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## 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.

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Articles

# Polygonal billiards

Polygonal billiards are easily-described dynamical systems, which are defined by the trajectories of a single particle in a polygonal region P of the plane by requiring the particle move in straight lines at constant velocity in the interior of P, and reflect off the boundary according to the familiar (“optical”) laws of reflection: the angle of incidence should equal to the angle of reflection.

(What happens if the particle hits a corner is undefined, but the set of trajectories for which this happens is vanishingly small, i.e. has zero measure.)

We can ask the usual questions that dynamicists do when they probe the behavior of their systems: what do the orbits of this system look like? Are they periodic / closed, or dense? How does that depend on the initial data (i.e. position and direction)? What does the generic orbit look like? How many closed orbits are there?  Is the system ergodic? Mixing? If so, at what rate? How do the answers to all of these questions depend on P?

## The great realm of irrational ignorance

Answering these questions can get fiendishly tricky surprisingly fast. For instance, very little is known in the case where the polygon P has angles which are not rational multiples of $\pi$.

Even more narrowly, and somewhat astonishingly, it is still an open question to determine if billiards on a general triangular P has a closed orbit. When P is acute, the triangular path formed by the line segments between the feet of the altitudes can be shown to be a billiards path (as demonstrated by Fagnano more than 240 years ago); when P is a right-angled there is a similarly explicit construction. What happens when P is obtuse is, still, anybody’s guess—although Rich Schwartz has at least partial, computed-assisted results.

## Rational enlightenment via translation surfaces

By contrast, when the angles of P are all rational multiples of $\pi$, there is great deal that can be said, using tools from such varied fields as Riemann surfaces, Teichmüller theory, hyperbolic geometry, and even algebraic geometry.

What allows us to start applying all of these diverse tools is a relatively simple device, the translation surface associated to the billiard system, which might be seen as akin to development maps for (G,X)-structures—the rough idea in both cases being to unroll transitions until we see everything at once on a single object, or, in slightly more technical language, to globalize the coordinate charts.

We can define translation surfaces more generally, as maximal atlases of coordinate charts into the Euclidean plane $\mathbb{R}^2$, with finitely many cone singularities and where the transition maps between coordinate charts are translations in $\mathbb{R}^2$. Note that we can obtain the genus from information about the cone singularities via (a discrete version of) Gauss-Bonnet.

We may equivalently define translation surfaces—although to show the equivalence takes a little work—as surfaces built from some finite collection of polygons embedded in the Euclidean plane with a distinguished direction (and so inheriting an Euclidean metric and a distinguished direction—by gluing maps between their sides which are translations.

For a translation surface coming from a billiard table, we may take of the polygons to be congruent. In this sense these translation surfaces have additional symmetries that we should not expect a generic translation surface to have, and will likely be rather special in this sense (although there might conceivably be some sort of rigidity result lurking somewhere.)

### Closed orbits: Masur’s theorem

The more general point of view of translation surfaces allows us more freedom of argument in proving things about our original, more restricted context.

For instance: a theorem of Masur states that periodic orbits exist on every translation surface; in fact, there are infinitely many. In fact, he proves even more: the number of periodic orbits of length N grows quadratically in N. If our translation surface did in fact come from a polygonal billiard system, these periodic orbits project down to closed trajectories.

The proof uses the idea of “changing the translation structure while preserving the affine structure”, as J. Smillie’s survey describes it:

Closed orbits can be detected geometrically via the presence of cylinders—subsets of our translation surface isometric to $S^1 \times I$. If we fix the genus, area, and singularity data for our surface, the translation surface can only have large diameter if it contains a (long) cylinder. In particular, if our translation surface had area greater than some universal constant D, we have our cylinder, and hence a closed orbit.

Otherwise, we observe that we have some freedom to change the translation structure, i.e. the coordinate charts and translations involved in the transitions—without changing the resulting affine geometry on the translation surface. Such a change takes cylinders to cylinders, but changes the metric geometry, and in particular the diameter. We can argue more carefully to find that there is always some such change of translation structure which takes the diameter to or beyond our universal constant D; and so the general case in fact reduces to our earlier, easier one.

(With a lot more care, we may also obtain the quadratic asymptotics stated above.)

In effect, the translation surface point of view tells us that polygonal billiards on families of polygons which produce the same translation surface have similar dynamical / geometric behavior—something which was not at all obvious just looking at the polygons themselves.

## Ergodicity of the billiard

A dynamical system $(X,T,\mu)$ is said to be ergodic (w.r.t. the given measure $\mu$) if any T-invariant subset of X has either zero or full measure w.r.t. $\mu$. Ergodic systems are, in some precise sense via the ergodic decomposition, the building blocks of all dynamical systems.

An example of an ergodic system is given by polygonal billiards on a rectangle, which is in fact equivalent to the geodesic flow on a flat torus. In fact, the flow / billiard trajectories in almost every direction are ergodic, and moreover equidistribute (this can then be shown to imply unique ergodicity—see below); the non-ergodic directions are precisely those with rational slopes, and are all periodic.

This especially nice state of affairs is a prototypical example of Veech dichotomy, which states that every direction for the constant-slope flow is either uniquely ergodic, or periodic. The former often form a small subset, but what “often” and “small” here entail precisely is still not entirely pinned down.

It follows from classical results that integrable polygons (whose corresponding translation surfaces are tori) satisfy Veech dichotomy. Gutkin proved that “almost-integrable” polygons also satisfy the dichotomy. Veech, in 1989, that all lattice surfaces (now also called Veech surfaces) satisfy the dicohotomy. Here, a lattice surface is one whose group of orientation-preserving affine automorphisms, or rather the image thereof under taking the derivative, forms a lattice in $\mathrm{SL}(2,\mathbb{R})$. All the previous examples are Veech surfaces, as well as those corresponding to regular n-gons, as well as certain triangles.

This is still not the most general class of translation surface satisfying Veech dichotomy, however: Smillie and Weiss proved in 2008 that there are non-lattice surfaces satisfying Veech dichotomy.

This result of Veech, plus the difficulty of the problem in its full generality, seems to have led to a focus on studying the ergodicity of the billiard flow in a fixed direction.

### Unique ergodicity and minimality

$(X,T,\mu)$ is uniquely ergodic if $\mu$ is the only invariant probability measure—in this case, since any invariant measure is a convex linear combination of ergodic measures, $\mu$ is necessarily ergodic.

The Bunimovich stadium is an example of a dynamical system—a planar billiard, in fact, although not polygonal—which is known to be ergodic but not uniquely ergodic.

Since the systems we are working with are naturally equipped with both a measure and a topology, we can also ask about minimality (a topological analogue of ergodicity—a minimal system has no proper closed T-invariant subsets), and how it relates to ergodicity.

It can be proven that a flow direction is minimal if there are no saddle connections (i.e. geodesic segments starting and ending at vertices, but not passing through any vertices in their interior) in that direction. The proof starts with the observation that it is useful to consider the first return map to a transverse interval in the surface. This turns out to be an interval exchange transformation (IET.) IETs are a well-studied class of dynamical systems, and criteria for the minimality of IETs lead to criteria for the minimality of directional flows, including this particular one.

From the combined / somewhat muddled viewpoint of dynamics both measurable and topological, minimal ergodic directions are especially nice, and non-minimal directions we might attempt to deal with using some sort of induction / topological reduction; so we might hope that there aren’t any minimal non-ergodic directions to deal with. It is a straightforward corollary of Veech dichotomy that Veech surfaces (or more generally surfaces which satisfy the dichotomy) do not have such directions.

However, Masur has produced examples of minimal non-ergodic directions for the geodesic flow on a translation surface of genus 2, by considering a translation structure given by a rectangular billiards table with two slits in the interior; these can be generalized to give (uncountably many) examples of minimal non-ergodic directions on any translation surface of genus 2 or greater.

### Enter the Teichmüller geodesic

However, it can be proven that set of such directions has Lebesgue measure zero; in fact it can be proved that rational billiards are uniquely ergodic in almost every direction .

The key step is a result of Masur that if $(X, \omega)$ is a translation surface for which flow in the vertical direction is not uniquely ergodic, then the Teichmüller geodesic associated to $(X, \omega)$ is divergent, i.e. it eventually leaves every compact set in the moduli space.

Here the Teichmüller geodesic associated to $(X, \omega)$ is the flowline of $(X,\omega)$ under the diagonal subgroup $\left\{ \left( \begin{array}{cc} e^t \\ & e^{-t} \end{array} \right) : t \in \mathbb{R} \right\}$ of $\mathrm{SL}(2,\mathbb{R})$ (the Teichmüller geodesic flow, for it really does correspond to flowing along geodesics in Teichmüller space), projected to moduli space.

There has been further work to figure out just “how big” the set of non-ergodic directions is, e.g. in terms of Hausdorff dimension.

## Aside: smooth non-polygonal billiards

From a slightly different perspective, billiard trajectories are akin to geodesic flow trajectories in the plane. With this in mind, perhaps the following result is not too surprising:

Any smooth surface in 3-space may be flattened to obtain something close to a smooth (not necessarily polygonal) billiard table in 2-space. Kourganoff showed that, under some mild hypotheses, the geodesic flow of this surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersive (i.e. any two distinct points have neighborhoods whose orbits are eventually separated) and has finite horizon (i.e. time between collisions remains bounded), then the geodesic flow of the corresponding surface is Anosov. This result can be applied to the theory of mechanical linkages and their dynamics, to provide e.g. novel examples of simple linkages whose physical behavior is Anosov.

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# Ergodicity of the Geodesic Flow

### The geodesic flow

Given any Riemannian manifold M, we may define a geodesic flow $\varphi_t$ on the unit tangent bundle $T^1M$ which sends a point (x, v) to the point $(\varphi_t x, \varphi_t^* v)$, where

• $\varphi_t x$ is the point distance from x along the geodesic ray emanating from x in the direction of v, and
• $\varphi_t^* v$ is the parallel transport of v along the same ray

(it’s a mouthful, isn’t it? It’s really simpler than all those words make it seem.) Note, at each point, we remember not just where we are—the point $x \in M$—, but also where we’re going—the direction vector $v \in T_x M$; if we were to forget this second piece of information, we would become a little unmoored: here we are … where should we go next?

### Ergodicity

When M is a closed (i.e. compact, no boundary) hyperbolic surface, or more generally closed with strictly negative curvature, this geodesic flow is ergodic, i.e. any subset of $\Sigma$ or M invariant under the flow has either zero measure, or full measure. Here the measure on our Riemannian manifold is the pushforward of the Lebesgue measure through the coordinate charts.

Since linear combinations of step functions are dense in the space of bounded measurable functions, we may equivalently define ergodicity as: any measurable function invariant under the flow is a.e. constant.

(Side note: with more assumptions on the curvature we may relax the compactness assumption to a finite volume assumption)

### The Hopf argument (for closed hyperbolic manifolds)

This is essentially due to the exponential divergence of geodesics in negative curvature , and the splitting of the tangent spaces $T_ v T^1M = E^s_v \oplus E^0_v \oplus E^u_v$ into stable, tangent (flowline), and unstable distributions; these give rise to three maximally transverse foliations, the stable foliation $W^s$, the unstable foliation $W^u$, and the foliation by flowlines $W^0$.

The flow is exponentially contracting in the forward time direction on the leaves of the stable foliation $W^s$, and on which the flow is exponentially contracting in the reverse time direction the leaves of the unstable foliation $W^u$. In other words, the flow is Anosov.

We may describe these foliations explicitly in the case of constant negative curvature—if we take $\gamma$ to be the geodesic tangent to $v \in T^1M$,

• $W^s$(v) is (the quotient image of) the unit normal bundle to the horosphere through $\pi(v) \in M$ tangent to the forward endpoint of $\gamma$ in $\partial_\infty\mathbb{H}^n \cong \partial_\infty\widetilde{M}$. “forward” here being taken with reference to how v is pointing along $\gamma$;
• $W^u(v)$ is (the quotient image of) the unit normal bundle to the horosphere through $\pi(v)$ tangent to the backward endpoint of $\gamma$ in $\partial_\infty\mathbb{H}^n \cong \partial_\infty\widetilde{M}$;
• $W^0(v) = \gamma$.

#### Step 1

Suppose f is a $\phi$-invariant function; by replacing f with min(f, C) if needed, WMA f is bounded. Since continuous functions are dense in the set of measurable functions on M, we may approximate f in $L^1$ by bounded continuous functions $h_\epsilon$.

By the Birkhoff ergodic theorem, forward time averages [w.r.t. $\phi$] exist for $h_\epsilon$.

By an argument involving the $\phi$-invariance of f and the triangle inequality, f is well-approximated (in $L^1$) by the forward time averages of $h_\epsilon$.

#### Step 2

The forward time averages of $h_\epsilon$ are constant a.e., since by invariance these averages are already constant a.e. on (each of) the leaves of $W^0$, and they are also constant a.e. on (each of the) unstable and stable leaves, by uniform continuity of $h_\epsilon$.

#### Step 3

To conclude that time averages, and hence our original arbitrary integrable function, are constant a.e. on M, we (would like to!) use Fubini’s theorem: locally near each $(x_0,v_0) \in T^1M$, the set of (x, v) along each of the foliation directions at which the time averages are equal to those at $(x_0,v_0)$ has full measure, by the previous Step.

By Fubini’s theorem applied to the three foliation directions, we (would) conclude that the set of nearby (x, v) at which the time averages are equal to those at $(x_0,v_0)$ has full measure. Hence the time averages are locally constant, and since $T^1M$ is connected we are done.

### But! (Also more generally, for K < 0)

The problem is that while our stable and unstable leaves are differentiable, the foliations need not be—i.e. the leaves may not vary smoothly in their parameter space.

To justify the use of a Fubini-type argument one instead shows that that these foliations are absolutely continuous.

The proof then immediately generalizes to all compact manifolds with (not necessarily constant) negative sectional curvature. For more general negatively-curved nanifolds, the stable and unstable foliations $W^s$ and $W^u$ may still be described in terms of unit normal bundles over horospheres, where horospheres are now described, more generally, as level sets of Busemann functions.

The proof of absolute continuity of the foliations proceeds as follows

1. Showing that the stable and unstable distributions $E^s$ and $E^u$ (also the “central un/stable” or “weak un/stable” distributions, i.e. $E^{s0} := E^s \oplus E^0$ and  $E^{u0} := E^u \oplus E^0$) (of any $C^2$ Anosov flow) are Hölder continuous—i.e. given $x, y \in M$, the Hausdorff distance in $TTM$ between the stable subspace $E^s(x)$ and the stable subspace $E^s(y)$ is $\leq A \cdot d(x,y)^\alpha$.
Roughly speaking, this is true because any complementary subspace to $E^s$ will become exponentially close to $E^s$ under the repeated action of the geodesic flow, by the same mechanism that makes power iteration tick; and the distance function on M is Lipschitz. Analyzing the situation more carefully, and applying a bunch of simplifying tricks such as the adjusted metric described in Brin’s section 4.3, yields the desired Hölder continuity.
2. Using this, together with the description of horospheres as limits of sequences of spheres with radii increasing to $+\infty$, to establish that between any pair of transversals for the un/stable foliation, we have a homeomorphism which is $C^1$ with bounded Jacobians, and hence absolutely continuous.
Very slightly less vaguely, Hölder continuity of $E^{u0}$, together with the power iteration argument as above, implies tangents to transversals to the stable foliation $W^s$ become exponentially close; given regularity of the Riemannian metric, this implies the Jacobians of the iterated geodesic flow on these transversals become exponentially close. By a chain rule argument and another application of the power iteration argument, this implies that the Jacobians of the map between transversals are bounded.
This condition on the foliations is known as transversal absolute continuity, and implies, by a general measure theoretic argument (see section 3 of Brin’s article), absolute continuity of the foliations.
3. Note that this last step, at least as presented in Brin, appears to require the use of pinched negative curvature.

### References

Eberhard Hopf, “Ergodic theory and the geodesic flow on surfaces of constant negative curvature.” Bull. Amer. Math. Soc. 77 (1971), no. 6, 863–877.

Yves Coudene, “The Hopf argument.

Misha Brin, “Ergodicity of the Geodesic Flow.” Appendix to Werner Ballman’s Lectures on Spaces of Nonpositive Curvature.

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