Given a taut depth-one foliation ℱ in a closed atoroidal 3-manifold transverse to a pseudo-Anosov flow ϕ without perfect fits, we show that the universal circle coming from leftmost sections associated to ℱ, constructed by Thurston and Calegari–Dunfield, is isomorphic to Fenley’s ideal boundary of the flow space associated to ϕ with natural structure maps. As a corollary, we use a theorem of Barthelmé–Frankel–Mann to show that there is at most one pseudo-Anosov flow without perfect fits transverse to ℱ up to orbit equivalence.
Mathematics Subject Classification 202057K30Keywordspseudo-Anosov flowsfoliationsuniversal circles3-manifoldsNSFDMS-2005328The author is partially supported by NSF grant DMS-2005328.Introduction
There has been an important theme in 3-manifold topology to study the interaction between flows and codimension-one foliations in 3-manifolds. The simplest examples of codimension-one foliations of a 3-manifold M are fibrations, which are exactly the foliations with all leaves compact. The theory of Thurston norm organizes different ways of fibration of M into a finite number of fibered faces, and there is a one-to-one correspondence between fibered faces and isotopy classes of suspension pseudo-Anosov flows [22, 31]. The aim of this paper is to study one of the next simplest classes of foliations, namely depth-one foliations, and their interaction with transverse pseudo-Anosov flows, by comparing the π1-actions on S1 that arise in both settings.
A foliation in a closed 3-manifold is called a depth-one foliation if the restriction to the complement of compact leaves is a fibration over the circle. More precisely, there are a finite number of compact leaves, called depth-zero leaves, and the rest of the leaves (namely the depth-one leaves) are infinite-type surfaces spiraling into the depth-zero leaves. One way to construct depth-one foliations is to “spin” a fibration around an embedded surface (see [6, Example 4.8]).
Given a taut depth-one foliation ℱ in a closed 3-manifold M, a result of Candel [8] shows that there exists a Riemannian metric on M such that the restrictions to the leaves of ℱ are hyperbolic, giving every ℱ-leaf a standard hyperbolic structure in the sense of [10] (see also Section 2.3). In particular, there is a natural circle at infinity associated to the universal circle of any leaf of ℱ. An unpublished construction of Thurston [32], which was later written down by Calegari–Dunfield [7], produces a circle 𝔖left associated to ℱ. We will call this circle a universal circle from leftmost sections. The circle 𝔖left is acted on faithfully by π1(M) and is equipped with a π1(M)-equivariant collection of monotone structure maps {Uλ}λ∈ℱ~ to the circles at infinity of all ℱ~-leaves, where ℱ~ is the lift of ℱ to the universal cover M~ of M.
In general, there is an axiomatized notion of a universal circle associated to a taut foliation (Definition 4.1). The universal circle 𝔖left is a universal circle of ℱ in this general sense but not a canonical one. However, when ℱ is a taut depth-one foliation transverse to a pseudo-Anosov flow without perfect fits ϕ (see Section 2.1 for discussions on pseudo-Anosov flows), we will see that it is possible to relate the 𝔖left to a more natural object, which is the ideal boundary of the flow space of ϕ.
For a pseudo-Anosov flow ϕ, the flow space𝒪 associated to ϕ is the space of orbits of the lifted flow ϕ~ in M~. It is homeomorphic to ℝ2 by [2, 14, 16], and there is a compactification 𝒪¯=𝒪∪∂𝒪 given by Fenley [13]. The ideal boundary ∂𝒪 is homeomorphic to a circle, and the π1(M)-action on 𝒪 extends continuously to ∂𝒪. If ϕ has no perfect fits, we will see that the shadow of any leaf λ of ℱ~ provides a natural structure map Iλ from ∂𝒪 to the circle at infinity of λ (see Section 3). For these structure maps, we prove the following theorem.
Theorem 1.1.
Let M be a closed atoroidal 3-manifold with a pseudo-Anosov flow ϕ without perfect fits, and let ℱ be a taut depth-one foliation in M transverse to ϕ. Then the circle ∂𝒪, together with the structure maps {Iλ}λ∈ℱ~, is a universal circle for ℱ.
While writing this paper, the author learned that Landry, Minsky and Taylor show a much more general version of Theorem 1.1 (recently appeared in [24]). More precisely, they prove that given a taut foliation ℱ almost transverse to a pseudo-Anosov flow ϕ in a closed hyperbolic manifold M, the boundary of the flow space naturally has the structure of a universal circle for ℱ.
Nevertheless, we show in our setting that the universal circles ∂𝒪 and 𝔖left are isomorphic, as made precise by the following theorem.
Theorem 1.2.
Let M be a closed atoroidal 3-manifold with a pseudo-Anosov flow ϕ without perfect fits, and let ℱ be a taut depth-one foliation in M transverse to ϕ. Then the π1(M)-actions on ∂𝒪 and on 𝔖left are conjugated by a homeomorphism T:𝔖left→∂𝒪. Moreover, for any leaf λ of ℱ~, we have Iλ∘T=Uλ.
Corollary 1.3.
Let M be a closed atoroidal 3-manifold, and let ℱ be a taut depth-one foliation in M. Then there is at most one pseudo-Anosov flow without perfect fits transverse to ℱ up to orbit equivalence.
Proof.
Suppose there are two pseudo-Anosov flows without perfect fits ϕ and φ that are transverse to ℱ. Since M is atoroidal, both ϕ and φ are transitive (we say a flow is transitive if it has an orbit that is dense in both positive and negative time) [27]. Since the construction of 𝔖left makes no use of the transverse flows, the actions of π1(M) on the ideal boundaries of the orbit spaces of ϕ and φ are conjugate by Theorem 1.2. By [3, Theorem 1.5], ϕ and φ are orbit equivalent.∎
Remark 1.4.
In personal conversations, Michael Landry told the author that using the construction by Gabai–Mosher for almost transverse pseudo-Anosov flows to finite depth foliations, one can construct different pseudo-Anosov flows transverse to the same depth-one foliation, but these flows have perfect fits. While there is currently no complete proof of Gabai–Mosher’s construction in the literature, the monograph [28] by Mosher contains an outline and the main ideas of the theory. See also the paper [25] by Landry–Tsang and their upcoming work [26], which are aimed at revisiting the theory using veering triangulations.
Remark 1.5.
In [30], Anna Parlak constructs examples of closed and cusped hyperbolic 3-manifolds with a non-fibered face dynamically represented by two topologically inequivalent pseudo-Anosov flows using mutations of veering triangulations. However, the properties of the resulting flows, for example, whether they have perfect fits or which foliations they are transverse to, are not so clear.
A conjectural picture of “pseudo-Anosov packages” is developed in [5] by Calegari in the hope that the different structures from taut foliations, laminations, universal circles and pseudo-Anosov flows are organized and compatible in the most natural way, and it is asked to what extent the picture is true.
In particular, given a universal circle 𝔖 for ℱ, he constructs a pair of π1(M)-invariant laminations Ξ± on 𝔖. In our case, one can apply the construction to the universal circle 𝔖left≅∂𝒪 and get a pair of laminations on ∂𝒪. On the other hand, the endpoints of the singular foliations ℱ𝒪u/s also induce a pair of laminations ℒ𝒪u/s on ∂𝒪 by taking the pairs of endpoints of regular leaves and faces of singular leaves. We partially verify Calegari’s picture by showing that Ξ± and ℒ𝒪u/s coincide.
Theorem 1.6.
In the setting of Theorem 1.2, the invariant lamination Ξ+ (resp. Ξ−) on the universal circle 𝔖left equals the induced stable lamination ℒ𝒪s (resp. ℒ𝒪u) under the isomorphism T:𝔖left→∂𝒪.
The organization of the paper is as follows. In Section 2, we briefly recall some knowledge about pseudo-Anosov flows, depth-one foliations and circle laminations. In Section 3, we summarize the structure of the shadows of leaves of ℱ in 𝒪 developed by [12, 19] and carefully study the infinity of shadows. From there, we introduce the restriction maps Iλ and a relative version of restriction maps. We start Section 4 with a brief review of the construction of the universal circle from leftmost sections 𝔖left following [7], and we relate 𝔖left to the universal circle structure of ∂𝒪 we developed in Section 3. We prove Theorem 1.2 in Section 5 by explicitly constructing the homeomorphism T and proving the desired properties. We conclude with a discussion of invariant laminations and the proof of Theorem 1.6 in Section 6.
PreliminariesConvention 2.1.
We consider a closed Riemannian atoroidal 3-manifold M with the Riemannian metric to be determined later. For a partition, Θ (e.g., a flow or a foliation) of a space X and a point x∈X, we let Θ(x) be the atom of Θ containing x. More generally, if A is a subset of C, we use Θ(A) to denote the saturation of A by Θ-atoms.
Pseudo-Anosov flows
We refer to [1, 14, 27] and the recent monograph [4] for detailed discussions of pseudo-Anosov flows in 3-manifolds. The following definition follows [13].
A flow ϕ:M×ℝ→M in M is a pseudo-Anosov flow if it has the following properties:
each flowline is C1 and not a single point;
the tangent line bundle Tϕ is continuous;
there are a finite number of singular closed orbits, and a pair of 2-dimensional singular foliations ℱu and ℱs in M so that
each leaf of ℱu or ℱs is a union of ϕ-orbits;
outside of the singular orbits ℱu and ℱs are regular foliations whose leaves intersect transversely along ϕ-orbits;
for each singular orbit ω, the leaf of ℱu or ℱs containing ω is homeomorphic to Pn×[0,1]/f where
Pn={re2kiπ/n∣r≥0, 0≤k≤n−1}⊂ℂ
is an n-prong and f is a homeomorphism from Pn to Pn. The orbit ω is the image of {0}×[0,1] and n is always greater than 2 in our case;
orbits in the same ℱs-leaf are forward asymptotic, and orbits in the same ℱu-leaf are backward asymptotic.
The singular foliations ℱs and ℱu are called the stable foliation and the unstable foliation of ϕ, respectively. When the set of singular orbits is empty, ϕ is an Anosov flow.
Fix a universal cover M~ of M, and let ϕ~,ℱ~u,ℱ~s be the lifts of ϕ,ℱu,ℱs to M~, respectively. The quotient of M~ by ϕ~ is the flow space𝒪, which is homeomorphic to ℝ2 [2, 14, 16]. We orient 𝒪 so that the coorientation coincides with the flow direction, and the pictures in this paper are drawn in a way that the flow is flowing toward the reader. The deck transformation on M~ descends to an orientation-preserving π1(M)-action on 𝒪, and the singular foliations ℱ~u and ℱ~s descend to a pair of π1(M)-invariant transverse singular foliations on 𝒪, denoted by ℱ𝒪u and ℱ𝒪s. The singular leaves of ℱ𝒪s and ℱ𝒪u are n-pronged with n≥3. The union of two adjacent prongs in a singular leaf of ℱ𝒪s or ℱ𝒪u is called a face.
A ϕ~-orbit ω is periodic if there exists a non-trivial deck transformation g of M~ with g(ω)=ω. In this case, g acts on ω by translation, and ω covers a closed orbit of ϕ in M. Similarly, a leaf of ℱ~s or ℱ~u is called periodic if it is fixed by some non-trivial deck transformation of M~. In particular, singular leaves are periodic. Finally, we call a point or a leaf of ℱ𝒪u or ℱ𝒪s in 𝒪 periodic if the corresponding ϕ~-orbit or ℱ~s/u leaf in M~ is periodic. The following fact is well known and can be found in [4, Proposition 1.4.3].
Lemma 2.2.
If ℓ is a leaf of ℱ𝒪s or ℱ𝒪u fixed by a non-trivial g in π1(M), then the g-action on ℓ has a unique fixed point. In other words, any periodic leaf of ℱ~s or ℱ~u contains a unique periodic orbit.
A ray of ℱ𝒪u or ℱ𝒪s is an embedded closed half-line contained in a leaf with the interior disjoint from singularities. Two rays l∈ℱ𝒪s and l′∈ℱ𝒪u are said to form a perfect fit if there is an (possibly orientation-reversing) embedding
ι:[0,1]×[0,1]−(1,1)→𝒪
mapping horizontal lines to ℱ𝒪s leaves, vertical lines to ℱ𝒪u leaves, [0,1)×{1} to l′ and {1}×[0,1) to l. We say a pseudo-Anosov flow is without perfect fits if no two rays in ℱ𝒪s and ℱ𝒪u form a perfect fit. The notion of perfect fits is introduced and studied by Fenley in [17, 18]. In particular, we have the following lemma which is an immediate consequence of [18, Theorem 4.8].
Lemma 2.3.
If ϕ has no perfect fits, then any non-trivial element of π1(M) has at most one fixed point in 𝒪.
Fenley introduces a compactification of 𝒪 in [13] by building an ideal boundary ∂𝒪 homeomorphic to S1, and the resulting compactified space 𝒪¯=𝒪∪∂𝒪 is homeomorphic to a closed 2-disk. We orient ∂𝒪 as the boundary of 𝒪, and the action of π1(M) on 𝒪 extends continuously to an orientation-preserving action on 𝒪¯. Each ray in ℱ𝒪s or ℱ𝒪u has a well-defined endpoint, and the endpoints of every leaf are distinct. When the flow has no perfect fits, a ray in ℱ𝒪s and a ray in ℱ𝒪u always have distinct endpoints. If we moreover assume that ϕ is not conjugate to an Anosov suspension flow (which is automatic when M is atoroidal), the action of π1(M) on ∂𝒪 is minimal [13, Main Theorem].
Convention 2.4.
We assume that ϕ is a pseudo-Anosov flow without perfect fits in M.
End-periodic automorphisms
We briefly recall the basics of end-periodic automorphisms of infinite-type surfaces, which arise naturally in the study of depth-one foliations. Readers are referred to [11] for a more complete treatment of the theory.
Let L be an infinite-type surface without boundary with finitely many ends, all of which are non-planar. Given a homeomorphism f:L→L, an end E of L is a contracting end of f if there is a neighborhood UE of E and an integer n>0 0\)]]> such that fn(UE)⊊UE and ⋂k≥0fnk(UE) is empty. Such a neighborhood UE is called a regular neighborhood of E. An end is a repelling end of f if it is a contracting end of f−1, and a regular neighborhood of E for f is just a regular neighborhood for f−1. A homeomorphism f is called end periodic if each end of L is either contracting or repelling. If f is end periodic, a multi-curve δ in L is called an f-juncture if δ is the boundary of a regular neighborhood of an end. If the end is contracting, δ is called a positive f-juncture. Otherwise, it is a negative f-juncture. An f-invariant choice of a positive (resp. negative) f-juncture for each contracting (resp. repelling) end is called a system of positive (resp. negative) f-junctures. An end-periodic homeomorphism is atoroidal if it does not preserve any essential multi-curve up to isotopy.
Given an end-periodic homeomorphism f, we fix a regular neighborhood UE for every end E. Let U+ be the union of UE of contracting ends, and let U− be the union of those of repelling ends. The positive escaping set𝒰+ and the negative escaping set𝒰− are defined as
𝒰±=⋃n≥0f∓n(U±).
In other words, 𝒰+ is the set of points whose positive iterations escape to contracting ends, and 𝒰− is the set of points whose negative iterations escape to repelling ends. The mapping torus Mf is non-compact, but it is topologically tame and possesses a nice compactification which we describe below.
The mapping torus Mf is the quotient of L×ℝ by an automorphism F where F is given by
F:L×ℝ→L×ℝ(x,t)↦(f−1(x),t+1).
We attach 𝒰+×{+∞} and 𝒰−×{−∞} to L×ℝ to obtain a manifold N with boundary. The transformation F extends to an automorphism of N by setting F(x,±∞)=(f−1(x),±∞). The ℤ-action on N generated by F is a covering action, and the quotient space Mf¯ is a compact 3-manifold with interior Mf and boundary ∂Mf¯=∂+Mf¯∪∂−Mf¯, where ∂±Mf¯ is a (possibly disconnected) closed surface homeomorphic to 𝒰±/f. See [21] for a more detailed discussion of the construction. In particular, [21, Lemma 3.3] shows that Mf¯ is atoroidal if and only if f is atoroidal.
Depth-one foliation
A foliation ℱ in M is a depth-one foliation if ℱ has finitely many compact leaves, whose union we denote by ℱ0, and ℱ restricted to M−ℱ0 is a fibration over a circle with non-compact fibers. A connected component of M−ℱ0 is called a fibered region of M. Any fibered region Ω is bounded by leaves in ℱ0, and we denote the collection of these leaves by ∂Ω. We say a leaf in ∂Ω is a positive (resp. negative) boundary leaf of Ω if it is on the ϕ-positive (resp. ϕ-negative) side of Ω. We denote the collection of positive/negative boundary leaves by ∂±Ω. Note that it is possible to have a compact leaf contained in both ∂+Ω and ∂−Ω. Let Ω¯ be the union of Ω and ∂Ω.
Let L be a fiber of the fibration ℱ|Ω. The leaf L limits on a compact leaf Σ⊂∂Ω in the following way [9]. Let N(Σ)≅Σ×[−1,1] be a regular neighborhood of Σ with Σ identified with Σ×{0}, and assume that L limits on Σ on the positive side. If N(Σ) is small enough, the intersection of L and N(Σ) is an infinite surface spiraling to Σ and covering Σ with infinite degree. More precisely, up to shrinking N(Σ), there is a multi-curve δ on Σ so that L∩N(Σ) is isotopic to an oriented cut-and-paste of δ×(0,1] and ⋃n≥2Σ×{1/n}. A fundamental domain for the spiraling is depicted on top of Figure 1a, and a schematic picture of the spiraling neighborhood is shown in Figure 1b.
We say a foliation ℱ is taut if for any leaf of ℱ, there is a transverse loop intersecting that leaf.
A schematic picture of the spiraling neighborhood of Σ.
Convention 2.5.
We fix a taut depth-one foliation ℱ of M and assume that ℱ is transverse to ϕ. By transversality, ℱ is coorientable, and we take the orientation to be consistent with the flow direction.
By transversality of ℱ to ϕ and by the compactness of M, the angle between Tϕ and Tℱ at any point is uniformly bounded away from zero. In particular, this implies that ϕ|Ω is a suspension flow of the fibration ℱ|Ω. The flow gives us a way to identify each ℱ-leaf in Ω with L. Since ℱ is taut, every leaf in ℱ0 is homologically non-trivial and incompressible by Novikov [29]. Since M is atoroidal, every compact leaf is a closed hyperbolic surface (the possibility of being a sphere is ruled out by the Reeb stability theorem). Therefore, the fundamental domains of the spiraling of L can be chosen to be non-planar. In a spiraling neighborhood of a compact leaf, ϕ looks like the product flow. One can see that the first return map induced by ϕ is an end-periodic homeomorphism f:L→L [15]. Since M is atoroidal, so is f. The metric completion of Ω with respect to the path metric induced by the metric in M gives a compactified mapping torus of f.
By [8], there is a Riemannian metric on M that restricts to hyperbolic metrics on leaves of ℱ. We fix such a metric on M. A hyperbolic metric on a surface is standard if there is no embedded half-space, following [10]. The induced hyperbolic metric on any depth-one leaf of ℱ has injectivity radius bounded from above. This is discussed in [23, Proof of Proposition 4.6], for instance, and the rough idea is that the junctures on any depth-one leaf have bounded length, and any point on the leaf is at a bounded distance from a juncture.
Let 𝒰±⊂L be the positive or negative escaping set of f. By definition, a point x∈L is in 𝒰+ if and only if the positive ray of ϕ(x) hits ∂+Ω, and it is in 𝒰− if and only if the negative ray of ϕ(x) hits ∂−Ω. If (Ω¯,∂−Ω,∂+Ω)≅(S×[0,1],S×{0},S×{1}) or (S×S1,S×{1},S×{1}) for some closed hyperbolic surface S, we say Ω is a trivial fibered region. This is equivalent to the monodromy f being a pure translation on L and to 𝒰−=𝒰+=L [11, Proposition 4.76].
Convention 2.6.
We assume that M has no trivial product region. In particular, any compact leaf of ℱ is not a fiber. All of our discussions and statements hold with trivial product regions and are readily checked, so we omit the related discussion for simplicity.
Let ℱ~ be the lift of ℱ in M~. The lifted foliation ℱ~ is a foliation by planes by [29]. A connected lift of a fibered region of M is called a product region of M~. Any product region Ω~ covering Ω is homeomorphic to L~×ℝ, and we fix a homeomorphism so that Ω~ is foliated by L~×{t} and the ϕ~-orbits are {x}×ℝ. A lift of a positive/negative boundary leaf of Ω is a positive/negative boundary leaf of Ω~, the collection of which is denoted by ∂±Ω~. Define
∂Ω~:=∂+Ω~∪∂−Ω~andΩ~¯:=Ω~∪∂Ω~.
Let 𝒰~± be the preimage of 𝒰± in L~. From the construction of the compactified mapping torus, we see that Ω~¯ is homeomorphic to
(L~×ℝ)∪(𝒰~+×{+∞})∪(𝒰~−×{−∞}),
where 𝒰~± is the preimage of 𝒰± in L~.
For any leaf μ∈∂+Ω~, there is a component 𝒰~μ+ of 𝒰~+ such that a ϕ~-orbit intersects μ if and only if it intersects L~ at a point contained in 𝒰~μ+. This gives a bijection between leaves in ∂+Ω~ and components of 𝒰~+. Similarly, there is a bijection between leaves in ∂−Ω~ and components of 𝒰~−.
A leaf of ℱ~ is called a type-0 leaf if it covers a leaf in ℱ0. Otherwise, we call it a type-1 leaf. Every type-1 leaf μ is contained in a unique product region in M~, denoted by Ω~(μ). Every type-0 leaf is the negative boundary of a unique production region, and the positive boundary of another different product region. A type-0 leaf λ and a type-1 leaf μ are called adjacent if λ⊂∂Ω~(μ). If moreover λ is in the positive side of μ, we say λ is positively adjacent to μ or μ≲λ; otherwise, we say λ is negatively adjacent to μ or μ≳λ.
Let Λ be the leaf space of ℱ~ obtained by collapsing each leaf to a point, which is a non-Hausdorff 1-manifold. Since each leaf of ℱ~ is properly embedded in M~ and hence separating, Λ is simply connected. Each product region Ω~ projects to an oriented open interval in Λ, with the orientation induced by ϕ~. Every such open interval has a countably infinite number of positive endpoints, each corresponding to a component of ∂+Ω~. See Figure 2 for an illustration of how leaves in the product regions in M~ limit to different type-0 leaves. The positive endpoints of the open intervals are non-separated from each other in Λ by (2.1). The same is true for negative endpoints. The closures of product regions are glued together along type-0 leaves in the boundary, so each point corresponding to a type-0 leaf is the negative endpoint of exactly one open interval associated to a product region, and the positive endpoint of exactly another different one.
Type-1 leaves (black) in a product region limit to type-0 leaves (blue) in the positive boundary and stay transverse to ϕ~ (red), creating non-Hausdorffness in Λ.
Left: a local picture near a product region Ω~ in Λ. Right: the corresponding parts in Λ∗. The red vertices represent type-0 leaves in both pictures, and the arrows indicate the direction of ϕ~.
It is sometimes useful to think about the dual graph Λ∗. The set of vertices is the set of product regions and type-0 leaves. For a product region or a type-0 leaf x, we denote the dual vertex in Λ∗ by x∗. The edges are the pairs (Ω∗,μ∗) where Ω is a product region, μ is a type-0 leaf and μ∈∂Ω. The dual graph Λ∗ is an infinite valence tree with an orientation given by the flow (see Figure 3).
Two leaves of ℱ~ are called comparable if they can be connected by an oriented path in Λ. Otherwise, they are incomparable. If λ is comparable to μ and μ is on the positive side of λ, we write λ<μ or μ>λ \lambda\)]]>. Similarly, we say two vertices are comparable if there is an oriented path connecting them in Λ∗ and are incomparable if otherwise. If two vertices v and w are comparable and v is on the positive side of w, we write w<v or v>w w\)]]>. Two product regions are comparable/incomparable if their dual vertices are comparable/incomparable.
Laminations on <inline-formula><alternatives><mml:math display="inline"><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math><tex-math><![CDATA[\(S^{1}\)]]></tex-math></alternatives></inline-formula>
We recall some definitions and constructions of abstract laminations on S1.
Let Symm2(S1):=S1×S1−Δ/∼ be the space of unordered pairs of distinct points in S1 endowed with the quotient topology, where the relation is given by (x,y)∼(y,x). Two pairs of points {x,y} and {z,w} on S1 are said to be unlinked if z and w lie in the same component of S1−{x,y}. A lamination on S1 is a closed pairwise unlinked subset of Symm2(S1).
By identifying S1 with ∂ℍ2, any lamination Ξ on S1 determines a geodesic lamination Ξgeod on ℍ2 by taking the union of geodesics that connect the pairs in Ξ. Conversely, any geodesic lamination in ℍ2 gives a lamination on S1 consisting of endpoint pairs of leaves.
Given any subset A⊂S1, the boundary of the convex hull of A¯ is a geodesic laminations on ℍ2, which can be viewed as a lamination ∂CH(A) on S1. Note that the lamination ∂CH(A) is independent of the choice of the identification S1≅∂ℍ2.
Infinity of shadows
Let p:M~→𝒪 be the projection map. For any subset A of M~, the image of A under p is called the shadow of A. In this section, we will study the shadow of leaves of ℱ~, especially the behavior of the shadow at infinity. The main results are Lemmas 3.3 and 3.5, which hint at a universal circle structure on ∂𝒪.
Let λ be a type-0 leaf of ℱ~, and let Σ be the leaf of ℱ it covers. Denote the covering map by π0. Since λ separates M~ and the flow ϕ~ crosses λ positively, each flowline intersects λ at most once. It follows that p restricted to λ is a continuous bijection to its image. This map p|λ:λ→p(λ) also has a continuous inverse, mapping a point in p(λ) to the intersection of the corresponding orbit with λ. Therefore, p|λ is a homeomorphism to p(λ). The map π:=π0∘(p|λ)−1:p(λ)→Σ is then a covering map, so we can view p(λ) as a universal cover of Σ. The deck transformation group action on p(λ) is simply the action of π1(M) on 𝒪 restricted to a subgroup in the conjugacy class of π1(Σ) that stabilizes p(λ). We identify this subgroup with π1(Σ). By transversality, the intersection of ℱs/u with Σ induces a singular foliation on Σ, denoted by ℱΣs/u. Similarly, let the intersection of ℱ~s/u with λ be denoted by ℱ~λs/u. These foliations are related by the relations p(ℱ~λs/u)=ℱ𝒪s/u|p(λ) and π(ℱ𝒪s/u|p(λ))=ℱΣs/u.
Shadows of type-0 leaves. Assume λ is a type-0 leaf, so Σ is an embedded closed surface in M which is not a fiber by Convention 2.6. The shape of the shadow of a non-fibered transverse closed embedded surface is carefully studied in [12] when ϕ is a pseudo-Anosov suspension flow, later generalized by [19] to general pseudo-Anosov flows. Each component of the topological boundary of p(λ) in 𝒪 is either a regular leaf of ℱ𝒪u or ℱ𝒪s or a face of a singular leaf that is regular on the side containing p(λ) [19, Proposition 4.3]. We call a boundary component of p(λ) in 𝒪 a side of p(λ). Consider a side e of p(λ), which we assume to be contained in a leaf of ℱ𝒪s for concreteness. We collect some useful facts about the local dynamics at e from [19] in the following proposition.
Proposition 3.1
([19]). The stabilizer of e in π1(M) is isomorphic to ℤ and contained in π1(Σ). There is a generator ge acting on p(λ) with the following dynamics (Figure 4):
The element ge acts as a contraction on e with a unique fixed point xe.
The element ge fixes and expands le:=ℱ𝒪u(xe)∩p(λ), which is the interior of a ray of ℱ𝒪u. We have that le projects via π to a closed leaf αe of ℱΣu, whose free homotopy class is represented by ge.
For any point x other than xe in e, the intersection lx:=ℱ𝒪u(x)∩p(λ) is connected, and it projects via π to a non-compact leaf of ℱΣu that spirals into αe. If we orient lx and le so that they point toward e and let π(lx) and αe inherit the orientation, then π(lx) and αe are asymptotic in the forward direction.
The dynamics near a side of the shadow of a type-0 leaf.
The 3-dimensional picture is the following (Figure 5). In M~, the orbit σe:=p−1(xe) is a periodic orbit disjoint from λ. The leaf p−1(e)⊂ℱ~s(σe) does not intersect λ, and p−1(le)⊂ℱ~u(σe) intersects λ transversely in a line α~e so that all the ϕ~-orbits in p−1(le) cross λ positively. The line α~e covers the simple closed curve αe in Σ. The element ge is a translation that fixes σe and λ.
For a side e contained in a leaf of ℱ𝒪u, we have a similar picture. From now on, given a side e of p(λ), we will continue to use the notations ge, le, xe and αe for the objects described in Proposition 3.1. In particular, we always take ge to be the generator in the stabilizer of e that contracts e.
The vertical arrowed line represents the periodic orbit σe (assumed to be regular), the red plane is ℱ~s(σe), the blue plane is ℱ~u(σe) and the green plane is λ.
Let ∂p(λ) be the boundary of p(λ)in 𝒪¯, consisting of sides and ideal boundary points at infinity of p(λ).
Lemma 3.2.
The intersection ∂p(λ)∩∂𝒪 is nowhere dense.
Proof.
First we observe that ∂p(λ)∩∂𝒪 is a closed subset of ∂𝒪. This is because the complement of ∂p(λ)∩∂𝒪 on ∂𝒪 is a union of disjoint open intervals bounded by the endpoints of sides of p(λ). Suppose that there is a maximal closed interval A contained in ∂p(λ)∩∂𝒪 with endpoints η1 and η2. Then there is a side e1 of p(λ) such that η1 is an endpoint of e1. Take ge1, xe1 and le1 as before. Let c be the endpoint of le1 at ∂𝒪.
There is another side e2 of p(λ) such that η2 is an endpoint of e2. If c does not lie in A, then e2 is between c and η1 (Figure 6). The action of ge1 fixes η1 and c, but it cannot fix e2. Otherwise, ge1 will have two fixed points on 𝒪 by Lemma 2.2, contradicting the assumption that ϕ has no perfect fits by Lemma 2.3. Then one of ge1±1(η2) will be in A, contradicting our choice of A.
Proof of Lemma 3.2.
Hence, c must lie in A. Now le1 divides p(λ) into two connected components, one of which contains no side of p(λ). In particular, this component contains no π1(Σ)-translation of le1. Translated to the hyperbolic plane by Qλ, this means there is a simple closed curve αe1∈Σ and a lift Qλ(le1) of αe1 in λ such that there is no other lift on one side of α~. But that is impossible. We conclude that such an interval A does not exist.∎
The boundary ∂p(λ) is homeomorphic to a circle, and p(λ)¯:=∂p(λ)∪p(λ) is the closure of p(λ) in 𝒪¯. Note that our choice of the metric on M restricts to a hyperbolic metric on Σ, so λ is isometric to ℍ2. We define λ¯ to be the usual compactified hyperbolic plane with ideal boundary ∂∞λ. The next lemma reveals how ∂p(λ) is related to ∂∞λ.
A map g from S1 to S1 is monotone if the preimage of any point on S1 is contractible. A gap of g is a maximal closed interval of positive length in S1 that is collapsed to a single point by g. The core of g is the complement of the union of the interiors of gaps, denoted by core(g). We remark that the gaps are sometimes taken to be open intervals in the literature, different from our convention.
Lemma 3.3.
Let λ be a type- 0 leaf of ℱ~. Then the homeomorphism p−1|p(λ):p(λ)→λ extends continuously to a map Qλ from p(λ)¯ to λ¯:=∂∞λ∪λ. The map restricted to ∂p(λ) is a monotone map to ∂∞λ with core p(λ)¯∩∂𝒪.
Proof.
To begin with, we define Qλ=p−1|p(λ) in the interior of p(λ). To extend Qλ to ∂p(λ), we will first define the map on the sides of p(λ) and then extend it to the entire ∂p(λ).
Let e be a side of p(λ). Take ge∈Stab(e), xe∈e and le as in Proposition 3.1. As an element of π1(Σ), ge acts as a hyperbolic element on λ¯ with a contracting fixed point ∂−ge and a repelling fixed point ∂+ge at infinity. We define Qλ on e as the constant map to ∂−ge. Different sides are sent to different points in ∂∞(λ) because of Lemma 2.3.
Since Qλ(le) projects to a simple closed curve αe in Σ, it is a quasi-geodesic in λ with well-defined endpoints in ∂∞λ. Moreover, αe represents the free homotopy class of ge up to taking the inverse, by Proposition 3.1. Therefore, the endpoints of Qλ(le) are ∂±ge. By the way we choose ge (item (2) of Proposition 3.1), it contracts le near e. Therefore, if we orient le to point toward e and give Qλ(le) the induced orientation, the forward endpoint of Qλ(le) is ∂−ge. This shows Qλ is continuous at xe when restricted to le∪xe. This further ensures that Qλ preserves the cyclic order of the boundary leaves, which is a consequence of the fact that Qλ is an orientation-preserving homeomorphism in the interior. For example, one can argue as follows (indicated in Figure 7). Consider three sides e1,e2 and e3 in clockwise order, and take xei and lei as before. Take a point yei∈lei, and let lei′⊂lei be the segment between xei and yei. We may ensure lei′ are pairwise disjoint by taking yei close enough to xei. Take any point z∈p(λ) that is not in the union of lei′, and connect yei to z to get an embedded 3-prong P with z as the center and xei as the endpoints. The image Qλ(P) is an embedded 3-prong in λ¯ with well-defined endpoints Qλ(ei). The cyclic order of the edges of R is preserved under Qλ, assuring that Qλ(e1),Qλ(e2) and Qλ(e3) are arranged clockwise.
Qλ preserves the cyclic order.
Moreover, if γ is an element of π1(Σ), it translates e to another side γe of p(λ). We have gγe=γgeγ−1, so Qλ(γe)=γQλ(e). By the minimality of the π1(Σ) action on ∂∞λ, the Qλ-images of all the sides of p(λ) are dense in ∂∞λ. Now there is a unique way to define Qλ on ∂p(λ) such that it is continuous when restricted to ∂p(λ). That is, for any point x∈∂p(λ), one can find a sequence of sides en converging to x and define Qλ(x) as the limit of Qλ(en) by Lemma 3.2. The limit exists and is independent of en because Qλ preserves the cyclic order of sides and the image of sides is dense in ∂∞λ.
To complete the proof of Lemma 3.3, what is left is to check that Qλ is continuous on p(λ)¯.
For a side e (which is assumed to be contained in a leaf of ℱ𝒪s for concreteness) of p(λ) and a point x∈e, we take a rectangular neighborhood of x as the image of an embedding ρ:(0,1)×[0,1)→p(λ)¯ such that
ρ(1/2,0)=x;
ρ((0,1)×{0}) is contained in e;
for any s∈[0,1), ρ((0,1)×{s}) is contained in a leaf of ℱ𝒪s;
for any t∈(0,1), ρ({t}×(0,1)) is contained in a leaf of ℱ𝒪u.
Such a neighborhood always exists because ℱ𝒪s is regular on the side of e that contains p(λ).
Fix a rectangular neighborhood RN(x) of x. By Proposition 3.1, we know that Qλ(ρ({t}×(0,1))) is asymptotic to Qλ(le), the latter being a quasi-geodesic ray in λ since it is a lift of a simple closed curve in Σ. This shows that Qλ(RN(x)) is a wedge-shaped region in λ with one ideal point Qλ(e) (Figure 8).
The Qλ-image of ℱ𝒪s-leaves near an unstable boundary leaf e.
Consider a sequence of points xn in p(λ) converging to x∈∂p(λ). If x is on a side e, we can take a rectangular neighborhood of x that will eventually contain xn. The shape of the rectangular neighborhoods under Qλ shows that Qλ(xn) converges to Qλ(x)=Qλ(e).
Continuity of Qλ at ∂p(λ)∩∂𝒪.
If x∈∂p(λ)∩∂𝒪, we can trap Qλ(xn) using segments of le, forcing them to converge to the right points. The argument is similar to the one we use to prove that Qλ preserves the cyclic order of the sides. See Figure 9 for an illustration. More precisely, take sequences of sides {em+} and {em−} of p(λ) that approximate x from two sides respectively (using Lemma 3.2). Fixing m, take a short segment lem±′⊂lem± with one endpoint at xem± so that lem+′ and lem−′ are disjoint. Connect the other endpoints of the two segments by a path in p(λ), and denote the resulting path by αm. The image Qλ(αm) is an embedded line in λ~ with disjoint endpoints, separating λ~ into two half-planes. We let the half-plane containing Qλ(x) be Hm(x). The fact that Qλ|p(λ) is an orientation-preserving homeomorphism guarantees that Qλ(xn) eventually enters Hm(x) for all m. We can arrange Hm(x) to be nested so that ⋂mHm(x) has exactly one ideal point Qλ(x). Since {xn}, hence {Qλ(xn)}, escapes every compact set, we know that {Qλ(xn)} must limit to Qλ(x).
We have proved that Qλ is continuous on p(λ)¯, finishing the proof of Lemma 3.3.∎
Shadows of type-1 leaves. In this subsection, we would like to study the shadow of type-1 leaves in ℱ~. Note that by [20, Proposition 4.1], the shadow of a leaf of a transverse foliation is always bounded by regular leaves or faces of singular leaves of the stable/unstable foliations. These boundary leaves are in general not periodic for leaves in an arbitrary transverse foliation. However, we will show in the next lemma that they are in fact periodic for type-1 leaves in a depth-one foliation.
We now assume λ is a type-1 leaf of F~ contained in a product region Ω~≅L~×ℝ and λ covers a non-compact leaf identified with L. Similar to the case of type-0 leaves, we will use ∂p(λ) to denote the boundary of p(λ) in 𝒪¯.
Lemma 3.4.
The shadow p(λ) is a proper open subset of 𝒪, bounded by periodic regular leaves of ℱ𝒪s or ℱ𝒪u or faces of singular leaves which are regular on the side containing p(λ). We call a component of the boundary of p(λ) (as a subset of 𝒪) a side of p(λ).
Moreover, if a side e of p(λ) is contained in a leaf of ℱ𝒪s, then there is a type-0 leaf μ negatively adjacent to λ (see Section 2.3 for the definition) such that e is also a side of p(μ). If e is contained in a leaf of ℱ𝒪u, then there is a type- 0 leaf μ positively adjacent to λ such that e is also a side of p(μ).
Proof.
Suppose z∈𝒪 is a boundary point of p(λ). Let σz=p−1(z) be the ϕ~-orbit that projects to z. The orbit σz cannot intersect any product region that is incomparable to Ω~(λ), which is the product region containing λ. This is because if σz intersects such a product region Ω~, z will have a neighborhood U such that every orbit that projects to U also intersects Ω~. This forces them to be disjoint from Ω~(λ). Then U is not in p(λ), and z is not in the boundary of p(λ). Similarly, it cannot intersect any type-0 leaf incomparable to Ω~(λ) either.
Since p(λ) is open, z is not contained in p(λ). The orbit σz induces an oriented path γz in the dual graph Λ∗ consisting of all the type-0 leaves and product regions that σz travels through (Section 2). Every vertex in γz is comparable to the dual vertex Ω~(λ)∗ by the previous paragraph, but Ω~(λ)∗ is not in γz. Take any vertex v in γz, and suppose v>Ω~(λ)∗ \tilde{\Omega}(\lambda)^{*}\)]]>. Since Λ∗ is a tree, any vertex w satisfying v>w>Ω~(λ)∗ w > \tilde{\Omega}(\lambda)^{*}\)]]> lies in the unique oriented interval from v to Ω~(λ)∗. So the path γz has to stay in the interval in the negative direction. The case where v<Ω~(λ)∗ is similar, and we conclude that the orbit σz will eventually stay in one product region in either the positive or the negative direction.
If γz is on the negative side of Ω~(λ)∗, then σz eventually stays in a product region, denoted by Ω~0, in the positive direction. Take the unique shortest oriented path γ from Ω~0∗ to Ω~(λ)∗ in Λ∗, and let Ω~1∗ be the last vertex in γ before Ω~(λ)∗. There is a unique type-0 leaf μ that is positively adjacent to Ω~1 and negatively adjacent to Ω~(λ). Any ϕ~-orbit intersecting both Ω~0 and λ has to enter Ω~(λ) via μ. Conversely, every ϕ~-orbit intersecting μ also meets λ. This shows that if an orbit σ intersects Ω~0, then p(σ)∈p(μ) if and only if p(σ)∈p(λ). Since p(Ω~0) is open, we have also shown that if x is in p(Ω~0), then x∈∂p(λ) if and only if x∈∂p(μ).
Since z is in the interior of p(Ω~0), the above reasoning shows that z is in the boundary of p(μ). Let ez be the side of p(μ) containing z. Note that all the orbits in the same ℱ~s-leaf as σz are positively asymptotic to σz. In particular, they intersect Ω~0 and are disjoint from μ. We see that ez is contained in a leaf of ℱ𝒪s, and using a similar argument to the last paragraph, we have ez⊂∂p(λ).
To summarize, we have shown the following: for a point z∈∂p(λ), if γz is on the negative side of Ω~(λ)∗, then there is type-0 leaf μ negatively adjacent to λ and a side ez∈∂p(μ) contained in a leaf of ℱ𝒪s such that z∈ez and ez⊂∂p(λ). One can apply the same argument to the case where γz is on the positive side of Ω~(λ)∗, finishing the proof.∎
Similar to the case of type-0 leaves, we define p(λ)¯ to be p(λ)∪∂p(λ). The following lemma is the type-1 version of Lemma 3.3.
Lemma 3.5.
Let λ be a type- 1 leaf of ℱ~. Then the homeomorphism (pλ)−1:p(λ)→λ extends continuously to a map Qλ from p(λ)¯ to ∂∞λ. The map restricted to ∂p(λ) is a monotone map to ∂∞λ with core p(λ)¯∩∂𝒪.
Proof.
We define Qλ=(p|λ)−1 in p(λ) and extend Qλ to the boundary following the same strategy as in the proof of Lemma 3.3.
Let e be any side of p(λ), and we assume it to be a leaf of ℱ𝒪s for concreteness. By Lemma 3.4, e is also a side of the shadow of a type-0 leaf μ negatively adjacent to λ. Take ge, xe∈e and le as in Proposition 3.1. We orient le to be pointing toward e. By [20, Theorem C], each ray of ℱ~s∩λ or ℱ~u∩λ has a well-defined endpoint at ∂∞λ. In particular, Qλ(le) has a well-defined forward endpoint at infinity. We define Qλ on each side e of p(λ) to be the constant map to the forward endpoint of Qλ(le).
Claim.
Different sides are mapped to different points by Qλ.
Proof of the claim.
Recall that Ω~(λ) is the product region containing λ, and it covers a fibered region Ω(λ) in M with fiber L and the monodromy h:L→L. Since M is atoroidal, h is an atoroidal end-periodic map. The fundamental group of Ω(λ) is isomorphic to the semidirect product ℤ⋉π1(L) with ℤ acting on π1(L) by h∗. The group π1(Ω(λ)) stabilizes p(λ), and we can define an action of π1(Ω(λ)) on λ by g(x):=Qλ∘g∘p(x) for g∈π1(Ω(λ)) and x∈λ. If the element g has a trivial ℤ-factor, then the action of g is a covering transformation. Otherwise, g acts as a lift of some power of the monodromy h.
Let e1 and e2 be different sides of p(λ), and take gei and xei as before for i=1,2. By Proposition 3.4, there is a type-0 leaf μi adjacent to λ so that ei is a side of p(μ1). The type-0 leaf μi covers a compact leaf Σi⊂∂Ω(λ). The element gei, being a deck transformation for the covering map μi→Σi, can be viewed as an element of π1(Ω(λ)). Therefore, gei stabilizes p(λ) and acts on λ either as a hyperbolic isometry or as a lift of some power of h. In both cases, the action extends continuously to an automorphism of ∂∞λ by [10]. Since gei stabilizes lei in p(λ), we know gei stabilizes Qλ(lei) in λ, hence also the point Qλ(ei) by the definition of Qλ(ei).
Suppose for contradiction that Qλ(e1)=Qλ(e2)=:q. Then ge1 and ge2 fix the same point q on ∂λ. Note that ge1 and ge2 represent different elements in π1(Ω(λ)) and do not share a non-trivial power by Lemma 2.3. Also note that for any lift h~:λ→λ of some power of the monodromy h, the action of h~ on ∂λ will not fix any fixed point of an element in π1(L) that acts as a hyperbolic isometry on λ. Otherwise, h will fix a closed geodesic α on L up to isotopy. A regular neighborhood U(α) of α will also be fixed by h up to isotopy, so a power of h fixes components of U(α), contradicting the assumption that h is atoroidal. Therefore, if one of the gei is a hyperbolic isometry, then ge1 and ge2 have no common fixed point on ∂∞λ. If both of them are lifts of some power of h, up to taking powers we may assume that they are lifts of the same power of h. Then there is an element γ∈π1(λ) such that ge1=γge2. The common fixed point q will also be fixed by γ, a contradiction. We have proved that ge1 and ge2 do not have common fixed points, which indicates that Qλ(e1)≠Qλ(e2).∎
We continue our proof of Lemma 3.5. The map Qλ preserves the cyclic order of the sides by the same reason as in the proof of Lemma 3.3, and the image is dense by the minimality of the π1(L)-action on ∂∞λ. Indeed, the hyperbolic structure on L has bounded injectivity radius, so the limit set of π1(L) is the entire circle at infinity.
We also have the following lemma, analogous to Lemma 3.2.
Lemma 3.6.
The intersection ∂p(λ)∩∂𝒪 is nowhere dense in ∂𝒪.
Proof.
Let e be a side of p(λ). By Lemma 3.4, e is a side of a shadow of a type-0 leaf. So we can take ge as in Proposition 3.1. By the discussion above, ge stabilizes p(λ), and the action near e has the desired expanding-contracting dynamics because of Proposition 3.1. The rest of the proof is the same as that of Lemma 3.2.∎
To continue the proof of Lemma 3.5, note that by Lemma 3.6, the map Qλ can be extended to ∂p(λ) by approaching any point in ∂p(λ)∩∂𝒪 by sides of ∂p(λ), as in the proof of Lemma 3.3. The extension is well defined and is continuous on ∂p(λ). To show that the extension is continuous on p(λ)∪∂p(λ), we again need the nice asymptotic property of rectangular neighborhoods of the sides. What we will show next is basically that Figure 8 is also a correct picture when λ is a type-1 leaf.
Again, let e be any side of p(λ), assumed to be a leaf of ℱ𝒪s. Let μ be a type-0 leaf negatively adjacent to λ such that e is also a side of p(μ), using Lemma 3.4. Finally, let ge, xe∈e and le be as in Proposition 3.1, and orient le to be pointing at xe. Let x′∈e be a point different from xe and set l′ to be a small segment of ℱ𝒪u(x′)∩p(μ) with an endpoint at x′. We orient l′ to be pointing toward e as well. By Proposition 3.1, Qμ(l′) is forward asymptotic to Qμ(le) under the orientation induced from l′ and le. Since ℱ𝒪s is regular near e on the side of p(λ), there is a pair of points pe∈le and p′∈l′ so that p′∈ℱ𝒪s(pe). Let σe and σ′ be the ϕ~-orbits corresponding to pe and p′, respectively. Since they lie in the same leaf of ℱ~s, σe and σ′ are positively asymptotic. It follows that the distance between σe∩λ and σ′∩λ is bounded by the distance between σe∩μ and σ′∩μ up to a uniform multiplicative constant. In other words, dλ(Qλ(pe),Qλ(p′)) is coarsely bounded by dμ(Qμ(pe),Qμ(p′)). Moreover, the pair {pe,p′} can be chosen arbitrarily close to e. Hence, Qλ(l′) and Qλ(le) are forward asymptotic. If RN(x′) is a rectangular neighborhood of x′ in p(μ), then we have shown that Qλ(RN(x′)) is a wedge-shaped domain in λ meeting ∂∞λ at exactly one point.
Since we have a similar description of the Qλ-image of a rectangular neighborhood, we can carry out the same argument and conclude that the extended Qλ is continuous on p(λ)∪∂p(λ), finishing the proof of Lemma 3.5.∎
The union of the shaded area is p(λ), and the heavily shaded area is p(μ), which is a subset of p(λ). The red arrows represent the map Qλ, and the blue ones represent Qλμ. One should think of each side of p(λ) or p(μ) as a single point at infinity.
For any leaf λ of ℱ~, either type-0 or type-1, we define a map Iλ:∂𝒪→∂∞λ as follows. For any ζ∈∂𝒪, if ζ is in p(λ)¯, set Iλ(ζ)=Qλ(ζ). If ζ is contained in an open interval Vζ in ∂𝒪\p(λ)¯, then there is a boundary leaf e of p(λ) with the same endpoints as Vζ. In this case, we define Iλ(ζ) to be Qλ(e). It is immediate from the definition that the maps Iλ are monotone surjections, therefore continuous, and π1(M)-equivariant, that is, for any g∈π1(M), λ∈ℱ~ and x∈∂𝒪, we have gIλ(x)=Igλ(gx), where the action g:∂∞λ→∂∞(gλ) is induced by the isometry g:λ→gλ.
If λ is a type-1 leaf and μ is a type-0 leaf adjacent to λ, then p(λ) contains p(μ), and the monotone quotient maps Iλ and Iμ satisfy the property that for any ξ1,ξ2∈∂𝒪, Iλ(ξ1)=Iλ(ξ2) implies Iμ(ξ1)=Iμ(ξ2). It follows that there is a continuous monotone surjection Iλμ:∂∞λ→∂∞μ such that Iμ=Iλμ∘Iλ. More precisely, for any point ξ∈∂∞λ, Iλμ(ξ) is defined to be Qμ(Qλ−1(ξ)). The maps Iλ and Iλμ can be visualized as in Figure 10.
Markers and universal circles
The outline of this section is the following. We first recall the definition of a universal circle (Definition 4.1) and prove Theorem 1.1. Then we review in Section 4.1 the construction from [7] of a particular universal circle 𝔖left, which we call the universal circle from leftmost sections, for any taut foliation. The circle 𝔖left arises from a collection of special sections, called the leftmost sections, of a circle bundle E∞ over Λ whose fibers are the circle at infinity of the leaves. The construction will then be examined carefully for our depth-one foliation ℱ in Section 4.2, where we study what a leftmost section looks like inside a product region, and in Section 4.3, where we analyze the behavior of a leftmost section at adjacent type-0 and type-1 leaves. The punchlines of this section are Lemmas 4.14 and 4.15, where we show that the leftmost sections can be determined by the structure of ∂𝒪 developed in Section 3.
The following axiomatic definition of a universal circle for ℱ first appears in [7]. It is worth remarking that although condition (2) seems not at all natural at first glance, it provides the universal circle with more interesting structures. In particular, it is necessary for the construction of invariant laminations in [5] (cf. Theorem 1.6).
Definition 4.1.
(Universal circle) A universal circle for ℱ is a circle 𝔖 with a faithful π1(M)-action and a monotone map Uλ:𝔖→∂∞λ, called a structure map, for any leaf λ of ℱ~ such that
for any leaf λ and any γ∈π1(M), the following diagram commutes:
if λ and μ are incomparable leaves, then the core of Uλ1 is contained in a single gap of Uλ2, and vice versa.
Two universal circles {𝔖,Uλ} and {𝔖′,Uλ′} are isomorphic if there is a π1(M)-equivariant homeomorphism h:𝔖→𝔖′ such that Uλ′∘h=Uλ for all λ.
Proof of Theorem <xref rid="j_GGD964_stat_001_001" ref-type="statement">1.1</xref>.
All the conditions of a universal circle in Definition 4.1 are obvious by properties of ∂𝒪 and the way we define Iλ except for condition (2). Suppose λ and μ are incomparable, then their shadows are disjoint. Otherwise, there is an orbit of the flow ϕ~ intersecting both leaves, contradicting their incomparability. Condition (2) is easily seen to be satisfied.∎
Calegari–Dunfield’s construction
In [7], Calegari–Dunfield describe an explicit construction of a universal circle 𝔖left for any taut foliation. We briefly review their construction below. For simplicity, we will stick to our ℱ instead of more general taut foliations.
The bundle E∞ is a circle bundle over Λ whose fiber at any leaf λ is the circle at infinity ∂∞λ. The topology of E∞ is defined as follows. For any transversal τ of ℱ~, τ embeds into Λ, and we identify τ with its embedding image in Λ. The unit tangent bundle of ℱ~ restricted to τ is the circle bundle UTℱ~|τ, and there is a natural map UTℱ~|τ→E∞|τ sending a tangent vector of a leaf to the ideal point it points toward. We require the map to be a homeomorphism. It is shown in [7] that this topology is well defined, that is, independent of the choice of τ.
Since ℱ is a taut foliation, there is an ε1>0 0\)]]> such that every leaf of ℱ~ is quasi-isometrically embedded in its ε1-neighborhood by [7, Lemma 2.4]. By the structure of depth-one foliations, there is a constant ε2>0 0\)]]> so that the ε2-neighborhood of ℱ0 is contained in a spiraling neighborhood. Fix ε0 to be min{ε1/3,ε2}.
Definition 4.2.
A marker for ℱ~ is an embedding
m:[0,1]×ℝ≥0→M~
such that
for any x∈[0,1], m({x}×ℝ≥0) is a geodesic ray in a leaf of ℱ~;
for any y∈ℝ+, m([0,1]×{y}) is a transversal with length bounded by ε0.
Any marker m gives a section s of E∞|τ, where τ is the image of m([0,1]×{y}) in Λ, such that for any leaf λ∈τ, s(λ) is the ideal endpoint of Image(m)∩λ. The image of τ under s is called the end of m.
Note that our choice of ε0 is different from but no larger than the constant chosen in [7]. Shrinking the constant will not affect the main results in their paper. In general, different ε might give rise to different universal circles, but in our case, it can be seen that for ε0 small enough (i.e., smaller than ε2 above), the leftmost universal circles are all isomorphic.
Given two marker ends, either they are disjoint or their union is an embedded closed interval in E∞ transverse to fibers.
A point ξ∈∂λ is called a marker endpoint if there is a marker m so that the end of m intersects ∂∞λ at ξ. The following theorem was originally announced by Thurston in an unpublished manuscript [32], and the proof is carefully written down in [7]. Heuristically, it says that the leaves of ℱ~ stay close in many directions.
Theorem 4.4.
(Thurston’s leaf pocket theorem) For any leaf λ of ℱ~, the set of marker endpoints in ∂∞λ is dense.
For any leaf λ of ℱ~ and any point ξ∈∂∞λ, there is a special section sξ of E∞, called the leftmost section starting fromξ, built as follows.
In Λ, there is a neighborhood τ of λ homeomorphic to a closed interval, and E∞|τ is a cylinder. We adopt the convention that the flow ϕ~ is flowing upward, and we are facing the cylinder E∞|τ from the outside. Take a finite collection C of marker ends in E∞|τ so that each fiber intersects at least one element in C. This is possible by Theorem 4.4. We build a path αC by starting from ξ, heading left horizontally in a fiber until we hit the first marker end in C and following the marker end to move upward. After we reach the top of the marker end, we turn left again, staying in a fiber until we hit the next marker end in C, and follow the same rules to move on until we reach the top of E∞|τ. We call this the leftmost-up rule, following [7]. We can also move downward from ξ, but in the rightmost-down way. This procedure gives us a staircase path αC in E∞|τ, which is an approximation to sξ (Figure 11).
Approximating the leftmost section on E∞|τ by starting from ξ and going leftmost up.
To go from the staircase approximations to the leftmost section sξ, we define sξ|τ to be the (rightmost) supremum above ξ and the (leftmost) infimum below ξ among all possible αC. To be precise, we view E∞|τ as τ×(ℝ/ℤ). For a leaf μ∈τ above λ, we define
sξ(μ)=supC(min(αC∩∂∞μ)).
For μ∈τ below λ, define
sξ(μ)=infC(max(αC∩∂∞μ)).
It was proved in [7] that the supremum and the infimum exist, and sξ is indeed a continuous section of E∞ over τ. We can define sξ for all leaves comparable to λ following this procedure.
Finally, we can branch out in Λ by turning around to reach incomparable leaves where sξ is not yet defined. More precisely, suppose μ is a leaf incomparable to λ. There is a sequence of leaves
λ0=λ,λ1,…,λn=μ
so that λ2i and λ2i+1 are comparable and λ2i+1 and λ2i+2 are non-separated. To illustrate the idea, we assume λ1 is above λ (see Figure 12 for the case when n=9). In this case, there is a product region Ω~1 so that λ1,λ2∈∂−Ω~1. Let l0 be the segment [λ,λ1] in Λ, and let l1 be the image of Ω~1 in Λ. By the above construction, sξ is already defined over l0, λ1 and l1. It is a consequence of Lemma 4.3 and Theorem 4.4 that sξ|l1 has a well-defined endpoint at ∂∞λ2 (see [7, Lemma 6.18]). We extend sξ to λ2 continuously, and follow the rightmost-down rule to define it over the segment l2:=[λ2,λ3]. We continue along the sequence λi until we have defined sξ(μ). Since the dual graph Λ∗ is a tree, there is a unique way to reach any incomparable μ from λ through such a sequence of λn. In the end, we have a section sξ that is well defined on the whole Λ.
The process of extending sξ is a process of branching out from λ and sweeping Λ. The values at leaves that are closer to λ are defined first, and the values at leaves farther away from λ are determined by the closer values. At each point of Λ, there is a direction of extension of sξ that points toward the direction away from λ, along which sξ is defined.
Extending a leftmost section to incomparable leaves.
There is a unique leftmost section starting from any point in E∞. The set of leftmost sections is denoted by LS. The images of two different leftmost sections might coalesce but can never cross each other. If ℓ is a line in Λ, the bundle E∞|ℓ is homeomorphic to a cylinder, and the leftmost sections restricted to ℓ give embedded lines on E∞|ℓ transverse to the fiber. For any three different leftmost sections, there is an embedded line in Λ so that the restrictions of the sections to this line have a well-defined cyclic order, and the cyclic order is independent of the choice of the line [7, Lemma 6.25]. The completion of LS with respect to the cyclic order is homeomorphic to a circle, denoted by 𝔖left. The fundamental group π1(M) acts naturally on LS, and the action extends to an action on 𝔖left. For any λ∈Λ, there is a valuation map Uλ:LS→∂∞λ given by
Uλ(s)=s(λ).
The map Uλ can be extended to a monotone map Uλ:𝔖left→∂∞λ.
The circle 𝔖left together with the π1(M)-action and the set of structure maps {Uλ}λ∈Λ is a universal circle for ℱ.
In order to prove Theorem 1.2, we need to analyze more carefully what marker ends and the leftmost sections look like on E∞. We will continue to use the terminologies from Section 3.
Markers contained in a product region
We first consider the ends of markers that are contained in a product region. The identification of Ω~ with L~×ℝ gives a canonical identification of E∞|Ω~ as ∂∞L~×ℝ. Here we implicitly use that for any homeomorphism between two infinite-type surfaces with standard hyperbolic structures, any lift to their universal covers extends continuously to a homeomorphism between their boundaries at infinity, and the extension is unique [10]. Denote the leaf L~×{t} by λt. Again, each λt is identified with L~.
Lemma 4.6.
For any ξ∈∂∞λt, there is an ε>0 0\)]]> so that {ξ}×[t,t+ε] is the end of a marker.
Proof.
We will use a tightening method described in [7, Section 5.3]. Take any point x∈λt and consider the geodesic ray γ from x to ξ. Since depth-one leaves in the same fibered region have asymptotic ends, for every small δ>0 0\)]]>, there is an ε such that any flowline of ϕ~ between λt and λt+ε has length <δ. Moreover, the map from λt to any λs with s∈(t,t+ε] induced by flowing λt to λs is K-bi-Lipschitz for some uniform K. Therefore, the flow image γs of γ in each such λs is a family of uniform quasi-geodesics with ideal endpoints (ξ,s). We can then tighten γs to geodesics γs∗ on λs. We claim that the union of the λs∗ for s∈[t,t+ε] is a continuous one-ended band with bounded width. That is because any pair of γs1 and γs2 are bounded Hausdorff distance from each other, and so are their geodesic tightenings γs1∗ and γs2∗. By the continuity of the leafwise hyperbolic metric, this only happens when γs∗ is a continuous family of geodesic rays. The union of γs∗ has bounded width because the tightening process only shifts the rays by a bounded amount. Finally, we can take ε even smaller to obtain a genuine marker with width <ε0 and with the end {ξ}×[t,t+ε].∎
The next corollary follows immediately from the construction of leftmost sections on comparable leaves.
Corollary 4.7.
(Leftmost sections on a product region are vertical) Suppose s is any leftmost section of E∞. If s(λt0)=(ξ,t0) for some t0∈ℝ, then s(λt)=(ξ,t) for all t.
Proof.
By the construction of leftmost sections, one can see that any leftmost section s has the following property: when extending s upward, if s meets the end e of a marker at a point x∈e, then s has to contain the part of e above x. The same is true if we are extending s downward: if s meets e at a point x, it must also contain everything in e below x. Since through any point in E∞|Ω~ there is a vertical marker end in both directions, the leftmost section s restricted to Ω~ is forced to be vertical.∎
Markers intersecting a type-0 leaf
We now consider the ends of markers intersecting a type-0 leaf. Let μ be a type-0 leaf covering a compact leaf Σ, and assume for the rest of this section that μ is in the positive boundary of Ω~. The case where μ is in the negative boundary of Ω~ is similar.
As in the previous discussion, we identify every depth-one leaf in Ω with L, and every type-1 leaf in Ω~ with L~. This gives us a homeomorphism between E∞|Ω~ and ∂∞L~×ℝ. Recall that in Section 3, we define a continuous map
Iλtμ:∂∞λt→∂∞μ
for any λt∈Ω~. Under the homeomorphism ∂∞λt≅∂∞L~, the map Iλtμ is the same map for any t when viewed as a map from ∂∞L~ to ∂∞μ. We denote this map by IΩ~μ.
In Λ, Ω~∪μ is a half-open interval. Let ℳμ,Ω~ be the set of markers m in μ∪Ω~ with one side lying in μ. By Corollary 4.7 and Lemma 4.3, the end of such an m intersects ∂∞μ at a single point ξμ(m) and intersects E∞|Ω~ in a vertical segment
{(ξΩ~(m),t)∣t>T}, T\},$$]]>
where T is a constant depending on m, and ξΩ~(m) is a point in ∂∞L~ depending on m and independent of t (Figure 13).
The end of a marker m (in red) in ℳμ,Ω~.
Define the set ξΩ~(ℳμ,Ω~):={ξΩ~(m)∣m∈ℳμ,Ω~}⊂∂∞L~. Intuitively, these are the directions on L~ in which μ does not diverge from λt.
Recall that the positive escaping set 𝒰+⊂L is the set of points whose ϕ-orbit escapes Ω in positive time. Let 𝒰~+ be the preimage of 𝒰+ in L~. There is a component 𝒰~μ+ of 𝒰~+ so that x∈λt is a point of 𝒰~μ+ if and only if ϕ~(x) hits μ. The hitting map Hμ:𝒰~μ+→μ defined by
Hμ(x)=ϕ~(x)∩μ
is a homeomorphism since it is an open bijection. It is tautological that
p|μ∘Hμ=p|𝒰~μ+,
where p is the projection to 𝒪 as in Section 3.
Lemma 4.8.
Let α∗ be an oriented simple closed geodesic in Σ with trivial ℱ-holonomy on the side of Ω, and let α~∗ be a lift of α∗ to μ. Denote the forward endpoint at infinity of α~∗ by ∂+α~∗. Then there is a marker m∈ℳμ,Ω~ so that ξμ(m)=∂+α~∗ and ξΩ~(m)∈IΩ~μ−1(∂+α~∗).
Proof.
Perform a homotopy of α∗ along ϕ into Ω for a short distance so that the final image is a simple closed curve on a depth-one leaf in Ω. This is possible because α∗ has trivial ℱ-holonomy on the side of Ω. The full homotopy image is an annulus, denoted by A. We lift A to M~ to get a two-ended infinite band A~ in Ω~∪μ with one side being α~∗ and the other side on λT for some T. Note that A has bounded width because it is a lift of an annulus.
Fix a base point x∈α~∗, and let α~∗(x) be the oriented ray in α~∗ starting from x toward ∂+α~∗. We restrict the lifted homotopy to the ray α~∗(x), and the restricted homotopy image is a one-ended infinite band B⊂A~ (Figure 14). The band B has finite width between μ and λT, and we can make the width arbitrarily small by cutting the homotopy at λT′ for large enough T′. When t>T′ T^{\prime}\)]]>, the ray B∩λt has a well-defined endpoint at infinity because it is contained in a lift of a simple closed curve on a hyperbolic depth-one leaf. Similar to the proof of Lemma 4.6 and the tightening method in [7, Section 5.3], by taking even larger T′ and pulling tight the intersection of B and λt, we get a marker m∈ℳμ,Ω~. Note that different from Lemma 4.6, here we do not have uniform bi-Lipschitz maps from μ to type-1 leaves in Ω~. However, we still have that the rays B∩λt are a family of uniform quasi-geodesics by the continuity of the leafwise metric, so the same tightening argument as in [7, Section 5.3] still works. It is clear from the construction that ξμ(m)=∂+α~∗. The point ξΩ~(m) is the endpoint of the ray
B∩λt=Hμ−1(α~∗(x)).
By (4.1), the projections of α~∗(x) and Hμ−1(α~∗(x)) to 𝒪 are identical, and both rays escape to infinity in the leaf. This implies ∂+α~∗∈∂p(μ)∩∂p(λt) and
ξΩ~(m)∈IΩ~μ−1(∂+α~∗).∎
An annulus without holonomy (shaded on the left) lifts to a band B (shaded on the right) that gives a marker.
To state the next lemma, we need one more definition. A simple closed curve β⊂L is called a Σ-juncture if β is a connected positive f-juncture and β covers a simple closed curve in Σ under the covering map 𝒰+→∂+Ω (note that Σ is only a component of ∂+Ω).
Corollary 4.9.
Let β⊂L be a Σ-juncture, and let β~ be a lift of β in L~ that lies in 𝒰~μ+. Then both endpoints of β~ are in ξΩ~(ℳμ,Ω~).
Proof.
Suppose β covers a simple closed curve α on Σ. Fix an orientation of β, which induces an orientation of α. The curve α has trivial ℱ-holonomy on the side of Ω because β is an f-juncture. For any T, we can view β~ as an embedded line in λT. There is a lift α~⊂μ of α given by α~=Hμ(β~). Let ∂+α~ be the forward endpoint of α~. Applying Lemma 4.8 to the geodesic tightening α∗ of α, we obtain a marker m∈ℳμ,Ω~. But since α~ is obtained by flowing β~ to μ, the proof of Lemma 4.8 implies that ξΩ~(m) is exactly the forward endpoint of β~. The same proof applies for the backward endpoint of β~, proving the corollary.∎
We define the limit set ℒ(𝒰~μ+) of 𝒰~μ+ as the intersection cl(𝒰~μ+)∩∂∞L~, where cl(𝒰~μ+) is defined to be the closure taken in L~∪∂∞L~≅ℍ3¯.
Lemma 4.10.
The closure in ∂∞L~ of ξΩ~(ℳμ,Ω~) is ℒ(𝒰~μ+).
Proof.
Suppose m is a marker in ℳμ,Ω~. The intersection of the image of m with any λt is a ray rt. After identifying λt with L, we claim that rt is contained in 𝒰~μ+. This is because the image of m is ε0-close to μ. By our choice of ε0, the ε0-neighborhood of Σ in M is contained in an ℱ-spiraling neighborhood N(Σ) of Σ, and every ϕ-orbit intersecting the spiraling neighborhood will hit Σ. So rt is contained in 𝒰~μ+, and the ideal endpoint of rt, which is exactly ξΩ~(m) by definition, will then be contained in ℒ(𝒰~μ+). This shows ξΩ~(ℳμ,Ω~)⊂ℒ(𝒰~μ+).
By Corollary 4.9, it now suffices to show that the endpoints of lifts of any Σ-juncture in 𝒰~μ+ are dense inside ℒ(𝒰~μ+). We first recall some known facts and constructions. By [11], the geodesic tightenings of a system of positive f-junctures limit to the negative Handel–Miller geodesic lamination ΛHM− under negative iterations of f, and ΛHM− is independent of the choice of junctures. On the other hand, the intersection of ℱs with L induces a singular foliation ℱLs on L. We define W− as the restriction of ℱLs to the complement of 𝒰+. The complement of 𝒰+ is saturated by leaves of ℱLs and W− is a singular sublamination of ℱLs by [23]. Let W~− be the lift of W− to L~. The singular lamination W~− determines an abstract lamination on ∂∞L~ by a standard construction (Section 2). It follows from [23, Theorem 8.4] that W~−=ΛHM− as abstract laminations (the paper proves it for circular pseudo-Anosov flows, but the same method applies to any pseudo-Anosov flow without perfect fits).
Now fix a Σ-juncture β. Suppose that β is the boundary of a contracting neighborhood of a contracting end ℰ, and n is the smallest positive integer so that fn(ℰ)=ℰ. The above facts imply that the endpoints of ∂𝒰~μ+ can be approximated by endpoints of lifts of {f−in(β)}i≥0. The limit set ℒ(𝒰~μ+) is nowhere dense by the next lemma (Lemma 4.11), so points in ℒ(𝒰~μ+) are approximated by endpoints of ∂ℒ(𝒰~μ+). Also note that if β is a Σ-juncture, so is f−in(β). Therefore, the endpoints of lifts of any Σ-juncture in 𝒰~μ+ are dense inside ℒ(𝒰~μ+), which completes the proof.∎
We give a proof of the following fact used in the proof of Lemma 4.10.
Lemma 4.11.
The limit set ℒ(𝒰~μ+) is nowhere dense in ∂∞L~.
Proof.
The subsurface 𝒰+⊂L is bounded by leaves of ℱLs [23], each of which has distinct well-defined endpoints at infinity. In particular, ℒ(𝒰~μ+) is a proper subset of ∂∞L~. If ℒ(𝒰~μ+) contains a non-trivial interval I, we can find a hyperbolic element g∈π1(L) with a fixed point in I. Here we are using the fact that L has bounded injectivity radius. But g(𝒰~μ+) is either disjoint from or equal to 𝒰~μ+ since 𝒰+ is embedded. So gn(I) is contained in ℒ(𝒰~μ+) for any integer n, a contradiction.∎
Lemma 4.12.
There is a subset ℳμ,Ω~′ of ℳμ,Ω~ such that
the set ξμ(ℳμ,Ω~′)is dense in∂∞μ;
the set ξΩ~(ℳμ,Ω~′)is dense inℒ(𝒰~μ+);
for any m∈ℳμ,Ω~′, we have
ξμ(m)=IΩ~μ(ξΩ~(m)).
Proof.
The subset ℳμ,Ω~′ can be taken to be the set of markers in ℳμ,Ω~ that arise from Lemma 4.8. Item (3) is exactly the content of Lemma 4.8. Item (1) is true because the endpoints of the lifts of a simple closed curve are dense in ∂∞μ. The second part of the proof of Lemma 4.10 only used markers in ℳμ,Ω~′, so we have already proved item (2).∎
Lemma 4.13.
If η is a point in ∂∞μ such that IΩ~μ−1(η) is a closed interval with positive length, then η∉ξμ(ℳμ,Ω~).
Proof.
Suppose there is a marker m∈ℳμ,Ω~ such that ξμ(m)=η. Let rμ be the geodesic ray m∩μ, and let rt be the geodesic ray m∩λt. Let e be the side of p(μ) corresponding to η. Then the interior of e is contained in p(λt) for all t. Take a quasi-geodesic ray r′ in μ such that p(r′) has an endpoint in the interior of e. For example, using the notations in Proposition 3.1, for any x∈e, we can take r′ to be a ray in lx with one end at x. Since r′ and rμ both have endpoint μ, they have bounded Hausdorff distance from each other on μ. By the choice of ε0 in Definition 4.2, each rt in m flows forward in bounded time to hit μ, and the image on μ is exactly Hμ(rt). Therefore, r′ has bounded Hausdorff distance from rt and Hμ(rt) as well. Since ε0 is smaller than a separation constant (Definition 4.2), r′ and Hμ(rt) are also a bounded Hausdorff distance from each other in μ.
Now fix a λt intersecting m, and for any y∈μ, define T(y) to be the time it takes for y to flow backward to λt. If we write ϕ~T for flowing along ϕ~ for time T, then T(y) satisfies ϕ−T(y)(y)∈λt. This function T is bounded on Hμ(rt) by the above. On the other hand, since we pick r′ to have an endpoint in the interior of p(λt), T(y) must go to +∞ as y travels along r′ to infinity. We will show that this is a contradiction. To see this, take sequences yn∈Hμ(rt) and zn∈r′ with dμ(yn,zn)≤C, where C is a constant, and with xn (and hence yn) going to infinity. Then T(yn) is bounded, while T(zn) goes to +∞. By acting with deck transformations in π1(μ), we can find translates (yn′,zn′) of (yn,zn) that stay in a compact subset of μ. Since the transformations preserve ℱ~, ϕ~−T(yn)(yn′) and ϕ~−T(zn)(zn′) are in the same type-1 leaf in Ω~. Assume yn′ converges to y and zn′ converges to z, and up to taking a subsequence, assume that T(yn) converges to a finite positive number T0. Then the flowline ϕ~(z) will never intersect the type-1 leaf containing ϕ~−T0(y), which is a contradiction because every flowline intersecting Ω~ will intersect every leaf in Ω~. This proves that such a marker m does not exist.∎
The following lemmas, Lemmas 4.14 and 4.15, together with Corollary 4.7, will give us the complete rules to build the leftmost section starting from a given point. Lemma 4.14 tells us how to extend the leftmost section from a product region to an adjacent type-0 leaf, and Lemma 4.15 tells us how to go from a type-0 leaf to an adjacent product region.
For any ξ∈∂∞L~, we use cξ to denote the vertical section of E∞|Ω~ given by cξ(λt)=(ξ,t).
Lemma 4.14.
For any ξ∈∂∞(L~), the vertical section cξ extends continuously to μ by setting cξ(μ)=IΩ~μ(ξ).
Proof.
First of all, we remark that as noted in [7, Proof of Lemma 6.18], the closure of cξ in E∞ is a closed interval transverse to the circle fibers, intersecting ∂∞μ in exactly one point. This is a consequence of the density of markers, and it implies the section cξ has a unique continuous extension to μ.
To determine the value of cξ at μ, we first consider the case where ξ is in a gap Gξ of IΩ~μ. Let ∂rGξ and ∂lGξ be the leftmost and the rightmost endpoints of Gξ, respectively. By Lemma 4.12, we have sequences {mn+} and {mn−} in ℳμ,Ω~′⊂ℳμ,Ω~ with the following properties:
the points ξΩ~(mn−) limit to ∂lGξ from the left and the points ξΩ~(mn+) limit to ∂rGξ from the right;
we have ξμ(mn±)=IΩ~μ(ξΩ~(mn±)).
These properties imply that the points ξμ(mn−) limit to IΩ~μ(ξ) from the left and the points ξμ(mn+) limit to IΩ~μ(ξ) from the right by item (3) of Lemma 4.12 and Lemma 4.3. The two sequences of marker ends pin down the endpoint of cξ to be IΩ~μ(ξ) (see the left-hand side of Figure 15).
The subset ℳμ,Ω~′ (red) pins down the way to extend the leftmost sections (blue), with the blue arrows indicating the direction of extension. The left-hand side corresponds to Lemma 4.14 and the right-hand side corresponds to Lemma 4.15.
When ξ is not in any gap of IΩ~μ, the proof can be done similarly to above by replacing Gξ, ∂rGξ and ∂lGξ all by the point ξ.∎
Lemma 4.15.
Let η be a point in ∂∞μ. Suppose ν is any leaf of ℱ~, ξ is any point in ∂∞ν and sξ is the leftmost section starting from ξ. If the direction of extension of sξ at μ points from μ to Ω~, that is, μ is closer to ν in Λ than Ω~, and sξ(μ)=η, then sξ(λt) is the rightmost endpoint of IΩ~μ−1(η). Here we view a singleton as a closed interval of length zero.
Remark 4.16.
We remind the readers that here we are assuming μ to be in the positive boundary of Ω~, as stated in the first paragraph of Section 4.3, so the direction of extension of sξ is the backward flow direction. If μ is in the negative boundary of Ω~ and the direction of extension of sξ is the forward flow direction pointing from μ to Ω~, the lemma remains true if we replace “the rightmost endpoint” by “the leftmost endpoint” with the same proof.
Proof.
We refer the readers to Figure 15 for an illustration of the situation. The value of sξ on Ω~ is determined by going down from η and following the rightmost-down rule, described in Section 4.1. Since, by Corollary 4.7, sξ can only go vertically down in Ω~, its value in Ω~ is the leftmost infimum of ξΩ~(m) over all m∈ℳμ,Ω~ such that ξμ(m) is in a small neighborhood U of η and ξμ(m) is not on the left of η (i.e., after identifying U with an interval in ℝ, η≤ξμ(m)). By Lemma 4.12, there is a sequence of markers mn∈ℳμ,Ω~′⊂ℳμ,Ω~ so that ξμ(mn) limits to η from the right. By the monotonicity of IΩ~μ and item (3) of Lemma 4.12, we have that ξΩ~(mn) limits to the rightmost endpoint of IΩ~μ−1(η) from the right. If IΩ~μ−1(η) is a single point, this implies that sξ(λt)=IΩ~μ−1(η). If IΩ~μ−1(η) is a closed interval, the existence of mn and Lemma 4.13 imply sξ(λt) is the rightmost endpoint of IΩ~μ−1(η).∎
Building the homeomorphism
Suppose λ is a leaf of ℱ and ξ is a point in ∂∞λ. Let Vξ denote Iλ−1(ξ)⊂∂𝒪, which is either a closed interval or a singleton. If Vξ is a closed interval, let ∂lVξ and ∂rVξ be the leftmost and the rightmost endpoints of Vξ, respectively. Note that by our convention, left means clockwise and right means counterclockwise. We say Vξ is a stable (resp. unstable) gap of λ if Qλ−1(ξ) is a boundary ℱ𝒪s-leaf (resp. ℱ𝒪u-leaf) of p(λ). Note that a closed interval of ∂𝒪 cannot be a stable gap of a leaf while being an unstable gap of another. Indeed, a leaf of ℱ𝒪s and a leaf of ℱ𝒪u cannot bound an ideal bigon. In fact, for ϕ without perfect fits, a stronger statement is true: any two rays in ℱ𝒪s∪ℱ𝒪u have distinct endpoints at infinity [13, Lemma 3.20]. We say a closed interval of ∂𝒪 is a stable (resp. an unstable) gap if it is a stable (resp. an unstable) gap of a leaf of ℱ~.
Since 𝔖left is the completion of the set LS of leftmost sections, to define the homeomorphism T:𝔖left→∂𝒪, it suffices to define T on LS and show that it admits an extension. For any point ξ in E∞, let sξ be the leftmost section starting from ξ. The set LS∗ of pointed leftmost sections is the set
{(s,ξ)∣s∈LS,ξ∈E∞,s=sξ}.
We point out that this definition is not redundant, for it is possible to have sξ=sξ′ with ξ≠ξ′. There is a natural forgetful map π:LS∗→LS given by forgetting the starting point. We define a map T∗:LS∗→∂𝒪 as follows. For a pointed leftmost section (s,ξ), consider Vξ. If Vξ is a single point, define T∗(s,ξ)=Vξ. If Vξ is an unstable gap, define T∗(s,ξ)=∂lVξ. If Vξ is a stable gap, define T∗(s,ξ)=∂rVξ.
Theorem 1.2 follows from the following theorem.
Theorem 5.1.
The map T∗ descends to a map T:LS→∂𝒪 that extends continuously to a map from 𝔖left to ∂𝒪, which we will again denote by T. The extension T is injective and preserves the cyclic order; hence it is a homeomorphism. Moreover, T is π1(M)-equivariant, and for any λ∈Λ and any s∈𝔖left, we have Uλ(s)=Iλ(T(s)). In other words, T is an isomorphism of universal circles between 𝔖left and ∂𝒪.
The rest of this section will be dedicated to proving Theorem 5.1.
Lemma 5.2.
Suppose (s,ξ1) is an element in LS∗. Let μ be any leaf of ℱ~ and suppose s(μ)=ξ. Then T∗(s,ξ1)∈Vξ.
Proof.
By definition, we have s=sξ1 is the leftmost section starting from ξ1, where ξ1∈∂∞λ1 for some λ1∈Λ. We assume that λ1 is a type-1 leaf for simplicity. The case when λ1 is type-0 is basically the same.
We first consider the case when λ1 and μ are comparable and λ1<μ. Take a sequence of ℱ~-leaves
λ1→λ2→λ3→⋯→λn=μ,
where λ2i is a type-0 leaf, λ2i+1 is a type-1 leaf (note that here we set up the notations so that λk is of type k mod 2) and λi≲λi+1. The sequence represents the shortest path from Ω~(λ1)∗ to μ∗ in Λ∗, after identifying a type-0 leaf with the dual vertex and a type-1 leaf with the vertex dual to the product region containing it. We record the value of s along this sequence by si=s(λi), and let Vi=Vs(λi). We have a sequence of closed intervals (possibly with length zero) Vi⊂∂𝒪. The goal is to show that for all i=1,…,n, we have T∗(s,ξ0)∈Vi. In particular, this implies T∗(s,ξ0)∈Vn=Vξ. We will show this by tracking how the intervals Vi vary along the sequence λi. By Lemma 4.14, we have V2i−1⊂V2i; by Lemma 4.15, we have V2i+1⊂V2i.
Lemma 5.3.
If V2i−1⊊V2i, then V2i is a stable gap. If V2i+1⊊V2i, then V2i is an unstable gap and V2i+1=∂lV2i.
Proof.
First, suppose V2i−1⊊V2i. The leaf e:=Qλ2i−1(Iλ2i(V2i)) is a side of p(λ2i) containing in the interior of p(λ2i−1). The fixed point xe in e under Stab(e) corresponds to a periodic ϕ~-orbit γ in M~ intersecting λ2i−1 but not intersecting λ2i. Since every ϕ~-orbit in ℱ~s(ϕ) is forward asymptotic to γ, we see that ℱ𝒪s(x) is not contained in p(λ2i). This means e=ℱ𝒪s(x) and V2i is a stable gap.
Now suppose V2i+1⊊V2i. A similar argument to the above shows V2i is an unstable gap. By Lemma 4.15 and Remark 4.16, if V2i+1 is a closed interval of positive length, there will be a boundary leaf e of p(λ2i) and a boundary leaf e′ of p(λ2i+1) different from e and sharing the leftmost endpoint with e. But this cannot happen because when ϕ has no perfect fits, no pair of leaves in ℱ𝒪s∪ℱ𝒪u can share an endpoint. Therefore, V2i+1 is a single point, and it is the leftmost endpoint of V2i by Lemma 4.15.∎
We continue the proof that T∗(s,ξ1)∈Vi for all 1≤i≤n. It is obvious that T∗(s,ξ1)∈V1 by the definition of T∗. If V1 is a single point or a stable gap of λ1, by Lemma 5.3 we have Vi⊂Vi+1 for all i. So we have T∗(s,ξ1)∈Vi.
If V1 is an unstable gap, then there are two cases. If for all 1≤i≤n, we have Vi=V1, then there is nothing to prove. If this is not the case, let N be the first positive integer so that VN≠V1. If V1⊊VN, then VN is a stable gap by Lemma 5.3, and we again have Vi⊂Vi+1 for all i≥N−1. If VN⊊V1, then VN=T∗(s,ξ1)=∂lV1 by definition and Lemma 5.3. We use Lemma 5.3 again to see that {Vi}i≥N is a monotone increasing sequence of closed intervals. In any case, we have T∗(s,ξ1)∈Vi for all i.
The case when μ<λ1 can be proved using a similar argument. Thus the lemma is proved for μ comparable to λ1.
Now suppose μ is not comparable to λ1. We again consider the shortest path from λ1 to μ in Λ∗ similar to above and track how Vi changes along the path. To illustrate the idea, we consider the following example. Suppose the shortest path from λ to μ in Λ∗ is of length five:
λ1→λ2→⋯→λ5=μ,
where λ1,λ3,λ5 are type-1 leaves, λ2,λ4 are type-0 leaves, and they satisfy λ1≲λ2≲λ3 and λ5≲λ4≲λ3. Let Vi=Vs(λi). We made a turn at λ3 from the positive flow direction to the negative direction. Our previous discussion shows that T∗(s,ξ1)∈Vi for i=1,2,3. Since λ2 and λ4 are incomparable, the core of Iλ2 is contained in a single unstable gap G of Iλ4. The interval V4 must be the gap G containing V3 by Lemmas 4.14 and 4.15 and the definition of T. In particular, we have T∗(s,ξ1)∈V4. Since λ4 is negatively adjacent to λ3 and V3⊊V4, a similar argument to the proof of Lemma 5.3 shows that V4 is an unstable gap, and an unstable gap will only become larger as we track Vi backward. Therefore, we have T∗(s,ξ1)∈V4⊂V5.
In general, the path from λ to μ has a finite number of turns. If we track the interval Vi along the path, at a turn from the positive direction to the negative direction, Vi will become a larger unstable gap and can only grow even larger until the next turn happens. Similarly, if we turn from the negative direction to the positive direction, Vi will become a larger stable gap and can only grow even larger until the next turn happens. Hence, Vi will be non-decreasing after we make the first turn. But we have shown that T∗(s,ξ1) is in Vi before we made any turn in the first part of the proof. So the proof of Lemma 5.2 is completed.∎
Corollary 5.4.
Let (s,ξ1)∈LS∗ be a pointed leftmost section. Then we have T∗(s,ξ1)=⋂λ∈ΛVs(λ).
Proof.
Lemma 5.2 already shows that T∗(s,ξ1)∈⋂λ∈ℱ~Vs(λ), so it suffices to prove that there is some λ with Vs(λ) being a singleton. Suppose ξ1 is at the infinity of the leaf λ1. If Vξ1 is a single point, it is trivial. If Vξ1 is a non-trivial closed interval, let e be the side of p(λ1) facing Vξ1, and let xe be the periodic point in e as in Proposition 3.1. The periodic orbit p−1(xe) intersects some leaf λ comparable to λ1. If we take a path in Λ∗ from λ1 to λ and record the intervals Vi as in the proof of Lemma 5.2, there must be some i so that Vi+1⊊Vi. Otherwise, we have V1⊂V2⊂⋯ and so Iλ has a gap containing Vξ1, but that contradicts xe∈p(λ). By Lemma 5.3, Vi+1 is a single point, so is the intersection ⋂λ∈ℱ~Vs(λ). The lemma is proved.∎
Corollary 5.5.
There is a map T:LS→∂𝒪 so that the following diagram commutes:
Proof.
For any s∈LS, pick a starting point ξ for s and define T(s)=T∗(s,ξ). By Corollary 5.4, we have T∗(s,ξ)=T∗(s,ξ′) for any (s,ξ),(s,ξ′)∈LS∗. Therefore, the map T is well defined.∎
Corollary 5.6.
The map T preserves the cyclic order of elements in LS. In particular, T is injective.
Proof.
The cyclic order of leftmost sections is determined by the cyclic order of their values on embedded lines in the leaf space ([7, Lemma 6.25]; see also Section 4). Their images under T must follow the same cyclic order by Lemma 5.2.∎
Lemma 5.7.
The image of LS under T is dense.
Proof.
It is clear that T is π1(M)-equivariant, so T(LS) is a π1(M)-invariant subset of ∂𝒪. The lemma follows from the minimality of the π1(M)-action on ∂𝒪 [20].∎
Lemma 5.8.
The map T:LS→∂𝒪 extends continuously to a homeomorphism T:𝔖left→∂𝒪.
Proof.
By Corollary 5.6, Lemma 5.7 and the fact that 𝔖left is the completion of 𝒬, there is a unique continuous extension of T to 𝔖left, and the extended T is a homeomorphism between 𝔖left and ∂𝒪.∎
Proof of Theorem <xref rid="j_GGD964_stat_005_052" ref-type="statement">5.1</xref>.
It suffices to show the “moreover” part about the map T defined above. The π1(M)-equivariance is automatic from the way we define T. The structure maps are intertwined by T because of Lemma 5.2.∎
It is also possible to define the universal circle from rightmost sections 𝔖right by considering the completion of rightmost sections (i.e., the sections of E∞ that go rightmost up and leftmost down). In general, there is no reason to expect 𝔖left=𝔖right. However, we have the following corollary of Theorem 1.2.
Corollary 5.9.
Under the assumption of Theorem 1.2, the universal circles 𝔖left and 𝔖right are isomorphic.
Proof.
Using the same proof of Theorem 1.2, it can be shown that 𝔖right is isomorphic to ∂𝒪, hence isomorphic to 𝔖left.∎
Corollary 5.9 suggests that one can view the sets of leftmost and rightmost sections as different dense subsets of the same circle. Indeed, the homeomorphisms from 𝔖left and 𝔖right to ∂𝒪 give embeddings of the sets of leftmost sections and rightmost sections into the ∂𝒪, both with dense image. Corollary 5.4 is true for both embeddings, so leftmost sections and rightmost sections never cross.
Invariant laminations
We conclude with a discussion of the invariant laminations on ∂𝒪≅𝔖left. See Section 2 for a discussion about laminations on a circle.
Any λ∈Λ separates Λ into two components, the one Λ+(λ) containing the flow positive side of λ and the one Λ−(λ) containing the flow negative side of λ. The leaf λ also separates M~ into two parts, denoted by M~+(λ) and M~−(λ) with the same sign convention. Define a subset Ξ±(λ) of Symm2(∂𝒪) by
Ξ±(λ)=∂CH(⋃μ∈Λ±(λ)core(Iμ)).
The set Ξ± is then defined as
Ξ±=⋃λ∈ΛΞ±(λ)¯.
It is proved in [5] that Ξ± is indeed a pair of π1(M)-invariant laminations on ∂𝒪.
We are now ready to prove Theorem 1.6. Recall that ℒ𝒪s/u is the lamination on ∂𝒪 induced by ℱ𝒪s/u.
Proof of Theorem <xref rid="j_GGD964_stat_001_007" ref-type="statement">1.6</xref>.
Take any type-0 leaf λ and consider the shadow p(λ). Since λ is not a fiber, [19, Proposition 4.6] shows that ∂p(λ) has leaves in both ℱ𝒪u and ℱ𝒪s. Suppose that e is a side of p(λ) that is contained in a leaf of ℱ𝒪s and consider the leftmost section s starting from Qλ(e). If μ is a leaf in Λ+(λ), we take a path in Λ∗ from λ to μ and track the closed interval Vi as in the proof of Lemma 5.2. The interval V0 is the stable gap of Iλ facing e, and the proof of Lemma 5.2 shows that Vi is monotone increasing along the path. This means the shadow of μ is on the same side of e as p(λ). The side e, viewed as an element of Symm2(∂𝒪), is therefore in Ξ+(λ), hence in Ξ+. By transitivity of ϕ, every leaf of ℱs is dense in M. This implies that the π1(M)-image of ∂e is dense in ℒ𝒪s because ϕ has no perfect fits. Since both Ξ+ and ℱ𝒪s are π1(M)-invariant and closed, we have Ξ+⊃ℒ𝒪s.
If Ξ+−ℒ𝒪s is not empty, the difference must be a union of diagonals of complementary regions of ℒ𝒪s. Note that these diagonals cannot be approximated by leaves in ℒ𝒪s∪ℒ𝒪u, so there must be such a diagonal d in Ξ+(λ) for some λ. The corresponding complementary polygon comes from a singular leaf l of ℱ𝒪s, and we denote the singularity in l by s.
Suppose d has endpoints ξ and ξ′. Then there is a sequence of leaves λn∈Λ+(λ), sides en of p(λn) and endpoints ξn of en so that ξn converges to ξ. Up to taking a subsequence, we can assume that all en are contained in leaves of ℱ𝒪s or in leaves of ℱ𝒪u. If all en are contained in leaves of ℱ𝒪u, then d cannot be a boundary component of the convex hull of ⋃μ∈Λ±(λ)core(Iμ), because en will eventually cross d by the absence of perfect fits. So all en are contained in leaves of ℱ𝒪s. In particular, the singularity s can be approximated by points in shadows of leaves in Λ+(λ). We will show that this is impossible, a contradiction.
If s is in p(λ), then there are points of core(Iλ) on both sides of d, contradicting the assumption that d∈Ξ+(λ). Therefore, the ϕ~-orbit p−1(s) is disjoint from λ. If p−1(s) is contained in M~+(λ), it contradicts our assumption that d∈Ξ+(λ) by a similar reason. So p−1(s) is contained in M~−(λ). Note that s is not in ∂p(λ), otherwise a face of l will be a side of p(λ), and the face will be a leaf of Ξ+(λ) as we showed above. This again contradicts the assumption that d∈Ξ+(λ). So orbits close enough to p−1(s) will stay in M~−(λ), giving the desired contradiction.
Therefore, Ξ+ must be the same as ℒ𝒪s. For the same reason, we have Ξ−=ℱ𝒪u, finishing the proof of Theorem 1.6.∎
Acknowledgements
The author is grateful to his advisor, Yair Minsky, for being inspiring and supportive throughout this project. The author would like to thank Hyungryul Baik, Ellis Buckminster, Danny Calegari, Sergio Fenley, Michael Landry, Anna Parlak and Sam Taylor for helpful comments and conversations. The author thanks the referee for carefully reading the paper and for giving numerous insightful comments and suggestions.
ReferencesAgolI.TsangC. C.Dynamics of veering triangulations: infinitesimal components of their flow graphs and applicationsAlgebr. Geom. Topol.2420246340134531552.57031481222210.2140/agt.2024.24.3401BarbotT.Caractérisation des flots d’Anosov en dimension 3 par leurs feuilletages faiblesErgodic Theory Dynam. Systems15199522472700826.58025133240310.1017/S0143385700008361BarthelméT.FrankelS.MannK.Orbit equivalences of pseudo-Anosov flowsInvent. Math.240202531119119208038253490216110.1007/s00222-025-01332-1BarthelméT.MannK.Pseudo-Anosov flows: a plane approach202520262509.15375v2CalegariD.Promoting essential laminationsInvent. Math.166200635836431106.57014225739210.1007/s00222-006-0004-3CalegariD.Foliations and the geometry of 3-manifoldsOxford Math. Monogr.Oxford University PressOxford20073631118.57002232736110.1093/oso/9780198570080.001.0001CalegariD.DunfieldN. M.Laminations and groups of homeomorphisms of the circleInvent. Math.152200311492041025.57018196536310.1007/s00222-002-0271-6CandelA.Uniformization of surface laminationsAnn. Sci. Éc. Norm. Super. (4)26199344895160785.57009123543910.24033/asens.1678CantwellJ.ConlonL.Poincaré–Bendixson theory for leaves of codimension oneTrans. Amer. Math. Soc.265198111812090484.57015060711610.2307/1998490CantwellJ.ConlonL.Hyperbolic geometry and homotopic homeomorphisms of surfacesGeom. Dedicata1772015127421359.37097337002010.1007/s10711-014-9975-1CantwellJ.ConlonL.FenleyS. R.Endperiodic automorphisms of surfaces and foliationsErgodic Theory Dynam. Systems4120211662121464.57022419005210.1017/etds.2019.56CooperD.LongD. D.ReidA. W.Bundles and finite foliationsInvent. Math.118199412552830858.57015129211310.1007/BF01231534FenleyS.Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometryGeom. Topol.162012111101279.37026287257810.2140/gt.2012.16.1FenleyS.MosherL.Quasigeodesic flows in hyperbolic 3-manifoldsTopology40200135035370990.53040183899310.1016/S0040-9383(99)00072-5FenleyS. R.Asymptotic properties of depth one foliations in hyperbolic 3-manifoldsJ. Differential Geom.36199222693130766.53018118038410.4310/jdg/1214448743FenleyS. R.Anosov flows in 3-manifoldsAnn. of Math. (2)13919941791150796.58039125936510.2307/2946628FenleyS. R.The structure of branching in Anosov flows of 3-manifoldsComment. Math. Helv.73199822592970999.37008161170310.1007/s000140050055FenleyS. R.Foliations with good geometryJ. Amer. Math. Soc.12199936196760930.53024167473910.1090/S0894-0347-99-00304-5FenleyS. R.Surfaces transverse to pseudo-Anosov flows and virtual fibers in 3-manifoldsTopology38199948238590926.57009167980110.1016/S0040-9383(98)00030-5FenleyS. R.Geometry of foliations and flows I: almost transverse pseudo-Anosov flows and asymptotic behavior of foliationsJ. Differential Geom.81200911891160.57026247789110.4310/jdg/1228400628FieldE.KimH.LeiningerC.LovingM.End-periodic homeomorphisms and volumes of mapping toriJ. Topol.1620231571051567.57023453249010.1112/topo.12277FriedD.Fibrations over S1 with pseudo-Anosov monodromyTravaux de Thurston sur les surfaces51266Astérisque 66–67Société Mathématique de FranceParis19790446.570230568308Landry,M. P.Minsky,Y. N.
and
Taylor,S. J.Endperiodic maps via pseudo-Anosov flows.2023,
arXiv:2304.10620v1,
to appear in Geom. Topol.LandryM. P.MinskyY. N.TaylorS. J.Simultaneous universal circlesJ. Topol.1920261e700542808143171500930110.1112/topo.70054LandryM. P.TsangC. C.Endperiodic maps, splitting sequences, and branched surfacesGeom. Topol.29202594531466308152702501775410.2140/gt.2025.29.4531LandryM. P. and
TsangC. C.,
Pseudo-Anosov flows from sutured hierarchies. In preparationMosherL.Dynamical systems and the homology norm of a 3-manifold, I: efficient intersection of surfaces and flowsDuke Math. J.65199234495000754.58030115417910.1215/S0012-7094-92-06518-5MosherL.,
Laminations and flows transverse to finite depth foliations. Preprint, 1996NovikovS. P.The topology of foliationsTr. Moskov. Mat. Obs.141965248278Trans. Moscow Math. Soc.1419652683040247.570060200938ParlakA.Mutations and faces of the Thurston norm ball dynamically represented by multiple distinct flowsGeom. Topol.29202542105217308084075492947410.2140/gt.2025.29.2105ThurstonW. P.A norm for the homology of 3-manifoldsMem. Amer. Math. Soc.591986339991300585.570060823443ThurstonW. P.,
Three-manifolds, foliations and circles, II. Unfinished manuscript, 1998