Ratner fails for ΩM 2

Ratner fails for ΩM 2 Curtis T. McMullen After Chaika, Smillie and Weiss 9 December 2018 Contents 1 Introduction............................ 2 2 Unipotent dynamics on ΩE 4................... 3 3 Shearing with respect to a measured foliation......... 4 4 Minimal but not ergodic foliations............... 5 5 Controlling excursions...................... 6 6 Measures............................. 7 Research supported in part by the NSF. Typeset 2019-02-06 00:31. 1

1 Introduction These informal notes sketch the proof by Chaika, Smillie and Weiss (as yet unpublished) that Ratner s theorem fails for the action of the unipotent group { ( ) } 1 t N = u t = : t R SL 2 (R) 0 1 on the moduli space ΩM 2 of holomorphic 1 forms (X, ω) of genus two. They are based on a talk by Chaika at Stanford in May of 2018, and my own seminar talks at Harvard in Fall of 2018. Let ΩE D ΩM 2 denote the eigenform locus of discriminant D (cf. [Mc1]). In the case D = 4, these are simply the forms that arise as connect sums (X, ω) = (E, dz)# (E, dz) I of an elliptic curve E = C/Λ with itself. (See [Mc1, 7] for the notion of connected sums of 1 forms.) Equivalently, there is a degree two holomorphic map π : X E such that π (dz) = ω. Let us say an orbit N p is uniformly distributed with respect to the probability measure µ if for all compactly supported continuous functions f, we have 1 T lim f(u t p) dt fµ. T T 0 Clearly µ is unique if it exists. Let Ω 1 M 2 denote the forms of genus two with X ω 2 = 1, and similarly for Ω 1 E D. Each of these loci carries a unique SL 2 (R) invariant probability measure that is absolutely continuous with respect to Lebesgue measure. Our aim is to prove two results. Theorem 1.1 There exists a form of genus two that does not belong to Ω 1 E 4, but nevertheless its N orbit is uniformly distributed with respect to the natural measure on Ω 1 E 4. Theorem 1.2 There exists an N orbit in Ω 1 M 2 which is not uniformly distributed with respect to any measure. In contrast, if we replace ΩM 2 with a finite volume homogeneous space M = G/Γ, then Ratner s theorem states (among other things) that every unipotent orbit N p is uniformly distributed with respect to a measure that includes p in its support. 2

2 Unipotent dynamics on ΩE 4 The proof that Ratner s theorem fails for ΩM 2 is based in part on the fact that it holds for the action of N on the eigenform locus ΩE 4. In fact the same is true for the N action on ΩE D for all D, as shown in [BSW]. The case D = 4 is simpler however; it follows directly from Ratner s theorem for the space M = G/Γ = ASL 2 (R)/ ASL 2 (Z), where ASL 2 (R) = R 2 SL 2 (R) is the special affine group of R 2. A point in M is specified by a triple (E, dz, p), where E = C/Λ, E dz = 1, and p E. In particular M is a torus bundle over the moduli space of 1 forms of genus one: we have a natural map M ΩM 1 = SL 2 (R)/ SL 2 (Z), and its fiber over (E, dz) is E itself. Put differently, there is an action of S 1 on M sending Λ to e iθ Λ, and M/S 1 is the universal elliptic curve over M 1. Let M M denote the locus where p 0 E. We then have a natural map M Ω 2 E 4, compatible with the action of N. This map sends (E, dz, p) to (X, ω), where (X, ω) = (E, dz) # (E, dz). [0,p] It allows us to identify M /Z/2 with Ω 2 E 4, where we have taken the quotient by p p. Using Ratner s theorem for M, one can then classify the orbits of N acting on Ω 2 E 4 as follows: 1. The branch point p is torsion in E. Then N (X, ω) is contained in a closed, SL 2 (R) invariant sublocus of square tiled surfaces. 2. The point p is in R. Then N (X, ω) is contained in the closed, N invariant sublocus where there is a horizontal geodesic (saddle connection) of length p joining the zeros of ω. 3. Neither of these conditions on p holds, and N (X, ω) is uniformly distributed in Ω 2 E 4. More precise statements can be made in case (2), but for our application the main point is that outside a countable union of explicit proper submanifolds, every N orbit in Ω 2 E 4 is uniformly distributed. 3

3 Shearing with respect to a measured foliation We now come to the main construction in the proof. This construction is quite general and works in any genus. It also makes contact with, and generalizes, some constructions in [Mc2], [Mar], [Wr] and [FR]. Let F(ω) denote the horizontal measured foliation of (X, ω) ΩM g. If we write ω = σ + iρ as a sum of real harmonic forms, then F(ω) is defined by ρ, in the sense that the form ρ vanishes along the leaves of F(ω) and the measure of a transversal is given by τ ρ. Let A X be a measurable subset of X that is saturated by leaves of the foliation F(ω), and let χ A be the indicator function of A. Then the current defined by χ A ρ satisfies d(χ A ρ) = (dχ A ) ρ = 0, since χ A is constant along leaves. Compare [Mc2, 2]. In particular, from A and ω we obtain a natural relative cohomology class [χ A ρ] H 1 (X, Z(ω)). (The span of all such cohomology classes is called the content of ρ in [Mc2].) The usual action of N is given in period coordinates by u t (X, ω) = [σ + (i + t)ρ]. From the measurable set A we obtain a similar flow, namely u A t (X, ω) = [σ + (i + tχ A )ρ]. This flow can be interpreted geometrically as shearing along the leaves of F(ω) using the transverse measure coming from ρ A. The flows determined by two different saturated sets A and B commute; and in particular, u A t commutes with the standard horocycle flow u t = u X t. Shearing in genus two. A simple case of this construction arises when (X, ω) = (A, dz)# (B, dz) Ω 2 E 4, I and I = [0, z] with z R. Then the two factors in the sum above give rise to a decomposition of X into a pair of slit tori, which we also denote by A and B. The condition that z R insures that both A and B are saturated with respect to the horizontal foliation F(ω). In this case we have and similarly for u B t. u A t (X, ω) = (u t (A, dz))# (B, dz), I 4

4 Minimal but not ergodic foliations By a standard construction, going back to Veech, a suitable limit of connected sums gives rise to foliations in genus 2 that are minimal but not ergodic. Start with (X, ω) = (E, dz)# I 1 (E, dz) where E = C/Λ and I 1 = [0, z 1 ]. Then the foliation F(s 1 ω) has leaves parallel to I 1, where s 1 = z 1 /z 1. Thus we get a decomposition X = A 1 B 1, as above, where A 1 and B 1 are slit tori saturated by the leaves of F(s 1 ω). Let I n = [0, z n ] where z n = z 1 + 2λ n and λ n Λ. Note that I 1 I n maps to a closed loop in E, and in fact I 1 I n is trivial in H 1 (E, Z/2). Because of this we also write (X, ω) = (E, dz)# (E, dz). I n In this way we obtain a countable sequence of different ways of presenting (X, ω) as a connected sum, along slits of varying slopes. Each one gives a decomposition X = A n B n saturated by the leaves for F(s n ω), where s n = z n /z n. Now suppose z 1 Q Λ. It is then easy to see that we can choose λ n in Λ such that det(in, I n+1 ) = Im(z n z n+1 ) <. In this case s n s S 1 for some s, and we can also arrange that R sλ = {0}; (4.1) equivalently, that F(sω) has dense leaves. (In fact one can construct a Cantor set of limiting s s, by varying the choices of λ n, and at most countably many of these correspond to the slopes of closed geodesics on (X, ω).) Let us now replace (X, sω) with (X, ω), i.e. reduce to the case where s = 1. By the discussion of the preceding section, we then know: The N orbit of (X, ω) is uniformly distributed in Ω 1 E 4. Next we observe that the Z/2 chain A n + A n+1 has mass a constant times det(i n, I n+1 ) and hence the decomposition X = A n B n converges, in measure, to a measurable decomposition X = A B. Provided the determinant sum is small, neither A or B has measure zero, and hence: 5

Let The foliation F(ω) is minimal but not ergodic. We can also arrange that: (X, ω ) = u A 1 (X, ω). The form (X, ω ) does not lie in ΩE 4. One way to do this is to first observe that, provided we normalize so z 1 R +, we have (X 1, ω 1) = u A 1 1 (X, ω) = (u 1 (E, dz))# I 1 (E, dz) (E, dz)# I 1 (E, dz), and hence (X 1, ω 1 ) is not in ΩE 4; and second, to observe that if we make the determinant sum very small, then (X, ω ) is very close to (X 1, ω 1 ), so it too lies outside of ΩE 4. 5 Controlling excursions To prove Theorem 1.1, it remains to show that the N orbit of (X, ω ) is uniformly distributed with respect to the natural probability measure on ΩE 4 even though (X, ω ) does not belong to this locus. Closed sets and distribution. To explain the mechanism of proof we recall some of its underlying principles. First, suppose an orbit u t p is uniformly distributed with respect to a probability measure µ, and µ(f ) = 0. Can we assert that the orbit spends almost all of its time outside of F? The answer is no in general e.g. we might have F = N p. But if F is closed, then the answer is yes. The reason is that for each ɛ > 0 we can find a continuous function f 0 such that f F = 1 and fµ < ɛ. Topology of non-unique ergodicity. We can make similar advantageous conclusion when F is a countable union of closed sets. So at this point it is useful to recall: Proposition 5.1 Let Z be a compact metric space. Then the set of non uniquely ergodic homeomorphisms f : Z Z is a countable union of closed sets. Proof. The key point is that the set of f n invariant probability measures M(Z) fn can only get larger in the limit as f n f. Thus the set F m of all f such that diam M(Z) f 1/m is closed, and the set of non uniquely ergodic f is given by F m. 6

Similarly, choose a metric on ΩM 2 and let F m Ω 1 E 4 consists of the (X, ω) such that for some measurable set A X, saturated by the leaves of F(ω), and some t [0, 1], we have d(u A t (X, ω), Ω 1 E 4 ) 1/m. It is then easy to see that F m is closed. Moreover, uniquely ergodic foliations have full measure, so F m has measure zero. Now consider the N orbits given by (X t, ω t ) = u t (X, ω) and (X t, ω t) = u t (X, ω ). Recall that (X, ω ) = u A 1 (X, ω). Because the flows u t and u A t commute, we have (X t, ω t) = u B 1 (X t, ω t ) for a suitable measurable set B (the image of A under shearing). But the flow line (X t, ω t ) is uniformly distributed in ΩE 4, so it spends almost none of its time in F m. Thus d((x t, ω t), (X t, ω t )) < 1/m for almost all t. (More precisely, the proportion of t [0, T ] such that the relation above holds tends to 1 as T.) Since the flow line N (X, ω) is uniformly distributed, so is N (X, ω ). This completes the proof of Theorem 1.1. 6 Measures Finally we give the idea of the proof of Theorem 1.2. Let µ 1 and µ 2 denote the natural probability measures on Ω 1 E 4 and Ω 1 M 2 respectively. Consider the set S i ΩM 2 consisting of all forms such that N (X, ω) is uniformly distributed with respect to µ i. By ergodicity, S 2 is dense (and of full measure) in Ω 1 M 2. But using the construction of the preceding section, one can show that S 1 is also dense. 7

It now follows by general principles that there is a dense G δ in Ω 1 M 2 consisting of forms whose N orbits are not uniformly distributed at all. More precisely, if we let G i denote the forms such that the natural measures along finite orbits u [0,T ] (X, ω) accumulate on µ i, then G i is a G δ ; and it clearly contains S i. Since both S 1 and S 2 are dense, G 1 G 2 is also dense; in particular it is nonempty, completing the proof. References [BSW] M. Bainbridge, J. Smillie, and B. Weiss. Horocycle dynamics: new invariants and eigenform loci in the stratum H(1, 1). Preprint, 2016. [FR] M. Fortier Bourque and K.Rafi. Non convex balls in the Teichmüller metric. J. Differential Geom. 110(2018), 379 412. [Mar] V. Markovic. Carathéodory s metrics on Teichmüller spaces and L- shaped pillowcases. Duke Math. J. 167(2018), 497 535. [Mc1] [Mc2] C. McMullen. Dynamics of SL 2 (R) over moduli space in genus two. Annals of Math. 165(2007), 397 456. C. McMullen. Cascades in the dynamics of measured foliations. Ann. scient. Éc. Norm. Sup. 48(2015), 1 39. [Wr] A. Wright. Cylinder deformations in orbit closures of translation surfaces. Geom. Topol. 19(2015), 413 438. 8