convex billiards

PETER M. G R U B E R

C O N V E X B I L L I A R D S

Dedicated to Professor Curt Christian on the occasion of his 70th birthday

AaS'ntACT. This article deals with (partly rather peculiar) properties of typical convex billiards. In particular, convex caustics, trajectories which terminate on the boundary and dense trajectories are investigated.

1. INTRODUCTION

A (convex) billiard table C in d-dimensional Euclidean space ~:d is a smooth convex body in ! :d. Here a convex body in •a is a compact convex subset of ~d with non-empty interior; if its boundary considered as a manifold is of differentiability class qfl, the body is called smooth. A billiard ball is a point in C which moves at unit velocity along a straight line in the interior of a billiard table C until it hits the boundary bd C of C where it is reflected in the usual way, that is, the component of the velocity vector parallel to the normal of bd C at the hitting point changes its sign. After the reflection the point again moves on a straight line, etc. The curve described by a billiard ball is a (billiard) trajectory in (the configurational space) C. Sometimes non-smooth billiard tables are considered but then only trajectories avoiding singular boundary points are investigated (see [3], [8], [11], [20]).

A countable union of nowhere dense subsets of a topological space is called meager or offirst Baire category. If the complement of any meager set is dense, the space is named after Baire and meager subsets may be considered 'small'. We say that some property holds for most elements of a Baire space or is generic if it holds for all elements except those in a meager set. Any complete metric or locally compact space is Baire. It is easy to prove that the complement of a meager set in a Baire space is also Baire (see [19, [31]).

The 'natural' topology on the space c~ of all convex bodies in ~d is induced by, for example, the Hausdorff metric ~ (see Section 2 below). The Blaschke selection theorem implies that q¢ is locally compact and thus Baire (see [4], [10], [15]). For surveys of results on Baire category in convexity we refer to [16], [35].

Since Klee [22] (see also [13]) proved that the non-smooth convex bodies form a meager set in ~, the subspace ~ c~ q¢1 of billiard tables in H :d is Baire.

A compact convex set K (~ J~) in the interior int C of a billiard table C is a (convex) caustic if any trajectory which touches K once touches it again after each reflection, that is, each line segment of the trajectory is contained in a supporting line of K. Caustics are important because of their relation to the

Geometriae Dedicata 33: 205-226, 1990. © 1990 Kluwer Academic Publishers. Printed in the Netherlands.

206 PETER M. G R U B E R

eigenfunction problem for the Laplace operator on C and to ergodic theory (see [81, [20,1, [251, [261, [341). In Section 2 it will be shown that the billiard tables containing caustics form a meager set.

A trajectory in a billiard table C terminates on the boundary of C if the sequence . . . . P-1, Po, Pl . . . . of its vertices, i.e. the points where it hits the boundary, satisfies

IPo - P d + IP~ -P2I + "'" < + ~ -

(11 denotes the Euclidean norm on ~:d.) Then clearly the sequence Po, Pl . . . . converges to a point of bd C. The phenomenon of trajectories terminating on the boundary was studied in the planar case by Halpern [17]. In Section 3 some of his results will be extended to higher dimensions. In addition we shall prove that in most billiard tables no trajectory terminates on the boundary, and in the exceptional cases when there are such trajectories the set of these is 'small' from the measure-theoretic as well as from the topological viewpoint.

The phase space ph C of a billiard table C is the set

{(p, v ) :p~bdC, y e S d-l , (v, n(p)) > 0} c b d C x S d-1 c ~:2d,

where S d- t is the (d-1)-dimensional Euclidean unit sphere, ( , ) denotes the inner product on ~:d and n(p) = nc(p) is the interior normal unit vector of bd C at p. The (discrete) billiard transformation T = T c on phC maps (p, v) e ph C onto (q, w) where q is the point on the trajectory starting at p in direction v where it first hits bd C and w is the direction after reflection at q. (Note that v = ( q - P) / lq - Pl and w = v - 2(v, n(q))n(q).) The dynamical system consisting of the phase space of a (convex) billiard table and the corresponding (discrete) billiard transformation is called a (convex) billiard. (There is a related second version of a billiard but with a continuous group of transformations instead of the discrete billiard transformation.) A trajectory in the phase space of a billiard table C is a sequence of the form . . . . T - ~(p, v), (p, v), T(p, v) . . . . which may terminate in one direction. Clearly there is a one- to-one correspondence between trajectories in C and trajectories in ph C. An important concept for billiards is the Borel measure fl = tic on the phase space of C defined by

= f s (v, n(p)) da ® ~(p, v) for Borel sets B c ph /~(8) C

where a and z denote the surface area measures on bd C and S ~-t, respectively./~ is invariant with respect to T, cf. Lemma 2 (see [31, [81, [11], [2o1).

Section 4 deals with the planar case and is related to work of Zemlyakov and Katok [37] and Mather [27,1. The main result of Section 4 says that most

CONVEX BILLIARDS 207

billiard tables C have the property that for most elements (p, v) of the phase space the trajectory (p, v), T(p, v) . . . . is dense in ph C (thus the corresponding trajectory in C is dense in C). As a consequence of this result and of the fact that in most billiard tables no trajectory terminates on the boundary (see Section 3) we shall prove that for most billiard tables C and most (p, v) e ph C the trajectory in C starting at p in direction v has the following peculiar property: For an arbitrarily long period of time it moves arbitrarily close to bd C in one direction and again later on, but then in the reverse direction.

In order to clearly show the basic scheme of the proofs of Baire type results in convexity, the proofs below are organized in a similar way. Sections 2, 3 and 4 each contain short surveys of related material.

2. CAUSTICS

For d = 2 caustics have been thoroughly investigated. By a result of Minasian [28] and Turner [34] a compact convex set K is a caustic of a billiard table C if and only if the following holds: There is a closed inelastic string such that bd C is the curve obtained by wrapping the string around K, pulling it tight at a point and moving the point around K while keeping the string tight. This will remind the reader of the gardener's construction of ellipses. On the other hand, given a compact convex set K, any curve constructed in the way just described is the boundary of a billiard table with caustic K. Lazutkin [24]-[26] showed that on each billiard table of class ~g553 and with positive curvature there exist 'many' caustics, his proof making use of ideas of Moser [29]. An approach of Riissmann [32] can be used to replace 553 by 8 (cf. [7]) and Douady [9] showed that 7 is sufficient.

Results for d = 2 essentially due to Jacobi (see [3, p. 170]) and for d = 3 due to Turner [34] show that for (solid) ellipses and (solid) ellipsoids in ~:3 the confocal ellipses, resp. ellipsoids, in their interior are caustics. It seems to be an open problem whether there are other caustics or not. See also Conjecture 1 below.

If K is a caustic of a billiard table C then the disjoint Borel sets

{(p, v)eph C: [p, q] n K = ~ where (q, w) = T(p, v)},

{(p, v) e ph C: [p, q] n K # O where (q, w) = T(p, v)},

are invariant with respect to T. Here [p, q] denotes the line segment with endpoints p, q. If K has dimension d - 1 or d both sets have positive fl- measure which implies that T is not ergodic (see [8]).

THEOREM 1. Most billiard tables in F a contain no caustic. Proof. First some preliminaries must be mentioned. The distance of two

sets A, B in E d is the infimum of Ix - Yl, where x e A, y e B. Using this, the

208 PETER M. GRUBER

Hausdorff metric ~ can be defined in the following way: Let A, B be (non- empty) compact sets in ~:d. Then ~(A, B) is the maximum distance that a point of one of the sets A, B has from the other set. Convergence of sequences of compact sets is always to be understood with respect to & The definition of immediately implies the following propositions:

(1) Let A, At, k = 1, 2 . . . . . be compact, Ak --* A. Then A consists of all

limits of converoent sequences qk, k = 1, 2 . . . . , where q~ ~ A~.

(2) Let A, B, At, Bk, k = 1, 2 . . . . . be compact, Ak --* A, B h - , B, and let t > O. I f the distance of A k, Bk is >>. 8for all k, then the distance of A, B is >~.

(3) Let C, Ck ~ ~ , k = 1, 2 . . . . . Ck ~ C. Then bd Ck --} bd C.

The next proposition requires a justification:

(4) Let C, C ~ r ~ r ~ 1, p ~ b d C , p ~ b d C k, k = 1, 2, . . . , Ck--*C,

Pk "-* P. Then n(pk ) = nck(pk) --* n(p) = nc(p).

(In general we shall omit the suffix C resp. Ck in nc(p) and nck(Pk).) To prove (4) suppose the contrary. By considering a suitable subsequence and renumbering (if necessary) we may assume that n(pk) -* m ~ n(p). The halfspace {x: ( x - Pk, n(pt)) >1 0} contains Ck. Since Ck --, C, Pk --} P, n(Pk) --} m, Propo- sition (1) shows that C is contained in the 'limiting' halfspace {x: ( x - p, m) /> 0}. Since m # n(p), this contradicts the smoothness assumption for C, thus proving (4).

As an immediate consequence of the definitions of billiard trajectories and of caustics the following obtains:

(5) Let C ~ r~ n ~ and let K c i n t C be compact and convex. Then K

is a caustic of C if and only if for each p ~ b d C the cone {p + ~ ( x - p ) :x~K, A >>. 0} is symmetric with respect to the line

through p havinff direction n(p).

The next proposition is formulated slightly more generally than is needed for the proof of the theorem.

(6) Let C E qf n r~l, let K ~ int C be compact and convex and let k e {2 . . . . . d - 1}. Then K is a caustic of C if and only if for any k- dimensional subspace S of F_ d the set K s is a caustic of C s, where (.)s

denotes the orthooonal projection of [_d onto S.

((6) is meaningful only for d >t 3.) Suppose first that K is a caustic of C and let S be a k-dimensional subspace of [d. Clearly C s is a billiard table in S and the compact convex set K s is contained in the relative interior of C s. Let q be a

C O N V E X B I L L I A R D S 209

relative boundary point of C s and choose p ~ bd C with pS = q. Then the lines L and M ( c S) containing p (resp. q) and orthogonal to bd C (resp. orthogonal to the relative boundary of C s) are parallel and L s = M. By (5) the cone { p + 2(x - p): x ~ K, 2 t> 0} is symmetric with respect to L. Its image in S, that is the cone {q + 2(x - q): x ~ K s, 2 >>, 0} then is symmetric with respect to L s = M. Hence the compact convex set K s in the relative interior of C s is a caustic of C s by (5). To prove the converse suppose that for any k- dimensional subspace S the set K s is a caustic of C s. This combined with (5) yields the following: Let p e bd C and let L be the line through p orthogonal to bd C. Then the orthogonal projection of the cone { p + 2(x - p): x E K, 2 >/0} into any k-dimensional plane containing L is symmetric with respect to L. Hence the cone itself is symmetric in L. Since this holds for any p ~ bd C, a further application of (5) shows that K in fact is a caustic of C, concluding the proof of (6).

(7) Let C E c~ c~ ~1 contain a caustic. Then C is strictly convex.

Using the string construction of planar billiard tables from their caustics, Turner [34] deduced that any billiard table with a caustic is strictly convex. This proves (7) for d = 2. In the case d >/3, Turner's result, together with (6), implies that the orthogonal projection of C into any 2-dimensional subspace is strictly convex. Hence C itself is strictly convex.

After these preparations the proof of Theorem 1 is rather short. For n = 1, 2 . . . . . let

~n = { CEcg c~ ~1: C contains a caustic having distance

>>. 1/n from bd C }.

We shall first show that

(8) c~n is closed in c~ n c~1.

Since the topology in ~ and thus in ~ c~ ~x is induced by a metric (~), closedness can be characterized in terms of sequences. It is thus sufficient to prove the following: Let CkeCg~, k = 1, 2 . . . . . converge to C ~ c~ ~ . Then C e ~ . Let Kk be a caustic of Ck having distance 1> 1/n from bd Ck. Since Ck ~ C the caustics Kk are all contained in some fixed bounded set. The Blaschke selection theorem (see, e.g., [4], [10], [15]) then shows that a suitable subsequence of Kk, k = 1 . . . . . converges to a compact convex set K. After renumbering we may suppose that Kk ~ K. Propositions (1)-(3) imply that

(9) the compact convex set K is contained in int C and has distance

>>. l /n f rom bd C.


In order to show that

(10) K is a caustic of C

we apply (5): Let p e bd C. By (1), (3) there exist points pk~bd Ck such that p~-,p. Then n(pk)--, n(p) by (4). Since K k is a caustic of Ck the cone { Pk + 2(x - Pk): x E K k, 2/> 0} is symmetric with respect to the line through Pk in direction n(pk). From Kk - , K, Pk ~ P, n(pk) --, n(p) and (1) it then follows that the cone {p + 2(x - p): x ~ K, 2 >/0} is symmetric with respect to the line through p having direction n(p). Since this holds for any p ~ bd C, Proposition (5) yields (10). From (9) and (10) we obtain that C ~qf,, concluding the proof of (8).

(11) ~, has empty interior in c~ n c~1,

for otherwise c~, would contain non-strictly convex billiard tables, contradicting (7).

The set u P , is meager by (8) and (11). Since it consists precisely of the billiard tables containing caustics, Theorem 1 is proved.

Using (6) and tools from projective geometry one can show that in any dimension the ellipsoids confocal to a given ellipsoid and contained in its interior are caustics. More generally we formulate the following:

CONJECTURE 1. The caustics of an ellipsoid in F_ d are precisely the confocal ellipsoids contained in its interior together with the intersection of all confocal ellipsoids.

Quite different to the situation in E 2 we have

CONJECTURE 2. For d >i 3 the only billiard tables in ~_d having caustics are

the ellipsoids.

Slightly more general is the following

CONJECTURE 3. A convex body K contained in the interior of a closed surface S in F_ d (d >1 3) is an ellipsoid iffrom any point p of S it appears centrally symmetric (see [2]).

Here K appears centrally symmetric from p if the cone {p + 2(x -- p): x ~ K, t> 0} is symmetric with respect to a suitable line through p. Since a caustic

K of a billiard table C appears centrally symmetric if looked at from any point of bd C (see (5)) the validity of this more general conjecture would imply that K is an ellipsoid. Then using (6) and the string construction of Minasian and Turner, we see that the orthogonal projection C s of C into any 2-


dimensional subspace S is an ellipse confocal to the ellipse K s. Finally, a theorem of Blaschke and Hessenberg [5], together with some arguments from projective geometry, show that K is an ellipsoid in the interior of the ellipsoid C and confocal with it. Thus the validity of this more general conjecture would prove Conjecture 1 for d t> 3 and Conjecture 2.

3. TRAJECTORIES TERMINATING ON THE BOUNDARY

A peculiar example of Halpern [17] exhibits a planar billiard table in which there is a trajectory terminating on the boundary. This example can easily be extended to any dimension >i 2. I fa planar billiard table is of class c~3 and has positive curvature there is no trajectory terminating on the boundary according to another result of Halpern. Finally, Halpern proves that for any planar billiard table of class c~2 for almost all (p, v) in the phase space of C the trajectory in C starting at p in direction v does not terminate on the boundary.

This section contains extensions of these theorems to d > 2 together with some adjunct category results. In the final part a remark is made on trajectories 'asymptotically terminating on the boundary'.

THEOREM 2. Let C be a billiard table in E d of class c63 having positive

Gaussian curvature. Then no trajectory in C terminates on the boundary of C.

The proof follows the same lines as Halpern's proof. Proof. We first mention some tools. If g is a real function let O(g) denote

any real function h such that Ihl ~< Ylgl, where 7 is a constant. If ak, k = 1, 2, . . . . is a real sequence we denote O(ak) any sequence bk such that Ibkl ~< yla~l, where the constant ~ is independent of k. For x e •d let (x 1 . . . . . x d) denote its coordinate array.

(1) Let K be a convex body in E d-i and let f be a convex function of class c63 on K such that the surface S: x d = f(x), x ~ K, has positive

Gaussian curvature. For s, t eK , s ~ t, let ~, i~ be the angles between S and the line segment [(s, f(s)), (t, f(t))] at the endpoints.

Then

/~ - • = 0 ( ~ 2 ) , ~ = O ( I t - s l ) ,

where the constants in the O-symbols depend only on f.

In the following the constants in all O-symbols depend only on an upper bound for the partial derivatives o f f up to third order. The summation over i, j is to be extended from 1 to d - 1. Since (-fxl(s) . . . . . -fx,-,(s), 1) is a normal vector of S at (s, f(s)) we have that


( - f x , ( s ) ) ( t I - s 1) + . . . + ( - f x , - , ( s ) ) ( t ~ -1 - s a - l ) + f ( t ) - f ( x ) sin a --- ((f.,(s))2 + " " + (fx,_,(s)) 2 + 1)1/2(i t _ s12 + ( f ( t ) - f ( s ) ) 2 ) 1/2

~iaf~,xs(s)uiuJlt - s] 2 + O(It - sl 3)

2 ( I + Z , ( L , ( s ) ) 2 ) '/z l + k ~ , / ] I t - s l

g i a f ~ , ~ ( s ) u i ~ l t - sl 2 + O(It - s[ 3)

= 2(1 + X:~ (fx,(s))2)~n(1 + (Z~ f~,(s)u~) 2 + O(It - sl))l/2lt - sl

~'id fx 'xJ(S)Uit t i It - s] + O(]t - s]2), = 2(1 + Y-.if~,(s))2)l/2(1 + (Zi fx,($)Ui)2) 1/2

where u = ( t - s ) / I t - sl ~ S d - 2. Expanding arcsin in a Taylor series yields

Z~'Jfx'~J(s)uiui It - s] + O(It - $12). (2) g = 2(1 + Y'i f~,(s))2)l/2( 1 + (~ifx,(S)Ui)2) 1/2

Since the surface S has positive Gaussian curvature, for each s the quadrat ic form

.~ L,~J(s)uiu j I ,J

is positive definite. Thus

m i n t ~ f~,x~(S)U'td: s ~ K , uE S d - 2 } > O, k',J

which combined with (2) implies that

(3) It - sl = O(~).

In analogy to (2) we have

Z i J f ~ ' ~ ( t ) ( - u i ) ( - u i ) It - sl + O(It - sl2). fl = 2(1 + Zi(fx ,( t ))2)l /2(1 + ( Z i f x , ( t X - U i ) ) 2 ) 1/2

Since

fx,xJ(t) =fx,~(s) + O(It - sl),

f~,(t) =fx,(s) + O(It - sl),

it follows that

~idf~'xJ(s)u"uJ It - s[ + O(It - sl2), fl = 2(1 + El (f~,(s))2)t/2(1 + (Zi fx,(s)ui)2) t/2

C O N V E X B I L L I A R D S 2 1 3

and therefore

fl - • = O(It - sl 2) = O(~t 2)

by (2) and (3), concluding the proof of (1). A theorem of Weierstrass (see, e.g., [23, p. 133]) implies the following:

(4) Let Y, tp~ be a series o f non-zero reals such that

tPk+ 1 = 1 + q/k, tPk

where 1~ d/k converges. Then Y~ tpk is divergent.

Using (1) and (4) the proof of Theorem 2 is quite easy. Suppose that Theorem 2 is false. Then there is a trajectory on C with ver t ices . . . , Po, Pl . . . . . say, such that

(5) ~ IPk+l -- Pkl converges. k=O

Hence Pk-'+ Poo E bd C, say. Denote by ct k the angle between the line segment [Pk, Pk+ 1] and bd C at Pk. In the supporting hyperplane H of C at p~ let a Cartesian coordinate system be chosen. Together with n(p=) it forms a coordinate system in E d. In this coordinate system the 'lower part ' of bd C is represented in the form x d = f(x): x e C H where f is a convex function on the orthogonal projection C n of C into H. Let K be a closed (d - 1)-dimensional ball in the relative interior of C H with center P~o. Since C is of class c~a and has positive Gaussian curvature, f l K satisfies the assumptions of (1). Since Pk -+ P~ we have p~ e K for all sufficiently large k. Hence (1) implies that

0~k+ 1 (6) ~k+l -- ak = 0( 52 ) or

CX k

say, where 6k = 0(1),

(7) ~q = O(Ipk+l - Pkl).

- - = 1 + O(~q) = 1 + 3k0tk,

(5) and (7) yield the convergence of X ~tk and thus of X 6k~ k. On the other hand, (6) and (4) with tp~ = aq, ~bk = 6k~tk show that E~q is divergent. This contradiction concludes the proof of Theorem 2.

For the proofs of Theorems 3, 5 and 6 we shall need the following technical

LEMMA 1. Let C, C k e ~ c~ C~ 1, k = 1, 2 . . . . , such that Ck ~ C. Let eo > 0

and let (pk, Vk)EphCk such that for n(pk)=nck(Pk) the inequality

(Vk, n(p~))>1 e o holds. Let p k ~ p , Vk--+V, say. Then (p, v ) e p h C and

214 P E T E R M . G R U B E R

(v, n(p)) >1 eo, where n(p) = nc(p). Further, there is a number el > 0 such that

for (q, w) = T(p, v), (qk, wk)= T(p~, vk) we have qk ~ q, w k ~ w and

(w~, n(qD) >>- ~i. Proof. From Propositions (I), (3), (4) in Section 2 it follows that p ~ bd C

and n(pk) --* n(p). Since (v~, n(pk) ) >i 80, V k --* V, n(pk) ~ n(p) we have (v, n(p))/> eo and thus (p, v )ephC, q (resp. q~) is the intersection point ~ p

(V~pk) of the line through p (Pk) having direction v (Vk) with bd C (bdCk). From Pk-'~ P, Vk "-~ V, C k --~ C and (v, n(p)) /> e o > 0 it thus follows that qk "-~q" A further application of Proposition (4) in Section 2 then

shows that n(qk) ~ n(q). Since w = v - 2 (v, n(q))n(q), and since Wk = VR -- 2

(Vk, n(qk))n(qk), VR --~ V, n(qk) ~ n(q), we obtain that Wk ~ W. Clearly (q, w) e ph C, (qk, Wk) e ph Ck and thus (w, n(q)) > O, (Wk, n(qk)) > 0. Since, on the other hand, w k ~ w, n(qk) ~ n(q) and thus ( w k, n(qk) ) ~ (w, n(q)), there is an el > 0 such that (Wk, n ( q k ) ) 1 > ~1 for all k, concluding the proof of Lemma 1.

Theorem 2 explicitly describes a 'small' set of billiard tables in which no trajectory terminates on the boundary. The next result shows that this is

typical for billiard tables.

T H E O R E M 3. In most billiard tables C in E d there is no trajectory terminatinff

on the boundary o f C.

Proof. For n = 1, 2 . . . . . let

c~, = {CeC~ c~ c~1: there exists an element (p, v ) e p h C such that

(v, n(p)) >>. 1/n and the trajectory in C startinff at p in

direction v has length <<. n}.

The first step of the proof consists of showing that

(8) ~ , is closed in c~ n c~1.

Let Ckeq~,, k = 1, 2, . . . , and suppose that C ~ C ~ f c ~ C ~ 1. Choose

(Pk, Vk) ~ ph Ck such that

(9) (vh, n(pk)) >t 1In

and the trajectory in C k starting at Pk in direction v k has length ~<n. If Pko : Pk, Pkl, ' ' " are its vertices, this is equivalent to

(10) IPko - P~ll + IP~I - Pk2[ q- " ' "

+ I Pkt -- Pu + 11 <<- n for each l = 1, 2 . . . .

Since C k --+ C the sequence Pk, k = 1, 2 . . . . . is bounded. The same holds for the sequence Vk, k = 1, 2 . . . . Hence by considering suitable subsequences of p~, v~ and renumbering we may assume that Pk ~ P, Vk ~ V, say. Note (9). The


repeated application of Lemma 1 yields the following:

(11) (p, v)~phC, (v, n(p)) >~ 1/n

and for the vertices Pko, Pkl . . . . . of the trajectory in C~ starting at Pk in direction Vk we have that Pkl - -~ Pol as k -~ ~ where Poo = P, Pol . . . . . is the trajectory in C starting at p in direction v. Thus (10) implies that

IPoo- Poll + IPol -Pozl + " "

+ lPo l -Po t+ l l~<n for e a c h l = l , 2 . . . .

Hence this trajectory has length ~<n which combined with (11) yields that C e c~,. This finishes the proof of (8).


(12) q(n has empty interior in c~ c~ ~ i ,

suppose the contrary. Since the convex bodies of class ~3 and with positive Gaussian curvature are dense in ~ (see, e.g., [14]) and thus in c~ c~ ~1 we may choose in ~ , a billiard table C of class ~3 having positive Gaussian curvature. This contradicts the fact that by Theorem 2 there is no trajectory in C terminating on the boundary and thus proves (12).

(8) and (12) together imply that urn, is meager in ~ c~ c~1. Since uc~, is precisely the set of all billiard tables on which there are trajectories terminating on the boundary, Theorem 3 is proved.

Given a billiard table in E d the Borel measure on its phase space defined in the introduction is invariant under the billiard transformation. Although this seems to be well known, the proofs available to us need additional differentiability assumptions. For this reason we insert a proof. A different argument that leads to a proof in the planar case was communicated to the author by Professor Jiirgen Moser.

LEMMA 2. Let C ~ n c~ be a billiard table in E d. Then the Borel measure

fl = tic on the phase space ph C of C is invariant under the billiard trans-

formation T = T c.

Proof. Let U be the open unit ball in E d- ~ and let p: U ~ bd C be a parametric representation of bd C (where one point of bd C is deleted) of class c~1. Define h: U x U ~ R and v: U x U \ { ( r , r ) : r ~ U } - ~ S d-1 by

h(s, t) = I/7(0 - p(s)] = (p(t) - p(s), p(t) - p(s)> 112

for (s, t ) • U x U,

p ( t ) - p ( s ) v ( s , t ) = for(s , t ) e U x U, s ~ t .

h(s, t)

216

Clearly,

PETER M. GRUBER

h,,(s, t ) = - - 1 2

2 h(s, t) <p(t)--p(s) , p,~(t)>

= <v(s, t), p,~(t)>,

1 v,s(s, t) = h(~, t) 2 { p,j(t)h(s, t) - ( tgt) - p(s))(v(s, t), p,j(t))}

_ p,j(t) v(s, t) h(s, t) (v(s , t), PtJ(t)) h(s, t ) '

and analogously for the derivatives with respect to s i. Fo r the definition and simple propert ies of the cross product x o fd - 1 vectors in [d, see [33, pp. 84,

85]. (The cross product is not to be confused with the Cartesian product of sets for which the same symbol is used.) The above formulas yield

v,,(s, t) × . . . x v~- , (s , t)

1 - h(s, t ) d - 1 p t l ( t ) x "'" x p ¢ - 1 ( t)

+ a sum of cross products, each containing the

factor v(s, t).

It thus follows that

(13a)

F r o m

<v(s, t), v,l(s, t )× . . . × v~- l (s , t)>

1 -h's,~ t) a-~ <v(s, t), p,,(t) x . . - x p~-,(t)>.

Iv(s, OI = <v(s, O, v(s, t)> 1/2 = 1

we deduce that

(v(s , t), v,j(s, t)) = 0 for j = 1 . . . . . d - 1.

Thus v(s, t) is parallel to vii(s, t) x . . . x v~-l(s , t). Using Iv(s, 01 = 1 again, gives

(13b) <v(s, O, v,l(s, t ) x "'" x vt~-,(s, t)>

= + Iv,,(s, t) x " " x vd- l (s , t)l

with suitable sign + . To prove the invariance of fl let B be a Borel set in ph C. Fo r (p, v) ~ B let

(q, w) = T(p , v). Then v = (q - P)/lq - P[ and we may choose (s, t) ~ U x


U \ {(r, r): r e U} such that p = p(s), q = p(t). Then (p, v) = (p(s), v(s, t)). The

set

V = {(s, t): (p(s), v(s, t)) e B} ~ V x V \ {(r, r): r e O }

is the inverse image of B under the continuous mapping (s, t) --. (Ms), v(s, t)) and thus is Borel (see [18, 10.41]). For p = p(s) the set

B , = {veS~-~: (p, v ) e B }

is a (possibly empty) Borel set (see [18, 21.19, 21.4]). Clearly

B , = {v(s, t): t e V~}

where V, = {te U: (s, t )e V} is Borel too. The definitions of/~, of surface integrals and of product measures, Fubini 's

theorem, Propositions (13) and the analogous formulas with s and t exchanged yield that

fl(B) = ~ (v, n(p)) dtr @ T(p~ v) J( p,v)~B

B t O

x vr,-,(s, t)[ dt 1"'" dt d- 1) Ip~,(s) x I l l

× p~-~(s)l ds 1"'" ds d- 1

= + ~ @(s, t), n(p(s)))lp~,(s)x D l l X p~-,(s)[ Jc s,t)eV

x (v(s, t), vt,(s, t) x "'" x vt,-,(s, t ) )

x ds l " ' ' d s d - t d t t . . . d t d-1

= +_~ h(s, t) d- ~(v(s, t), v,,(s, t )x . . . x v~,-~(s, t)) .1( s,t)e~V

x (v(s , t), v,l(s, 0 x . . . x v~-,(s, t))

x ds t "'" ds d- 1 dt I . . . dt d- 1.

For (q, w ) e T B let (p, v ) = T - l ( q , w). Then w is the reflection of - v with respect to the line through the origin o having direction n(q). Choose (s, t)e U x U \ {(r, r): r e U} such that p = p(s), q = p(t). Then (q, w) = (p(t),

218 P E T E R M. G R U B E R

v(t, s)') where (-)' indicates the reflection in the line through o having direction

n(q). Let

(TB)~ = {w ~ S d- 1: (q, w) ~ TB},

1, n = {s~ U : (s, t)~ V}.

Then

f fl(TB) = | <w, n(q)) do ® z(q, w)

J{ q,w)¢TB

cbd C ¢(TB)~

= ftcv (fs~v, (v(t' s)' n(p(t))>'Vs'(t' s) × " " × vd='(t' s)[ dsl

• . . ds d - l ) Ipt,(t)× . . . ×pt,_,(t)l dt 1 . . . dtd-1

= + f h(s, t)d-l<v(s, t), vs,(s, t)+ " " x Vs,-~(s, t)> J( s,t)EV

x <v(s, t), vt,(s, t) x " '" x v¢-,(s, t)>

x ds l " " " ds d- l dt 1 . . . t i t a- l ,

concluding the proof of Lemma 2. Theorem 3 shows that only in exceptional billiard tables there are

trajectories which terminate on the boundary. The following result says that even in these cases the set of trajectories terminating on the boundary is small in measure as well as in the topological sense.

THEOREM 4. Let C be a billiard table in E d. Then the set of elements (p, v) in the phase space of C for which the trajectory in C starting at p in direction v terminates on the boundary of C has measure zero and is meager.

The proof of this result also makes use of ideas of Halpern. Proof. For n = 1, 2 , . . . , let

Bn = {(p, v)~phC: <v, n(p)> ~> 1/n and the trajectory in C starting at p in direction v has length ~< n}.

Clearly B = uB , consists of all elements (p, v)eph C for which the trajectory starting at p in direction v has finite length. A simplified version of the proof


of (8) shows that

(14) B, is closed in ph C.

(14) yields in particular that B = wB, is Borel. To prove the first assertion of Theorem 4 we assume on the contrary that fl(B) > 0. For e > 0 let

A~ = {(p, v)ephC: (v, n(p)) < e}.

Since fl(A,) ~ 0 as e ~ 0, we may choose e so small that

#(n) (15) /~(A3 < -

2

A simple compactness argument yields the following: For any $ > 0 there is an q > 0 such that for each (p ,v )~phC with ( v , n ( p ) ) > ~ we have

[P - ql >1 7, where (q, w) = T(p, v). This shows that for any trajectory terminating on bd C the angles of the trajectory at its vertices with bd C converge to 0. Hence for each (p,v)6B we have that Tl(p,v)~A~ or (p, v)E T-~(A~) for all sufficiently large i. Thus

B c 0 ~ T-'(A~). k = l l = k

This, together with a simple measure-theoretic result on increasing unions of sets (see 1"18, 10.17]), implies that

for all sufficiently large k. Since

~(T-~(A3) = ~(A,) < 8(8) 2

by Lemma 2 and (15) we arrive at a contradiction. Hence fl(B)= 0, concluding the proof of the measure assertion.

To show the category assertion note that

(16) B. has empty interior in ph C.

Otherwise fl(B.)> 0 and hence fl(B)> 0, in contrast to the result already proved.

From (14) and (16) we obtain that B -- u B . is meager in ph C. This proves

the category part of Theorem 4. We say that a trajectory in a billiard table C asymptotically terminates on

the boundary of C if for its vertices . . . . P- 1, Po, Pl . . . . . we have that

[pk-- Pk+l['-~O as k-~ +oo.


Clearly, if a trajectory terminates on the boundary of C it asymptotically terminates on the boundary. Let

D = {(p, v)ephC: the trajectory starting at p in direction v asymptotically terminates on bd C}.

The same proof that led to the measure-theoretic result of Theorem 4 but with D instead of B and the measure/~ replaced by the corresponding outer measure (note that it is not clear a priori that D is Borel) yields the following: For almost all elements (p, v) in the phase space of C the trajectory in C starting at p in direction v does not terminate asymptotically on the boundary of C.

4. DENSE TRAJECTORIES

In this section d = 2. Zemlyakov and Katok [37], Boldrighini et al. [6] and Kerckhoff et al. [21] specified (convex) 'polygonal' billiard tables P in which there are 'many' trajectories (all of them avoiding the vertices of P) which are dense in P. Moreover, Zemlyakov and Katok proved the following: There is a set of 'polygonal' billiard tables which is dense in cg such that for each such billiard table P there is a trajectory in P which approaches any point p of P and any direction v at p arbitrarily closely. Using this result we shall prove

THEOREM 5. For most billiard tables C in E 2 the trajectory (p, v), T(p, v) . . . . is dense in the phase space ph C of C for most (p, v) ~ ph C.

Given a dense trajectory in the phase space of a billiard table C, the corresponding trajectory on C is dense in C but not conversely.

Birkhoff [3] proved that in a plane billiard table there are always trajectories of period n for any n = 2, 3, . . . Birkhoff's theorem can easily be extended to d > 2. A recent result of Galperin [12] exhibits 'polygonal' billiard tables in which there are non-dense non-periodic trajectories. These results are complemented by the following

COROLLARY 1. For most billiard tables C in E 2 the sets of elements (p, v) in the phase space of C for which the trajectory in C starting at p in direction v is periodic resp. non-dense in C are both meager.

Corollary 1 is an immediate consequence of Theorem 5.

PROBLEM 1. Extend Theorem 5 to all dimensions d >>, 2.

Proof of Theorem 5. The proof consists of two steps. In the first step a preliminary version of Theorem 5 is proved which then is used to produce the desired result.

CONVEX B I L L I A R D S 221

Let C be a billiard table and let e > 0. A trajectory (p, v), T(p, v) . . . . in ph C is called 8-double-dense if there are integers 0 < l < m such that the sets {Ti(p, v): i = 0 . . . . . l} a n d {TJ(p, v), j = l + 1 . . . . , m} both are e-nets in clph C = {(q, w): q e bd C, w ~ S d- 1, (w, n(q)} 1> 0} c bd C × S d- 1 c [2d. (Here

'cr stands for closure.)

Our first aim is to show that

(1) for most billiard tables C there is an e-double-dense trajectory in

ph C for any e > O.

For n = 1, 2 . . . . . let

c~, = { C e ~ c~ c~1: there is no (1/n)-double-dense

trajectory in ph C}.


(2) ~ is closed in ~ c~ ~1,

Let us assume the contrary. Then there is a billiard table C with a (1/n)- double-dense trajectory (p, v), T(p, v), . . . , in ph C such that for suitable C1, C2, . . . e c~, we have that Ck ~ C. Since the trajectory (p, v), T(p, v) . . . . . is (1/n)-double-dense and cl ph C is compact, there is a ~ > 0 with the property that

(3) {T'(p, v), i = 0 . . . . . l}, {TJ(p, v),j = l + 1 , . . . , m} are

((l/n) - 28)-nets in cl ph C

for suitable integers 0 < l < m. Since Ck ~ C, Propositions (1), (3), (4) in Section 2 and Lemma 1 together imply that

(4) for all sufficiently large k for any (r, u)ec lphCk there is (q, w) E clph C such that I(r, u) - (q, w)[ <

and there are elements (qk, wk)E ph Ck, k = 1, 2 . . . . , such that

(5) for all sufficiently large k the inequality IT~k(qk, Wk)- T~(p, v)l < 8 holds for i = 0 . . . . . m.

(3), (4) and (5) combined yield that the trajectory (qk, Wk), Tck(qk, Wk) . . . . . in ph Ck is (1/n)-double-dense for all sufficiently large k contradicting Cke--~ ~. This proves (2).

We next prove that

(6) ~, has empty interior in c~ c~ c~1.

222 PETER M. GRUBER

If (6) does not hold, we can choose a billiard table C in the interior of ft.. By the result of Zemlyakov and Katok 1371 referred to in the introduction of this section, there is a 'polygonal' billiard table P with a trajectory as described and such that 6(C, P) is less than the distance (in the sense of the metric 6) between C and the boundary of ft . in ff c~ c~1. By smoothing P slightly at its vertices one may obtain a billiard table D with the following properties: In ph D there is a (1/n)-double-dense trajectory (an initial segment of which corresponds to an initial segment of the trajectory in P considered before); ~(C, D) is still less than the distance between C and the boundary of c~.. Hence D e ~ . , which presents a contradiction. This confirms (6).

By (2) and (6) the set u ~ . is meager. Since, on the other hand, uc~. consists precisely of those billiard tables C with no e-double-dense trajectory in ph C for some e > 0, Proposition (1) follows.

Considering (1) it is sufficient for the proof of Theorem 5 to establish the following proposition:

(7) Let C ~ c~ n ~1 contain an e-double-dense trajectory for any e > O. Then for most (q, w) ~ ph C the trajectory (q, w), T(q, w) . . . . is dense in ph C.

We start by proving the following:

(8) Let ~ ~ N c ph C be open. Then the set B(N) consisting of all (q, w)~ ph C for which the trajectory (q, w). T(q, w) . . . . . does not meet N is closed and has empty interior in ph C.

If B(N) were not closed there would exist a sequence (qk, Wk) E B(N), k = 1, 2, . . . . converging to (q, w)e(ph C)\ B(N), say. Since (q, w)¢B(N) there is an index i/> 1 such that T~(q, w) ~ N. Then T~(q~, wk) ~ N for all sufficiently large k by Lemma 1, contradicting (qk, Wk) ~ B(N). Thus B(N) is closed. Assume now that the interior of B(N) is non-empty. Now choose interior elements (q, w)~ B(N) and (r, u)~ N. Let e > 0 be so small that the e-neighborhoods of (q, w) and (r, u) are still contained in B(N) resp. N. By our assumption (see (7)) there is an e-double-dense trajectory, say (p, v), T(p, v) . . . . Choose integers 0 < l < m such that both {Ti(p, v):i = 0 . . . . . l} and {T~(p, v):j = l + 1 . . . . . m} are z-nets on clph C. Thus there are indices i~ {0 . . . . . l) a n d j ~ {l + 1 . . . . . m} such that IT~(p, v) - (q, w)l < e and ITS(p, v) - (r, u)l < e. This implies that Ti(p, v)e B(N) and that the trajectory Ti(p, v), Ti+ l(p, v) . . . . intersects N, which by definition of B(N) is impossible. Hence B(N) has empty interior and (8) is proved. Using (8), Proposition (7) can be proved in the following way: Let N., n = 1, 2 . . . . . be an open basis for the topology on ph C. Thus by (8), B(N.) is nowhere dense in ph C. Hence UB(N.) is meager. Since UB(N.) is the


set of all (q, w) ~ ph C for which the trajectory (q, w), T(q, w) . . . . is not dense in

ph C, this proves (7). Now Theorem 5 is an immediate consequence of Propositions (1) and (7). Let 1:2 be oriented counter-clockwise. An appealing result of Mather [27]

says the following: Let C be a plane billiard table of class c~2 the curvature of which vanishes at least at one point of bd C. Then for any e > 0 there is a trajectory in C which at some vertex p has direction v such that the angle between v and the positively oriented tangent unit vector t(p) of bd C at p is less than e and later on at some vertex q has direction w such that the angle between w and - t (q ) is again less than e. In Theorem 6 a related, slightly stronger property is used. For e > 0 the e-channel of a plane billiard table C is the space between bd C and the inner parallel body of C at distance e,

C~ = {xeC: {y~:2: Ix - Yl ~< e} c C}. It is clear what is meant when we say that a trajectory in C circles m times in the positive or negative direction in the e-channel. Using Theorems 3 and 5 we shall show

T H E O R E M 6. Most billiard tables C in F_ 2 have the following property: For

most elements (p, v) in the phase space of C the trajectory in C starting at p in

direction v circles, in a certain period of time, m times in the positive direction in

the e-channel of C and, at a later period of time, m times in the negative

direction for any e > 0 and any positive integer m.

Proof. The main step of the proof consists of showing the following:

(9) Let C ~ c~ c~ c~1 and suppose that no trajectory in C terminates on

bd C, let e > 0 and let m be a positive integer. Then there is a 3 > 0

such that any trajectory starting at a point p E bd C in direction v

where the angle between v and t(p) (resp. - t (p) ) is less than

circles m times in the positive (resp. negative) direction in the e-

channel of C.

By choosing a smaller e, if necessary, we may assume that a trajectory in the e- channel does not change from the positive to the negative direction or vice versa unless it leaves the e-channel. Suppose now that (9) is not true. Then there are elements (Pk, VR) ff ph C for which the angle ~k between vk and t(pk) converges to 0 and the trajectory on C starting at Pk in direction Vk penetrates into C~ at a point r k such that the length of the segment of the trajectory between Pk and rk is less than 2, say, where 2 is m times the perimeter of C. Let q~0 be the last vertex on the trajectory before rk. Consider the trajectory from qko back to Pk in the phase space representation:

(10) (qko, Wko), (qkl, WRI) = T(qkO, Wko) . . . . . (qkl~, Wklk)

= TZk(qko, W~o) = (Pk, W9 . . . .

224 P E T E R M. G R U B E R

where w k = -Vk + 2(Ok, n(pk))n(pk). Since the line segment [rk, qko] and Wko

make equal angles with n(qko) and rke Ca, qkO e bd C, there is an go/> 0 independent o fk such that (wko, n(qko)) >>- eo for all k. Since bd C is compact, we may assume by considering a subsequence and renumbering that qko ~ qo, Wk0 --* W0, say. Repeated application of Lemma 1 then implies that there are numbers e~, 82 . . . . > 0 such that

Further

(11)

(Wko, n(qko)) >>- %, (Wkl, n(qkl)) >1 ~1 . . . .

q~o -* qo, qkl -'* ql . . . . for suitable qo, ql . . . . ~ bd C.

Since (v k, n(pk)) = (Wkzk, n(qk~k)) = sin ~)k >i 81k and 3~ ~ 0, we have 81k --~ 0 and thus lk ~ +go. From (10) and from the choice of(p~, Vk), k = 1, 2 , . . . , and 2 it follows that for all k

Iqko- q~ll + Iqkl - - qk2] + " ' "

+ Iqu-1 - qul ~< 2 for each l ~< lk.

Taking into account that I k ~ +go and observing (11) we thus obtain that

Iqo - ql[ -I- Iql - q2l + " " ~< 2.

Hence the trajectory starting at qo in direction w 0 terminates on bd C. This contradicts our assumption and thus proves (9).

Having proved (9) the next step is simple:

(12) Let C ~ :~ :~ ~1 and suppose that no trajectory in C terminates on bd C and let (p, v)~ ph C be given such that (p, v), T(p, v) . . . . . is

dense in ph C. Then for any e > 0 and any positive integer m the following holds: The trajectory in C starting at p in direction v

circles, during a certain period of time, m times in the e-channel of

C in the positive direction and, at a later time, m times in the

negative direction.

Choose e, m and then ~ according to (9). Since (p, v), T(p, v) . . . . is dense in phC, there are 0 < i < j with the property that for ( q , w ) = T~(p,v), (r, u) = TJ(p, v) the angles between w and t(q), resp. between u and - t ( r ) , are less than ~). Then (9) yields (12).

Theorem 6 now follows from (12) and Theorems 3 and 5.

FINAL REMARK. In recent years the concept of a-porous sets has attracted interest in convexity, see, e.g., Zamfirescu [36]. A a-porous set is meager but the converse does not hold generally. If some property holds for all elements of a metric space with a a-porous set of exceptions, it holds for nearly all

elements.


PROBLEM 2. Determine the results of this article in which 'most' may be replaced by 'nearly all'.

A C K N O W L E D G E M E N T

For their many valuable hints I am obliged to Professors Philipp and Schnitzer and to Dr Buchta.

R E F E R E N C E S

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Author's address:

Peter M. Gruber, Abteilung fiir Analysis, Technische Universit~it Wien, Wiedner HauptstraBe 8-10/1142, A-1040 Vienna, Austria.

(Received, September 14, 1989; revised version, November 27, 1989)

convex billiards

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