Dynamics and Geometry of Surface Homeomorphisms

Dynamics and geometry of surface homeomorphisms

Harry Baik

Berstein Seminar

Nov 1, 2011 I Automorphisms of tori

I Geometric Structures on mapping tori

I Geodesic laminations

I Train tracks

I Singular foliations

I quadratic holomorphic diﬀerentials

I Pseudo-Anosov automorphisms of surfaces

I Geometric structures on general mapping tori

I Peano curves

What will be covered in this talk

I Surfaces and mapping class groups I Geometric Structures on mapping tori

I Geodesic laminations

I Train tracks

I Singular foliations

I quadratic holomorphic diﬀerentials

I Pseudo-Anosov automorphisms of surfaces

I Geometric structures on general mapping tori

I Peano curves

What will be covered in this talk

I Surfaces and mapping class groups

I Automorphisms of tori I Geodesic laminations

I Train tracks

I Singular foliations

I quadratic holomorphic diﬀerentials

I Pseudo-Anosov automorphisms of surfaces

I Geometric structures on general mapping tori

I Peano curves

What will be covered in this talk

I Surfaces and mapping class groups

I Automorphisms of tori

I Geometric Structures on mapping tori I Train tracks

I Singular foliations

I quadratic holomorphic diﬀerentials

I Pseudo-Anosov automorphisms of surfaces

I Geometric structures on general mapping tori

I Peano curves

What will be covered in this talk

I Surfaces and mapping class groups

I Automorphisms of tori

I Geometric Structures on mapping tori

I Geodesic laminations I Singular foliations

I quadratic holomorphic diﬀerentials

I Pseudo-Anosov automorphisms of surfaces

I Geometric structures on general mapping tori

I Peano curves

What will be covered in this talk

I Surfaces and mapping class groups

I Automorphisms of tori

I Geometric Structures on mapping tori

I Geodesic laminations

I Train tracks I quadratic holomorphic diﬀerentials

I Pseudo-Anosov automorphisms of surfaces

I Geometric structures on general mapping tori

I Peano curves

What will be covered in this talk

I Surfaces and mapping class groups

I Automorphisms of tori

I Geometric Structures on mapping tori

I Geodesic laminations

I Train tracks

I Singular foliations I Pseudo-Anosov automorphisms of surfaces

I Geometric structures on general mapping tori

I Peano curves

What will be covered in this talk

I Surfaces and mapping class groups

I Automorphisms of tori

I Geometric Structures on mapping tori

I Geodesic laminations

I Train tracks

I Singular foliations

I quadratic holomorphic diﬀerentials I Geometric structures on general mapping tori

I Peano curves

What will be covered in this talk

I Surfaces and mapping class groups

I Automorphisms of tori

I Geometric Structures on mapping tori

I Geodesic laminations

I Train tracks

I Singular foliations

I quadratic holomorphic diﬀerentials

I Pseudo-Anosov automorphisms of surfaces I Peano curves

What will be covered in this talk

I Surfaces and mapping class groups

I Automorphisms of tori

I Geometric Structures on mapping tori

I Geodesic laminations

I Train tracks

I Singular foliations

I quadratic holomorphic diﬀerentials

I Pseudo-Anosov automorphisms of surfaces

I Geometric structures on general mapping tori What will be covered in this talk

I Surfaces and mapping class groups

I Automorphisms of tori

I Geometric Structures on mapping tori

I Geodesic laminations

I Train tracks

I Singular foliations

I quadratic holomorphic diﬀerentials

I Pseudo-Anosov automorphisms of surfaces

I Geometric structures on general mapping tori

I Peano curves Basic Deﬁnitions

Surfaces A surface S is a 2-dimensional topological manifold. In this talk, they will be always assumed to be orientable, and often even closed. The theory I presented here is easily applied to the surfaces of ﬁnite type, ie., closed surfaces with ﬁnitely many points removed. The compact-open topology The compact-open topology on Map(X , Y ) is the one generated by open sets of the form UK,U := {φ ∈ Map(X , Y )|φ(K) ⊂ U} where K ⊂ X is compact and U ⊂ Y is open. The compact-open topology The group of self-homeomorphisms of S is denoted Homeo(S). We would like to think of Homeo(S) as a topological group. The topology came by regarding Homeo(S) as a subspace of Map(S, S) with compact-open topology. Smooth and PL category

Without a proof, I state the following fact: The inclusions

Diﬀeo(S) ,→ Homeo(S), PL(S) ,→ Homeo(S)

induce isomorphisms on π0. So up to isotopy, we could work with whichever category is most convenient for our purposes. Essential loops and hierarchies Essential loops An embedded loop α ⊂ S is essential if it does not bound a disk or cobound an annulus together with a component of ∂S. A properly embedded arc β ⊂ S is essential if there is no other arc γ ∈ ∂S such that β ∪ γ is an embedded circle which bounds a disk in S.

Figure: This shows which are essential and which are not. Homotopy gives Isotopy

Lemma Let α, β be essential loops or arcs in S. If α and β are properly homotopic, they are properly isotopic. Proof. We only show for loops. Whenever there is a bigon, remove it by isotopy. The we may assume that α and β intersect eﬃciently. Since α is essential, it is a nontrivial element in π1(S). Moreover any power of α is nontrivial too. Thus π1 has a subgroup ∼ < α >= Z. Consider the cover Sˆ of S corresponding to this subgroup. Note that Sˆ is a cylinder. Homotopy gives Isotopy Choose liftsα, ˆ βˆ of the loops α, β. Note that they intersect at the even number of points, since α ' β implies that their algebraic intersection number is 0.

Figure: If they are disjoint (above), they are isotopic. Otherwise (below), we get a contradiction, since we get an embedded bigon. Hierarchy Let S be a connected surface of ﬁnite type and non-positive Euler characteristic. Let α1 be an essential loop or arc on S. Then we can get a new surface S1, possibly disconnected, by cutting S along α1. We achieve either smaller genus or bigger Euler characteristic.

Figure: Cutting along the red separating curve gives us two once punctures torus T . Note χ(T ) = −1 > χ(Σ2) = −2. Cutting along the blue non-separating one gives a smaller genus surface. Hierarchy

After ﬁnitely many such cuts, one is left a collection of disks and get a hierarchy

α1 α2 αn S S1 ··· Sn = ∪i Di The Mapping Class Group

MCG(S)

The quotient group MCG(S) := Homeo(S)/Homeo0(S) is called the Mapping Class Group of S. MCG +(S) will denote the subgroup consisting of orientation-preserving homeomorphisms. Lemma Suppose φ1, φ2 ∈ Homeo(S) represents the same element in MCG(S). Then the induced (outer) automorphisms (φ1)∗, (φ2)∗ of π1(S) are equal. Proof Homotopic maps induce isomorphic actions on π1. It follows that there is an induced homomorphism ρ : MCG(S) → Out(π1(S)). The Mapping Class Group

Lemma Let φ : S → S be a proper homotopy equivalence. If α is an essential loop or arc in S, then φ(α) is properly homotopic to an essential loop or arc. Proof Omitted. The Mapping Class Group

Dehn-Nielsen Theorem Suppose S is a closed orientable surface of genus g ≥ 1. Then ρ is an isomorphism. Otherwise, ρ is an injection, with image equal to the subgroup of Out(π1(S)) consisting of automorphisms which permute the peripheral subgroups. Proof Omitted. Coarse Geometry

Proper/geodesic metric spaces A complete metric space is proper if it is locally compact. A metric space is geodesic if any two points may be joined by an isometrically embedded arc. δ-hyperbolicity A geodesic metric space X is δ-hyperbolic for δ ≥ 0 if for all geodesic triangles pqr, every point on the edge pq is within distance δ from the union of the edges qr and rp. A geodesic space is Gromov hyperbolic if it is δ-hyperbolic for some δ. e.g. Trees are 0-hyperbolic. Coarse Geometry

Morse Lemma Let X be a δ-hyperbolic space. Then for every k, there is a universal constant C(δ, k, ) such that every (k, )-quasigeodesic segment with endpoints p, q ∈ X lies in the C-neighborhood of any geodesic joining p and q. Gromov boundary To a δ-hyperbolic proper metric space X , we may associate the ideal boundary ∂X , deﬁned as follows. As a set, ∂X is a set of equivalence classes of quasigeodesic rays, where r ∼ r 0 if each is contained in the T-neighborhood of the other for some T (which might depend on r, r 0). Let [r] be an arbitrary equivalence class. Let x0 be any point in X . Then [r] contains a geodesic ray s which can be taken to satisfy s(0) = x0. Automorphisms of tori

Let T denote the standard 2-dimensional torus. Then π1(S) = Z ⊕ Z. Since this groups is Abelian, we have an equality

Out(π1(T )) = Aut(π1(T )) = GL(2, Z)

provided that we pick a basis, and MCG+(T ) = SL(2, Z). (Since we have integral entries, the elements of GL(2, Z) has det = ±1, so the orientation preserving elements form SL(2, Z).) Let φ ∈ MCG +(T ). Under this identiﬁcation, φ corresponds to a 2 × 2 a b matrix where a, b, c, d ∈ and det(φ) = 1. The c d Z −1 eigenvalues of φ are λ, λ for some λ ∈ C, where tr(φ) := a + d = λ + λ−1. Automorphisms of tori

Since the trace of φ is real, if λ, λ−1 are not real, they are both on the unit circle, and therefore |tr(φ)| < 2. Since a and d are integers, in this case we have tr(φ) = 0, ±1. a b When tr(φ) = 0, φ = with a2 + bc = −1. Then I c −a −1 0 φ2 = . Thus φ4 = Id. 0 −1 6 I When tr(φ) = ±1, similar computation shows that φ = Id. Such φ has ﬁnite order. Automorphisms of tori

Let’s consider the case λ = λ−1 = 1. Then either φ = Id or φ is 1 n conjugate to a matrix of the form T := . One observation n 0 1 we can make is, φ has only one eigenvector (up to scaling). By conjugation, we can make the vector (1, 0) as the eigenvector. Then the result follows easily. (Conjugation could be done constructively. Let bz/(1 − a) bw/(1 − a) − bnz/(1 − a)2 M = with choice of z w n, z, w so that M has integer entries and have determinant 1. Then φM = MTn.) Automorphisms of tori Topologically, such an element φ preserve the isotopy class of one of the loops which generates π1(T ). Such a φ is said to be reducible. If λ = λ−1 = −1, then either φ = −Id or conjugate to a transformation which takes the vector (1, 0) to its inverse. We call this is reducible too.

Figure: φ is a Dehn-twist along the preserved curve. Automorphisms of tori

Finally, if the eigenvalues are real and λ > 1 > λ−1, then say φ has two eigenvectors e±. Let F ± be the linear foliations of T by lines parallel to e±. Then the linear representative of φ takes leaves of F ± to themselves, stretching the leaves of F + by a factor of λ, and stretching the leaves of F − by a factor of λ−1. For some choice of Euclidean structured on T , these two foliations may be taken to be perpendicular. Such a homeomorphism is said to be Anosov. Automorphisms of tori

We summarize what we have observed as the following theorem. Classiﬁcation of toral homeomorphisms Let T be a torus, and let φ ∈ Homeo+(T ). Then one of the following holds: 1. φ is periodic; that is, some ﬁnite power of φ is isotopic to the identity. 2. φ is reducible; that is, there is some simple closed curve in T which is taken to itself by φ, up to isotopy. 3. The linear representative of φ is Anosov. PSL(2, Z) and Euclidean structure on tori

Every Riemannian metric on T is conformally equivalent to a unique flat metric, up to similarity. The set of all such flat metrics is parametrized by Teichmüllerspace. Now we defined it first as a set. Teichmüllerspace The Teichmüllerspace of the torus, denoted T (T ), is the set of equivalence classes of pairs (f , Σ), where Σ is a torus with a flat metric, f : T → Σ is an orientation-preserving homeomorphism, and (f1, Σ1) ∼ (f2, Σ2) iff there is a similarity i :Σ1 → Σ2 for which the composition i ◦ f1 is homotopic to f2. PSL(2, Z) and Euclidean structure on tori

The map f in a pair (f , Σ) is called a marking. MCG +(T ) acts on T (T ) by changing the marking: φ(f , Σ) = (f ◦ φ−1, Σ). Geometric structures on mapping tori

Given a homeomorphism φ : T → T , we can form the mapping torus Mφ by Mφ = T × I /(s, 1) ∼ (φ(s), 0).

We study the relationship between geometry of Mφ and the dynamics of φ. Suppose that φ has ﬁnite order. We saw that we could have either φ4 = Id or φ6 = Id. So the order is one of 2, 3, 4 and 6. Thus it preserves either a square or a hexagonal Euclidean metric on T . Hence the gluing map can be realized as an isometry of T × I , giving the mapping torus Mφ a Euclidean structure. Geometric structures on mapping tori Suppose φ : T → T is reducible, preserving a simple closed curve γ. Then γ × I ⊂ T × I glues up under φ to give a closed, π1-injective torus or Klein bottle in Mφ. Finally, let’s consider the Anosov case with invariant foliations F ±. In this case, the automorphism φ of Z ⊕ Z extends linearly to an automorphism of R ⊕ R which is conjugate to the diagonal λ 0 automorphism φ ∼ . Let Sol denote the 3-dimensional 0 λ−1 solvable Lie group which is an extension of abelian groups

2 0 → R → Sol → R → 0

where the conjugation action of the generator t of the R factor 2 acts on the R factor by the matrix

t −1 v1 e 0 v1 t t = −t v2 0 e v2 Geometric structures on mapping tori

The fundamental group π1(Mφ) is the extension

2 0 → Z → π1(Mφ) → Z → 0

2 where the generator t of the Z factor acts on the Z factor ( coming from π1(T )) as φ∗. Since φ linearly extends to an 2 automorphism of R , this short exact sequence includes into the previous one which deﬁnes Sol in such a way that the generators of 2 ± R become the eigenvectors e of the automorphism φ, which includes into R by φ → log(λ). This exhibits π1(Mφ) as a lattice of Sol. Geometric structures on mapping tori

Summarizing, we have the following theorem. Geometric structures on mapping tori of tori Let φ : T → T be a homeomorphism of the torus. Then the mapping torus Mφ satisﬁes the following: 3 1. If φ is periodic, then Mφ admits an E geometry. 2. If φ is periodic, then Mφ contains a reducing torus or Klein bottle.

3. If φ is Anosov, then Mφ admits an Sol geometry. Now we are ready to study the analogue theory for the surfaces S with χ(S) < 0. Dynamics of surface homeomorphisms

The basic idea in studying the action of an automorphism φ of a surface S is to ﬁnd some kind of essential 1 dimensional object in S which is preserved (up to some suitable equivalence relation) by φ. There several more or less equivalent objects of this kind; amongst the most important are geodesic laminations, train tracks and singular foliations. We start from geodesic laminations.

Let S be a surface of finite type with χ(S) < 0. Then by the uniformization theorem, we can find a hyperbolic structure on S in every conformal class of metric, which is complete with finite area. The set of all marked hyperbolic structures on S is parametrized by a Teichmüllerspace, just as in the case of tori. Hyperbolic structures on surfaces

Teichm¨ullerspace Let S be a closed surface of genus ≥ 2. The Teichm¨ullerspace of S, denoted T (S), is the set of equivalence classes of pairs (f , Σ) where Σ is a hyperbolic surface, f : S → Σ is an orientation-preserving homeomorphism, and

(f1, Σ1) ∼ (f2, Σ2)

iﬀ there is an isometry i :Σ1 → Σ2 for which the composition i ◦ f1 is homotopic to f2.

One definition of the topology on T (S) is that (fi , Σi ) → (f , Σ) if there are a sequence of 1 + i bilipschitz maps ji :Σi → Σ such t hat ji ◦ fi is homotopic to f where i → 0. With respect to this topology, T (S) is path-connected, and homeomorphic to an open ball of dimension 6g − 6. Teichmüllerspace There are a number of natural ways to give local parameters on T (S). We introduce one briefly here. Fenchel-Nielsen Coordinate

Figure: This is an example of a decomposition of a closed surface of genus 6. For genus g, we need 3g − 3 simple closed geodesics to cut the surface into 2g − 2 hyperbolic pairs of pants. Simple closed curve on hyperbolic surface

Let Σ be a closed hyperbolic surface of genus ≥ 2, and let α ⊂ Σ be an essential simple closed curve. Letα ˜ denote the preimage of α in the universal cover Σ.˜ Then Σ˜ is homeomorphic to a plane, andα ˜ is a locally ﬁnite union of properly embedded lines in Σ.˜ 2 Since Σ is hyperbolic, Σ˜ is isometric to H . We denote the ideal ˜ 1 ˜ circle of Σ by S∞(Σ) to stress the dependence on Σ. The hyperbolic structure and the orientation on Σ determine a faithful homomorphism ρ : π1(Σ) → PSL(2, R) unique up to conjugacy. Since Σ is closed, for every nontrivial α ∈ π1(Σ), the image ρ(α) has two distinct real eigenvalues, and the associated 1 eigenspaces correspond to two distinct ﬁxed points of α in S∞. Simple closed curve on hyperbolic surface

Since det2(ρ(α)) = 1, one of the eigenvalue has absolute value > 1, and one has absolute value < 1. The corresponding fixed points of α are attracting and repelling respectively. Every connected componentα ˜i ofα ˜ is stabilized by some 2 gi ∈ Isom(H ) in the deck transformation group of the covering ˜ ± 1 Σ → Σ, and gi fixed two points pi in S∞.(gi is simply ρ([α]), and gi cannot be parabolic, sinceα is essential and S is closed.) ± Let li be the unique hyperbolic geodesic joining pi . Take a point n i onα ˜i . Then the orbit {gi (p)} lies onα ˜ . Since gi is an isometry, n d(gi (p), li ) is a finite constant. Then by compactness of α, we know thatα ˜i is in a bounded distance from li , which implies that ± α˜i itself limits to the points pi . In this way, one sees that the system of properly embedded linesα ˜ determines a family of pairs 1 of distinct unordered points in S∞. Call the set Kα. Simple closed curve on hyperbolic surface

Figure: The space of pairs of distinct unordered points in S 1 is topologically open M¨obiusband. Two pairs are unlinked if and only if the geodesics in H2 which span them are disjoint. It follows that the 2 geodesics in H with endpoints in Kα is embedded. Thus this family of geodesics covers a simple closed geodesic on Σ which we denote by αg . Since αg is homotopic to α, they are isotopic as we proved. Simple closed curve on hyperbolic surface

1 Figure: This construction can be generalized as follows: if α , . . . , αn is ﬁnite collection of disjoint, nonparallel essential simple closed curves in 1 n Σ, then the geodesic representatives αg , . . . , αg of their homotopy classes are embedded and disjoint, and isotopic as a family to the αi . Geodesic laminations

The concept of a geodesic lamination is a ntatural generalization of the concept of a simple closed geodesic. Geodesic lamination Let Σ be a hyperbolic surface, A geodesic lamination Λ on Σ is a union of disjoint embedded geodesics which is closed as a subset of Σ. The geodesics making up Λ are called the leaves of Λ. Since the leaves of a geodesic lamination are disjoint, pairs of leaves which are close at some point are almost parallel there. It follows that if Λ is a geodesic lamination, there is a cover of Σ by open sets Ui called product charts such that for each i, there is a product structure on the intersection Ui ∩ Λ ≈ Ci × I where Ci is a closed subset of I . In each product chart, the closed subset Cj is called leaf space of Λ ∩ Ui , or more informally local leaf space of Λ. Minimal laminations

Minimal lamination A geodesic lamination Λ is minimal if every leaf is dense in Λ. e.g. A simple closed geodesic is a minimal geodesic lamination. Lemma Let Σ be a closed surface of genus g. Then Σ admits at most 3g − 3 disjoint geodesic laminations, with equality only if all laminations are simple closed geodesics. Minimal laminations

Proof. Let Λ = ∪i Λi be a finite union of geodesic laminations in Σ. We may assume that each Λi is minimal, and then Λ itself is a geodesic lamination. For each component C of Σ − Λ, let C be the completion w.r.t. path metric, and call it path closure of C. Doubling C along ∂C, we get a finite area hyperbolic surface with punctures. (Punctures come from infinite geodesic components of ∂C.) The area is −2πχ by Gauss-Bonnet. Each C is homeomorphic to the interior of a closed surface with n boundary components.

I If n = 3, then the area of C is at leats 2π(n − 2) with equality iﬀ C is genus 0 with n closed boundary components.

I If n = 1, 2, then the area is at least 2πn with equality iﬀ C is genus 1 with 1 or 2 closed boundary components. Minimal laminations

I If n = 3, then the area of C is at leats 2π(n − 2) with equality iﬀ C is genus 0 with n closed boundary components.

I If n = 1, 2, then the area is at least 2πn with equality iﬀ C is genus 1 with 1 or 2 closed boundary components.

Let’s say we have k number of minimal laminations Λi . Each Λi contains at least two connected components of the boundary of the path closure of Σ − Λ. Since the area of Σ is (4g − 4)π, we have

(4g − 4)π = Area sum ≥ (2k/3) × 2π

which is same as 3g − 3 ≥ k Space of all geodesic laminations

Let L(Σ) denote the set of geodesic laminations in a hyperbolic surface Σ. If Σ is compact, each geodesic lamination is also compact. Giving a Hausdorﬀ topology to L(Σ) is compact and totally disconnected. Lemma Let Σ, Σ0 be two hyperbolic surfaces, and φ :Σ → Σ0 be a homeomorphism. Then there is an induced homeomorphism between the respective spaces of geodesic laminations 0 φ∗ : L(Σ) → L(Σ ) which depends only on the homotopy class of φ. Space of all geodesic laminations

0 0 Sketch. We have φ :Σ → Σ so that φ∗ : π1(Σ) → π1(Σ ). Take 0 point x0 ∈ Σ and let y0 = φ(x0). Let φ˜ : Σ˜ → Σ˜ be the one covers φ and φ˜(x ˜0) =y ˜0. The ideal boundary of Σ˜ could be identiﬁed with the set of geodesic rays starting from x0. Similar things is 0 ˜ 1 ˜ 1 ˜ 0 true for Σ . Then φ gives a map φ∞ : S∞(Σ) → S∞(Σ ). This 0 intertwines the actions of π1(Σ, x0) and π1(Σ , y0). Space of all geodesic laminations Since L(Σ) lifts to a geodesic lamination L(Σ),˜ they form a closed subset KL(Σ) of the set of pairs of distinct unordered points whose points are unlinked. The order structure of this set, together with 1 the action of π1(Σ) on S∞, allows one to reconstruct the lamination. I omit the detail.

Full lamination A geodesic lamination Λ ⊂ Σ is full if complementary regions are all finite-sided ideal polygons. Now we introduce an intermediate step toward the classification theorem of surface homeomorphisms, which we already went though for tori. Theorem Let φ ∈ MCG(Σ). Then one of the following three possibilities must hold: 1. φ has finite order in MCG(Σ). 2. There is some finite disjoint collection of simple geodesics γ1, . . . , γn which are permuted by φ∗. (In this case we say φ is reducible.)

3. φ∗ preserves a full minimal geodesic lamination Λ. Invariant Lamination

Proof Suppose φ does not have finite order in MCG(Σ). Then there is some simple closed geodesic γ such that the iterates i γi := φ∗(γ) do not form a periodic sequence. Otherwise, we can take a finite collection of simple closed geodesics which fills Σ. Then some power of φ∗ fixes the curves and complementary regions, thus homotopic to the identity. So, we can choose γ such that γi are not periodic. For any fixed hyperbolic structure on Σ, and for any constant T , there are only finitely many simple closed geodesics on Σ with length ≤ T . Invariant Lamination

As a consequence, the length of the γi eventually increases without bound. On the other hand, for any ﬁxed n, we have

#{γi ∩ γi+n} = Kn < ∞

where Kn is independent of i. Since L(Σ) is compact, we can extract a subsequence ni for which

0 γni → Λ

in the Hausdorﬀ topology. We let Λ be a minimal sublamination of Λ0. Invariant Lamination

Suppose Λ is a simple closed geodesic. Then for suﬃciently large i,

γni contains arbitrarily long segments which spiral around a tubular 0 0 n 0 neighborhood of Λ . Thus Λ cannot intersect φ∗(Λ ) transversely for any n. Then the lemma we showed before implies that i φ∗(Λ) = Λ for some 0 < i ≤ 3g − 3. Thus Λ is the desired invariant full minimal lamination. Invariant Lamination

Now let’s assume that Λ is not a simple closed geodesic. Suppose φ∗(Λ) intersects Λ transversely. Since Λ is minimal, there is no isolated leaf. This implies that φ∗(Λ) ∩ Λ has no isolated points. Hence it is uncountable. In this case, since Λ is contained in the

Hausdorﬀ limit of γni , and since transverse intersections of geodesics are stable under perturbation, the cardinality of

γni ∩ γ1+ni is unbounded as i → ∞ contradicting to our previous observation. Invariant Lamination

This contradiction shows that φ∗(Λ) and Λ does not meet transversely (though they might be equal) and by similar n argument, the same is true of Λ and φ∗(Λ) for any n. Again by the i same lemma,φ∗(Λ) = Λ for some 0 < i ≤ 3g − 3. If Λ is not full, then some boundary curve of a tubular neighborhood of Λ is essential. By construction, this boundary curve is periodic and disjoint from its translates. So we are done. (it’s reducible case.) Otherwise, since no two full geodesic laminations have nonempty intersection, we should have φ∗(Λ) = Λ. Invariant Lamination

Though the proof is now complete, we can talk more about this. Since Λ is minimal and full, γ is transverse to Λ. Indeed, it must cross every leaf. Let P be a complementary polygon to Λ and let l be a boundary leaf of l. Let P˜ be a lift of P and ˜l be a corresponding lift of l to Σ.˜ Letγ ˜ be a component of the preimage of γ which crosses ˜l. Since φ∗ fixes Λ, after replacing φ∗ by a finite power if necessary, we can assume that φ∗ fixes every boundary leaf of Λ. Invariant Lamination

1 ˜ ˜ Let φ∞ ∈ Homeo(S∞(Σ)) be some lift of φ which fixes l. Note that φ∞ fixes the vertices of P˜ pointwise. Since Λ is minimal, there is li converging to ˜l. We will later prove that li do not share endpoints with l. By definition, we also have a leaf m ∈ Λ such i that φ∞(˜γ) → m. See the picture in the next slide. Invariant Lamination

Figure: Note that m should share an end point with ˜l but li do not. So we must have m = ˜l. Thus that shared ﬁxed point is going to be an attracting ﬁxed point. 1 Dynamics on S∞

Suppose the other end point of ˜l is repelling. Then there exists i, j such that φ∞(li ) intersect lj transversely, but the leaves are disjoint. Thus both endpoints are attracting. Hence, φ∞ must have another ﬁxed points in the interior of I with (partially) repelling dynamics. Lemma φ∞ has a single repelling ﬁxed point in each such interval I . There is a cute proof. 1 Dynamics on S∞

Proof. Suppose not, so that there exists a maximal closed interval J contained in the interior of I whose endpoints are ﬁxed by φ∞. Since J is maximal one, by applying negative powers of φ∞ to li , we see that the endpoints of J are the endpoints of another leaf n of Λ.˜ If we apply the previous theorem to φ−1 in place of φ, then we see that φ∗ preserves two geodesic laminations, and in fact that some sequence of iterates of φ−1 applied to γ converges to a full minimal 0 −i geodesic lamination. Temporarily call it Λ . As before, φ∞ (˜γ) converges to a leaf m0 of Λ˜0 which is asymptotic at one end to n. But the minimal laminations which contain leaves which are asymptotic are equal, which contradicts to the fact that m0 crosses leaves li transversely. This shows J must consist of at most a single point. 1 Dynamics on S∞

Now we use new notation; Λ+ = Λ and Λ− = Λ0.Λ+ is the Hausdorﬀ limit of the images of γ under the forward iterates of φ, so we call it the stable lamination. Similarly, Λ− is called the unstable lamination. Dynamical picture

i + For any simple closed geodesic γ, we know φ∗(γ) converges to Λ as i → ∞ possibly together with finitely many proper leaves which are diagonals of complementary regions. Since the length of the i φ∗(γ) increases without bound as i → ∞, it follows that φ∗ stretches long subarcs in Λ+ by a definite amount. In fact, iterates + of φ∗ stretch all sufficiently long arcs in Λ at a constant rate λ > 1. For any essential simple closed curve γ,

i+1 length(φ∗ (γ)) lim i → λ i→∞ length(φ∗(γ)) Similar fact is true for the negative powers with 1/λ.

This is the most purely topological description of the pseudo-Anosov dynamics of an automorphism φ which is neither ﬁnite order nor reducible. Measured laminations

Transverse measure An (invariant) transverse measure µ for a geodesic lamination Λ is a nonnegative Borel measure on the local leaf space of Λ in each product chart which is compatible on the overlap of distinct charts.

Example Let γ be a simple closed geodesic. Then γ admits an atomic measure called hitting measure, for which µ(τ) is equal to the cardinality of µ ∩ τ for each transversal τ. Measured laminations

Example Let Λ be a geodesic lamination, and let γ ⊂ Λ be an inﬁnite embedded geodesic in Σ. Pick a base point p ∈ γ, and for each T > 0 let NT (p) denote the geodesic subarc of γ with length 2T and center p. For each transversal τ and each T > 0 deﬁne

#{τ ∩ N (p)} µ (τ) = T T 2T

Then for some subsequence Ti → ∞ the µT converges to a nontrivial transverse measure on Λ. Measured laminations

Deﬁne the support of µ to be the set of leaves contained in the support of some (and therefore every) Borel measure in some product chart. For any Λ, µ the support of µ is a sublamination of Λ. We say µ has full support if the support is equal to the entire Λ.

Example If Λ is minimal, it admits a transverse measure which is necessarily of full support. Missing part

I owe you to show that li do not share the endpoints with ˜l when we develop the stable and unstable laminations. The following lemma now ﬁlls the gap. Lemma Let Λ be a minimal lamination on a compact hyperbolic surface Σ. If γ, γ0 leaves of Λ are asymptotic, then they are boundary leaves of the same complementary subsurface. Missing part

2 0 0 Proof. Lift to H . Then γ, γ lift toγ, ˜ γ˜ which share a common endpoint. Suppose they are not boundary leaves of the same complementary subsurface. Then we have a transversal τ with endpoints onγ, ˜ γ˜0, which meets uncountably many leaves of Λ.˜ If µ is an invariant transverse measure for Λ, then µ(τ) > 0. Since γ,˜ γ˜0 are asymptotic, we can find an arbitrarily short transversal τ 0 with endpoints onγ, ˜ γ˜0 and intersects the same leaves as τ. But since Λ is minimal, µ has no atom, and therefore by the compactness of Σ, for every there exists δ such that all transversals σ with length(σ) ≤ δ satisfy µ(σ) ≤ . Intersection number For two essential simple closed curves α, β, the intersection number i(α, β) is the number of points where they intersect if they are isotoped so that α, β meet efficiently. How could we measure such a quantity between two laminations? For j = 1, 2, let (Λj , µj ) be a pair of transverse measured geodesic laminations. the intersection number For the transverse measured laminations, we define by Z 1 2 i(Λ1, Λ2) := d(µ × µ ) Λ1∩Λ2

If Λ1, Λ2 contain a common sublamination Λ, defnie

i(Λ1, Λ2) := i(Λ1/Λ, Λ2/Λ)

The case of two simples closed geodesics could be regarded as a special case with hitting measures. Intersection number

Intersection numbers gives a bilinear map

+ i : ML(Σ) × ML(Σ) → R

Equipped with the weakest topology for which this map is 6g−6 continuous, ML(Σ) is homeomorphic to a positive cone in R . Fact The set of measured laminations consisting of finitely many simple closed geodesics is dense in ML. This is not hard to see using Poincarérecurrence theorem, but we skip the argument here. It will be more directly visible once we get to the weighted train tracks. Warning The set of laminations consisting of finitely many simple closed geodesics is not dense in L(Σ). Projective measured laminations

For a homeomorphism φ :Σ → Σ0, there is an induced map 0 φ∗ : ML(Σ) → ML(Σ ) as before, and this agrees with the previous φ∗ on the support. Thus we can talk about ML(S) for any topological surface of genus ≥ 2. The space PML(S) of projective measured laminations is the quotient + PML(S) := (ML(S) \{0})/R which is homeomorphic to a sphere of dimension 6g − 7. (We don’t prove it here.) Projective measured laminations

Fact. For φ ∈ MCG(S) which is not ﬁnite order or reducible has an invariant transverse measure µ+, µ− for Λ+, Λ− respectively. Thus it has at least two ﬁxed points in PML(S). Lemma Let Λ± be the stable and unstable laminations of φ with ± + projectively invariant measures µ . Suppose φ∗ multiplies µ by − λ > 1. Then φ∗ multiplies µ by 1/λ. Proof + − + − + 0 − 0 + − i(Λ , Λ ) = i(φ∗(Λ ), φ∗(Λ )) = i(λ·Λ , λ ·Λ ) = λλ ·i(Λ , Λ ) Thurston boundary of T (S)

For an essential simple closed curve α on S, we can deﬁne + lα : T (S) → R such that (f , Σ) 7→ length of (f (α))g . Extend linearly to the set of simple closed curves and extend by continuity to entire ML, then we get

+ l : T (S) × ML(S) → R

ML(S) ML(S) This gives an embedding T (S) → R → PR . i gives an ML(S) embedding PML(S) → PR , where it compactiﬁes l(T (S)). Then T (S) ∪ PML(S) is homeomorphic to a closed ball of dimension6g − 6. Train tracks

Branched submanifold A k dimensional subcomplex N of a m-manifold M is a branched submanifold if each point x ∈ N has a neighborhood which is a finite union of k-disks, all tangent at x. A train track is an one-dimensional example of this sort. Train tracks A train track τ is a finite embedded C 1 graph in a surface S with a well-defined tangent space at each vertex. Train tracks Train tracks

Figure: One example on the genus 2 surface Train tracks Since a train track τ is C 1, it has a well-deﬁned normal bundle, and a regular neighborhood N(τ) of τ in S can be foliated by intervals which are transverse to τ.

Figure: Does it look more like a real train track? Train tracks One train track τ is said to carry another train track τ 0 if τ 0 can be isotoped (preserving the C 1 structure) so that at the end of the isotopy, τ 0 is transverse to the intervals in this I -bundle structure 0 on N(τ). We write τ - τ in this case. Collapsing the I -ﬁbers gives one a map from N(τ) to τ. If τ carries τ 0, then we get an induced immersion from τ 0 to τ which we call a carrying map. Splitting and shifting Weights Recurrence A train track τ is recurrent if for evert edge e of τ there is a simple closed curve c ⊂ S which is carried by τ such that under the carrying map, e is contained in the image of c. Note that any muticurve α determines a weight on τ which is a just a map wα : E → N such that wα(e) is the number of preimages in α of a point in e under the carrying map. Also note that wα satisﬁes the switch condition. Construction

Let τ be a train track, and let w be an integral weight, satisfying the switch conditions at each vertex. For each edge e of τ, put w(e) parallel copies of e in a tubular neighborhood N(e), transverse to the I -ﬁbers. At each vertex, glue the ends of these intervals together in pairs, in such a way that the resulting 1-manifold is embedded. The result is a multicurve which is carried by τ. In general, we can deﬁne a weight using nonnegative real numbers. Thus the set of weights W (τ) carried by τ is a convex cone in a real vector space, whose integral points correspond to multicurves carried by τ. Non-recurrent example

Notice that τ admits a weight w such that w(e) > 0 iﬀ it is recurrent. Induced Weights

Suppose that τ carries τ 0 with a weight w 0. Then w 0 descends to a weight w on τ: Let c : τ 0 → τ be the carrying map. For an edge e −1 of τ and a generic point p ∈ e, the preimage c (p) = q1,..., qn 0 0 0 consists of points in the interiors of edges e1,..., en of τ . Then deﬁne X 0 0 w(e) = w (ei ) i For each complementary region R of τ, R is a surface possibly with cusps. χ(R) is deﬁned to be the Euler characteristic of the underlying topological surface minus 1/2 the number of cusps. Essential train tracks

τ is said to be essential of every complementary region has negative Euler characteristic, and full if every complementary region is a disk with at least three cusps. Any simple closed curve c carried by an essential train track is (homotopically) essential in S. For, otherwise, c bounds a disk D which can be tiled by a ﬁnite collection of complementary regions to τ. But then X 1 = χ(D) = χ(R) < 0 R⊂D which is a contradiction. Essential train tracks

It is a theorem of Thurston that for any closed, oriented surface S with χ(S) < 0, one can construct ﬁnitely many recurrent, essential train tracks τ1, . . . , τn which carry every essential multicurve. Now we start talking about the relationship between laminations and train tracks by considering which laminations are carried by train tracks. Laminations carried by train tracks

Every geodesic lamination Λ is carried by some train track. Laminations carried by train tracks

Let Λ be an abstract geodesic lamination on Σ and ﬁx a hyperbolic structure on Σ. Then for any suﬃciently small > 0, the open neighborhood N(Λ) of Λ in Σ can be foliated by intervals transverse to Λ as follows: For p ∈ N(Λ) \ Λ, we say p ∼ q if q is the point on Λ closest to p. If there are two closed points q1, q2, then q1 ∼ p ∼ q2. The leaves of the foliations of N(Λ) are the equivalence classes generated by ∼. Laminations carried by train tracks

Collapsing the closures of these intervals to points collapses N(Λ) to a graph, which admits the structure of a train track τ in such a way that the collapsing map is a carrying map on any geodesic segment contained in Λ. τ is not unique, since it depends on the hyperbolic metric and . But any two train tracks obtained in this way are defer by ﬁnitely many shiftings and splittings we saw. Laminations carried by train tracks

Figure: Diﬀerent choices of give diﬀerent train tracks. Laminations carried by train tracks

If Λ is measured, then we have a natural weight on τ defined by −1 wµ(e) = µ(c (p)) where c :Λ → τ is the carrying map. Easy to check that this is well-defined and if µ has full support, then wµ is positive on each edge e so that τ is recurrent. Using this construction, one can prove the following theorem. Invariant train tracks Let φ be a homeomorphism of Σ. Suppose that φ is not reducible, and does not have finite order in MCG(Σ). Then there is some full essential train track τ so that φ(τ) is equivalent to τ, up to splitting and shifting. Invariant train tracks

Idea We know there are two invariant laminations Λ±. The hyperbolic metric g on Σ pushes forward to a hyperbolic metric + φ∗(g) on Σ for which φ(Λ ) is a geodesic lamination, carried by φ(τ). We deform φ∗(g) back to g through a family of hyperbolic + + metric gt , t ∈ [0, 1]. Then lamination φ(Λ ) deforms back to Λ through a family of (gt )-geodesic laminations. For each t, we associate τt as we constructed before. As t goes from 0 to 1, φ(τ) deforms back to τ, undergoing splits, shifts and their inverses at a discrete set of intermediate values of t. Since Λ+ is minimal, it admits an invariant transverse measure µ of full support, which pushes forward to a weight wµ on τ. Since wµ is positive on each edge, τ is recurrent. Invariant train tracks Singular foliation

A singular foilation F on a surface S is a 1-dimensional foliation in the complement of ﬁnitely many points pi , called the singularities. Each pi has an open neighborhood such that the leaves of F look n /2 like the level sets in C of the function Im(z i ) = constant for some natural number ni ≥ 3, where we choose the local coordinate so that the singular point is at 0. Singular foliation

Note that Im(zn/2) = r n/2 sin((n/2) · θ). Thus this local model of singular foliation includes the locus sin((n/2) · θ) = 0, or equivalently, θ = 2mπ/n for m = 0, 1,..., n − 1. Thus we get n-prong singularity. The picture in the case n = 3 will look like the followings. Singular foliation

Figure: The locus Im(z3/2) = constant. Mathematica : ContourPlot[Im[(x + Iy)(3/2)], {x, −5, 5}, {y, −5, 5}, Contours → 50] Singular foliation

Figure: The locus Im(z4/2) = constant. Singular foliation

Figure: The locus Im(z5/2) = constant. Singular foliation

How many singularities do we have? The following lemma answers. Lemma For a closed surface S one can must have

X 2 − ni = χ(Σ) 2 i

Proof Omitted. Singular foliation

Suppose we have a surface Σ with genus g. Then χ(Σ) = 2 − 2g. Since we require each ni ≥ 3, there are at most 4g − 4 singular points. If we start to consider the pants-decomposition, each pair of pants P has χ(P) = −1 so that it has either two 3-prong singular points or one 4-prong singular point. Singular foliation

Figure: Two possible conﬁgurations if F t ∂P From foliation to lamination

One way to do that is: ﬁrst we remove the singular leaves. Singular foliation

Figure: Nonsingular leaves of some singular foliation on the surface with genus 2 From foliation to lamination

Then pull tight. Each nonsingular leaf is isotopic to a unique embedded (typically noncompact) geodesic representative. The closure of the union of these geodesics is a geodesic lamination Λ. If you are suspicious about potential technical diﬃculties, just go up to the universal cover, where the lamination is just the set of pairs of unordered distinct points on the circle. From weighted train tracks to Measured foliations

We could consider a transverse measure for the singular foliations too. Deﬁned in the exactly same way as we did for laminations. We only consider those measures with full support and no atoms.

Let τ be a full, recurrent, essential train track and let w ∈ W (τ) be positive on every edge. We show how to associate naturally a measured singular foliation (F, µ) to the pair (τ, w). From weighted train tracks to Measured foliations

For each edge e of τ, let Re be a Euclidean rectangle with height w(e) and arbitrary width. The switch condition guarantees that the sum of the heights of the rectangles for the incoming edges is equal to one for the outgoing edges. So we may glue these rectangles along those vertical edges. Think each rectangle as foliated by horizontal line segments. From weighted train tracks to Measured foliations

Since τ is full, after gluing, each boundary component of the resulting surface is a ﬁnite-sided polygon; such an n-sided polygon can be collapsed to an n-prong singularity. Thus we get a singular foliation which is transversely measured via the height coordinate in each rectangle. More generally, if τ is not full, then some complementary regions might be more complicated than polygons. Then we need some more thought. From weighted train tracks to Measured foliations

If S is a connected surface with nonempty boundary, S deformation retracts to a graph called a spine. Any two spines for T are related by collapsing or expanding a sequence of embedded arcs joining two vertices. This operation is called a Whitehead move, and the equivalence relation it generates is called Whitehead equivalence. From weighted train tracks to Measured foliations

If S has cusp singularities along its boundary, we can deﬁne a spine for S to be a properly embedded graph in S \ cusps to which S deformation retracts. Now we do the same construction. Glue and rectangles and now sew in a copy of a spine for each complementary region. If (τ, w) gives rise to a measured foliation (F, µ), then the set of singular leaves of F form a graph, made from spines of complementary regions to τ. From weighted train tracks to Measured foliations

Observation If τ ∼ τ 0 via shifts and splits then (F, µ) ∼ (F 0, µ0) via Whitehead equivalence.

Similarly with ML, we deﬁne MF, the space of measure singular foliations up to Whitehead equivalence, and PML. From weighted train tracks to Measured foliations

The following diagram shows the relationship among; 1) measured laminations, 2) weighted train tracks up to equivalence, 3) measured singular foliations up to equivalence.

Figure: Note that ’Collapsing’ and ’straightening’ do not need to invoke the notion of a transverse measure. Transverse pair

For F +, choose complex coordinates so that near each regular point, the leaves are the level sets of Im(z) and near each singular point of index n, the leaves are the level sets of Im(zn/2). Similarly, deﬁne F − using real parts. Thurston’s classiﬁciation theorem

It’s time to deﬁne pseudo-Anosov automorphisms. pseudo-Anosov maps A map φ ∈ MCG(S) is pseudo-Anosov if there are a transverse pair of transversely measured singular foliations F ±, µ± of S where µ± have no atoms and full support, and there is a real number λ > 1 such that φ takes leaves of F + to leaves of F + and similarly for F −, and multiplies the µ+ length of curves by λ, and the µ− length by λ−1.

Example Anosov automorphisms on tori are examples of pseudo-Anosov maps with no singularities. Thurston’s classiﬁciation theorem

Let φ ∈ MCG(S) be a map which does not have ﬁnite order and is not reducible. We proved that there exists a pair of invariant laminations and they admits some transverse measures. Collapse -neighborhood to get a weighted train tracks and glue rectangles to get a transverse pair of measured singular foliations. These (roughly) show the Thurston’s theorem in the next slide. Thurston’s classiﬁciation theorem

Thurston classification of surface homeomorphisms Let Σ be a closed, orientable surface of genus at least 2, let φ ∈ Homeo+(Σ). Then one of the following three alternatives holds: 1. φ is periodic; that is, some finite power of φ is isotopic to the identity. 2. φ is reducible; that is, there is some finite collection of disjoint essential simple closed curves in Σ which are permuted up to isotopy by φ. 3. φ is pseudo-Anosov; that is, some ψ isotopic to φ acts on Σ by a pseudo-Anosov automorphism. Geometric structures on general mapping tori

I just state the results. Thurston, Geometrization of surface bundles Let Σ be a closed, orientable surface of genus at least 2, let + φ ∈ Homeo (Σ). The mapping torus Mφ satisﬁes the following: 2 1. If φ is periodic, then Mφ admits an H × R geometry. 2. If φ is reducible, Mφ contains some embedded essential tori or Klein bottles. 3 3. If φ is pseudo-Anosov, then Mφ admits an H geometry.

It could be rephrased as ”Mφ admits a hyperbolic metric if and only if φ is pseudo-Anosov”. Peano Curves

˜ 3 Let φ be pseudo-Anosov. Then Mφ is naturally identified with H . 2 The covering group action of π1(Mφ) extends continuously on S∞, ie., we have 2 ρgeom : π1(Mφ) → Homeo(S∞) Another way of seeing the universal cover is identifying with ˜ Σ × R. This comes from the foliated structure of Mφ. The 2 universal cover of each fiber Σθ is quasi-isometric to H so it can 1 be compactified by its ideal boundary S∞. This circle can just be identified with the ideal boundary of π1(Mφ) with its word metric. There is a natural action

1 ρfol : π1(Mφ) → Homeo(S∞) Peano Curves

Cannon-Thurston, Continuity of Peano map 1 Suppose Mφ is a hyperbolic surface bundle over S with ﬁber Σ and monodromy φ. Then there exists a continuous surjective map 1 2 P : S∞ → S∞ which is a semiconjugacy between two actions of π1(Mφ), ie., for each α ∈ π1(Mφ),

P ◦ ρfol (α) = ρgeom(α) ◦ P Peano Curves