the lemma. The second part follows in exactly the same way. ∎
As a consequence we obtain
for every non-orientable recurrent train track of topological type .
for every orientable recurrent train track of topological type .
The disc components of a non-orientable recurrent train track of topological type can be subdivided in steps into trigons by successively adding small branches. A repeated application of Lemma 2.2 shows that the resulting train track is maximal and recurrent. Since for every maximal recurrent train track we have (see [PH92]), the first part of the corollary follows.
To show the second part, let be an orientable recurrent train track of type . Then is even for all . Add a branch to which cuts some complementary component of into a trigon and a second polygon with an odd number of sides. The resulting train track is not recurrent since a trainpath on can only pass through at most once. However, we can add to another small branch which cuts some complementary component of with at least 4 sides into a trigon and a second polygon such that the resulting train track is non-orientable and recurrent. The inward pointing tangent of is chosen in such a way that there is a trainpath traveling both through and . The counting measure of any simple closed curve which is carried by gives equal weight to the branches and . But this just means that (see the proof of Lemma 2.2 for a detailed argument). By the first part of the corollary, we have which completes the proof. ∎
A train track of topological type is fully recurrent if carries a large minimal geodesic lamination .
Note that by definition, a fully recurrent train track is connected and fills up . The next lemma gives some first property of a fully recurrent train track . For its proof, recall that there is a natural homeomorphism of onto the subspace of of all measured geodesic laminations carried by .
A fully recurrent train track of topological type is recurrent.
A fully recurrent train track of type carries a minimal large geodesic lamination . The carrying map induces a bijection between the complementary components of and the complementary components of . In particular, a carrying map is surjective. Since a minimal geodesic lamination supports a transverse measure, there is a positive transverse measure on . In other words, is recurrent. ∎
There are two simple ways to modify a fully recurrent train track to another fully recurrent train track. Namely, if is a mixed branch of then we can shift along to a new train track . This new train track carries and hence it is fully recurrent since it carries every geodesic lamination which is carried by [PH92, H09a].
Similarly, if is a large branch of then we can perform a right or left split of at as shown in Figure A below. A (right or left) split of a train track is carried by . If is of topological type , if is minimal and is carried by and if is a large branch of , then there is a unique choice of a right or left split of at such that the split track carries . In particular, is fully recurrent. Note however that there may be a split of at such that the split track is not fully recurrent any more (see Section 2 of [H09a] for details).
The following simple observation is used to identify fully recurrent train tracks.
Let be a large branch of a fully recurrent non-orientable train track . Then no component of the train track obtained from by splitting at and removing the diagonal of the split is orientable.
Let be a large branch of a fully recurrent orientable train track . Then the train track obtained from by splitting at and removing the diagonal of the split is connected.
Let be a fully recurrent non-orientable train track of topological type . Let be a large branch of and let be a switch on which the branch is incident. Let be the train track obtained from by splitting at and removing the diagonal branch of the split. Then the train tracks obtained from by a right and left split at , respectively, are simple extensions of .
Now assume that contains an orientable connected component (not necessarily distinct from ). Let be a diagonal of the split connecting to . If is a trainpath with and and if for some then equals the branch traveled through in opposite direction. Since is orientable, this is impossible. Therefore and hence once again, is not recurrent. On the other hand, since is fully recurrent, it can be split at to a fully recurrent and hence recurrent train track. This is a contradiction. The first part of the corollary is proven. The second part follows from the same argument since a split of an orientable train track is orientable. ∎
Example: 1) Figure B below shows a non-orientable recurrent train track of type on a closed surface of genus two. The train track obtained from by a split at the large branch and removal of the diagonal of the split track is orientable and hence is not fully recurrent. This corresponds to the fact established by Masur and Smillie [MS93] that every quadratic differential with a single zero and no pole on a surface of genus is the square of a holomorphic one-form (see Section 3 for more information).
2) To construct an orientable recurrent train track of type which is not fully recurrent let be a surface of genus and let be an orientable fully recurrent train track on with complementary components. Choose a complementary component of in , remove from a disc and glue two copies of along the boundary of to a surface of genus . The two copies of define a recurrent disconnected oriented train track on which has an annulus complementary component .
Choose a branch of in the boundary of . There is a corresponding branch in the second boundary component of . Glue a compact subarc of contained in the interior of to a compact subarc of contained in the interior of so that the images of the two arcs under the glueing form a large branch in the resulting train track . The train track is recurrent and orientable, and its complementary components are topological discs. However, by Lemma 2.6 it is not fully recurrent.
To each train track which fills up one can associate a dual bigon track (Section 3.4 of [PH92]). There is a bijection between the complementary components of and those complementary components of which are not bigons, i.e. discs with two cusps at the boundary. This bijection maps a component of which is an -gon for some to an -gon component of contained in , and it maps a once punctured monogon to a once punctured monogon contained in . If is orientable then the orientation of and an orientation of induce an orientation on , i.e. is orientable.
There is a notion of carrying for bigon tracks which is analogous to the notion of carrying for train tracks. Measured geodesic laminations which are carried by the bigon track can be described as follows. A tangential measure on a train track of type assigns to a branch of a weight such that for every complementary -gon of with consecutive sides and total mass (counted with multiplicities) the following holds true.
(The complementary once punctured monogons define no constraint on tangential measures). The space of all tangential measures on has the structure of a convex cone in a finite dimensional real vector space. By the results from Section 3.4 of [PH92], every tangential measure on determines a simplex of measured geodesic laminations which hit efficiently. The supports of these measured geodesic laminations are carried by the bigon track , and every measured geodesic lamination which is carried by can be obtained in this way. The dimension of this simplex equals the number of complementary components of with an even number of sides. The train track is called transversely recurrent if it admits a tangential measure which is positive on every branch.
In general, there are many tangential measures which correspond to a fixed measured geodesic lamination which hits efficiently. Namely, let be a switch of and let be the half-branches of incident on and such that the half-branch is large. If is a tangential measure on which determines the measured geodesic lamination then it may be possible to drag the switch across some of the leaves of and modify the tangential measure on to a tangential measure . Then is a multiple of a vector of the form where denotes the function on the branches of defined by and for .
A train track of topological type is called fully transversely recurrent if its dual bigon track carries a large minimal geodesic lamination . A train track of topological type is called large if is fully recurrent and fully transversely recurrent. A large train track of type is called complete.
For a large train track let be the set of all measured geodesic laminations whose support is carried by . Each of these measured geodesic laminations corresponds to a tangential measure on . With this identification, the pairing
is just the restriction of the intersection form on measured lamination space (Section 3.4 of [PH92]). Moreover, is naturally homeomorphic to a convex cone in a real vector space. The dimension of this cone coincides with the dimension of .
Denote by the set of all isotopy classes of large train tracks on of type .
Remark: In [MM99], Masur and Minsky define a large train track to be a train track whose complementary components are topological discs or once punctured monogons, without the requirement that is generic, transversely recurrent or recurrent. We hope that this inconsistency of terminology does not lead to any confusion.
The goal of this subsection is to relate components of strata in to large train tracks.
For a closed oriented surface of genus with punctures let be the bundle of marked area one holomorphic quadratic differentials with a simple pole at each puncture over the Teichmüller space of marked complex structures on . For a complete hyperbolic metric on of finite area, an area one quadratic differential is determined by a pair of measured geodesic laminations which jointly fill up (i.e. we have for every measured geodesic lamination ) and such that . The vertical measured geodesic lamination for corresponds to the equivalence class of the vertical measured foliation of . The horizontal measured geodesic lamination for corresponds to the equivalence class of the horizontal measured foliation of .
A tuple of positive integers with defines a stratum in . This stratum consists of all marked area one quadratic differentials with simple poles and zeros of order which are not squares of holomorphic one-forms. The stratum is a real hypersurface in a complex manifold of dimension
The closure in of a stratum is a union of components of strata. Strata are invariant under the action of the mapping class group of and hence they project to strata in the moduli space of quadratic differentials on with a simple pole at each puncture. We denote the projection of the stratum by . The strata in moduli space need not be connected, but their connected components have been identified by Lanneau [L08]. A stratum in has at most two connected components.
Similarly, if then we let be the bundle of marked area one holomorphic one-forms over Teichmüller space of . For a tuple of positive integers with , the stratum of marked area one holomorphic one-forms on with zeros of order is a real hypersurface in a complex manifold of dimension
It projects to a stratum in the moduli space of area one holomorphic one-forms on . Strata of holomorphic one-forms in moduli space need not be connected, but the number of connected components of a stratum is at most three [KZ03].
We continue to use the assumptions and notations from Section 2. For a large train track let
be the set of all measured geodesic laminations whose support is carried by and such that the total weight of the transverse measure on defined by equals one. Let
be the set of all marked area one quadratic differentials whose vertical measured geodesic lamination is contained in and whose horizontal measured geodesic lamination is carried by the dual bigon track of . By definition of a large train track, we have . The next proposition relates to components of strata. For the purpose of its proof and for later use, define the strong unstable manifold of a quadratic differential to consist of all quadratic differentials whose horizontal measured geodesic lamination coincides with the horizontal measured geodesic lamination of . The strong stable manifold is defined to be the image of under the flip . For a component of a stratum in and every , define the strong unstable (or strong stable) manifold (or ) to be the connected component containing of the intersection (or ). Then is a manifold of dimension (). The manifolds (or ) define a foliation of which is called the strong unstable (or the strong stable) foliation.
For every large non-orientable train track
there is a component of the stratum
such that for every the set is the closure in of an open subset of .
For every large orientable train track there is a component of the stratum such that for every the set is the closure in of an open subset of .
Let be a marked quadratic differential. A saddle connection for is a geodesic segment for the singular euclidean metric defined by which connects two singular points and does not contain a singular point in its interior. A separatrix is a maximal geodesic segment or ray which begins at a singular point and does not contain a singular point in its interior.
Let be the support of the vertical measured geodesic lamination of . By [L83], the geodesic lamination can be obtained from the vertical foliation of by cutting open along each vertical separatrix and straightening the remaining leaves with respect to a complete finite area hyperbolic metric on . In particular, up to homotopy, a vertical saddle connection of is contained in the interior of a complementary component of which is uniquely determined by .
Let . Consider first the case that is non-orientable. Let , with support contained in . Then is non-orientable since otherwise inherits an orientation from . If then the measured geodesic laminations jointly fill up (since the support of is different from the support of and fills up ) and hence if is normalized in such a way that then the pair defines a point . Our first goal is to show that .
Since , the orders of the zeros of the quadratic differential are obtained from the orders by subdivision. There is a non-trivial subdivision, say of the form , if and only if has at least one vertical saddle connection.
Choose a complete finite area hyperbolic metric on . This choice identifies the universal covering of with the hyperbolic plane , and it identifies the fundamental group of with a group of isometries of . Assume to the contrary that has a vertical saddle connection . Let be the lift of to a quadratic differential on and let be a preimage of .
The preimage of is a closed -invariant set of geodesic lines in . Since fills up , the complementary components of are finite area ideal polygons and half-planes which are the components of the preimages of the once punctured monogons. As discussed in the second paragraph of this proof, up to homotopy the saddle connection of is contained in a complementary component of . This component is an ideal polygon with finitely many sides, and it is determined by .
For the singular euclidean metric defined by , each cusp of is a cone point with cone angle . Let be the complement of a standard neighborhood of the cusps. A smooth geodesic arc for the singular euclidean metric on defined by (i.e. an arc which does not pass through singular points) with endpoints in can be modified near the cusps to a homotopic arc of roughly the same length which is entirely contained in . Moreover, can be chosen to be without transverse self-intersections. In particular, the hyperbolic geodesic with the same endpoints does not enter deeply into the cusps of , i.e. it is contained in a compact subset of .
On , the singular euclidean metric and the hyperbolic metric are uniformly quasi-isometric. Therefore a lift to of a biinfinite vertical or horizontal geodesic is a uniform quasi-geodesic for the hyperbolic metric. Such a quasi-geodesic has well defined endpoints in the ideal boundary of (see also [L83, PH92]).
Choose an orientation for the saddle connection . There are two oriented vertical geodesic lines for the metric defined by which contain the saddle connection as a subarc and which are contained in a bounded neighborhood of a side of . The geodesics are determined by the requirement that their orientation coincides with the given orientation of and that moreover at every singular point , the angle at to the left of (or to the right of ) for the orientation of the geodesic and the orientation of equals (see [L83] for details of this construction).
The ideal boundary of the closed half-plane of which is bounded by (or ) and which is disjoint from the interior of is a compact subarc (or ) of bounded by the endpoints of (or ). The arcs are disjoint (or, equivalently, the sides of are not adjacent). A horizontal geodesic line for which intersects the interior of the saddle connection is a quasi-geodesic in with one endpoint in the interior of the arc and the second endpoint in the interior of the arc . Since the horizontal length of is positive, this means that the support of the horizontal measured geodesic lamination of contains geodesics with one endpoint in the arc and the second endpoint in .
Since the topological types of the support of and of coincide, a carrying map is surjective and induces a bijection between the complementary components of and the complementary components of . In particular, the projections to of the geodesics determine two non-adjacent sides of a complementary component of which is the image of the projection of to .
On the other hand, by construction of the dual bigon track of (see [PH92]), if is any trainpath which intersects the complementary component of then every component of is a compact arc with endpoints on adjacent sides of . In particular, a lift to of such a trainpath is a quasi-geodesic in whose endpoints meet at most one of the two arcs . Now the support of the horizontal measured geodesic lamination of is carried by . Therefore every leaf of the support of determines a biinfinite trainpath on and hence a lift to of such a leaf does not connect the arcs . However, we observed above that the support of the horizontal measured geodesic lamination of contains geodesics connecting to . This is a contradiction and shows that indeed .
Let be the open set of all projective measured geodesic laminations whose support is distinct from the support of . Then the assignment which associates to a projective measured geodesic lamination the area one quadratic differential with vertical measured geodesic lamination and horizontal projective measured geodesic lamination is a homeomorphism of onto a strong stable manifold in .
By Corollary 2.3, the projectivization of is homeomorphic to a closed ball in a real vector space of dimension , and this is just the dimension of a strong stable manifold in a component of . Therefore by the above discussion and invariance of domain, there is a component of the stratum such that the restriction of the map to is a homeomorphism of onto the closure of an open subset of a strong stable manifold .
The above argument also shows that if is defined by and if the support of is contained in then we have . If denotes the component of containing then for every point in the projectivization of , the pair defines a quadratic differential which is contained in the strong unstable manifold . The set of these quadratic differentials equals the closure of an open subset of .
The set of quadratic differentials with the property that the support of the vertical (or of the horizontal) measured geodesic lamination of is minimal and of type is dense and of full Lebesgue measure in [M82, V86]. Moreover, this set is saturated for the strong stable (or for the strong unstable) foliation. Thus by the above discussion, the set of all measured geodesic laminations which are carried by (or ) and whose support is minimal of type is dense in (or in ). As a consequence, the set of all pairs with which correspond to a quadratic differential is dense in the set of all pairs with . Thus the set is contained in the closure of a component of the stratum . Moreover, by reasons of dimension, contains an open subset of this component. This shows the first part of the proposition.
Now if is orientable and if is a geodesic lamination which is carried by , then inherits an orientation from an orientation of . The orientation of together with the orientation of determines an orientation of the dual bigon track (see [PH92]). This implies that any geodesic lamination carried by admits an orientation, and if jointly fill up and if is carried by , is carried by then the orientations of determine the orientation of . As a consequence, the quadratic differential of is the square of a holomorphic one-form. The proposition follows. ∎
The next proposition is a converse to Proposition 3.1 and shows that train tracks can be used to define coordinates on strata.
For every there is a large non-orientable train track and a number so that is an interior point of .
For every there is a large orientable train track and a number so that is an interior point of .
Fix a complete hyperbolic metric on of finite volume. Define the straightening of a train track to be the immersed graph in whose vertices are the switches of and whose edges are the geodesic arcs which are homotopic to the branches of with fixed endpoints.
The hyperbolic metric induces a distance function on the projectivized tangent bundle of . As in Section 3 of [H09a], we say that for some a train track -follows a geodesic lamination if the tangent lines of the straightening of are contained in the -neighborhood of the tangent lines of in the projectivized tangent bundle of and if moreover the straightening of any trainpath on is a piecewise geodesic whose exterior angles at the breakpoints are not bigger than . By Lemma 3.2 of [H09a], for every geodesic lamination and every there is a transversely recurrent train track which carries and -follows .
Let . Assume first that the support of the vertical measured geodesic lamination of is large of type . This is equivalent to stating that does not have vertical saddle connections. For let be a train track which carries and -follows . If is sufficiently small then a carrying map defines a bijection of the complementary components of onto the complementary components of . The transverse measure on defined by the vertical measured geodesic lamination of is positive.
Let be a complementary component of the preimage of in the hyperbolic plane . Then is an ideal polygon whose vertices decompose the ideal boundary into finitely many arcs ordered counter-clockwise in consecutive order. Since does not have vertical saddle connections, the discussion in the proof of Proposition 3.1 shows the following. Let be a leaf of the preimage in of the support of the horizontal measured geodesic lamination of . Then the two endpoints of in either are both contained in the interior of the same arc or in the interior of two adjacent arcs . As a consequence, for sufficiently small the geodesic lamination is carried by the dual bigon track of (see the characterization of the set of measured geodesic laminations carried by in [PH92]). Moreover, by the explicit construction of a measured geodesic lamination from a measured foliation [L83], for any two adjacent subarcs of cut out by , the transverse measure of the set of all leaves of the preimage of connecting these sides is positive. Therefore for sufficiently small , the horizontal measured geodesic lamination of defines an interior point of .
Now the set of quadratic differentials so that the support of the horizontal measured geodesic lamination of is large of type is dense in the strong stable manifold of . The above reasoning shows that for such a quadratic differential and for sufficiently small , the horizontal measured geodesic lamination of is carried by . But this just means that . Moreover, if is the total weight which the vertical measured geodesic lamination of puts on then is an interior point of . Thus satisfies the requirement in the proposition. Note that is necessarily non-orientable.
If is such that the support of the vertical measured geodesic lamination of is large of type then the above reasoning also applies and yields an oriented large train track with the required property.
Consider next the case that the support of the vertical measured geodesic lamination of fills up but is not of type . Then has a vertical saddle connection. The set of all vertical saddle connections of is a forest, i.e. a finite disjoint union of finite trees. The number of edges of this forest is uniformly bounded. For let be a train track which -follows and carries