on orbit closures in moduli space

Overview

Eskin, Mirzakhani, and Mohammadi proved that every $SL(2,\mathbb{R})$ orbit closure in a stratum of Abelian differentials is an affine invariant submanifold (Eskin, Mirzakhani & Mohammadi, 2015). In period coordinates this means that the closure is cut out locally by real linear equations and carries the natural affine Lebesgue measure. The result is the flat-surface analogue of Ratner-type orbit-closure and equidistribution theorems for unipotent flows on homogeneous spaces (Ratner, 1991a; Ratner, 1991b).

The Annals paper uses the measure classification theorem of Eskin and Mirzakhani as an input: every probability measure invariant under the upper triangular subgroup $P<SL(2,\mathbb{R})$ is $SL(2,\mathbb{R})$-invariant and affine (Eskin & Mirzakhani, 2018). Its new work is the isolation and non-escape mechanism that turns measure classification into orbit-closure, closed-set, compactness, and equidistribution statements. The proof is a powerful template: classify all possible stationary limits, then build height functions that prevent limits from losing mass or accumulating on unwanted smaller invariant pieces.

Translation Surfaces and Strata

Let $M$ be a compact Riemann surface of genus $g$, and let $\omega$ be a holomorphic one-form on $M$. Away from the zeros of $\omega$, local integration

$$z(p)={\int }^p\omega$$

gives charts whose transition maps are translations. Thus $(M,\omega )$ is a translation surface: it is locally Euclidean except at the zeros of $\omega$, where the flat metric has conical singularities.

If the zeros of $\omega$ have multiplicities

$$\alpha =({\alpha }_1,\dots ,{\alpha }_n),\sum _{i=1}^n{\alpha }_i=2g-2,$$

then the corresponding stratum is

$$\mathcal{H}(\alpha )=\{(M,\omega ):{\mathrm{ord}}_{p_i}(\omega )={\alpha }_i\}.$$

The area is

$$\mathrm{area}(M,\omega )=\frac{i}{2}{\int }_M\omega \wedge \overline{\omega }.$$

The area-one stratum is denoted

$${\mathcal{H}}_1(\alpha ).$$

Masur and Veech proved that the natural Lebesgue measure on ${\mathcal{H}}_1(\alpha )$ has finite total mass (Masur, 1982; Veech, 1982). This measure is now called the Masur—Veech measure.

The group $SL(2,\mathbb{R})$ acts on translation surfaces by postcomposing the flat coordinate charts with real linear maps. If locally

$$\omega =dx+idy,$$

then a matrix

$$g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$

sends the real coordinate pair $(x,y)$ to

$$(ax+by,cx+dy).$$

Since $\det g=1$, area is preserved. This action generalizes the familiar action of $SL(2,\mathbb{R})$ on flat tori

$$G{L}^{+}(2,\mathbb{R})/SL(2,\mathbb{Z}).$$

For genus at least two, however, strata are not homogeneous spaces. This is why a Ratner-type theorem here is surprising.

Period Coordinates

Let $\Sigma \subset M$ be the zero set of $\omega$. Choose a basis

$${\gamma }_1,\dots ,{\gamma }_k$$

of the relative homology group

$$H_1(M,\Sigma ;\mathbb{Z}).$$

The period map is

$$\Phi (M,\omega )=\left({\int }_{{\gamma }_1}\omega ,\dots ,{\int }_{{\gamma }_k}\omega \right)\in {\mathbb{C}}^k.$$

These periods give local coordinates on the stratum. Equivalently, a point of the stratum is locally represented by the relative cohomology class

$$[\omega ]\in H^1(M,\Sigma ;\mathbb{C}).$$

The $SL(2,\mathbb{R})$ action is linear in these coordinates once one identifies

$$H^1(M,\Sigma ;\mathbb{C})\simeq H^1(M,\Sigma ;\mathbb{R})\oplus i{H}^1(M,\Sigma ;\mathbb{R}).$$

Writing

$$\omega =\mathrm{Re}\omega +i\mathrm{Im}\omega ,$$

the matrix $g$ acts on the pair

$$(\mathrm{Re}\omega ,\mathrm{Im}\omega )$$

by the standard two-dimensional representation. This linearity is the source of the word affine in the theorem.

A useful mental model is the following. The ambient stratum is nonlinear as a moduli space, but in small period-coordinate charts it looks like an open subset of a complex vector space. The EMM theorem says that orbit closures are not wild subsets in those coordinates. They are locally linear.

Affine Invariant Manifolds

An ergodic $SL(2,\mathbb{R})$-invariant probability measure ${\nu }_1$ on ${\mathcal{H}}_1(\alpha )$ is called affine if its support ${\mathcal{M}}_1$ is, up to a measure-zero self-intersection set, an immersed submanifold whose cone

$$\mathbb{R}{\mathcal{M}}_1=\{(M,t\omega ):(M,\omega )\in {\mathcal{M}}_1,t\in \mathbb{R}\setminus \{0\}\}$$

is locally a linear subspace in period coordinates, and if the lifted measure is Lebesgue measure on that subspace (Eskin, Mirzakhani & Mohammadi, 2015). Such a support is called an affine invariant submanifold.

Thus an affine invariant submanifold has three simultaneous meanings:

• it is dynamically invariant under $SL(2,\mathbb{R})$,
• it is geometrically linear in period coordinates,
• it carries a canonical affine probability measure ${\nu }_{\mathcal{M}}$.

The full stratum is an improper affine invariant submanifold. Closed $SL(2,\mathbb{R})$ orbits, also called Teichmuller curves, are the smallest familiar examples. The theorem asserts that every orbit closure is built from the same linear-period-coordinate structure.

The Main Theorems

Let

$$A=\left\{\left(\begin{array}{cc}{e}^t& 0\\ 0& e^{-t}\end{array}\right):t\in \mathbb{R}\right\},N=\left\{\left(\begin{array}{cc}1& s\\ 0& 1\end{array}\right):s\in \mathbb{R}\right\},$$

and let

$$P=AN$$

be the upper triangular subgroup of $SL(2,\mathbb{R})$.

The measure-classification input is:

Eskin-Mirzakhani Measure Classification

Every $P$-invariant probability measure on ${\mathcal{H}}_1(\alpha )$ is $SL(2,\mathbb{R})$-invariant and affine (Eskin & Mirzakhani, 2018).

The Annals paper proves the corresponding topological and equidistribution consequences.

Orbit Closure Theorem

For every $x\in {\mathcal{H}}_1(\alpha )$,

$$\overline{Px}=\overline{SL(2,\mathbb{R})x}$$

is an affine invariant submanifold (Eskin, Mirzakhani & Mohammadi, 2015).

The equality of $P$- and $SL(2,\mathbb{R})$-closures is part of the conclusion. The theorem says that no intermediate fractal closure appears between a single orbit and the ambient stratum.

Closed Invariant Set Theorem

Every closed $P$-invariant subset of ${\mathcal{H}}_1(\alpha )$ is a finite union of affine invariant submanifolds.

This is stronger than just describing individual orbit closures. It rules out closed invariant sets assembled from infinitely many unrelated orbit closures inside a fixed compact region.

Mozes-Shah Type Compactness

If ${\mathcal{N}}_n$ is a sequence of affine invariant submanifolds and

$${\nu }_{{\mathcal{N}}_n}\to \nu$$

weakly, then $\nu$ is the affine measure ${\nu }_{\mathcal{N}}$ on the smallest affine invariant submanifold $\mathcal{N}$ that contains all but finitely many of the ${\mathcal{N}}_n$.

In particular, the set of ergodic $P$-invariant probability measures is compact in the weak-star topology. This is the moduli-space analogue of the Mozes-Shah compactness theorem for unipotent flows (Mozes & Shah, 1995).

Equidistribution Statements

The orbit-closure theorem is accompanied by averaged equidistribution. Let

$$a_t=\left(\begin{array}{cc}{e}^t& 0\\ 0& e^{-t}\end{array}\right),r_{\theta }=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right).$$

If $\mathcal{M}$ is the minimal affine invariant submanifold containing $x$, then for every compactly supported continuous function $\phi$ and every interval $I\subset [0,2\pi )$,

$$\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_0^T\frac{1}{|I|}{\int }_I\phi (a_t{r}_{\theta }x)d\theta dt={\int }_{\mathcal{M}}\phi d{\nu }_{\mathcal{M}}.$$

There is also a uniform version. For a fixed affine invariant submanifold $\mathcal{M}$, a test function $\phi$, and an error tolerance $\epsilon >0$, there are finitely many proper affine invariant submanifolds

$${\mathcal{N}}_1,\dots ,{\mathcal{N}}_{\ell }⊊\mathcal{M}$$

such that the same equidistribution holds uniformly on compact subsets of

$$\mathcal{M}\setminus \bigcup _{j=1}^{\ell }{\mathcal{N}}_j.$$

The paper proves parallel statements for random walks and for Folner-type averages along sets

$$\{a_t{u}_s:0\le t\le T,0\le s\le r\},u_s=\left(\begin{array}{cc}1& s\\ 0& 1\end{array}\right).$$

The random-walk form is technically central. Sector and Folner averages are then recovered from general averaging statements.

Measure Classification as an Input

The measure classification theorem supplies the possible weak limits. Consider a compactly supported probability measure $\mu$ on $SL(2,\mathbb{R})$, absolutely continuous with respect to Haar measure and bi-invariant under $SO(2)$. A probability measure $\nu$ on the stratum is $\mu$-stationary if

$$\mu \ast \nu =\nu ,\mu \ast \nu =\int g_{\ast }\nu d\mu (g).$$

Furstenberg’s correspondence, in the form used by Eskin-Mirzakhani-Mohammadi, relates $P$-invariant measures to stationary measures (Furman, 2002). Therefore the measure-classification theorem implies that stationary limits are affine.

For a point $x$, define the Cesaro random-walk averages

$${\overline{\mu }}^{(n)}\ast {\delta }_x=\frac{1}{n}\sum _{k=1}^n{\mu }^{(k)}\ast {\delta }_x.$$

Every weak limit of these averages is stationary. Hence every such limit is a convex combination of affine measures:

$$\nu =\sum _{\mathcal{N}}{a}_{\mathcal{N}}{\nu }_{\mathcal{N}},\sum _{\mathcal{N}}{a}_{\mathcal{N}}=1,$$

where the sum ranges over affine invariant submanifolds. This is already a powerful structural restriction, but it does not yet prove orbit closure. One must still show that the only affine component that can appear is the minimal one containing $x$.

That is the isolation problem.

Isolation and Height Functions

The stratum is noncompact. Translation surfaces can degenerate when saddle connections become short. Even inside a compact part of the stratum, a trajectory can spend time near a proper affine invariant submanifold. To convert measure classification into orbit closure and equidistribution, EMM build functions that penalize both types of bad behavior.

For every affine invariant submanifold $\mathcal{M}$, including the empty set, the paper constructs an $SO(2)$-invariant function

$$f_{\mathcal{M}}:{\mathcal{H}}_1(\alpha )\to [1,\infty ]$$

with the following properties:

• $f_{\mathcal{M}}(x)=\infty$ exactly on $\mathcal{M}$,
• $f_{\mathcal{M}}$ is bounded on compact subsets of ${\mathcal{H}}_1(\alpha )\setminus \mathcal{M}$,
• sublevel sets of $f_{\mathcal{M}}$ have compact closure away from $\mathcal{M}$,
• for the circle average

$$(A_{t}f)(x)=\frac{1}{2\pi }{\int }_0^{2\pi }f(a_t{r}_{\theta }x)d\theta ,$$

there are constants $b>0$ and, for every $0<c<1$, a time $t_0$ such that

$$(A_t{f}_{\mathcal{M}})(x)\le c{f}_{\mathcal{M}}(x)+b$$

for all $t>t_0$ and all $x\notin \mathcal{M}$,

• near the identity of $SL(2,\mathbb{R})$, the value of $f_{\mathcal{M}}$ changes by at most a fixed multiplicative factor.

The inequality

$$A_t{f}_{\mathcal{M}}\le c{f}_{\mathcal{M}}+b$$

is a drift inequality. It says that after a long geodesic stretch followed by angular averaging, the expected height contracts up to a bounded error. This is the noncompact analogue of saying that the dynamics returns to controlled regions.

For $\mathcal{M}=\varnothing$, such height functions are related to quantitative recurrence for Teichmuller geodesic flow, developed by Eskin-Masur and Athreya (Eskin & Masur, 2001; Athreya, 2006). For general $\mathcal{M}$, the function must also blow up near the affine submanifold. The proof uses recurrence estimates, Forni’s hyperbolicity estimates for the Hodge bundle, and a careful notion of complexity for affine invariant manifolds (Forni, 2002; Eskin, Mirzakhani & Mohammadi, 2015).

The same construction gives countability: each stratum contains at most countably many affine invariant submanifolds. Wright later gave another proof by showing that affine invariant submanifolds are defined over number fields (Wright, 2012).

Random-Walk Proof of Orbit Closure

Let $\mathcal{M}$ be an affine invariant submanifold of minimal dimension containing $x$. Consider a weak limit

$${\overline{\mu }}^{(n_j)}\ast {\delta }_x\to \nu .$$

As above, $\nu$ is stationary and hence decomposes into affine measures. Since $\mathcal{M}$ is invariant and contains $x$, the support of $\nu$ lies in $\mathcal{M}$.

The height function excludes all proper affine submanifolds of $\mathcal{M}$. If

$$\mathcal{N}⊊\mathcal{M},$$

then $x\notin \mathcal{N}$ by minimality. The drift inequality for $f_{\mathcal{N}}$ implies that the random-walk averages starting from $x$ spend arbitrarily small mass near $\mathcal{N}$, after passing far enough along the Cesaro averages. Therefore

$$\nu (\mathcal{N})=0.$$

Applying the same argument with $\mathcal{N}=\varnothing$ prevents escape of mass and shows that $\nu$ is a probability measure. Since countability lets one write the ergodic decomposition as a countable sum over affine invariant submanifolds, all coefficients on proper submanifolds vanish. The only remaining component is

$${\nu }_{\mathcal{M}}.$$

Thus the random-walk averages equidistribute in the minimal affine invariant submanifold containing $x$.

This proves the random-walk equidistribution theorem. The orbit-closure theorem follows because the support of the limiting affine measure is $\mathcal{M}$, and the orbit closure must contain that support. Conversely $\mathcal{M}$ was chosen to contain the orbit. Hence

$$\overline{SL(2,\mathbb{R})x}=\mathcal{M}.$$

Mozes-Shah Compactness and Finite Exceptions

Suppose

$${\nu }_{{\mathcal{N}}_n}\to \nu$$

where each ${\mathcal{N}}_n$ is affine invariant. The same isolation functions show that no mass escapes. Measure classification then gives an affine ergodic decomposition for $\nu$.

The key conclusion is stronger than mere compactness. If $\mathcal{N}$ is an affine invariant submanifold with positive $\nu$-mass and no smaller affine piece carrying that mass, then almost all sufficiently late ${\mathcal{N}}_n$ must lie inside $\mathcal{N}$. Otherwise the height function $f_{\mathcal{N}}$ would force the random-walk averages from generic points of ${\mathcal{N}}_n$ to avoid a neighborhood of $\mathcal{N}$, contradicting the positive mass of the weak limit on $\mathcal{N}$.

This proves that the weak limit is the affine measure on the smallest affine invariant submanifold that eventually contains the sequence. The finite-exception equidistribution theorem is then a compactness consequence: if infinitely many bad proper submanifolds avoided every finite exceptional list, their affine measures would have a convergent subsequence, and the Mozes-Shah theorem would force them eventually into a smaller affine submanifold, contradicting how the list was chosen.

This is the exact point where countability and isolation simplify the argument compared with homogeneous dynamics. In homogeneous spaces, continuous families of closed invariant manifolds can appear through normalizers and centralizers. In the EMM setting, affine invariant manifolds are isolated enough to produce finite exceptional collections.

Rational Billiards and Siegel-Veech Constants

A rational polygon can be unfolded into a translation surface. Periodic billiard trajectories become cylinders of closed geodesics on that surface. If $N(Q,T)$ counts cylinders of periodic billiard trajectories of length at most $T$, Masur proved upper and lower quadratic bounds (Masur, 1988; Masur, 1990). EMM obtain a weak asymptotic formula:

$$\underset{t\to \infty }{\lim }\frac{1}{t}{\int }_0^{t}N(Q,e^s)e^{-2s}ds=c(Q).$$

The constant $c(Q)$ is the Siegel—Veech constant of the affine invariant submanifold generated by the unfolded surface (Veech, 1998; Eskin, Masur & Zorich, 2003).

The extra Cesaro averaging in $s$ is important. Removing it would require stronger control of measures invariant under the unipotent subgroup

$$U=\left\{\left(\begin{array}{cc}1& s\\ 0& 1\end{array}\right):s\in \mathbb{R}\right\},$$

which is beyond the general method of the paper. Thus the billiards application shows both the strength and the boundary of the current Ratner-type theory in moduli space.

Scope and Limitations

The theorem classifies $SL(2,\mathbb{R})$ orbit closures in strata of Abelian differentials, not arbitrary dynamical systems on moduli space. It depends crucially on period coordinates, the linear $SL(2,\mathbb{R})$ action on real and imaginary parts of cohomology, and deep recurrence estimates for Teichmuller flow.

The result does not classify all affine invariant submanifolds explicitly. In genus two, such classifications were known through work of McMullen and Calta; in higher genus the list of possible affine invariant submanifolds remains a major problem. EMM proves that any orbit closure must be one of these affine objects, but it does not enumerate them.

Finally, the paper classifies $P$-invariant and stationary measures through the Eskin-Mirzakhani input, and derives averaged equidistribution. It does not give a full classification of measures invariant only under the one-parameter unipotent subgroup $U$. This is why some counting consequences retain an additional averaging operation.

Transferable Mechanisms

• Period coordinates can turn a nonlinear moduli problem into a linear local problem. The EMM theorem is powerful precisely because it proves that dynamical orbit closures respect these coordinates rather than forming arbitrary closed sets (Eskin, Mirzakhani & Mohammadi, 2015; Zorich, 2006).

• Classifying invariant measures is often the right first step toward classifying orbit closures. This is the same broad logic as Ratner theory: measure rigidity constrains weak limits, then topological conclusions follow after non-divergence and isolation are established (Ratner, 1991a; Ratner, 1991b; Eskin & Mirzakhani, 2018).

• Height functions convert noncompactness into inequalities. The drift estimate

$$A_{t}f\le cf+b$$

plays the role of a Lyapunov function: it prevents escape of mass and keeps averages away from prescribed singular sets (Athreya, 2006; Forni, 2002; Eskin, Mirzakhani & Mohammadi, 2015).

• Compactness plus isolation produces finite exceptional sets. The Mozes-Shah type theorem says that limits of affine measures are affine measures on the eventual container; the uniform equidistribution theorem then follows by excluding finitely many proper affine submanifolds (Mozes & Shah, 1995; Eskin, Mirzakhani & Mohammadi, 2015).

• Random walks can replace deterministic orbit averages during the proof. Stationary-measure classification is technically cleaner, and sector or Folner equidistribution can be recovered after the random-walk statement is established (Furman, 2002; Benoist & Quint, 2011; Eskin, Mirzakhani & Mohammadi, 2015).

See Also

• on Billiards --- Rational billiards unfold to translation surfaces, and the EMM equidistribution theorem gives averaged counting results for periodic billiard cylinders on the associated affine invariant submanifold.

References

🐻  Athreya, J.S. 2006. Quantitative recurrence and large deviations for Teichmuller geodesic flow. Geometriae Dedicata 119, 121–140.

🐻  Benoist, Y. & Quint, J.-F. 2011. Mesures stationnaires et fermés invariants des espaces homogènes. Annals of Mathematics 174(2), 1111–1162.

🐻  Eskin, A. & Masur, H. 2001. Asymptotic formulas on flat surfaces. Ergodic Theory and Dynamical Systems 21(2), 443–478.

🐻  Eskin, A. & Mirzakhani, M. 2018. Invariant and stationary measures for the SL(2,\mathbb R) action on moduli space. Publications Mathématiques de l’IHÉS 127, 95–324.

🐻  Eskin, A., Masur, H. & Zorich, A. 2003. Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants. Publications Mathématiques de l’IHÉS 97, 61–179.

🐻  Eskin, A., Mirzakhani, M. & Mohammadi, A. 2015. Isolation, equidistribution, and orbit closures for the SL(2,\mathbb R) action on moduli space. Annals of Mathematics 182(2), 673–721.

🐻  Forni, G. 2002. Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Annals of Mathematics 155(1), 1–103.

🐻  Furman, A. 2002. Random walks on groups and random transformations. In Handbook of Dynamical Systems, Volume 1A, pp. 931–1014. , North-Holland.

🐻  Masur, H. 1982. Interval exchange transformations and measured foliations. Annals of Mathematics 115(1), 169–200.

🐻  Masur, H. 1988. Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential. In Holomorphic Functions and Moduli, Volume 1, pp. 215–228. , Springer.

🐻  Masur, H. 1990. The growth rate of trajectories of a quadratic differential. Ergodic Theory and Dynamical Systems 10, 151–176.

🐻  Mozes, S. & Shah, N.A. 1995. On the space of ergodic invariant measures of unipotent flows. Ergodic Theory and Dynamical Systems 15(1), 149–159.

🐻  Ratner, M. 1991a. On Raghunathan’s measure conjecture. Annals of Mathematics 134, 545–607.

🐻  Ratner, M. 1991b. Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Mathematical Journal 63(1), 235–280.

🐻  Veech, W.A. 1982. Gauss measures for transformations on the space of interval exchange maps. Annals of Mathematics 115(1), 201–242.

🐻  Veech, W.A. 1998. Siegel measures. Annals of Mathematics 148, 895–944.

🐻  Wright, A. 2012. The field of definition of affine invariant submanifolds of the moduli space of Abelian differentials. arXiv preprint arXiv:1210.4806.

🐻  Zorich, A. 2006. Flat surfaces. In Frontiers in Number Theory, Physics, and Geometry I, pp. 437–583. , Springer.