Articles

## Structure of orbits: a geometric Ratner’s theorem?

The ergodicity of the $\mathrm{SL}(2,\mathbb{R})$ action on the moduli space $\mathcal{H}$ of translation surfaces (= moduli space of Abelian differentials, under the identification we made earlier) allows us to understand generic orbits, but not of arbitrary orbits. In particular, for example, a family of flat surfaces correspond to a fixed rational polygonal billiard forms a positive-codimension subspace, about which ergodicity allows us to say nothing.

There are, however, results in dynamics / ergodic theory which classify not just almost all, but all orbits, the prototypical example being Ratner’s Theorem(s) on unipotent flows:

Theorem/s (Ratner) Let G be a connected Lie group and U be a connected subgroup generated by unipotents. Then

• for any lattice $\Gamma \subset G$ and any $x \in G / \Gamma$, the closure of the orbit $Ux \in G / \Lambda$ is an orbit of a closed algebraic subgroup of G.
• every ergodic invariant probability measure is homogeneous;
• every unipotent orbit is equidistributed in its closure.

A basic example is given by a horocycle flow on a hyperbolic manifold. These are ergodic, and so we know that almost every orbit is dense; but Ratner’s theorem tells us that in fact we have a strict dichotomy: every orbit is either closed or dense.

The hope here is for a similar result: one precise formulation of this is the following

Conjecture (“Magic Wand”) The closure of a $\mathrm{SL}(2,\mathbb{R})$-orbit of any flat surface is a complex-algebraic suborbifold. (By a theorem of Kontsevich, any $\mathrm{SL}(2,\mathbb{R})$-invariant complex suborbifold is represented by an affine subspace in cohomological period coordinates.)

## Aspects of Teichmüller theory

Recall that we have identified $\mathcal{H}$ as a space of pairs (complex structure, holomorphic 1-form). Recalling some of the plumbings of Teichmüller theory, we consider also the space of pairs (complex structure, holomorphic quadratic differential), and identify it with the cotangent bundle to the moduli space $\mathcal{M}$ of complex structures. $\mathcal{H}$ can be identified with a subspace of $\mathcal{Q}$ consisting of those quadratic differentials which can be represented as global squares of holomorphic 1-forms.

This subspace may be considered as a “unit cotangent bundle”, being invariant under the Teichmüller geodesic flow (i.e. the diagonal subgroup action induced by the $\mathrm{SL}(2,\mathbb{R})$ action on $\mathcal{H}$.)

We may check that $\mathrm{SL}(2,\mathbb{R})$ orbits in (the image of) $\mathcal{H}$ in $\mathcal{Q}$ descend to isometric maps of $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) \cong \mathbb{H}^2$ to$\mathcal{M}$—i.e. the projections of these orbits are Teichmüller discs, also known as complex geodesics.

Complex geodesics may be described more directly in terms of the language of flat surfaces as follows: recall $\mathrm{SL}(2,\mathbb{R})$ orbits in $\mathcal{H}$ correspond to translation surfaces with a distinguished direction, encoded by the holomorphic 1-form; the $\mathrm{SL}(2,\mathbb{R})$ action changes the translation structure, i.e. the fundamental polygon, but not the affine structure, i.e. the resulting translation surface. Then we obtain a complex geodesic by forgetting the 1-form, i.e. forgetting the distinguished direction.

The classification of orbits is then closely related to the classification of these complex geodesics, which allows us to potentially use (even more) language and tools from Teichmüller theory.

## “Revolution in genus 2”

Kontsevich-Zorich classified strata of the moduli space using spin structures and hyperellipticity. In genus 2, the two stratum $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$ are each connected and consist entirely of hyperelliptic surfaces.

Smilie proved that closed $latex\mathrm{SL}(2,\mathbb{R})$-orbits—orbits of flat surfaces which are in some sense exceptionally symmetric—correspond exactly to orbits of Veech surfaces (see first post on polygonal billiards for a description of Veech surfaces.) The identification of closed orbits thus reduces to (or, at any rate, is equivalent to) the classification of Veech surfaces, about which some things, but not very many, are known.

McMullen proved that there is (up to ramified coverings) only one Veech surface in the stratum $\mathcal{H}(1,1)$, given by the regular decagon with identified opposite sides.

Calta and McMullen, using different methods, described all Veech surfaces in $\mathcal{H}(2)$—there is a countable family even up to ramified coverings—and gave efficient algorithms to recognize and classify these.

They also describe invariant submanifolds of intermediate dimension—intermediate between the full stratum and but larger than single closed orbits.

Finally, McMullen shows, using all of this, plus more subtle tools, that our “magic wand” conjecture is true in genus 2; the classification is in fact rather more precise, and he also obtains results about invariant measures in the spirit of Ratner’s theorems.

Mirzakhani and collaborators (Eskin and Mohammadi), together with Filip, in spectacular (relatively) recent work, proved the “magic wand” conjecture, plus measure rigidity results, for all genera.

The measure rigidity result of Eskin-Mirzakhani states that any ergodic P-invariant measure (where P is a maximal parabolic subgroup, e.g. the Borel subgroup) is in fact a Lebesgue class measure on a manifold cut out by linear equations, and must be $\mathrm{SL}(2,\mathbb{R})$-invariant. This uses considerable machinery from ergodic theory: “almost 100 pages of delicate” entropy arguments, plus ideas of Benoist-Quint.

The theorem of Eskin-Mirzakhani-Mohammadi then builds on this to state that the $\mathrm{SL}(2,\mathbb{R})$-orbit closure of a translation surface is always a manifold. Moreover, the manifolds that occur are locally defined by linear equations in period coordinates, with real coefficients and zero constant term.

The proof proceeds, given the measure rigidity result, by constructing a P-invariant measure on every P-orbit closure. Here the use of P, as opposed to $\mathrm{SL}(2,\mathbb{R})$, is crucial—the former is amenable whereas the latter is not, and this allows us to use averaging methods in our construction.

(Filip’s result is needed to go from analyticity, which Eskin-Mirzakhani-Mohammadi actually gives us, to algebraicity.)

## Where can the magic wand take us?

These results allow us to say things about specific families of translation surfaces—e.g. a rational billiard table, whose orbit under the $\mathrm{SL}(2,\mathbb{R})$-action forms a high-codimension family in $\mathcal{H}$—rather than just “almost all” translation surfaces

Thus, for instance, we can prove quadratic asymptotics (exact, not just lower and upper bounds as was previously the case) for the number of generalized diagonals, etc. in polygonal billiards.

There are many other instances where some problem may be naturally (re)formulated in terms of translation surfaces coming from polygonal billiards; then the magic wand implies additional structure on a relevant family of translation surfaces, which yields insight into the original problem. Below we outline two concrete examples of this:

### The illumination problem

Given a room, how many light-bulbs are required to light it? Or, to abstract the problem a little: given a polygonal domain P (or really any planar domain, but let’s stick to polygons for now) and a point $x \in P$, which points in P can (or cannot) be reached by billiard trajectories through x? A point y which can be reached from x is said to be  illuminated from x.

Billiard trajectories very much resemble light-ray trajectories (at least locally)—indeed the word “optical” appeared in our description of billiard systems—and so it should be no surprise that the study of billiard systems and hence of translation surfaces yields insight into this and related problems. Indeed, as this wonderfully-named paper notes, the illumination problem “elementary properties which can be fruitfully studied using the dynamical behavior of the $\mathrm{SL}(2,\mathbb{R})$-action on the moduli space of translation surfaces.”

Using the magic wand theorem, and that the geometric properties considered in the illumination problem produce closed sets of the moduli space $\mathcal{H}$Lelièvre-Monteil-Weiss have proved that, for any P and any $x \in P$, there are finitely many $y \in P$ which are not illuminated from x.

### The wind-tree model

The wind-tree model was originally formulated by statistical physicists Paul and Tatiana Ehrenfest as a model for a Lorenz gas: in this model, particles (the “wind”) travel in straight-line trajectories in the plane $\mathbb{R}^2$, reflecting off rectangular obstacles (“trees”) placed along a $\mathbb{Z}^2$ lattice in a billiards-like fashion. One can also describe it, precisely, as billiards in the plane with these rectangles removed.

One can form a translation surface by restricting to some suitable subset of the plane and obstacles, and gluing the sides together: (figure taken from the also wonderfully-named “Cries and whispers in wind-tree forests“)

The result is a genus-5 flat surface in the stratum $\mathcal{H}(2^4)$.

One can then describe the behavior of the trajectories in terms of properties of the translation surface, e.g. Delecroix-Hubert-Lelièvre have computed the diffusion of divergent trajectories, for rectangular obstacles of any size, in terms of the Lyapunov exponents of  a natural dynamical system (the Kontsevich-Zorich cocycle) on a certain stratum of genus-5 translation surfaces—not the one specified above, but a quotient thereof.

## References

Alex Wright’s article describes Eskin-Mirzakhani-Mohammadi result, the context for it, as well as applications and connections to nearby areas.