Totally periodic pseudo-Anosov flows in graph manifolds

Classification and Rigidity of totally periodic pseudo-Anosov flows in graph manifolds

Thierry Barbot  and  Sérgio R. Fenley Thierry Barbot
Université d’Avignon et des pays de Vaucluse
LANLG, Faculté des Sciences
33 rue Louis Pasteur
84000 Avignon, France.
thierry.barbot@univ-avignon.fr Sérgio Fenley
Florida State University
Tallahassee
FL 32306-4510, USA   and   Princeton University
Princeton
NJ 08544-1000, USA
fenley@math.princeton.edu

Abstract In this article we analyze totally periodic pseudo-Anosov flows in graph three manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show that each such flow is topologically equivalent to one of the model pseudo-Anosov flows which we previously constructed in [Ba-Fe]. A model pseudo-Anosov flow is obtained by glueing standard neighborhoods of Birkhoff annuli and perhaps doing Dehn surgery on certain orbits. We also show that two model flows on the same graph manifold are isotopically equivalent (ie. there is a isotopy of mapping the oriented orbits of the first flow to the oriented orbits of the second flow) if and only if they have the same topological and dynamical data in the collection of standard neighborhoods of the Birkhoff annuli.

1. Introduction

Pseudo-Anosov flows are extremely common amongst three manifolds, for example111We also mention a recent work in progress by F. Béguin, C. Bonatti and Bin Yu, constructing a wide family of new Anosov flows; which can be seen as an extension of the construction in [Ba-Fe] ([BBB]).:  1) Suspension pseudo-Anosov flows [Th1, Th2, Th3], 2) Geodesic flows in the unit tangent bundle of negatively curved surfaces [An], 3) Certain flows transverse to foliations in closed atoroidal manifolds [Mo3, Cal1, Cal2, Cal3, Fe4];  flows obtained from these by either 4) Dehn surgery on a closed orbit of the pseudo-Anosov flow [Go, Fr],  or 5) Shearing along tori [Ha-Th];  6) Non transitive Anosov flows [Fr-Wi] and flows with transverse tori [Bo-La].

The purpose of this article is to analyse the question: how many essentially different pseudo-Anosov flows are there in a manifold? Two flows are essentially the same if they are topologically equivalent. This means that there is a homeomorphism between the manifolds which sends orbits of the first flow to orbits of the second flow preserving orientation along the orbits. In this article, we will also consider the notion of isotopic equivalence, i.e. a topological equivalence induced by an isotopy, that is, a homeomorphism isotopic to the identity.

We will restrict to closed, orientable, toroidal manifolds. In particular they are sufficiently large in the sense of [Wald3], that is, they have incompressible surfaces [He, Ja]. Manifolds with pseudo-Anosov flows are also irreducible [Fe-Mo]. It follows that these manifolds are Haken [Ja]. We have recently extended a result of the first author ([Ba2]) to the case of general pseudo-Anosov flows: if the ambient manifold is Seifert fibered, then the flow is up to finite cover topologically equivalent to a geodesic flow in the unit tangent bundle of a closed hyperbolic surface [Ba-Fe, Theorem A]. In addition we also proved that if the ambient manifold is a solvable three manifold, then the flow is topologically equivalent to a suspension Anosov flow [Ba-Fe, Theorem B]. Notice that in both cases the flow does not have singularities, that is, the type of the manifold strongly restricts the type of pseudo-Anosov that it can admit. This is in contrast with the strong flexibility in the construction of pseudo-Anosov flows that is because many flows are constructed in atoroidal manifolds or are obtained by flow Dehn surgery on the pseudo-Anosov flow, which changes the topological type of the manifold. Therefore in many constructions one cannot expect that the underlying manifold is toroidal.

In this article we will mainly study pseudo-Anosov flows in graph manifolds. A graph manifold is an irreducible three manifold which is a union of Seifert fibered pieces. In a previous article [Ba-Fe] we produced a large new class of examples in graph manifolds. These flows are totally periodic. This means that each Seifert piece of the torus decomposition of the graph manifold is periodic, that is, up to finite powers, a regular fiber is freely homotopic to a closed orbit of the flow. More recently, Russ Waller [Wa] has been studying how common these examples are, that is, the existence question for these type of flows. He showed that these flows are as common as they could be (modulo the necessary conditions).

In this article we will analyse the question of the classification and rigidity of such flows. In order to state and understand the results of this article we need to introduce the fundamental concept of a Birkhoff annulus. A Birkhoff annulus is an a priori only immersed annulus, so that the boundary is a union of closed orbits of the flow and the interior of the annulus is transverse to the flow. For example consider the geodesic flow of a closed, orientable hyperbolic surface. The ambient manifold is the unit tangent bundle of the surface. Let be an oriented closed geodesic - a closed orbit of the flow - and consider a homotopy that turns the angle along by . The image of the homotopy from to the same geodesic with opposite orientation is a Birkhoff annulus for the flow in the unit tangent bundle. If is not embedded then the Birkhoff annulus is not embedded. In general Birkhoff annuli are not embedded, particularly in the boundary. A Birkhoff annulus is transverse to the flow in its interior, so it has induced stable and unstable foliations. The Birkhoff annulus is elementary if these foliations in the interior have no closed leaves.

In [Ba-Fe, Theorem F] we proved the following basic result about the relationship of a pseudo-Anosov flow and a periodic Seifert piece : there is a spine for which is a connected union of finitely many elementary Birkhoff annuli. In addition the union of the interiors of the Birkhoff annuli is embedded and also disjoint from the closed orbits in . These closed orbits, boundaries of the Birkhoff annuli in , are called vertical periodic orbits. The set is a deformation retract of , so is isotopic to a small compact neighborhood of . In general the Birkhoff annuli are not embedded in : it can be that the two boundary components of the same Birkhoff annulus are the same vertical periodic orbit of the flow. It can also occur that the annulus wraps a few times around one of its boundary orbits. These are not exotic occurrences, but rather fairly common. For every vertical periodic orbit in , the local stable leaf of is a finite union of annuli, called stable vertical annuli, tangent to the flow, each realizing a homotopy between (a power of) and a closed loop in . One defines similarly unstable vertical annuli in .

We first analyse periodic Seifert pieces. The first theorems (Theorem A and B) are valid for any closed orientable manifold , not necessarily a graph manifold. The first result is (see Proposition 3.2):

Theorem A Let be a pseudo-Anosov flow in . If is the (possibly empty) collection of periodic Seifert pieces of the torus decomposition of , then the spines and neighborhoods can be chosen to be pairwise disjoint.

We prove that the can be chosen pairwise disjoint. Roughly this goes as follows: we show that the vertical periodic orbits in cannot intersect for , because fibers in different Seifert pieces cannot have common powers which are freely homotopic. We also show that the possible interior intersections are null homotopic and can be isotoped away.

The next result (Proposition 3.4) shows that the boundary of the pieces can be put in good position with respect to the flow:

Theorem B Let be a pseudo-Anosov flow and be periodic Seifert pieces with a common boundary torus . Then can be isotoped to a torus transverse to the flow.

The main property used to prove this result is that regular fibers restricted to both sides of (from and ) cannot represent the same isotopy class in .

Finally we prove the following (Proposition 3.5): Theorem C Let be a totally periodic pseudo-Anosov flow with periodic Seifert pieces . Then neighborhoods of the spines can be chosen so that their union is and they have pairwise disjoint interiors. In addition each boundary component of every is transverse to the flow. Each is flow isotopic to an arbitrarily small neighborhood of .

We stress that for general periodic pieces it is not true that the boundary of can be isotoped to be transverse to the flow. There are some simple examples as constructed in [Ba-Fe]. The point here is that we assume that all pieces of the JSJ decomposition are periodic Seifert pieces.

Hence, according to Theorem C, totally periodic pseudo-Anosov flow are obtained by glueing along the bondary a collection of small neighborhoods of the spines. There are several ways to perform this glueing which lead to pseudo-Anosov flows. The main result of this paper is that the resulting pseudo-Anosovs flow are all topologically equivalent one to the other. More precisely (see section 5.1):

Theorem D Let , be two totally periodic pseudo-Anosov flows on the same orientable graph manifold . Let be the Seifert pieces of , and let , be spines of , in . Then, and are topologically equivalent if and only if there is a homeomorphism of mapping the collection of spines onto the collection and preserving the orientations of the vertical periodic orbits induced by the flows.

Theorem D is a consequence of the following Theorem, more technical but slightly more precise (see section 5.2):

Theorem D’ Let , be two totally periodic pseudo-Anosov flows on the same orientable graph manifold . Let be the Seifert pieces of , and let , be spines of , in , with tubular neighborhoods , as in the statement of Theorem C. Then, and are isotopically equivalent if and only if, after reindexing the collection , there is an isotopy in mapping every spine onto , mapping every stable/unstable vertical annulus of in to a stable/vertical annulus in and preserving the orientations of the vertical periodic orbits induced by the flows.

The main ideas of the proof are as follows. It is easy to show that, if the two flows are isotopical equivalent, the isotopy maps every onto a spine of , every onto a neighborhood , so that vertical stable/unstable annuli and the orientation of vertical periodic orbits are preserved.

Conversely, assume that up to isotopy and admit the same decomposition in neighborhoods of spines , so that they share exactly the same oriented vertical periodic orbits and the same stable/unstable vertical annuli. Consider all the lifts to the universal cover of the tori in for all . This is a collection of properly embedded topological planes in , which is transverse to the lifted flows and . We show that an orbit of or (if not the lift of a vertical periodic orbits) is completely determined by its itinerary up to shifts: the itinerary is the collection of planes it intersects. One thus gets a map between orbits of and orbits of . This extends to the lifts of the vertical periodic orbits. This is obviously group equivariant. The much harder step is to prove that this is continuous, which we do using the exact structure of the flows and the combinatorics. Using this result we can then show that the flow is topologically equivalent to . Since the action on the fundamental group level is trivial, this topological equivalence is homotopic to the identity, hence, by a Theorem by Waldhausen ([Wald3]), isotopic to the identity: it is an isotopic equivalence.

We then show (section 6.2) that for any totally periodic pseudo-Anosov flow there is a model pseudo-Anosov flow as constructed in [Ba-Fe] which has precisely the same data , that has. This proves the following:

Main theorem Let be a totally periodic pseudo-Anosov flow in a graph manifold . Then is topologically equivalent to a model pseudo-Anosov flow.

Model pseudo-Anosov flows are defined by some combinatorial data (essentially, the data of some fat graphs and Dehn surgery coefficients; see section 6.1 for more details) and some parameter . A nice corollary of Theorem D’ is that, up to isotopic equivalence, the model flows actually do not depend on the choice of , nor on the choice of the selection of the particular glueing map between the model periodic pieces.

In the last section, we make a few remarks on the action of the mapping class group of on the space of isotopic equivalence classes of totally periodic pseudo-Anosov flows on .

2. Background

Pseudo-Anosov flows definitions

Definition 2.1.

(pseudo-Anosov flow) Let be a flow on a closed 3-manifold . We say that is a pseudo-Anosov flow if the following conditions are satisfied:

- For each , the flow line is , it is not a single point, and the tangent vector bundle is in .

- There are two (possibly) singular transverse foliations which are two dimensional, with leaves saturated by the flow and so that intersect exactly along the flow lines of .

- There is a finite number (possibly zero) of periodic orbits , called singular orbits. A stable/unstable leaf containing a singularity is homeomorphic to where is a -prong in the plane and is a homeomorphism from to . In addition is at least .

- In a stable leaf all orbits are forward asymptotic, in an unstable leaf all orbits are backwards asymptotic.

Basic references for pseudo-Anosov flows are [Mo1, Mo2] and [An] for Anosov flows. A fundamental remark is that the ambient manifold supporting a pseudo-Anosov flow is necessarily irreducible - the universal covering is homeomorphic to ([Fe-Mo]). We stress that in our definition one prongs are not allowed. There are however “tranversely hyperbolic” flows with one prongs:

Definition 2.2.

(one prong pseudo-Anosov flows) A flow is a one prong pseudo-Anosov flow in if it satisfies all the conditions of the definition of pseudo-Anosov flows except that the -prong singularities can also be -prong ().

Torus decomposition

Let be an irreducible closed –manifold. If is orientable, it has a unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of obtained by cutting along the tori is either atoroidal or Seifert-fibered [Ja, Ja-Sh] and the pieces are isotopically maximal with this property. If is not orientable, a similar conclusion holds; the decomposition has to be performed along tori, but also along some incompressible embedded Klein bottles.

Hence the notion of maximal Seifert pieces in is well-defined up to isotopy. If admits a pseudo-Anosov flow, we say that a Seifert piece is periodic if there is a Seifert fibration on for which, up to finite powers, a regular fiber is freely homotopic to a periodic orbit of . If not, the piece is called free.

Remark. In a few circumstances, the Seifert fibration is not unique: it happens for example when is homeomorphic to a twisted line bundle over the Klein bottle or is . We stress out that our convention is to say that the Seifert piece is free if no Seifert fibration in has fibers homotopic to a periodic orbit.

Orbit space and leaf spaces of pseudo-Anosov flows

Notation/definition:  We denote by the universal covering of , and by the fundamental group of , considered as the group of deck transformations on . The singular foliations lifted to are denoted by . If let denote the leaf of containing . Similarly one defines and in the universal cover . Similarly if is an orbit of define , etc… Let also be the lifted flow to .

We review the results about the topology of that we will need. We refer to [Fe2, Fe3] for detailed definitions, explanations and proofs. The orbit space of in is homeomorphic to the plane [Fe-Mo] and is denoted by . There is an induced action of on . Let

be the projection map: it is naturally -equivariant. If is a leaf of or , then is a tree which is either homeomorphic to if is regular, or is a union of -rays all with the same starting point if has a singular -prong orbit. The foliations induce -invariant singular -dimensional foliations in . Its leaves are as above. If is a leaf of or , then a sector is a component of . Similarly for . If is any subset of , we denote by the set . The same notation will be used for any subset of : it will just be the union of all flow lines through points of . We stress that for pseudo-Anosov flows there are at least -prongs in any singular orbit (). For example, the fact that the orbit space in is a -manifold is not true in general if one allows -prongs.

Definition 2.3.

Let be a leaf of or . A slice of is where is a properly embedded copy of the reals in . For instance if is regular then is its only slice. If a slice is the boundary of a sector of then it is called a line leaf of . If is a ray in then is called a half leaf of . If is an open segment in it defines a flow band of by . We use the same terminology of slices and line leaves for the foliations of .

If and then and intersect in at most one orbit.

We abuse convention and call a leaf of or periodic if there is a non trivial covering translation of with . This is equivalent to containing a periodic orbit of . In the same way an orbit of is periodic if is a periodic orbit of . Observe that in general, the stabilizer of an element of is either trivial, or a cyclic subgroup of .


Perfect fits, lozenges and scalloped chains

Recall that a foliation in is -covered if the leaf space of in is homeomorphic to the real line [Fe1].

Definition 2.4.

([Fe2, Fe3]) Perfect fits - Two leaves and , form a perfect fit if and there are half leaves of and of and also flow bands and , so that the set

separates and forms an a rectangle with a corner removed: The joint structure of in is that of a rectangle with a corner orbit removed. The removed corner corresponds to the perfect of and which do not intersect.

We refer to fig. 1, a for perfect fits. There is a product structure in the interior of : there are two stable boundary sides and two unstable boundary sides in . An unstable leaf intersects one stable boundary side (not in the corner) if and only if it intersects the other stable boundary side (not in the corner). We also say that the leaves are asymptotic.

Figure 1. a. Perfect fits in , b. A lozenge, c. A chain of lozenges.
Definition 2.5.

([Fe2, Fe3]) Lozenges - A lozenge is a region of whose closure is homeomorphic to a rectangle with two corners removed. More specifically two points define the corners of a lozenge if there are half leaves of defined by and half leaves of defined by , so that and form a perfect fit and so do and . The region bounded by the lozenge does not have any singularities. The sides of are . The sides are not contained in the lozenge, but are in the boundary of the lozenge. There may be singularities in the boundary of the lozenge. See fig. 1, b.

There are no singularities in the lozenges, which implies that is an open region in .

Two lozenges are adjacent if they share a corner and there is a stable or unstable leaf intersecting both of them, see fig. 1, c. Therefore they share a side. A chain of lozenges is a collection , where is an interval (finite or not) in ; so that if , then and share a corner, see fig. 1, c. Consecutive lozenges may be adjacent or not. The chain is finite if is finite.

Definition 2.6.

(scalloped chain) Let be a chain of lozenges. If any two successive lozenges in the chain are adjacent along one of their unstable sides (respectively stable sides), then the chain is called s-scalloped (respectively u-scalloped) (see fig. 2 for an example of a -scalloped chain). Observe that a chain is s-scalloped if and only if there is a stable leaf intersecting all the lozenges in the chain. Similarly, a chain is u-scalloped if and only if there is an unstable leaf intersecting all the lozenges in the chain. The chains may be infinite. A scalloped chain is a chain that is either -scalloped or -scalloped.

Figure 2. A partial view of a scalloped region. Here are stable leaves, so this is a s-scalloped region.

For simplicity when considering scalloped chains we also include any half leaf which is a boundary side of two of the lozenges in the chain. The union of these is called a scalloped region which is then a connected set.

We say that two orbits of (or the leaves ) are connected by a chain of lozenges , if is a corner of and is a corner of .

Remark 2.7.

A key fact, first observed in [Ba3], and extensively used in [Ba-Fe], is the following: the lifts in of elementary Birkhoff annuli are precisely lozenges invariant by some cyclic subgroup of (see [Ba3, Proposition ] for the case of embedded Birkhoff annuli). It will also play a crucial role in the sequel. More precisely: let be an elementary Birkhoff annulus. We say that lifts to the lozenge in if the interior of has a lift which intersects orbits only in . It follows that this lift intersects every orbit in exactly once and also that the two boundary closed orbits of lift to the full corner orbits of . This uses the fact that a lozenge cannot be properly contained in another lozenge.

In particular the following important property also follows: if and are the periodic orbits in (traversed in the flow forward direction), then there are positive integers so that is freely homotopic to . We emphasize the free homotopy between inverses.

Remark 2.8.

According to remark 2.7, chains of lozenges correspond to sequences of Birkhoff annuli, every Birkhoff annulus sharing a common periodic orbit with the previous element of the sequence, and also a periodic orbit with the next element in the sequence. When the sequence closes up, it provides an immersion (or ) , which is called a Birkhoff torus (if the cyclic sequence contains an even number of Birkhoff annuli), or a Birkhoff Klein bottle (in the other case).

3. Disjoint pieces and transverse tori

In [Ba-Fe] section 7, we proved that if is a periodic Seifert fibered piece of with a pseudo-Anosov flow then: there is a connected, finite union of elementary Birkhoff annuli, which is weakly embedded this means that restricted to the union of the interiors of the Birkhoff annuli it is embedded and the periodic orbits are disjoint from the interiors. In addition a small neighbourhood is a representative for the Seifert fibered piece , that is, is isotopic to . We call such a a spine for the Seifert piece . In this section we prove several important results concerning the relative position of spines of distinct periodic Seifert pieces (if there are such), and also how the boundaries of such interact with the flow .

Lemma 3.1.

Let be two elementary Birkhoff annuli which lift to the same lozenge in and so that the cores of are freely homotopic. Then is flow homotopic to in the interior. That is, there is a homotopy from to so that for any in the interior of ,   is contained in a flow line of . In addition if is a point where does not self intersect, then is a set of no self intersections, of the homotopy.

Proof.

Choose fixed lifts so that the interiors intersect exactly the orbits in . Let in so that it generates . The fact that a single generates both stabilizers uses the condition on the cores of .

Let be the interior of . For any in , there is a unique real number so that is a point in . This map is continuous and clearly equivariant under : . Since this is equivariant, it projects to a map from the interior of to the interior of . The linear homotopy along the orbits in the required homotopy. The homotopy is an isotopy where is embedded. ∎

Proposition 3.2.

Let be a pseudo-Anosov flow and let (where may be ) be the periodic Seifert pieces of . Then we may choose the spines of so that they are pairwise disjoint.

Proof.

The construction in [Ba-Fe] shows that the periodic orbits of each are uniquely determined by . The Birkhoff annuli in are not unique and can be changed by flow homotopy. We prove the proposition in several steps.

I) For any , a periodic orbit in does not intersect .

In this part there will be no need to make adjustments to the Birkhoff annuli. Suppose is a closed orbit in which intersects with distinct. The first situation is that intersects a closed orbit in , in which case . Recall that in a periodic Seifert piece some power represents a regular fiber in and similarly some power represents a regular fiber in . But then the regular fibers in have common powers this is impossible for distinct Seifert fibered pieces of [He].

The second situation is that intersects the interior of a Birkhoff annulus in . Since is a spine for the Seifert piece , there is an immersed Birkhoff torus in containing . In addition choose to be -injective. This can be achieved by looking at a lift to and the sequence of lozenges intersected by . If there is no backtracking in the sequence of lozenges then is -injective. It is easy to choose one such with no backtracking.

Fix a lift of contained in a lift of and let be a lift of intersecting . Since is incompressible, is a properly embedded plane in . The topological plane is contained (except for the lifts of the periodic orbits) in a bi-infinite chain of lozenges . Any orbit in the interior of one of the lozenges in intersects exactly once.

Since corresponds to a closed curve in and is isotopic into , then can be homotoped to be disjoint from . Lift this homotopy to from to a bi-infinite curve in disjoint from . Recall that intersects in a single point. This implies that a whole ray of has to move cross by the homotopy. Hence this ray is at bounded distance from . As is compact this implies that is freely homotopic into . Let be the covering translation which is a generator of . Then with the correct choices we can assume that is in , which we assume is contained in .

This is now a contradiction because leaves invariant the bi-infinite chain of lozenges . In addition so leaves invariant the lozenge of containing . But then it would leave invariant the pair of corners of , contradiction to leaving invariang . We conclude that this cannot happen. This finishes part I).

II) Suppose that for some there are Birkhoff annuli so that .

Notice that the intersections are in the interior by part I). Recall also that the interiors of the Birkhoff annuli are embedded. By a small perturbation put the collection in general position with respect to itself. Let be a component of .

Suppose first that is not null homotopic in . Since is -injective, then the same is true for in and in . Then is homotopic in to a power of a boundary of , which itself has a common power with the regular fiber of . This implies that the fibers in have common powers, contradiction as in part I).

It follows that is null homotopic in and hence bounds a disc in . Notice that is embedded as both and have embedded interiors. We proceed to eliminate such intersections by induction. We assume that is innermost in : the interior of does not intersect any (switch if necessary). In addition also bounds a disc in whose interior is disjoint from by choice of . Hence is an embedded sphere which bounds a ball in because is irreducible. We can use this ball to isotope to replace a neighborhood of in by a disc close to and disjoint from , eliminating the intersection and possibly others. Induction eliminates all intersections so we can assume that all are disjoint (for a more detailed explanation of this kind of argument, see [Ba3, section ]).

Notice that the modifications in in part II) were achieved by isotopies. This finishes the proof of the proposition. ∎

Recall that we are assuming the manifold to be orientable, so that we can use [Ba-Fe, Theorem F]; however the following Lemma holds in the general case, hence we temporally drop the orientability hypothesis.

Lemma 3.3.

(local transversality) Let be an immersed Birkhoff torus or Birkhoff Klein bottle with no backtracking this means that for any lift of to , the sequence of lozenges associated to it has no backtracking. Let be a fixed lift of to and let a lift to of a closed orbit in . There are well defined lozenges in which contain a neighborhood of in (with removed):   is a corner of both and . If are adjacent lozenges then can be homotoped to be transverse to near . Conversely if can be homotoped to be transverse to near then and are adjacent. This is independent of the lift of and of .

Proof.

Formally we are considering a map (or ) so that the image is the union of (immersed) Birkhoff annuli. The homotopy is a homotopy of the map and it may peel off pieces of which are glued together. This occurs for instance if the orbit is traversed more than once in , the image of . An example of this is a Birkhoff annulus that wraps around its boundary a number of times. Another possibility is that many closed curves in or may map to and we are only modifying the map near one of these curves.

Let be the domain of which is either the torus or the Klein bottle . There is a simple closed curve in and a small neighborhood of in so that is also the projection of a small neighborhood of in to . Notice that may be an annulus or Mobius band. The statement “ can be homotoped to be transverse to in a neighborhood of ” really means that can be homotoped so that its final image is transverse to . We will abuse terminology and keep referring to this as “ can be homotoped …”.

Let a covering translation associated to . It follows that . In addition since is associated to a loop coming from (and not just a loop in ), then preserves and more to the point here preserves the pair . It may be that switches , for example if is one sided in a Mobius band. This is crucial here: if we took associated to for instance, then , but could scramble the lozenges with corner in an unexpected manner and one could not guarantee that would be preserved by . We also choose so that .

Suppose first are adjacent and wlog assume they are adjacent along a half leaf of . The crucial fact here is that since preserves the pair then leaves invariant. Let be a neighborhood of in . Choose it so the intersection with is either an annulus or Mobius band (in general only immersed). Using the image of the half leaf we can homotope the power of corresponding to (that is, corresponding to as a parametrized loop) away from so that its image in is transverse to the flow and closes up. In the universal cover preserves the set then the pushed curve from returns to the same sector of and this curve can be closed up when mapped to . Once that is done we can also homotope a neighborhood of in as well to be transverse to .

In the most general situation that neighborhood of could be an annulus which is one sided in , then the push away from could not close up. In our situation it may be that this annulus goes around say twice over and going once around sends the lozenges to other lozenges. But going around twice over (corresponding to ) returns to itself. If the neighborhood is a Mobius band, we want to consider the core curve as it generates the fundamental group of this neighborhood. This finishes the proof of the first statement of the lemma.

Suppose now that can be homotoped to be transverse to is a neighborhood of . We use the same setup as in the first part. Let be a neighborhood of in so that the pulled back neighborhood of in is either an annulus or Mobius band. We assume that can be perturbed near to in , keeping it fixed in , and to be transverse to in a neighborhood of . Let be the the part of which is the part of perturbed near .

Consider all prongs of . By way of contradiction we are assuming that the lozenges are not adjacent. Then there are at least 2 such prongs as above separating from in on either component of . Let be the lift of near .

We first show that is empty. Suppose not and let in the intersection. Since is transverse to then is transverse say to so we follow the intersection from . This projects to a compact set in , contained in the interior of as does not intersect . This is because is contained in the union of the lozenges and they are disjoint from any prong of . So the original curve in has to return to and looking at this curve in this transverse curve has to intersect twice, which makes it impossible to be transverse to the flow.

Since cannot intersect and it has boundaries in and then it has to intersect at least two prongs from , at least one stable and one unstable prong in . Project to . Then cannot be transverse to the flow . This is because in a stable prong of the flow is transverse to in one direction and in an unstable prong of the flow is transverse to in the opposite direction. This finishes the proof of lemma 3.3. ∎

Proposition 3.4.

(transverse torus) Let be a pseudo-Anosov flow in . Suppose that are periodic Seifert fibered pieces which are adjacent and let be a torus in the common boundary of . Then we can choose neighborhoods of so that the components of isotopic to are the same set and this set is transverse to .

Proof.

By proposition 3.2 we may assume that are disjoint. Since is a spine for , the torus is homotopic to a Birkhoff torus contained in . We assume that has no backtracking. Quite possibly is only an immersed torus, for example there may be Birkhoff annuli in which are covered twice by . The torus lifts to a properly embedded plane which intersects a unique bi-infinite chain of lozenges . With appropriate choices we may assume that corresponds to a subgroup of covering translations leaving invariant. The corners of the lozenges in project to closed orbits in . These have powers which are freely homotopic to the regular fiber in because is a periodic Seifert piece. Similarly produces a Birkhoff torus homotopic to with contained in and a lift contained in a bi-infinite chain of lozenges , which is also invariant under the same . The corners of the lozenges in project to closed orbits of the flow with powers freely homotopic to a regular fiber in . If these two collections of corners are the same, they have the same isotropy group, which would imply the fibers in have common powers, impossible as seen before.

We conclude that are distinct and both invariant under . This is an exceptional situation and proposition 5.5 of [Ba-Fe] implies that both chains of lozenges are contained in a scalloped region and one of them (say ) is s-scalloped and the other () is u-scalloped. The lozenges in the s-scalloped region all intersect a common stable leaf, call it and the corners of these lozenges are in stable leaves in the boundary of the scalloped region.

Let then be a periodic orbit in with lift to and lozenges of which have corner . Then are adjacent along an unstable leaf. Further if is the curve as in the proof of the previous lemma, which is homotopic to a power of , then is in and so the covering translation associated to preserves and hence also . By the previous lemma we can homotope slightly near to make it transverse to the flow near . When lifting to the universal cover, the corresponding lift of the perturbed torus will not intersect , but will intersect all orbits in the half leaf of which is in the common boundary of and . Do this for all closed orbits of in . Notice we are pushing along unstable leaves. Consider now a lozenge in and the Birkhoff annulus contained in the closure of which is contained in . Both boundaries have been pushed away along unstable leaves. The unstable leaves are on the same side of the Birkhoff annulus . Therefore one can also push in the same direction the remainder of to make it disjoint from . This produces a new torus satisfying

  • is a contained in a small neighborhood of and is transverse to the flow ,

  • is disjoint from every (including ),

  • There is a fixed lift which is invariant under and that it intersects exactly the orbits in the scalloped region.

The much more subtle property to prove is the following:

Claim can be chosen embedded.

To prove this claim we fix the representative of and a Seifert fibration so that is a union of fibers: each Birkhoff annulus of is a union of fibers and it is embedded in the interior. Since is orientable, the orbifold is a surface with a finite number of singular points, which are the projections by of the vertical periodic orbits. Moreover, is a fat graph, i.e. is a graph embedded in which is a deformation retract of . One can furthermore select so that the stable and unstable vertical annuli in are Seifert saturated, ie. project to arcs in with one boundary in , the other being a vertex of . Finally, one can assume that the retraction is constant along stable and unstable arcs, mapping each of them on the vertex of lying in their boundary.

Since the Birkhoff annuli in are transverse to the flow, one can distinguish the two sides of every edge of , one where the flow is “incoming”, and the other “outgoing”. The stable arcs are contained in the incoming side, whereas the unstable arcs are contained in the outgoing side. It follows that the set of boundary components of can be partitioned in two subsets so that for every edge of , the two sides of in lie in different sets of this partition.

The immersed Birkhoff torus is a sequence of Birkhoff annuli , , … , , . It corresponds to a sequence , … , , of edges in . As described above, since is -scalloped, is obtained by pushing every along the unstable annuli so that intersects no stable annulus. It follows that we always push on the “outgoing” side. Let be the unique segment in the outgoing boundary of whose image by the retraction is : it follows that the sequence of segments , , … , describe an outgoing component of . In other words, is the retraction of a boundary component of .

Hence, if we have for some , we have , and so on, so that the sequence , … , is the repetition of a single loop in . Then, is homotopic to the boundary component of repeated at least twice. But it would mean that the JSJ torus is homotopic to the JSJ torus repeated several times, which is a clear contradiction.

Therefore, , … , is a simple loop: can pass through a Birkhoff annulus in at most once. Then the homotopy from to does the following: the interiors of the Birkhoff annuli are homotoped to an embedded collection. The neighborhoods of the periodic orbits also satisfy that. We conclude that can be chosen embedded. This finishes the proof of the claim.

As is embedded and homotopic to and is irreducible, then is in fact isotopic to [He]. The same is true for to produce with similar properties. Notice that and intersect exactly the same set of orbits in . Hence their projections to bound a closed region in with boundary , homeomorphic to and so that the flow is a product in . We can then isotope and along flow lines to collapse them together.

In this way we produce representatives of respectively; with boundary components , isotopic to (they are the same set) and transverse to the flow . This finishes the proof of proposition 3.4. ∎

Proposition 3.5.

(good position) Suppose that is a totally periodic pseudo-Anosov flow in a graph manifold . Let be the Seifert fibered spaces in the torus decomposition of . Then there are spines made up of Birkhoff annuli for and compact neighborhoods so that:

  • is isotopic to ,

  • The union of is and the interiors of are pairwise disjoint,

  • Each is a union of tori in all of which are transverse to the flow .

Proof.

If and are adjoining, the previous proposition explains how to adjust the corresponding components of and to satisfy the 3 properties for that component without changing any of the or the other components of . We can adjust these tori in boundary of the collection one by one. This finishes the proof. This actually shows that any component of is homeomorphic to . ∎

This proposition shows that given any boundary torus of (the original) it can isotoped to be transverse to . Fix a component of the inverse image of in and let be a lozenge with a corner in and so that contains a lift of an (open) Birkhoff annulus in . Let be the projection of to . The proof of proposition 3.4 shows that for each side of in there is a torus which is a boundary component of a small neighborhood and which contains an annulus very close to . Going to the next Birkhoff annulus on each torus beyond , proposition 3.4 shows that the corresponding lozenge is adjacent to . Hence we account for the two lozenges adjacent to . This can be iterated. This shows that for any corner , every lozenge with corner contains the lift of the interiof of a Birkhoff annulus in . Hence there are no more lozenges with a corner in . Hence the pruning step done in section 7 of [Ba-Fe] is inexistent: the collection of lozenges which are connected by a chain of lozenges to any corner in is already associated to . This shows the assertion above.

4. Itineraries

In the previous section, we proved that admits a JSJ decomposition so that every Seifert piece is a neighborhood of the spine and whose boundary is a union of tori transverse to . Denote by , … , the collection of all these tori: for every , there is a Seifert piece such that points outward along , and another piece such that points inward along (observe that we may have , and also and may have several tori in common). It follows from the description of that the only full orbits of contained in are closed orbits and they are the vertical orbits of .

From now on in this section, we fix one such JSJ decomposition of associated to the flow . In this section when we consider a Birkhoff annulus without any further specification we are referring to a Birkhoff annulus in one of the fixed spines . In the same way a Birkhoff band is a lift to of the interior of one of the fixed Birkhoff annuli.

Let , be the foliations induced on by , . It follows from the previous section (and also by the Poincaré-Hopf index theorem) that and are regular foliations, i.e. that the orbits of intersecting are regular. Moreover, admits closed leaves, which are the intersections between and the stables leaves of the vertical periodic orbits contained in . Observe that all the closed leaves of are obtained in this way: it follows from the fact that is an approximation of an union of Birkhoff annuli.

Hence there is a cyclic order on the set of closed leaves of , two successive closed leaves for this order are the boundary components of an annulus in which can be pushed forward along the flow to a Birkhoff annulus contained in the spine . We call such a region of an elementary -annulus of .

Notice that an elementary annulus is an open subset of the respective torus the boundary closed orbits are not part of the elementary annulus.

Similarly, there is a cyclic order on the set of closed leaves of so that the region between two successive closed leaves (an elementary -annulus) is obtained by pushing forward along a Birkhoff annulus appearing in . The regular foliations and are transverse one to the other and their closed leaves are not isotopic. Otherwise have Seifert fibers with common powers, contradiction. Hence none of these foliations admits a Reeb component. It follows that leaves in an elementary or -annulus spiral from one boundary to the other boundary so that the direction of “spiralling” is the opposite at both sides. It also follows that the length of curves in one leaf of these foliations not intersecting a closed leaf of the other foliation is uniformly bounded from above. In other words:

Lemma 4.1.

There is a positive real number such that any path contained in a leaf of (respectively ) and contained in an elementary -annulus (respectively -annulus) has length .

The sets and Let be the collection of all the lifts in of the tori . Every element of is a properly embedded plane in . We will also abuse notation and denote by the union of the elements of . Let be the union of the lifts of the vertical orbits of . Finally let .

Observe that there exists a positive real number such that the -neighborhoods of the are pairwise disjoint. Therefore:

(1)

Here is the minimum distance between a point in and a point in .

What we have proved concerning the foliations , implies the following: for every , the restrictions to of and are foliations by lines, that we denote by , . These foliations are both product, i.e. the leaf space of each of them is homeomorphic to the real line. Moreover, every leaf of intersects every leaf of in one and only one point. Therefore, we have a natural homeomorphism , identifying every point with the pair of stable/unstable leaf containing it (here, denotes the leaf space of ).

Bands and elementary bands Some leaves of are lifts of closed leaves: we call them periodic leaves. They cut in bands, called (stable or unstable) elementary bands, which are lifts of elementary annuli (cf. fig. 3). Observe that the intersection between a stable elementary band and an unstable elementary band is always non-trivial: such an intersection is called a square. Finally, any pair of leaves of the same foliation or bounds a region in that we will call a band (elementary bands defined above is in particular a special type of band). A priori bands and elementary bands can be open, closed or “half open” subsets of .

Remark 4.2.

We arbitrarily fix a transverse orientation of each foliation , . It induces (in a -equivariant way) an orientation on each leaf space . Since every leaf of is naturally identified with , the orientation of induces an orientation on every leaf of . When one describes successively the periodic leaves of , this orientation alternatively coincide and not with the orientation induced by the direction of the flow. This is because such leaves are lifts of closed curves isotopic to periodic orbits.

Figure 3. Bands and elementary bands. Nearly vertical lines are leaves of , and nearly horizontal lines are leaves of . Thicker lines are periodic leaves. The shaded region is a band which is not an elementary band.

For every in , let be the list of the elements of successively met by the positive -orbit of (including an initial if is contained in an element of ). Observe that this sequence can be finite, even empty: it happens precisely when the positive orbit remains trapped in a connected component of , i.e. the lift of a Seifert piece . In this case, the projection of the orbit lies in the stable leaf of a vertical periodic orbit of . In other words, lies in where is a lift of . In that case, we denote by the sequence , where is the last element of intersecting the positive -orbit of , and all the following terms are all equal to . We say then that is finite. In the other case, i.e. when the sequence is infinite, will denote this infinite sequence. In both situations, is called the positive itinerary of .

Similarly, one can define the negative itinerary has the sequence of elements of successively crossed by the negative orbit of . Once more, such a sequence can be finite if lies in the unstable leaf of the lift of a periodic vertical orbit