Reading Seminar: Translation Surfaces and Rational Billiards - The Veech Alternative

Seminar notes exploring Vorobets' approach to the Veech alternative — the dichotomy in geodesic flow behavior on translation surfaces: trajectories are either periodic or uniformly distributed.

This post serves as lecture notes from our reading seminar at MPI MiS, where we are exploring the dynamical behaviors of geodesic flows on translation surfaces through a study of:

Ya. B. Vorobets, “Planar structures and billiards in rational polygons: the Veech alternative,” Russian Mathematical Surveys, 1996, Volume 51, Issue 5, 779-817.

The Veech alternative presents a remarkable dichotomy in the behavior of geodesic flows on certain translation surfaces: for any fixed direction, either all infinite trajectories are uniformly distributed, or all trajectories are periodic.

Note: These notes reflect my understanding of our discussions – if it were a game of telephone, consider this the final whisper. All errors and clunky phrasings are surely mine; all good ideas belong to the speakers and authors of the reference material.

Session 1 (Jiajun Shi), February 26th: Translation Structures and their connection to Billiards in Rational Polygons

Session 2 (Fabian Lander), March 5th: Interval Exchange Transformations and Elementary Translation surfaces

Last time we saw that the geodesic flow on the phase space $\Phi \subset M \times S^1$ preserves the measure $\mu \times \lambda$ where $\mu$ is the Lebesgue measure on $M$ (induced by the flat structure $\omega$) and $\lambda$ is the Lebesgue measure on $S^1$. In particular, if we fix a direction, we have a measure-preserving dynamical system on a full measure subset of $M$. For simplicity, we will continue to denote this subset as $M$.

Today we will connect this dynamical system to another closely related family of dynamical systems called Interval Exchange Transformations, or IETs for short.

Defining Interval Exchange Transformations

Given an interval $I = [a,b] \subset \mathbb{R}$ and points $S= \{x_1, \dots, x_m \}$ where $a < x_1 < x_2 < \dots < x_m < b$, an IET is a map $T: I\setminus S \rightarrow I$ that acts as a translation (shift) on each of the components of $I\setminus S$.

Various definitions appear in the literature. Some use half-open intervals (either left- or right-open) and require $T$ to be left- or right-continuous, meaning $T$ shifts the half-open “components”. This makes $T$ a bijection on the half open interval.

Regardless of these technical differences, the fundamental concept remains the same: We partition the interval into finitely many pieces and then rearrange them via translations.

We call the set $S$ the set of singularities, break points, saddles, or discontinuities. By including the preimages of break points back into $S$, we can see that powers of an IET are themselves IETs.

A crucial element in analyzing the dynamical properties of IETs is the set $C(T)$, which consists of points $x\in I$ that are mapped into $S$ in both forward and backward time:

This set is clearly finite, because any sequence $s_1, T(s_1), ..., T^n (s_1) = s_2$ that connect two break points $s_1, s_2$ is unique (here we view $T$ as a bijection on the half open interval) and there are only finitely many possible combinations of breakpoints.

Another useful point of view is the following: This set is finite because of the connection between IETs and translation surfaces - specifically, points in $C(T)$ correspond to saddle connections in the suspension of the IET. Since a translation surface has finitely many saddle connections in any given direction, $C(T)$ must be finite as well.

Foundational Results on IETs

The following key results about IETs come from:

• [1] M. Keane, “Interval exchange transformations”, Math. Z. 141 (1975), 25-31.
• [2] M. D. Boshernitzan, “Rank two interval exchange transformations”, Ergodic Theory Dynamical Systems 8 (1988), 379-394.

This directly implies:

Lastly, we have a theorem regarding ergodic measures:

In the next section, we’ll explore how these interval exchange transformations relate to the geodesic flows on translation surfaces, providing a powerful tool for understanding their dynamics.

IETs and Translation Surfaces

We will now discuss how we can obtain an IET from a Translation surface $(M, \omega)$.

Ingredients for an IET:

• A geodesic segment $I$ on $M$ (which may contain a singular point)
• A direction $v\in S^1$

We define $T$ as the first return map of the flow from $I$ in direction $v$. Before proving that this is indeed well-defined, let’s examine a simple example:

Consider a translation surface (e.g., a torus) together with a horizontal geodesic segment. The first return map is not defined if we flow into a singularity. So we take the “preimages” of all possible flow lines in direction $v$ emanating from singularities as well as the preimages of the boundary points of our segment (careful there is a mistake in the picture, however a posteriori the idea still works):

Note that since we only have removable singularities in this example, there is only one preimage per singularity. In general, the number of preimages could be up to the sum of all multiplicities of singularities. This partitions the segment into components which get mapped back to $I$ via translation:

Now we will state and prove a general result about the first return map on translation surfaces, which will formally establish that our map is indeed an IET (and in particular well-defined).

Let $S = \{ x_1, \dots, x_m \}$ be all the points in $I$ such that their trajectory hits a saddle point or one of the two boundary points of $I$ without intersecting the segment itself before. Equivalently, we can send $m_i$ - many rays in direction $v$ from every saddle point $s_i$ and collect the first hit with the segment (if there is one). Take a component $J$ of the partition $I\setminus{S}$.

By Poincaré’s recurrence theorem, this set will eventually return to $I$ (I will elaborate on this application below). So the first return map is well-defined. By construction, all of $J$ must return to the interior of $I$ at the same time (otherwise this would contradict the definition of $S$). Since our flow preserves the length and orientation of our segment, $J$ has to be mapped back to $I$ by a translation.

The trajectory of $J$ forms a rectangle which has area at most that of the whole surface. Therefore, the time until the first return (or equivalently, the distance traveled) is less than $\text{Area}(M)/|J|$.

Application of Poincaré’s Recurrence Theorem to IETs

Poincaré’s recurrence theorem is a fundamental result in measure-preserving dynamical systems. It states that for any measure-preserving transformation $T$ on a probability space $(X, \mathcal{B}, \mu)$ and any measurable set $A \in \mathcal{B}$, almost every point in $A$ returns to $A$ under iterations of $T$. More precisely, for almost every $x \in A$, there exists $n > 0$ such that $T^n(x) \in A$.

In the context of geodesic flows on translation surfaces, we can apply this theorem because the geodesic flow in a fixed direction preserves the Lebesgue measure on the translation surface $(M, \omega)$, and $M$ has finite area.

We can apply the theorem to our interval $I$ because it has positive measure with respect to the measure transverse to the flow direction $v$. More specifically, we can thicken our interval in direction $v$ to form a rectangle. Since the geodesic flow preserves rectangles, returning to our initial rectangle with base $J$ implies that the interval $J$ must return to itself at some point. Therefore, the whole component $J \subset I \setminus S$ returns to $I$ under flow. This leads us to the following important observation:

Saddle Connections and Minimal Components

A saddle connection of a translation surface is a geodesic segment joining two singular points which doesn’t contain singular points in its interior. Notice that in our previous construction of IETs on translation surfaces, saddle connections in the direction $v$ play precisely the same role as the set $C(T)$ we discussed earlier.

In essence, this means we can fully understand the geodesic flow on $M$ in any given direction $v$ by examining an appropriate IET on a transverse segment (or collection of segments). This powerful connection allows us to transfer our intuition between these two perspectives.

Here’s a key insight worth highlighting: When constructing an IET from a geodesic flow, we can actually reconstruct the original translation surface if we carefully track additional data—specifically, which segment neighbors which segment after the first return (permutation data), how long each segment travels along its trajectory until it returns, and the distances between the saddle connections and the break points. This “suspension” of the IET effectively recreates the original surface geometry.

Now that we can analyze the dynamical behavior of the geodesic flow in any direction by examining a suitable IET, we can reformulate our earlier theorems in the context of translation surfaces.

A beautiful consequence of this analysis is the following theorem:

Finally, we can adapt Theorem 3 to the setting of translation surfaces:

Translation Surfaces out of IETs (Suspensions)

The reverse construction—building a translation surface from an IET—is remarkably straightforward. We can create a translation surface with a segment such that the vertical flow induces the same IET on that segment. The following illustration captures the essence of this construction:

Session 2.5 (Fabian Lander), March 12th: “Decomposing” a translation surfaces into IETs and defining elementary translation surfaces

Elementary Translation Surfaces

Now that we understand the connection between translation surfaces and interval exchange transformations, let’s explore some fundamental properties of translation surfaces, particularly the relationship between their topology and the singularities they contain.

This theorem has several interesting consequences:

1. There do not exist translation structures on the sphere. This is because for a sphere, $\chi = 2$, which would require $\sum_{i=1}^{k}(m_i - 1) = -2$. Since $m_i \geq 1$ for all $i$ (as singularities must have multiplicity at least 1), this equation cannot be satisfied.

2. Translation structures on a torus can have removable singularities only. For a torus, $\chi = 0$, so we need $\sum_{i=1}^{k}(m_i - 1) = 0$. This is only possible if all singularities have multiplicity exactly 1, making them removable.

3. A translation structure on a surface of genus $g > 1$ must have at least one non-removable singularity, as $\chi = 2-2g < 0$ requires at least one $m_i > 1$.

Let’s explore the second consequence in more detail. Let $v_1, v_2$ be linearly independent vectors in $\mathbb{R}^2$. By $\mathbb{T}_{v_1, v_2}$ we denote the quotient space of $\mathbb{R}^2$ by the subgroup $\mathbb{Z}v_1 \oplus \mathbb{Z}v_2$. Then $\mathbb{T}_{v_1, v_2}$ is a torus, the canonical projection $\pi: \mathbb{R}^2 \to \mathbb{T}_{v_1, v_2}$ is a local homeomorphism, and the continuous maps from domains in $\mathbb{T}_{v_1, v_2}$ into $\mathbb{R}^2$ that are right inverse to $\pi$ define on $\mathbb{T}_{v_1, v_2}$ a translation structure without singular points. This structure is called a flat torus.

It turns out that every translation structure on a torus can be obtained in this manner, which is formalized in the following proposition:

By Theorem 9 it suffices to prove that a translation structure $\omega$ without singular points on a torus $M$ is isomorphic to a planar torus. We will first show that $\omega$ has a periodic trajectory.

Let $x \in M$ and let $I$ be a geodesic interval starting at $x$ in an arbitrary direction $v_1$. The trajectory emitted from $x$ in a direction $v_2$ perpendicular to $v_1$ intersects $I$ at a point $x'$. We denote by $s_1$, $s_2$ the distances to be travelled along $I$ and along the trajectory, respectively, from $x$ to $x'$.

As can be readily seen, the trajectory emitted from $x$ in a direction $e_1$ parallel to $-s_1v_1 + s_2v_2$ is periodic. By Proposition 7, the whole surface $M$ is a single pencil of periodic trajectories in the direction $e_1$, having the same length $l_1$.

We draw the geodesic interval $J$ in a direction $e_2$ perpendicular to $e_1$ whose length is the width $w_1$ of the pencil of periodic trajectories in the direction $e_1$. The end-points $x$ and $x''$ of this interval belong to the same trajectory of the pencil; all other trajectories intersect $J$ just once.

Let $l_2$ be the distance from $x''$ to $x$ when moving along the direction $e_1$. Then the trajectories parallel to $v_2 = w_1e_2 + l_2e_1$ form a pencil of periodic trajectories of length $|v_2|$. The trajectories from the pencils parallel to $e_1$ and $v_2$ intersect one another just once. This implies that the translation structure $\omega$ is isomorphic to the planar torus $\mathbb{T}_{l_1e_1, v_2}$.

It is well known that on $\mathbb{T}_{v_1, v_2}$, flows parallel to vectors in $\mathbb{Z}v_1 \oplus \mathbb{Z}v_2$ are periodic, while flows in all other directions are strongly ergodic. This dichotomy represents the simplest possible dynamical behavior for geodesic flows on translation surfaces.

This definition captures a key aspect of the Veech alternative that we’re working toward—the dichotomy in the behavior of geodesic flows, where for any given direction, the flow is either completely periodic or completely ergodic with no “intermediate” behavior.

Session 3 (Magali Jay), March 19th: The Stabilizer of a Planar Structure

Let $\omega$ be a translation surface. Given an element $a\in \mathrm{SL}(2,\mathbb{R})$ we define $a\omega$ as the translation surface obtained by post-composing charts $(U_\alpha, f_\alpha)$ of $\omega$ with $a$, i.e. $(U_\alpha, a\circ f_\alpha)$. This defines an action on the space of translation structures and we denote by $\Gamma(\omega)$ the stabilizer subgroup of $\omega$, i.e.,

The goal for this talk is to prove:

Given a saddle connection, we can cover its interior with a single chart. The image of the saddle connection will be a straight (open) line segment. A development of a saddle connection is a vector $v\in\mathbb{R}^2$ that corresponds to the difference of endpoints of the oriented line segment. So every saddle connection has two developments: $v$ and $-v$. We define $SC(\omega)$ to be the multiset containing all developments of all saddle connections. (A multiset is a set where we also remember how often an element occurs. This can be encoded using a map from the set into the positive integers.)

Take $v\in \mathbb{S}^1$ and let $\epsilon >0$. Consider a singularity $A$ and a segment of length $s>0$ going from $A$ in a direction orthogonal to $v$. We call the endpoint $A'$ and denote the segment by $I$. We can send a ray in direction $v$ from $A$. We assume that this ray never hits a singularity (otherwise we would be done since $v$ would be the direction of a saddle connection). In particular, we can record the length of the trajectory until the first return to $I$ (see the Remark in the last session). The first return time from any point on $I$ has a lower bound $L$. So if we look at the $n$-th return of $A$, which we denote by $A_n$, the distance $l_n$ travelled by $A$ is at least $nL$.

Now imagine dragging the point $A_n$ to $A$. This results in a saddle connection from $A$ to itself in a direction that differs from $v$ at most by $\theta = \arctan (\frac{s}{nL})$ which is arbitrarily small for large $n\in\mathbb{N}$. Note that if we would encounter a saddle point $B$ somewhere on the segments whilst sliding, this would give us a saddle connection between $A$ and $B$ which will have a direction even closer to $v$.

This proves the first claim.

For the second claim, we first note that $0 \in \mathbb{R}^2$ is not an accumulation point of $SC(\omega)$ because the length of saddle connections of $\omega$ is bounded from below. Now let $v\in SC(\omega)$. We draw a segment in direction $v$ of length $|v|$ from every saddle point. Since there are only finitely many saddle points, the flow in an orthogonal direction $v^\perp$ needs time at least $t>0$ to go from one segment to another one (in forward or backward time). This gives us the existence of a neighborhood $U$ of the union of all the segments that doesn’t contain any saddle points other than the ones already contained in the segments themselves.

This in turn implies that a development $u$ that is close to $v$ from a saddle point is contained in this neighborhood. This concludes the proof.

We denote the minimal length of saddle connections in a translation structure $\omega$ by $m(\omega)$. For fixed $\omega$, we define the map

Let $v\in\mathbb{R}^2$ be a development of the shortest saddle connection of $a\omega$, so $|v| = d(a)$. Then for $b\in \mathrm{GL}(2,\mathbb{R})$ we have $$(ba^{-1})v \in SC(b\omega)$$

and

$$d(b) \le |ba^{-1} v| \le \Vert ba^{-1} \Vert \cdot d(a),$$

where $\Vert \cdot \Vert$ denotes the operator norm on $\mathrm{GL}(2,\mathbb{R})$. Exchanging the roles of $a$ and $b$, we get

$$\Vert ab^{-1} \Vert^{-1} \cdot d(a) \le d(b) \le \Vert ba^{-1} \Vert \cdot d(a).$$

So if $a\rightarrow b$ then $\Vert ab^{-1} \Vert \rightarrow 1$ and $d(b) \rightarrow d(a)$ which proves the continuity of $d$. Acting on $\omega$ by an element of $\mathrm{SL}(2,\mathbb{R})$ doesn’t change the area $S$ of $\omega$. To show that $d$ is bounded on $\mathrm{SL}(2,\mathbb{R})$, we will prove that $m(a\omega) \le \sqrt{2\text{Area}(a\omega)} = \sqrt{2\text{Area}(\omega)}$. For this, take a segment $I$ of length $\sqrt{S}$ in an arbitrary direction, and starting in some singular point $A$. We assume that this segment doesn’t contain a saddle point besides $A$. Drawing another segment of length $\sqrt{S}$ orthogonal to $I$, also with no other singularities, gives rise to a rectangle of area $S$. We are going to assume that this rectangle contains no singularity since otherwise we would have a saddle connection of length at most $\sqrt{S}$. Now consider a maximal segment $J$ in $I$ on which the first return time of the flow orthogonal to $I$ is constant. This return time is at most $\sqrt{S}$. The boundary of $J$ maps back to $I$.

Since by assumption $I$ doesn’t contain another singularity, the “leftmost” point of $J$ has to be mapped to $A$ which give a saddle connection from $A$ to $A$ of length less than $\sqrt{S}$.

By non-uniform we mean that the quotient $\mathrm{SL}(2,\mathbb{R})/\Gamma(\omega)$ is non-compact.

Session 4 (Jiajun Shi), March 24th: Veech’s Theorem

We will assume that the translation structure $\omega$ has singular points (see the discussion in the section about Elementary translation surfaces after Theorem 9). The goal is to prove the following theorem:

To prove this theorem, we will need the following key lemma originally from

In Vorobets it is cited as follows:

Later we will show how this result follows from another Theorem from the late 80’s.

(There is more to write for this session.)

Session 5 (Fabian Lander), March (?): Masur’s Lemma

Let $T$ be an IET on $I = [a,b]$ and $a_0 = a, a_1, \dots, a_k = b$ be the break points of $T$. Denote by $\varepsilon (T)$ the length of the shortest interval(s) of this partition. For each $n\in \mathbb{N}$ we write $\varepsilon_n(T) = \varepsilon(T^n)$.

For a proof we refer to: