THE MATHEMATICS OF BILLIARDS: SUMMER COURSE

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1 THE MATHEMATICS OF BILLIARDS: SUMMER COURSE MOON DUCHIN 1. Introduction to Billiards 1.1. Billiard trajectories and unfolding. The mathematics of billiards might be considered an abstraction of the game of billiards, but the sense in which this is true is (sadly) guaranteed not to improve your pool game. First, we remove the pockets and consider a single ball s motion to be modeled by a point moving in straight lines. Then, we replace the rectangular boundary of the table by a Euclidean polygon. And of course, we neglect friction and spin. What we preserve is the boundary rule: angle of incidence equals angle of reflection for trajectories on this idealized table. 1 Figure 1. Part of a trajectory on a regular octagonal table The first construction in the theory is that of a translation surface derived from a billiard on a polygon via unfolding it to a collection of polygons with gluings. That is, from a polygon P we develop a polygonal system Q with sides identified by translation to get a surface X = Q/. The idea is to associate a closed, orientable surface to the billiard table which has the same geodesics (the trajectories, that is). To see this for an example, let s consider the billiard in a square. First we orient and label the edges. Next, instead of drawing the trajectory reflecting within the square when it hits an edge, we will draw a reflected copy of the square in such a way that the trajectory continues in a straight line. To make a closed surface, continue reflecting in this way until a figure is obtained which has each of its edges glued to a parallel edge elsewhere. In the square example, 1 For now, we will say that the reflection is undefined at the corners of the polygon note that from any fixed starting point, only countably many trajectories would ever encounter a corner. Later, the corners will be discussed in more detail. 1

2 2 BILLIARDS Figure 2. The bold line segment is the beginning of a trajectory; the dashed line shows it continuing in the square billiard table P, while the solid line is the continuation on the unfolded quadruplesquare Q. The surface is achieved by gluing opposite sides of Q to obtain the torus X. after the three reflections shown in Figure 2, the opposite edges of the quadruplesquare are glued, making a torus. This indicates the procedure in general we continue developing the figure by unfolding until each exterior edge is paired (identified by translation) to another one parallel to itself which has the same label and orientation. Remarks. For a billiard on a rational polygon one where the interior angles are rational multiples of π a closed surface will always be accomplished after finitely many reflections. (See Exercise 4.) In general, the unfolding process may produce too many copies around a point to draw contiguously in the plane. (For example, consider the right triangle whose smallest angle is 3π/10, reflected repeatedly around the vertex at that angle. This is discussed below in Example 4.) We can accommodate this by drawing the next reflection non-contiguously and marking the repeated side by gluing data. The reason for this obstacle to contiguous reflections is that these unfolded polygons may have cone points: the corners of the reflected polygons may be points on the surface with an angle around them of more than 2π. This can happen at (at most) finitely many points on a surface derived from a rational billiard because there are only finitely many vertices in the reflections. Such cone points always have angles 2πk for k N. We will sometimes call these points cone-type singularities when k > 1. Definition 1. For any point and direction (p, v) on a billiard polygon P R 2, the billiard flow φ t will be the unit-speed movement of the point according to the rules of motion described above. The flow will be considered to terminate at corners of the billiard. Remark. That is: for sufficiently small t, we have φ t (p, v) = (p + tv, v). Also note that the direction vector can take on only finitely many values in a rational billiard. Definition 2. On any length space, a geodesic is a locally length-minimizing path.

3 SECTION I 3 The translation surface (punctured at the singularities, say) can be endowed with the metric inherited from R 2 in its realization as X = Q/. In that metric, the images of the billiard orbits from P are geodesics. Certain geodesics also exist which change direction at cone-type singularities (though the billiard flow stops there). More on this will be discussed in later sections. Definition 3. A geodesic between two singularities (or possibly from one singularity to itself) will be called a saddle connection if it has no singularities in its interior. Example 1. Rectangular billiard. Just as in the square case, the translation surface is a torus and there are no cone points. This does not depend on the lengths of the sides. Example 2. Octagonal surface. Consider a regular octagon with opposite sides identified. This produces a two-holed torus (a genus-two surface). (See Exercise 5.) By the gluing rule which associates opposite sides, it is easy to check that the eight vertices are all associated to the same point on the surface. The angle around that point is 6π, so it is a cone point. Remark. It s easy to get confused between two different setups: one one hand a polygon viewed as a billiard (here, called P ), and on the other hand a polygon (or collection of polygons) with gluings (here, called Q), which can arise from unfolding a billiard. In the previous example, we considered the regular octagon with opposite sides associated. This arises from rational billiard because it is obtained from a (π/2, π/8, 3π/8) triangle by unfolding. This is rather different from considering billiard trajectories in a regular octagon itself. (To see this, convince yourself that the octagon billiard has a shorter closed geodesic than any on the octagonal surface.) Example 3. Double pentagon surface. Begin with the billiard on a (π/2, π/5, 3π/10) triangle. One way of unfolding produces the shape of two pentagons glued along an edge this figure is bounded by four pairs of parallel edges. Notice that something different would occur from unfolding a pentagon itself, and even from unfolding the (2π/5, 3π/10, 3π/10) triangle, though each one tiles the double pentagon. Example 4. The other n-gons and general triangular billiards. The class of triangular billiards includes all of the previous examples, as well as many examples which behave quite differently. Similarly to the octagon and the (double) pentagon, any regular n-gon can be achieved by starting with the appropriate triangular billiard. Whether the n-gon is doubled or not in the unfolded figure Q only depends on the parity (even- or odd-ness) of n. The number and type of singularities and the genus of X all vary as n increases. The regular 10-gon, for example, has two distinct singularities of cone angle 4π, as can be easily checked. Example 5. Swiss cross surface and other L-shaped billiard tables. Billiard tables need not be convex, and one interesting class of examples is the L-shaped tables that is, (non-regular) hexagons with six right angles. Four tiles of this polygon P unfold to make a surface.

4 4 BILLIARDS Figure 3. The Swiss cross surface is obtained by gluing opposite sides in the figure. Here, the vertices are marked by equivalence on the surface. Only one of the three actually marks a cone-type singularity. Additionally pictured: the dotted line shows the billiard P, with suitable dimensions to make the edges of Q unit-length. The vertical cylinder given by ( 0 1 ) is also marked. Example 6. Billiard with barrier. One interesting variation is to begin with a square billiard and insert a barrier, which is to be treated like an boundary edge in terms of the billiard flow when approached from either side. Consider the unit square with a vertex at the origin, and let the barrier be drawn vertically from the lower edge of the square to the point (1/2, α). Just as with the usual square, this will produce a closed surface after three reflections: it will be a torus with two slits for which the left-hand side of one is glued to the right-hand side of the other. After this association, the surface X has genus two and has two singularities of cone angle 4π Holonomy: Vector representation of edges, saddle connections, and cylinders. Given a collection of planar polygons, each (oriented) edge has a corresponding vector in R 2. For example, the standard octagon is given by the edges v 1 = ( 1 0 ), v 2 = ( ) 2/2, v 3 = 2/2 ( 0 1 ), v 4 = ( ) 2/2, 2/2 and then the same four repeated reversed ( v 1, v 2, v 3, v 4 ), with the eight laid end-to-end. R 2 -coordinates for edges and other curves on billiards will sometimes be called the (affine or linear or translation) holonomy. There will be more discussion of these coordinates in Section 5. Example 7. Perturbed polygons. Some properties of billiards are preserved by small perturbations. For instance, in the octagon example, perturbing the four v i (that is, changing the vectors by no more than a small amount componentwise) would still yield an octagon (no longer

5 SECTION I 5 regular) by laying them, then their negatives, end-to-end. In this example as with any sufficiently small perturbation of a billiard, the number and type of cone points are preserved. Remark. Perturbing a polygon may make a great difference to the trajectories! For instance, in a perturbation of the standard octagon, the vertical direction will likely no longer be closed. On a fixed billiard table, some directions may be periodic. For example, on the square, any direction with rational slope produces closed orbits from any starting point. On the Swiss cross, the vertical direction from any starting point is periodic. Any curve on the surface can be straightened to a geodesic in an essentially unique way. Usually, even if a curve misses the singularities on the surface, its geodesic representative will go through singularities. When this does not occur, however, the geodesic can be varied parallel to itself in either direction, producing new closed curves. This parallel movement can continue until a singularity blocks the way. A collection of all of the parallel variants of a closed nonsingular geodesic is called a cylinder on the surface so called because it is an actual isometrically embedded Euclidean cylinder and its boundary components are unions of saddle connections. As with the edge vectors, the saddle connections and nonsingular geodesics can each be represented by vectors in R 2. So, to rephrase the last paragraph, most curves can be represented up to homotopy 2 by a sequence of saddle connections v 1, v 2,..., v k, while the nonsingular ones can be represented by a single vector v, and in that case a whole parallel family of curves is described by the vector v. Sometimes, in a fixed direction, the entire surface will decompose into a union of cylinders. This is the case for many of the examples above, but keep in mind that this is in general quite rare. The octagon, for instance, decomposes in the vertical direction into two cylinders. Same for the double-pentagon and the Swiss cross. One cylinder suffices for the square billiard in fact, this is true not just in the vertical direction, but in any rational direction Orbits and counting. Many natural questions arise about billiards. Some that sound the most simple are still open. For instance: does every rational billiard have a closed orbit? Amazingly, this is even unknown for (general) obtuse triangles. Another natural question is whether the infinite orbits are equidistributed on the surface this is clearly true for the torus, where the density of the infinite orbits in a region on the subsurface depends only on the area of the region. However, this may not always be the case. Definition 4. An orbit parameterized by γ(t) is said to be equidistributed if n ( ) χ A γ(t) dt 0 area A n area Q for every open region A Q. (A similar definition would work for equidistribution on P or X.) 2 Homotopy is the change of a curve through continuous deformations. See a topology reference.

6 6 BILLIARDS Figure 4. Here, a billiard trajectory gets trapped in a chimney on the surface, which illustrates one way of failing to equidistribute. Figure from McMullen[12]. We will say that a billiard behaves like a square 3 if for every direction, either all the orbits in that direction are closed or they are all equidistributed. Theorem 5 (Kenyon-Smillie[5], Puchta[13], Ward). The following is a complete list of the acute and right triangles that behave like a square : the isosceles acute triangles with apex angle of the form π/n for n 3; the right triangles with smallest angle π/n for n 4; the three exceptional cases ( π 4, π 3, 5π ) ( π, 12 5, π 3, 7π ) ( 2π, and 15 9, π 3, 4π ). 9 Remark. This generalizes Veech s remarkable result that the regular n-gons all behave like squares ([15],[16]). Note also that this theorem provides information about obtuse isosceles triangles, since they can be unfolded from right triangles. Now we define symbols to use for counting trajectories in billiards. V S := {v R 2 : cylinder on S given by v}; N(S, T ) := VS (0, T ] V sc := {v R 2 : saddle connection on S given by v}; N sc (S, T ) := V S (0, T ] Theorem 6 (Masur[8]). For any translation surface X, the asymptotics of these quantities are quadratic in T : there exist positive constants c 1 and c 2 such that c 1 T 2 N(S, T ) N sc (S, T ) c 2 T 2. In special cases (as in the behaving-like-a-square theorem), these asymptotics can be nailed down more precisely, but in general the actual coefficients of growth are quite hard to find. 3 This is made precise in Section 5 when we discuss the Veech dichotomy.

7 SECTION I 7 Exercises. Exercise 1. Show that a direction is periodic on the unit-square billiard if and only if the slope is rational. Show that the orbit of a point in a non-periodic direction is dense on the surface. (Dense means the orbit enters every open set.) Exercise 2. Write the billiard flow φ t for the unit square with a vertex at the origin as a function of an initial point p and direction v. Exercise 3. Formalize the claim about straightening : show that every homotopy class of curves on a rational billiard has a geodesic representative. Explain what the conditions are on a geodesic changing angle at a cone point. Exercise 4. Show that, for any rational billiard, there is at least one way of unfolding it to a closed surface. Show that the number of tiles and the number and type (total angle) of singularities are invariants of the billiard. Exercise 5. Prove that the octagon with opposite sides identified is a genus-two surface. Do this two ways: with the Euler formula (2 2g = V E + F ) and without. What about the other n-gons? Exercise 6. Find the cylinder decomposition for the 10-gon (with one edge ( 1 0 )) in the vertical direction. What can you say about other directions? Exercise 7. On every acute triangle T, the Fagnano trajectory follows the inscribed triangle between the endpoints of the altitudes (dropped from the vertices of T to the opposite sides). Show that this is a billiard orbit. Harder: There is a parallel family of closed curves which are approximate doubles of the Fagnano trajectory. Describe this family. Exercise 8. Code or draw by hand a small perturbation from one of the examples above and examine the vertical trajectories. Exercise 9. It is a fact that Theorem 7. N 2c (S, T ) ct 2, where N 2c (S, T ) = {v R 2, v T : two disjoint cylinders with vector v}. (This is surprising: a seemingly much smaller quantity than N(S, T ) also has quadratic asymptotics!) Your task: for g > 2, find a 2g-dimensional family of translation surfaces which have two cylinders with the same holonomy. References [1] M. Bachir Bekka and Matthias Mayer, Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces, London Mathematical Society Lecture Note Series 269. Cambridge University Press: [2] Alex Eskin, Counting problems in moduli space, Lectures from Luminy, France. (preprint) [3] Pascal Hubert and Thomas Schmidt, Affine diffeomorphisms and the Veech dichotomy, Lectures from Luminy, France. (preprint) [4] S. Katok, Fuchsian groups, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, [5] Richard Kenyon and John Smillie, Billiards on rational-angled triangles, Comment. Math. Helv. 75 (2000), no. 1, [6] Steven Kerckhoff, Howard Masur, and John Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2) 124 (1986), no. 2,

8 8 BILLIARDS [7] John M. Lee, Riemannian Manifolds, Graduate Texts in Mathematics Springer-Verlag, New York, [8] Howard Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory and Dynamical Systems 10 (1990), [9] Howard Masur, Teichmüller space, dynamics, probability, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), , Birkhauser, Basel, [10] Howard Masur, Ergodic theory of flat surfaces, Lectures from Luminy, France. (preprint) [11] Howard Masur and Serge Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, Vol. 1A, , North-Holland, Amsterdam, [12] Curtis McMullen, Gallery, [13] Jan-Christoph Puchta, On triangular billiards, Comment. Math. Helv. 76 (2001), no. 3, [14] John Stillwell, Geometry of surfaces, Universitext. Springer-Verlag, New York, [15] W. Veech, Teichmuller curves in moduli space, Eisenstein series, and an application to triangular billiards, Inventiones Math. 97 (1989), [16] W. Veech, The billiard in a regular polygon, Geom. Func. Anal. 2 (1992),