Diophantine and ergodic foliations on surfaces

This paper gives a topological characterization of Diophantine and recurrent laminations on surfaces. It also establishes an upper bound for the number of ergodic components of a measured foliation, in terms of limits of the corresponding geodesic ray in M¯g,n . Taken together, these results give a new approach to Masur's theorem on unique ergodicity.


Introduction
This paper gives a topological characterization of Diophantine and recurrent laminations on surfaces. It also establishes an upper bound for the number of ergodic components of a measured foliation, in terms of limits of the corresponding geodesic ray in M g,n . Taken together, these results give a new approach to Masur's theorem on unique ergodicity. Teichmüller rays. Let M g,n denote the moduli space of Riemann surfaces of genus g with n punctures. Consider the Teichmüller ray γ : [0, ∞) −→ M g,n generated by a holomorphic quadratic differential q = q(z) dz 2 on X ∈ M g,n . If there exists a compact set K ⊂ M g,n such that γ(t) ∈ K for all t, we say that γ is Diophantine. If γ(t i ) ∈ K for a sequence t i → ∞, we say that γ is recurrent.
In § 2 we will show:

It is recurrent if and only if
Here S ranges over all simple closed curves in ML g,n , and |S| is a combinatorial measure of its length (for example, the number of 1-cells required to describe S in a fixed triangulation of Σ g,n ).
This corollary is a restatement of [15,Theorem 1.1]; see also [16,Theorem 3.8]. The latter survey also describes its striking applications to polygonal billiards. The approach given here avoids technical issues that may arise from singular measures and the zeros and poles of limits (Y, q) . Veech examples. A well-known construction of Veech provides examples of quadratic differentials (X 0 , q 0 ) of genus 2 such that F(q 0 ) is minimal but not ergodic [20], [16, § 3]. To conclude, we show that Theorem 1.4 yields the sharp bound k = 2 for the number of ergodic components of these examples. Theorem 1.6. The flow line in Q 1 M 2 generated by a Veech example accumulates on a stable differential (Y, q) such that Y * has two components, each of genus 1.
Proof. By Veech's construction, we have a differential (E 0 , r 0 ) of genus 1 and a degree 2 holomorphic map π 0 : X 0 → E 0 such that q 0 = π * 0 (r 0 ). Applying the Teichmüller flow to both (X 0 , q 0 ) and (E 0 , r 0 ), we obtain a family of maps π t : X t → E t with π * t (r t ) = q t . Since F(q 0 ) is minimal, so is F(r 0 ), and thus (E t , r t ) is recurrent in Q 1/2 M 1 ; say (E tn , r tn ) → (E, r). The differentials arising from 2-fold branched covers of (E, r) have compact closure K ⊂ Q 1 M 2 , so (X t , q t ) has some accumulation point (Y, q) ∈ K. Since F(q 0 ) is not ergodic, q must be supported on k > 1 components of Y * ; hence Y * has exactly two components, each of genus 1.
Notation. The notation A B means that A/B and B/A are both bounded above by an implicit constant C.

Diophantine and recurrent laminations
In this section, we discuss quadratic differentials, measured foliations and laminations, and the Teichmüller geodesics they generate. For more on these topics, see, for example, [5,8,10,1].
Proofs of the statements in Theorem 1.1 follow. We conclude with a discussion of harmonic forms with two periods, and various other sources of Diophantine and recurrent geodesics.
Quadratic differentials. Let M g,n denote the moduli space of Riemann surfaces of genus g with n punctures. For each X ∈ M g,n we let Q(X) denote the space of holomorphic quadratic differentials on X whose total mass m(X, q) = X |q| is finite. This means that q has at worst simple poles at the punctures of X. The zeros of q form a finite set Z(q) ⊂ X, provided q = 0.
The moduli space of quadratic differentials QM g,n −→ M g,n consists of pairs (X, q) with 0 = q ∈ Q(X). The corresponding bundle over Teichmüller space will be denoted by QT g,n . We let Q 1 M g,n denote the locus where m(X, q) = 1. There is a natural action of PSL 2 (R) on QM g,n , preserving m(X, q), with the diagonal matrices giving the Teichmüller geodesic flow.
Laminations. Let ML g,n denote the space of (nonzero) measured geodesic laminations on a standard topological surface Σ g,n . The intersection number of a pair of laminations will be denoted by i(λ, μ).
By straightening the vertical foliation F(q) of a quadratic differential, we obtain a measured lamination Λ(q) (cf. [14]). The map q → (Λ(q), Λ(−q)) gives an embedding QT g,n → ML g,n × ML g,n , which sends a Teichmüller geodesic (X t , q t ) to a path of the form (e t λ, e −t μ). A pair of laminations is in the image if and only if i(λ, ξ) + i(μ, ξ) > 0 for all ξ ∈ ML g,n . Remarks.
1. One might expect X t to converge to [λ] ∈ PML g,n as t → ∞. This is true often, but not always [13].
2. The process of straightening a foliation to a lamination is similar to the process of resolving an interval exchange transformation f : I → I, by replacing all points of discontinuity of all iterates f n with two points. The result is a measure-preserving homeomorphism F : K → K, usually on a Cantor set, to which the usual methods of topological dynamics can be applied.
Lengths. It is useful to introduce a measure |S| of the combinatorial length of a simple closed curve S ∈ ML g,n . Many definitions are possible; for concreteness, we fix a triangulation of Σ g,n , and we let |S| denote the minimum number of 1-cells in a cycle homotopic to S.
For any nonzero q ∈ Q(X) we let L(S, |q|) denote the length of S in the conformal metric with area form |q|. That is, If the vertical and horizontal foliations of q are given by the laminations (λ, μ), then we have Indeed, if X is compact, we can replace S by a geodesic in the |q|-metric; then, in local coordinates, where q = dz 2 , z = x + iy, we have and (2.1) follows from the inequality |z| |x| + |y| 2|z|. In the noncompact case, S need not be represented by a geodesic, but the same argument applies to a length-minimizing sequence of representatives.
Since |S| L(S, |q|), we also have where the implicit constants depend on q. Compactness. As is well known, a sequence X i ∈ M g,n tends to infinity if and only if the length of the shortest closed hyperbolic geodesic on X i tends to zero [19]. The metrics coming from quadratic differentials also detect divergence.
Here the infimum is over all simple closed curves S ∈ ML g,n .
Proof. The function inf S L(S, |q|) is positive and continuous, so Diophantine geodesics. Let γ : [0, ∞) → M g,n be the Teichmüller ray generated by (X, q) ∈ QM g,n . The ray γ is Diophantine if it is contained in a compact subset of moduli space.

Theorem 2.2. The ray γ generated by q is Diophantine if and only if the vertical lamination
Proof. Let (λ, μ) be the vertical and horizontal laminations of q, and let γ(t) = (X t , q t ) be the corresponding geodesic ray. Then the laminations of q t are (e t λ, e −t μ). By the proposition above, γ is Diophantine if and only if Consider any fixed S and let (A, The same reasoning shows the following corollary.

n generated by q is bounded if and only if its vertical and horizontal laminations satisfy
Remark (Diophantine slopes on the torus). Up to scale, a lamination λ on the square torus Σ 1 = R 2 /Z 2 is given by a slope α, say 0 < α < 1. A simple closed curve S is determined by a pair of relatively prime integers (p, q). Their intersection number is given by which can only be small if p < q. Under this assumption, we can take |S| = |q|; then (2.4) becomes the standard bounded type condition or equivalently |α − p/q| > C/q 2 . Thus, we recover the well-known fact that α is the endpoint of a bounded geodesic ray in M 1 if and only if its continued fraction expansion is bounded.
. We will show that γ diverges if and only if δ(T ) → 0. Reasoning as in the Diophantine case, we find γ is divergent if and only if as T → ∞. Suppose this is the case. Then (T ) 1 for all T large enough. For each such T , we have a simple curve S with |S| (T )T T and T · i(S, λ) (This is always possible.) Then we have δ(α(T )T ) 1 for all T sufficiently large. For each such T , there is an S such that |S| α(T )T and α(T )T · i(S, λ) δ(α(T )T ). This shows Since the right-hand side tends to zero as T → ∞, γ is divergent.
Harmonic forms with two periods. The following criterion is sometimes useful for giving examples. Consider a nonzero holomorphic 1-form ω on a Riemann surface X ∈ M g with vertical foliation F = F(ω 2 ). This foliation depends only on the harmonic form ρ = Re(ω) and the smooth structure on X. The relative periods of ρ are given by Proof. This result is immediate for the foliation E of the square torus E = R 2 /Z 2 determined by the 1-form ξ = α dx + β dy. To handle the general case, choose a smooth map f : X → E such that f * (ξ) = ρ and f sends Z(ρ) to a single point p ∈ E. (Such a map exists by our assumptions on Per(ρ); indeed, we can write ρ = αξ 1 + βξ 2 as a linear combination of two integral cohomology classes, and construct the two coordinates of f by integrating ξ 1 and ξ 2 .) To complete the proof, it suffices to show that F is Diophantine (respectively, recurrent) whenever the same is true of E.
Consider an essential simple closed curve S on X. Up to isotopy, S is represented by a geodesic on (X, |ω|) consisting of a chain of saddle connections S 1 , . . . , S n . This representative satisfies i(S, F) = | Si ρ| and Combining these bounds we find where C > 0 is independent of S. Since F has no loop made up of saddle connections, i(T i , E) = 0, and thus each T i is isotopic to a multiple of a nontrivial simple loop T on E. Consequently, we have

|T |i(T, E).
This shows that F is Diophantine whenever E is, by Theorem 2.2. The same reasoning works for recurrence as well, using Theorem 2.4.
A similar result, expressed in the language of interval exchange transformations, appears in [2]. Example (genus 2). Let x denote the Galois conjugate of x ∈ Z( such that α > α and β > β . Consider the L-shaped polygon P shown in Figure 1. By gluing opposite sides with vertical and horizontal translations, we obtain a Riemann surface X of genus 2. We claim that the diagonal foliation F of X, defined by the harmonic form ρ = dx − dy, is Diophantine. Since the periods of ρ lie in Q( √ D), we only need to rule out saddle connections. Consider the first return map to the horizontal edges of P/∼ under the diagonal flow. Locally, this map has the form x → x + n + β or x + m + α + β, with n, m ∈ Z. Both of these forms strictly increase x − x , so the first return map has no periodic points and hence F has no saddle connections. Further examples: Schottky groups and winning sets. Any lamination in the limit set of a convex cocompact subgroup of the mapping-class is necessarily Diophantine. Examples of such subgroups, similar to Schottky groups acting on H 3 , are discussed in [4].
In [12] it is shown that, for any quadratic differential q ∈ QM g,n , the vertical foliation of e iθ q is Diophantine for all θ in a set of Hausdorff dimension 1. In fact the Diophantine directions form an absolute winning set [3].

Ergodicity of pseudo-Anosov foliations
Let γ ⊂ M g,n be a closed Teichmüller geodesic generated by (X, q) ∈ QM g,n . Then we have an associated pseudo-Anosov mapping f : X → X, locally of the form f (x + iy) = Kx + iK −1 y, K > 1 in coordinates where z = x + iy and q = dz 2 . (Such coordinates exist away from the zeros of q.) The length of γ is log K in the Teichmüller metric.
In this section, we will show the following theorem.

Theorem 3.1. The stable foliation F(q) of a pseudo-Anosov mapping is ergodic.
This result is well known, and in fact F(q) is uniquely ergodic (see, for example, [5,Exposé 12]). The proof we give below generalizes readily from closed geodesics to recurrent geodesics, after which it too will yield unique ergodicity (see the proof of Corollary 1.5).
Distribution of leaves. We begin with the setup for a general foliation. Let (X, q) ∈ Q 1 M g,n be a unit-area quadratic differential, and let F = F(q) be the vertical foliation. The conformal metric |q| determines a probability measure This measure is given locally by the product |dz| 2 = |dx| · |dy| of arclength measure |dy| along the leaves of F with the transverse invariant measure |dx|.
The foliation is ergodic if any measurable saturated set satisfies m(A) = 0 or 1. Since F has only finitely many singular leaves (through the zeros and poles of q), these play no role in the definition of ergodicity.
Let X denote the compact surface obtained by filling in the punctures of X. For any continuous function h ∈ C(X) and any x ∈ X, we can average the values of h with respect to arclength along the leaves of F through x. If these averages converge (for leaves in both directions through x), they define a probability measure ν x on X. Note that ν x = ν y whenever x and y lie on the same leaf of F.
The measure m = |q| on X is invariant under the (locally defined) unit speed flow along the leaves of F. By the ergodic theorem, this implies that ν x exists for all x in a set E ⊂ X with m(E) = 1. In fact, for almost every x and h ∈ C(X) we have where H is the projection of h to the closed subspace of L 2 (X, m) consisting of functions that are constant on the leaves of F. In particular, Thus ν x is supported in X for almost every x. Formula 3.1 implies the following:

Proposition 3.2. The foliation F is ergodic if and only if ν x is constant almost everywhere on X.
In the ergodic case, ν x = m almost everywhere, and hence almost every leaf of F is uniformly distributed.

Proposition 3.3. Suppose x, y ∈ E and there is an index set S such that the limits
both exist and lie in the same rectangle R of (X, q). Then ν x = ν y .
Proof. Consider, for each s ∈ S, the rectangle R s = f −s (R). Since f −s expands the vertical sides of R by K s and contracts the horizontal sides by K −s , the region R s is close to a long leaf of F through x. Any h ∈ C(X) is uniformly continuous in the metric |q|, and thus (1/m(R s )) Rs h dm → h dv x . The same reasoning applies to y, and hence ν x = ν y .
Since f preserves the measure m, a general result in ergodic theory (see Appendix A) guarantees the following.  m(ω(S, B)) > 0. Thus there exists an x ∈ B and an index set T ⊂ S such that y = lim T f t (x) is not a zero or pole of q. Then y lies in a rectangle R for (X, q). But R meets F (A), so ν y = ν a = μ for some a ∈ A, which is a contradiction.
Thus m(B) = 0, and hence F is ergodic.

Ergodic components and stable curves
In this section, we will discuss quadratic differential on stable curves, and prove the bound on the number of ergodic components of F(q 0 ) stated in Theorem 1.4. For more on the stable curves and their quadratic differentials, see, for example, [8,6,7]. Stable quadratic differentials. Let M g,n denote the moduli space of stable curves Y . Let Q(Y ) denote the holomorphic quadratic differentials on the smooth points Y * of Y , with at worst simple poles at punctures and at worst double poles, with equal residues, at nodes. The moduli space QM g,n consists of pairs (Y, q) with 0 = q ∈ Q(Y ). Note that m(Y, q) = |q| is finite if and only if all poles of q are simple.
As in the case of holomorphic 1-forms, the bundle of projective spaces PQM g,n → M g,n is compact. However, mass can be lost in the limit, so the locus Q 1 M g,n of differentials with m(Y, q) = 1 is not compact.
Teichmüller maps and area-preserving maps. To set up the proof of Theorem 1.4, we first relabel (X ti , q ti ) as (X i , q i ). The natural Teichmüller mapping F i : X 0 → X i sends the measure |q 0 | to |q i |, while shrinking the leaves of F(q 0 ) by a factor of K i = exp(−t i ).
Let W ⊂ Y denote the support of q, that is, the union of the irreducible components of Y where q is not identically zero. Since m(X i , q i ) = m(Y, q) = 1, we can find a sequence of piecewise smooth, bijective maps H i : X i → W sending the measure |q i | to the measure |q|.
(We do not require that H i be continuous). We can also arrange that the differentials (H i ) * (q i ) converge to q smoothly on compact subsets of W * − Z(q).
(Here is one way to construct the maps H i . Choose a compact set K ⊂ W * − Z(q) carrying most of the mass of |q|. Then, for all i 0, we have holomorphic maps h i : K → X i such that h * i (q i ) → q on K. Thus ρ i = |h * i (q i )|/|q| → 1 smoothly on K, and in particular K ρ i |q| < W |q| for all i 0. We can therefore find area-preserving maps s i : W → W , converging smoothly to the identity on K, such that s i |K sends the measure ρ i |q| to |q| (cf. [18]). The map H i = s i • h −1 i is then smooth and area-preserving on h i (K), and (H i ) * (q i ) = (s i ) * (q) → q on K, as required. It is now easy to extend these maps to the rest of X i and let K grow with i to complete the construction.) The compositions f i = H i • F i : X 0 → W then give a sequence of area-preserving maps, with respect to the area measures |q 0 | and |q| on their domain and range.
Proof of Theorem 1.4. Let k be the number of components of W * . We wish to show that F(q 0 ) has no more than k ergodic components.
x ∈ X 0 we have a probability measure ν x describing the distribution of the leaf of F(q 0 ) through x. Again we make two observations.
1. If x = lim S f s (x) and y = lim S f s (y) both lie in an open rectangle R of (W, q), then ν x = ν y .
2. There exist a countable set A ⊂ X 0 and an index set S such that F (x) = lim S f s (x) exists for all x ∈ A, and F (A) is dense in W * .
To see the first observation, use the fact that H −1 i (R) is contained in a nearly isomorphic rectangle R i ⊂ X i for all i 0. The second observation follows from Appendix A. Now let W 1 , . . . , W k denote the connected components of W * , and let A i = {x ∈ A : F (x) ∈ W i }. Since any two points of W i are joined by a chain of rectangles, ν a is constant for a ∈ A i . Call this constant value μ i .
Each of these sets is saturated, and F|Z i is ergodic. It remains only to show that B = X − k 1 Z i has measure zero. But if m(B) > 0, then (again by Lemma A.1) there is an x ∈ B and an index set T ⊂ S such that y = lim T f t (x) is neither a zero of q nor a node of W . Then y lies in a rectangle R meeting F (A i ) for some i, which implies ν x = μ i , which is a contradiction.
So, in fact X 0 = k 1 Z i gives the ergodic decomposition of F.
Examples. Let (E i , q i ) ∈ QM 1 , i = 1, 2 be a pair of quadratic differentials of genus 1 with Diophantine vertical foliations. Slit each surface open along a unit vertical arc, and then glue them together to obtain a new differential (X 0 , q 0 ) ∈ QM 2 .
The slits give a pair of vertical saddle connections forming a loop α 0 ⊂ X 0 . The corresponding loop α t ⊂ X t shrinks to a node as t → ∞.
It easy to see that all accumulation points (Y, η) of (X t , q t ) in QM g consist of 2-forms (F 1 , η 1 ), (F 1 , η 2 ) of genus 1 joined at a node, and that many such accumulation points exist. Indeed, each component (F i , η i ) is simply an accumulation point of the bounded flow line in QM 1 generated by (E i , q i ). In particular, there is no loss of mass; we have Thus we have an instance of Theorem 1.4 with k = 2. And indeed F(q 0 ) has two ergodic components, separated by α 0 . Now repeat the construction, but replace α 0 with a cylinder C 0 of positive area (Figure 2) foliated by closed leaves. Then the accumulation points of (X t , q t ) remain the same, but the vertical foliation F(q 0 ) has infinitely many ergodic components. The difference is that the mass of C 0 now disappears in the limit, so we no longer have an instance of Theorem 1.4.
This example shows that recurrence in Q 1 M g,n , unlike recurrence in M g,n , is not a monotone property of the vertical foliation.

Appendix: Orbits and measures
This section gives an elementary result in ergodic theory, used in § 3 and § 4.
Let X be a compact metric space equipped with a complete probability measure m of full support, and let f n : X −→ X be a sequence of measure-preserving mappings. Assume that f −1 n sends open sets to Borel sets; this regularity certainly holds if f n is piecewise continuous.
Let ω(x) ⊂ X denote the set of all accumulation points of the sequence f n (x) , and let ω(E) = x∈E ω(x). The projection of a Borel set is not always Borel, but it is analytic and hence measurable for any complete measure [11].
Corollary A.2. If E ⊂ X has full measure, then ω(E) is dense.
As in § 3, given S = {s 1 < s 2 < · · · } ⊂ N, we let lim S f s (x) = lim i→∞ f si (x). Corollary A. 3. If E has full measure, then there exist an index set S and a countable set A ⊂ E such that F (x) = lim S f s (x) exists for all x ∈ A, and F (A) is dense in X.
Proof. Let (U i ) be a countable base for X. Since ω(E) is dense, there is an a 1 ∈ X and an S 1 ⊂ N such that lim S1 f s (a 1 ) ∈ U 1 . Restricting attention now to {f s : s ∈ S 1 }, we find that there is an a 2 ∈ X and an S 2 ⊂ S 1 such that lim S2 f s (a 2 ) ∈ U 2 . Continuing inductively and then diagonalizing, we obtain the required set A = {a 1 , a 2 , . . .} and index set S.
Remark. These results apply just as well to a sequence of measure-preserving maps f n : Y → X.