топология / Farb, Margalit, A primer on mapping class groups

rewrite it combinatorially, that is, as a collection of foliated rectangles with side identifications.

Let τ be a small arc transverse to F. If τ contains any singularities, we assume that it contains exactly one, at an endpoint. We subdivide τ by placing finitely many extra vertices at the points found by th e following procedures.

1.From each endpoint of τ , and for each of the two directions along F, follow F to the point of first return on τ .

2.From each singularity, follow each half-leaf of F to the first point of intersection with τ .

Denote the closed segments of the subdivision {τi}.

By Poincar´e Recurrence for foliations (Lemma 14.13), any l eaf starting at a point of τ eventually returns to τ . By the assumptions on τ , we can “push” each τi along F until it “hits” some τj . The result of this process is a union of rectangles in S, foliated horizontally. What is more, the union of these rectangles cover S—otherwise, the boundary of the union of rectangles would be a cycle of leaves of F (apply Lemma 14.11). Thus, we have obtained the desired rectangle decomposition.

An example of such a rectangle decomposition on the torus is given by the left side (and also the right side) of Figure 14.7.

STABLE AND UNSTABLE LEAVES ARE DENSE

Since stable and unstable foliations of T 2 have irrational slope, the following fact is well-known in the case of the torus.

Corollary 14.14 Let L be a leaf of a stable or unstable foliation F of a pseudo-Anosov homeomorphism of S. Then L is dense in S.

Proof. Let τ be an arbitrary arc in S that is transverse to F. It suffices to show that L intersects τ . Construct the rectangle decomposition of S obtained from τ as in the previous subsection. Since L is homeomorphic to either R or R≥0, and since L is contained in the horizontal foliations of the rectangles, it follows that L must hit one face of at least one rectangle. But

The following is a much stronger theorem. A foliation is uniquely ergodic if it only admits one measure, up to scale.

Theorem 14.15 If (Fs, µs) and (Fu, µu) are the stable and unstable foliations for a pseudo-Anosov mapping class, then Fs and Fu are uniquely ergodic.

The proof of Theorem 14.15, which relies on the theory of Markov partitions, can be found in [54, Expos´e 12, Th´eor`eme I]

14.4 ORBITS

A basic feature of any dynamical system is its set of orbits. Pseudo-Anosov homeomorphisms have the following property, which is an indication of a kind of “mixing” behavior.

Theorem 14.16 If f Homeo+(S) is a pseudo-Anosov homeomorphism, then f has a dense orbit in S.

Proof. We first show that if U is a nonempty open set that is invariant under f , then U is dense in S.

By taking a power of f , we may assume without loss of generality that f fixes the singular points of the stable and unstable foliatio ns Fs and Fu of f . Let L be a leaf of Fs departing from a singularity s (i.e., a separatrix). By Corollary 14.14, L is dense, and so U contains a point x of L. We may choose a segment J U of a leaf of Fu so that x J. Since x L, we have

lim f n(x) = s.

n→∞

Further, for each n, f n(J) is a subset of a leaf of Fu and is also contained in U . Thus, since f is stretching along Fu, we see that as n → ∞, the f n(J) approach the union of the separatrices of Fu bounding the sector containing the {f n(x)}. Since the leaves of Fu are dense in S (Corollary 14.14), it follows that U is dense in S.

Now, to finish the proof, let {Ui} be a countable basis for S. Each set

[

Vi = f n(Ui)

n Z

is a nonempty open set that is invariant under f , and hence is dense in S. By the Baire Category Theorem, the set ∩Vi is nonempty (dense, in fact). Let

xbe any point in this intersection. Then, for each i, there is an ni so that

xf ni (Ui), or, f −ni (x) Ui. Since there is an orbit point of x in each basis element Ui, the orbit of x is dense.

2

We will need the following standard tool; see, e.g., [156, Page 7].

Theorem 14.17 (Poincare´ Recurrence) Let M be a finite measure space, and let T be a measure-preserving self-map of M. For every set A with nonzero measure, and for almost every x A, there is an infinite increasing sequence of integers {ni} so that T ni (x) A for every i.

The following theorem gives another property that pseudo-Anosov homeomorphisms share with chaotic dynamical systems.

Theorem 14.18 For any pseudo-Anosov homeomorphism φ of the surface S, the periodic points of φ are dense in S.

For an Anosov mapping class of the square torus T 2 = R2/Z2, the periodic points are exactly the rational points, which are, of course, dense.

Proof. Let U be a “standard square” in S with respect to the stable and unstable foliations for φ. By assumption U does not contain a singularity of the foliations. It suffices to show that U contains a periodic point for φ.

Let V be a similar rectangle, contained strictly inside U . As the homeomorphism φ leaves invariant the area measure associated to the pair of foliations fixed by φ, we may apply the principle of Poincar´e Recurrence

(Theorem 14.17). This gives that for any N , there is an n > N so that

φn(V ) ∩ V 6= .

Theorem 14.19 A pseudo-Anosov homeomorphism f has the minimum num- ber of periodic points, for each period, in its homotopy class.

Proof. Suppose that h is homotopic to f . Then there is a canonical bijection between the set of lifts of f to S and the set of lifts of h to S; two lifts are

We first prove that | Fix(f )| ≤ | Fix(h)|. For x Fix(f ) and y Fix(h),

similar fashion—that is, set h = f —so the Nielsen equivalence classes for

˜

f are the projections to S of sets of the form Fix(f ) for different lifts of f .

We next show that if f is pseudo-Anosov, and h is homotopic to f , then every fixed point of f is Nielsen equivalent to some fixed point of h. To see

number L(F ), that is, the sum of the indices of the fixed points of F (see Section 8.2) is

homeomorphism as constructed above. Now, both L and L∞ are homotopy invariants of f (homotopic maps have the same action at infinity). So since

2L(f ) + L∞(f ) = 2L(h) + L∞(h) = 2L(h) + L∞(f )

(actually, by the formula in Section 8.2, all terms are equal to 2, but we don't need this). Since f is pseudo-Anosov, each of its fixed points must

˜ 6 ˜

have nonzero index. It follows that L(h) = 0, and in particular h has a fixed point. In other words, h has a fixed point Nielsen equivalent to the original fixed point for f .

We now claim that each Nielsen class of fixed points for f has at most one element. Indeed, if some Nielsen class of fixed points for f had at

least two elements, we would have some lift ˜ fixing two points. But the f

metric. Thus, if ˜ fixes two points, it would have to fix pointwise the unique f

geodesic between these two points, which contradicts the fact that ˜is acting f

by expansion by λ 6= 1. It follows immediately that | Fix(f )| ≤ | Fix(h)|.

Let k ≥ 1. Since f k and hk are homotopic, and f k is a pseudo-Anosov homeomorphism, it follows that | Fix(f k)| ≤ | Fix(hk )|. However, this does not prove the theorem, since points of Fix(hk ) might have “minimal

period” strictly less than k, that is, points of Fix(hk ) might be points of Fix(hj ) for some j < k.

So suppose that x has minimal period k with respect to f , and let h be any map homotopic to f . What we have shown is that hk has at least one fixed point y that is Nielsen equivalent to x. We need to show that y has

As above, a lift of a pseudo-Anosov homeomorphism can only fix one point

14.5 LENGTHS AND INTERSECTION NUMBERS UNDER ITERATION

As noted above, if A SL2(Z) has two distinct real eigenvalues λ and λ−1, then for almost any vector v the number ||An(v)|| grows like λn||v|| as n → ∞. On further inspection, we see that the absolute values of the individual coordinates of An(v) go to infinity as n → ∞. These facts imply that the length of any simple closed curve on T 2 grows exponentially under iteration

of an Anosov homeomorphism, and the intersection number of any curve with any fixed curve grows exponentially as well. Our goal in t his section is to prove that similar phenomena hold for pseudo-Anosov homeomorphisms of higher genus surfaces.

For both theorems, we will need the following concept. Let a be an isotopy class of simple closed curves in S, and let (F, µ) be a measured foliation on S. We define

I((F, µ), a)

as the infimum of the µ-measures of (not necessarily simple) representatives of a.

Theorem 14.20 Suppose that f Mod(S) is pseudo-Anosov, with dilatation λ. If ρ is any Riemannian metric on S, and a is any isotopy class of simple closed curves in S, then

Proof. Let µs and µu be the measures associated to the stable and unstable foliations for f , and let

p

µ = (µs)2 + (µu)2 be the corresponding singular Euclidean metric.

We start by proving the analogue of the theorem for the metric µ. Let ℓu denote the length function with respect to µ. For an isotopy class b, the number ℓµ(b) is taken to be an infimum, as usual.

Let α be a representative simple closed curve for the isotopy class a, and let φ be a pseudo-Anosov homeomorphism representing f .

From the definitions, we have

Z Z Z Z

ℓµ(f n(a)) ≤ dµs + dµu = λn dµs + λ−n dµu

φn (α) φn(α) α α

and

lµ(f n(a)) ≥ I((Fs, µs), φn(a)) = λnI((Fs, µs), a).

By Corollary 14.12, we know I((Fs, µs), a) > 0. We thus have

q

lim n lµ(φn(α)) = λ.

n→∞

To complete the proof, we now need to relate the metric µ to the arbitrary metric ρ. We know that any another Riemannian metric can only differ from ρ by a bounded amount, but we will also see that ρ only differs from the singular metric µ by a bounded amount. In particular, we will show that there are constants m and M so that

m ≤ lµ(c) ≤ M.

for any isotopy class of curves c.

Let {si} be the set of singularities of the foliation Fs. Choose a radius r small enough so that the closed balls B(si, r) are embedded and pairwise disjoint, and small enough so that the geodesic (in either metric) between two points on ∂B(si, r) lies entirely in B(si, r).

There are constants m′ and M ′ so that if β is any rectifiable curve, then, in the complement of the union of the B(si, r/2), we have

m′ ≤ lρ(β) ≤ M ′. lµ(β)

Claim: There exist constants m′′ and M ′′ so that if x and y are any distinct points in the same ∂B(si, r), we have

m′′ ≤ dρ(x, y) ≤ M ′′.

dµ(x, y)

If x and y are sufficiently close, then the previous inequality applie s. For x and y outside an open neighborhood of the diagonal of ∂B(si, r)×∂B(si, r), the fraction is a well-defined, continuous, positive functi on on a compact set. This completes the proof of the claim.

Set m = min{m′, m′′} and M = max{M ′, M ′′}.

Let γ be a ρ-geodesic representative for c, and let γ′ be the curve obtained from γ by replacing each segment of the intersection B(si, r) ∩ γ with the

corresponding µ-geodesic segment. Combining the previous two inequalities, we see

m ≤ lµ(γ′) ≤ M.

This gives

lρ(c) = lρ(γ) ≥ mlµ(γ′) ≥ mlµ(c).

For the other direction, choose γ to be a µ-geodesic for c, and let γ′ be the curve obtained by substituting ρ-geodesics inside the B(si, r). We have:

lρ(c) ≤ lρ(γ′) ≤ M lµ(γ) = M lµ(c).

In the case of the torus T 2, an Anosov mapping class “pulls” all curves in the direction of the eigenvector. We will now give a version of this statement for higher genus surfaces. We should think of Theorem 14.21 and Corollary 14.22 as saying that pseudo-Anosov mapping classes “pu ll” curves in the “direction” of the unstable foliation.

Theorem 14.21 Let f be a pseudo-Anosov element of Mod(S) with stable and unstable foliations (Fs, µs) and (Fu, µu), and suppose that the area of S with respect to the area form µu µs is 1. For any two isotopy classes of curves a and b in S, we have

lim i(f n(a), b) = I((Fs, µs), a)I((Fu , µu), b).

n→∞ λn

Let MF denote the set of equivalence classes of measured foliations on a surface S, where the equivalence relation is generated isotopy and Whitehead moves (see Figure 15.12). Next, let PMF denote the set of projective classes of (equivalences classes of) measured foliations. Since Whitehead moves do not affect the function I, we can think of I as giving a map

PMF → P (RS),

where S is the set of isotopy classes of simple closed curves in the surface S, and P (RS) is the set of projective classes in RS.

Via the geometric intersection number i, we can also identify S with a subset of P (RS).

The thrust of Thurston's proof of the Nielsen–Thurston Clas sification is to also think of Teich(S) as sitting in P (RS); with the induced topologies, Teich(S) PMF is homeomorphic to a closed ball of dimension 6g − 6. See Chapter 15 for more on this.

In the statement of Corollary 14.22, we use brackets to denote projective classes in P (RS).

Corollary 14.22 Let f be a pseudo-Anosov element of Mod(S) with stable and unstable foliations (Fs, µs) and (Fu, µu), and let a be any isotopy class of simple closed curves. We have

lim [f n(a)] = [(Fu, µu)]

n→∞

in P (RS).

14.6 MARKOV PARTITIONS

Consider the mapping class f of T 2 given by the matrix

2 1 .

1 1

If we take τ to be a small arc of the stable foliation, and construct the corresponding rectangle decomposition as in Section 14.3, we obtain a decomposition as in the left hand side of Figure 14.7.1

The linear map in the class of f take the picture on the left hand side of Figure 14.7 to the picture on the right hand side. We see that a lot of the structure is preserved. In particular, rectangles get taken to rectangles, the horizontal direction is preserved, and sides of rectangles in the stable foliation get sent to other such sides.

If we decompose T 2 into its constituent rectangles, we get a picture like the top of Figure 14.8 (note the identifications).

1The construction explained earlier really gives three rectangles, two of which combine to form the red rectangle. One can modify the construction in the case of orientable foliations so that, in this case, one immediately gets the 2 rectangles: subdivide τ along all backwards images of endpoints of τ and singularities, and consider those segments the left sides of the rectangles.