Periodic trajectories in 3-dimensional convex billiards

Michael Farber^{∗} and Serge Tabachnikov

April 5, 2001

Abstract

We give a lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n − 1) such trajectories. Convex plane billiards were studied by G. Birkhoff, and the case of higher dimensional billiards is considered in our previous papers. We apply a topological approach based on the calculation of cohomology of certain configuration spaces of points on 2-sphere.

MSC: Primary 3Dxx; Secondary 58Exx

Keywords: Convex billiards, cohomology of configuration spaces, Morse-Lusternik-Schnirelman critical point theory

1 Introduction

Given a smooth strictly convex closed hypersurface X^{m} ⊂ R^{m+1}, consider the billiard system inside the convex body T (the billiard table), bounded by X. One views the billiard ball as a point which moves freely inside T with the elastic reflections in the boundary X = ∂T subject to the familiar law of geometric optics: the incoming and the outgoing trajectories lie in the same 2-plane with the normal at the impact point, and the angle of reflection equals the angle of incidence. Billiard trajectories are extrema of the length functional.

∗Partially supported by the US - Israel Binational Science Foundation; part of this work was done while M.F. was visiting Max-Planck Institute for Mathematics in Bonn

The problem of periodic billiard trajectories is related to the closed geodesics problem on Riemannian manifolds. Unlike the thoroughly studied latter problem (see [8]), the problem of estimating the number of periodic trajectories in convex billiards did not receive much attention. The first results are due to G. Birkhoff [2] who studied periodic billiard trajectories in convex plane billiards (when m = 1), known also as ”Birkhoff billiards”. The next step was made by I. Babenko [1] about 10 years ago who considered convex billiards in 3-space. This case is technically the hardest, and the paper [1] contained an error in cohomological computations (to quote R. Bott [3] on the closed geodesics problem: “For some reason, the large literature on this subject teems with mistakes”). The case of convex billiards in m-dimensional spaces with m ≥ 3 was studied in our recent paper [5]. A relevant result is as follows.

Theorem 1 ([5]) Let n ≥ 3 be an odd number and X^{m} ⊂ R^{m+1}, m ≥ 3, be a generic smooth strictly convex closed hypersurface. Then the number of distinct n-periodic billiard trajectories inside X is not less than (n − 1)m.

Note that the dihedral group D_{n} acts on n-periodic billiard trajectories in a natural way, and the above theorem, as well as the results below, estimates the number of distinct D_{n}-orbits of n-periodic trajectories.

The techniques of [5] do not apply to the case m = 2.

A weaker estimate (which however holds for all strictly convex billiard tables without any genericity assumptions) is obtained in [7]:

Theorem 2 ([7]) Let n ≥ 3 be an odd prime and X be a smooth strictly convex closed surface in 3-space. Then the number of distinct D_{n}-orbits of n-periodic billiard trajectories inside X is not less than (n + 1)/2.

The goal of the present paper is to show that the result of Theorem 1 holds for m = 2 as well (it does, for m = 1, as follows from Birkhoff’s work [2]). Our main result is as follows.

Theorem 3 Let n ≥ 3 be an odd number and let X be a generic smooth strictly convex closed surface in 3-space. Then the number of distinct D_{n}-orbits of nperiodic billiard trajectories inside X is not less than 2(n − 1).

The meaning of the genericity assumption is explained in the next section.

2 Cyclic configuration spaces

Let X be a topological space and n be a positive integer. Denote by G(X, n) the subspace of the Cartesian power X^{×n} = X × X × ⋯ × X consisting of all configurations (x_{1}, x_{2}, …, x_{n}) such that x_{i} ≠ x_{i+1} for i = 1, 2, …, n − 1 and x_{n} ≠ x_{1}. The space G(X, n) is called the cyclic configuration space, see [5].

The dihedral group D_{n} acts naturally on the cyclic configuration space G(X, n). One may think of D_{n} as the group of symmetries of a regular n-gon. Identifying D_{n} with the subgroup of permutations of the set of vertices 1, 2, …, n, one describes the action of D_{n} on G(X, n) as follows:

τ · (x_{1}, …, x_{n}) = (x_{τ(1)}, …, x_{τ(n)}) for τ ∈ D_{n}, (x_{1}, …, x_{n}) ∈ G(X, n).

The action of D_{n} is free if and only if n is prime.

The strategy of the proof of Theorem 3 (as well as two other theorems stated in the Introduction) is as follows. As was already mentioned, n-periodic billiard trajectories inside the billiard table T are critical points of the perimeter length function L_{X} on the space of n-gons, inscribed in X = ∂T. The space of inscribed n-gons is the cyclic configuration space G(X, n). The genericity assumption in Theorem 3 means that L_{X} is a Morse function of G(X, n). This function is invariant under the action of the dihedral group D_{n}. Although the space G(X, n) is not compact, one can show that the number of critical D_{n}-orbits of L_{X} is bounded below by the rank of the D_{n}-equivariant cohomology space of the cyclic configuration space – see [5], §4.

More precisely, we will use Proposition 4.5 from [5]. Since X is topologically the 2-sphere, our goal is to find the dimension of the graded vector space H_{D_{n}}^{∗}(G(S^{2}, n); Z_{2}). The answer is given in the next theorem which, therefore, implies Theorem 3.

Theorem 4 For odd n ≥ 3, the Poincaré polynomial of H_{D_{n}}^{∗}(G(S^{2}, n); Z_{2}) equals

(1 + t^{2}) · (1 + t + t^{2} + ⋯ + t^{n−2}),

and the sum of the respective Betti numbers is 2(n − 1).

This result complements the computation of the rings H_{D_{n}}^{∗}(G(S^{m}, n); Z_{2}) for m ≥ 3 and odd n in [5] and the rings H^{∗}(G(S^{m}, n); k) for various values of m and various fields k in [7]. However, the results in [7] do not cover the case m = 2 and k = Z_{2}. The calculations of [5], [6], [7] are based on a spectral sequence computing the cohomology algebra of the cyclic configuration space G(X, n) for an arbitrary manifold X, developed in [5]; this spectral sequence is similar to the Totaro spectral sequence [9] for the usual configuration spaces.

3 Cohomological computations

In this section we prove Theorem 4. We will use some results from our previous papers [5, 6, 7]. Recall that n is assumed to be odd.

Let T : G(S^{2}, n) → G(S^{2}, n) be the reflection with respect to the first point

T (x_{1}, x_{2}, …, x_{n}) = (x_{1}, x_{n}, x_{n−1}, …, x_{2}), T^{2} = Id.

Clearly T has no fixed configurations. Denote by G′ = G(S^{2}, n)/T the quotient manifold.

Let Z_{2} ⊂ D_{n} denote the cyclic subgroup of the dihedral group D_{n} generated by T. We want to prove that H_{D_{n}}^{∗}(G(S^{2}, n); Z_{2}) is isomorphic to H_{Z_{2}}^{∗}(G(S^{2}, n); Z_{2}) ≃ H^{∗}(G′; Z_{2}), and we will compute below the latter space. Consider the following commutative diagram

EZ_{2} ×_{Z_{2}} G(S^{2}, n) → ED_{n} ×_{D_{n}} G(S^{2}, n)
↓ G(S^{2}, n)  ↓ G(S^{2}, n)
BZ_{2}  →  BD_{n}

and the Serre spectral sequences with Z_{2} coefficients of the the fibrations represented by the vertical arrows (where EG → BG denotes the universal bundle of a group G). We obtain a homomorphism of spectral sequences which induces a homomorphism

H_{D_{n}}^{∗}(G(S^{2}, n); Z_{2}) → H_{Z_{2}}^{∗}(G(S^{2}, n); Z_{2}).  (3.1)

Lemma 5 The homomorphism (3.1) is an isomorphism.

Proof. On the level of E_{2}-terms, one has a homomorphism

E_{2}^{p,q} = H^{p}(D_{n}; H^{q}(G(S^{2}, n); Z_{2})) → E′_{2}^{p,q} = H^{p}(Z_{2}; H^{q}(G(S^{2}, n); Z_{2})). (3.2)

We claim that (3.2) is an isomorphism. The claim would follow from Proposition 10.4 of Chapter 3, [4] if one shows that the dihedral group D_{n} acts trivially on the vector space

H^{p}(Z_{2}; H^{q}(G(S^{2}, n); Z_{2})).

According to Theorem 4 and Remark 3.3 from [5], the cyclic group Z_{n} ⊂ D_{n} acts trivially on H^{q}(G(S^{2}, n); Z_{2}) and the dihedral group D_{n} acts trivially on H^{q}(G(S^{2}, n); Z_{2}) for q equal 0, 1, n − 1, n. Moreover, for 1 < q < n − 1 the cohomology space H^{q}(G(S^{2}, n); Z_{2}) is two-dimensional and D_{n} acts trivially on a one-dimensional subspace and on its quotient space. It follows, that for any value of q between 1 and n − 1 there exist two possibilities: either D_{n} acts trivially on M = H^{q}(G(S^{2}, n); Z_{2}), or M, viewed as a Z_{2}[Z_{2}]-module, is the unique nontrivial extension 0 → Z_{2} → M → Z_{2} → 0. One checks directly, using the definition of group cohomology, that in the second case H^{p}(Z_{2}; M) = 0 for all p > 0. In the second case H^{0}(Z_{2}; M) is isomorphic to the subspace of Z_{2}-invariants in M, which clearly coincides with the subspace of D_{n}-invariants in M. Hence (3.2) is an isomorphism for all p and q.

The comparison theorem for spectral sequences implies an isomorphism

H_{D_{n}}^{∗}(G(S^{2}, n); Z_{2}) → H_{Z_{2}}^{∗}(G(S^{2}, n); Z_{2}),

as claimed. □

Now we will study the space H^{∗}(G′; Z_{2}). Fix a point A ∈ S^{2}. Along with the cyclic configuration space G(S^{2}, n), consider its subspace G_{A} consisting of the configurations (x_{1}, x_{2}, …, x_{n}) ∈ G(S^{2}, n) with x_{1} = A. Clearly, G_{A} is T -invariant; set G′_{A} = G_{A}/T.

The standard action of SO(3) on S^{2} by rotations induces an action of SO(3) on the cyclic configuration space G(S^{2}, n)

SO(3) × G(S^{2}, n) → G(S^{2}, n).

The restriction of this map to G_{A} gives a continuous map

p : SO(3) × G_{A} → G(S^{2}, n).  (3.3)

and we claim that it is a fibration. To prove this we will show that (3.3) is induced from the standard fibration

q : SO(3) → S^{2}, where q(R) = R(A), R ∈ SO(3),  (3.4)

by the map f : G(S^{2}, n) → S^{2} given by f(x_{1}, …, x_{n}) = x_{1}. The total space E of the induced fibration is the space of all pairs (R, (x_{1}, …, x_{n})) with R(A) = x_{1}, where R ∈ SO(3) and (x_{1}, …, x_{n}) ∈ G(S^{2}, n). The map (R, (x_{1}, …, x_{n})) ↦ (R, (A, R^{−1}(x_{2}), …, R^{−1}(x_{n})) is a homeomorphism E → SO(3) × G_{A}. This proves that (3.3) is the induced fibration.

Let c = (A, x_{2}, …, x_{n}) ∈ G_{A} be a fixed configuration. If a rotation R ∈ SO(3) preserves c then R is identical (if n is even the configuration (A, −A, A, −A, …) is invariant under rotations about the axis (A, −A)). The fiber p^{−1}(c) is identified with the set of pairs (R_{ϕ}, R_{−ϕ}c), where R_{ϕ} ∈ SO(3) denotes the rotation through angle ϕ about A. Hence the fiber of (3.3) is the circle S^{1}. Factorizing by T, one obtains the fibration

p′ : SO(3) × G′_{A} → G′  (3.5)

with fiber S^{1}. As above, fibration (3.5) is induced from (3.4) by the map G′ → S^{2} sending orbit of any configuration (x_{1}, x_{2}, …, x_{n}) ∈ G(S^{2}, n) to x_{1} ∈ S^{2}.

Consider the Serre spectral sequence of (3.5) with Z_{2} coefficients. The cohomology of the total space

H^{∗}(SO(3) × G′_{A}; Z_{2}) ≃ H^{∗}(SO(3); Z_{2}) ⊗ H^{∗}(G′_{A}; Z_{2})

is as follows.

It is well known that H^{∗}(SO(3); Z_{2}) = Z_{2}[v]/(v^{4}) is the truncated polynomial algebra with a single 1-dimensional generator v satisfying the relation v^{4} = 0. The cohomology algebra H^{∗}(G′_{A}; Z) with integral coefficients was described in Theorem 12 of [7]. It has generators

δ_{i} ∈ H^{4i}(G′_{A}; Z) ≃ Z, where i = 1, 2, …,

and two other generators

a ∈ H^{1}(G′_{A}; Z) ≃ Z, and b ∈ H^{3}(G′_{A}; Z) ≃ Z_{2},

which satisfy the following defining relations:

δ_{i}δ_{j} = ( \begin{matrix}2i + 2j \\ 2i \end{matrix} ) · δ_{i+j},  δ_{[(n+1)/4]} = 0,
b^{2} = 0  2b = 0,
a^{2} = 0,  ab = 0,
δ_{k}b = 0 (only for n = 4k + 3).

An additive basis of H^{∗}(G′_{A}; Z_{2}) is given by mod 2 reductions of the classes δ_{i} of degree 4i and classes δ_{i}a of degree 4i + 1; moreover, according to the universal coefficient formula, each integral class δ_{i}b of order 2 produces two classes in Z_{2}- cohomology, one of degree 4i + 2 and one of degree 4i + 3.

We claim that the Poincarè polynomial of H^{∗}(G′_{A}; Z_{2}) equals

1 + t + t^{2} + ⋯ + t^{n−2}.  (3.6)

To prove this, consider two cases: n = 4k + 3 and n = 4k + 1. In the first case the mod 2 reductions of the generators δ_{i} and δ_{i}a produce the Poincaré polynomial

(1 + t)(1 + t^{4} + t^{8} + ⋯ + t^{4k}),

and the classes δ_{i}b contribute

(t^{2} + t^{3})(1 + t^{4} + t^{8} + ⋯ + t^{4k−4})  (3.7)

to the Poincaré polynomial. Summing these two polynomials gives (3.6). In the second case, when n = 4k + 1, the contribution of the generators δ_{i} and δ_{i}a is

(1 + t)(1 + t^{4} + ⋯ + t^{4k−4});

the contributions of the classes δ_{i}b is (3.7), and the total is again (3.6).

Next, consider the cohomological spectral sequence of the fibration (3.5). The term E_{2} consists of two rows, and the only possibly non-trivial differential is d_{2}. We claim that this differential vanishes. Assuming this claim, the Poincarè polynomial of the base equals the Poincarè polynomial of the total space divided by the Poincarè polynomial of the fiber. We obtain the Poincarè polynomial of H^{∗}(G′; Z_{2}):

\frac{(1 + t + t^{2} + t^{3}) · (1 + t + t^{2} + ⋯ + t^{n−2})}{1 + t} =

= (1 + t^{2}) · (1 + t + t^{2} + ⋯ + t^{n−2})

as stated in Theorem 4.

It remains to prove that d_{2} = 0. Let s ∈ E_{2}^{0,1} = H^{1}(S^{1}; Z_{2}) be the generator. It suffices to show that d_{2}(s) = 0. The class s is transgressive, and d_{2}(s) is the mod 2 reduction of the Euler class of fibration (3.5). Fibration (3.5) is induced from fibration (3.4), cf. above. The latter is the unit tangent bundle of the 2sphere. The Euler number of S^{2} equals 2, and the Euler class of (3.4) vanishes in H^{2}(S^{2}; Z_{2}). By functoriality of characteristic classes, the mod 2 reduction of the Euler class of (3.5) is trivial.

This completes the proof of Theorem 4.

References

[1] I. Babenko, Periodic trajectories in three-dimensional Birkhoff billiards, Math. USSR Sbornik, 71 (1992), 1–13.

[2] G. Birkhoff, Dynamical systems. Amer. Math. Soc. Coll. Publ., 9, 1927.

[3] R. Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc., 7 (1982), 331–358.

[4] K.S. Brown, Cohomology of groups, Springer, 1982

[5] M. Farber and S. Tabachnikov, Topology of cyclic configuration spaces and periodic trajectories of multidimensional billiards, ArXiv preprint math.DG/9911226, to appear in ”Topology”

[6] M. Farber, Topology of billiard problems, I, ArXiv preprint math.DG/0006049

[7] M. Farber, Topology of billiard problems, II, ArXiv preprint math.AT/0006085

[8] W. Klingenberg, Lectures on closed geodesics, Springer-Verlag, 1978.

[9] B. Totaro, Configuration spaces of algebraic varieties, Topology, 35 (1996), 1057–1067.

Michael Farber, Department of Mathematics, Tel Aviv University, Ramat Aviv 69978, Israel farber@math.tau.ac.il

Serge Tabachnikov Department of Mathematics, Penn State University, University Park, PA 16802, USA tabachni@math.psu.edu