February 21, 1998

Cylindric Billiards and Transitive Lie Group Actions

Nándor Simányi^{1} Bolyai Institute, University of Szeged,

6720 Szeged, Aradi Vértanuk tere 1, Hungary. E-mail: simanyi@math.u-szeged.hu

Domokos Szász^{1}

Mathematical Institute of the Hungarian Academy of Sciences

H-1364, Budapest, P. O. B. 127, Hungary E-mail: szasz@math-inst.hu

Abstract. A conjecture is formulated and discussed which provides necessary and sufficient condition for the ergodicity of cylindric billiards (this conjecture improves a previous one of the second author). This condition requires that the action of a Lie-subgroup G of the orthogonal group O(d) (d being the dimension of the billiard in question) be transitive on the unit sphere S^{d−1}. If C_{1}, …, C_{k} are the cylindric scatterers of the billiard, then G is generated by the embedded Lie-subgroups G_{i} of O(d), where G_{i} consists of all orthogonal transformations of ℝ^{d} that leave the points of the generator subspace of C_{i} fixed (1 ≤ i ≤ k). In this paper we can prove the necessity of our conjecture and we also formulate some notions related to transitivity. For hard ball systems, we can also show that the transitivity holds in general: for arbitrary number N ≥ 2 of balls, arbitrary masses m_{1}, …, m_{N} and in arbitrary dimension ν ≥ 2. This result implies that our conjecture is stronger than the Boltzmann-Sinai ergodic hypothesis for hard ball systems. As a by-remark, we can give a somewhat surprising characterization of the positive subspace of the second fundamental form for the evolution of special orthogonal manifold (wavefront), namely for the parallel beam of light. Thus we obtain a new characterization of sufficiency of an orbit segment.

1Research supported by the Hungarian National Foundation for Scientific Research, grants OTKA-7275 and OTKA-16425.

1. Introduction.

Semi-dispersive billiards is a class of billiards evidently without any ellipticity, actually with more or less manifest hyperbolicity; their study was initiated by Chernov and Sinai in 1987 [S-Ch(1987)]. Cylindric billiards, a much interesting subfamily of semi-dispersive billiards were introduced in 1992 by the second author, [Sz(1993)]. Cylindric billiards are interesting for,

on one hand, they contain hard ball systems, a fundamental model from the aspects of statistical physics and a much beautiful one, we believe, from the point of view of mathematics; and

on the other hand, this is apparently the widest subclass of semi-dispersive billiards where the search for transparent necessary and sufficient conditions of ergodicity is promising.

In words, cylindric billiards are toric billiards where the scatterers are cylinders. In our discussion, the bases of the cylinders will be assumed to be stricly convex, a property ensuring that the scatterers be convex, and thus the arising billiard be semi-dispersive. Because of the simplicity of our model, let us immediately start with a formal definition.

The configuration space of a cylindric billiard is Q = 𝕋^{d} \ (C_{1} ∪ ⋯ ∪ C_{k}), where 𝕋^{d} = ℝ^{d}/ℤ^{d} (d ≥ 2) is the unit torus. Here the cylindric scatterer C_{i} (i = 1, …, k) is defined as follows:

Let A_{i} ⊂ ℝ^{d} be a so called lattice subspace of the Euclidean space ℝ^{d}, which means that the discrete intersection A_{i} ∩ ℤ^{d} has rank dim A_{i}. In this case the factor A_{i}/(A_{i} ∩ ℤ^{d}) naturally defines a subtorus of 𝕋^{d}, which will be taken as the generator of the cylinder C_{i} ⊂ 𝕋^{d}. Denote by L_{i} = A_{i}^{⊥} the orthocomplement of A_{i}. Under the above conditions the subspace L_{i} must also be a lattice subspace. Throughout this article we will always assume that dim L_{i} ≥ 2. Let, moreover, B_{i} ⊂ L_{i} be a convex, compact domain with a C^{2}-smooth boundary ∂B_{i} containing the origin as an interior point. We will assume that B_{i} is strictly convex in the sense that the second fundamental form of its boundary ∂B_{i} is everywhere positive definite. Furthermore, in order to avoid unnecessary complications, we assume that the convex domain B_{i} does not contain any pair of points congruent modulo ℤ^{d}. The domain B_{i} will be taken as the base of the cylinder C_{i}. Finally, suppose that a translation vector t_{i} ∈ ℝ^{d} is given, playing an essential role in positioning the cylinder C_{i} in the ambient torus 𝕋^{d}. Set

C_{i} = {a + l + t_{i} | a ∈ A_{i}, l ∈ L_{i}, } /ℤ^{d}.

In order to avoid further unnecessary complications, we also assume that the interior of the configuration space Q = 𝕋^{d} \ (C_{1} ∪ ⋯ ∪ C_{k}) is connected.

The phase space M of our billiard will be the unit tangent bundle of Q, i. e. M = Q × 𝕊^{d−1}. (Here, as usual, 𝕊^{d−1} is the d − 1-dimensional unit sphere.)

The dynamical system (M, S^{ℝ}, μ), where S^{ℝ} is the dynamics defined by uniform motion inside the domain and specular reflections at its boundary (the scatterers!) and μ is the Liouville measure, is a cylindric billiard we want to study. (As to notions and notations in connection with semi-dispersive billiards the reader is recommended to consult the work [K-S-Sz(1990)].)

In 1994, the first general result: necessary and sufficient conditions of ergodicity, was obtained in [Sz(1994)] for the class of the so-called orthogonal cylindric billiards, where the generator of each cylinder is parallel to some coordinate subspace of the orthogonal system of coordinates in which the underlying torus is given. Then, in [S-Sz(1994)], sufficient conditions were found for the ergodicity, in fact, for a hyperbolic one, of a billiard with two non-orthogonal cylindric scatterers. The method of the proof already revealed the intimate role which the transitivity on 𝕊^{d−1} of the action of a Lie-subgroup G of the orthogonal group O(d) would play in ensuring the hyperbolicity, and consequently the ergodicity, too, of the cylindric billiard in question. To be more precise, for any cylinder C_{i} (i = 1, …, k) consider the group G_{i} consisting of all orthogonal transformations U of ℝ^{d} that leave the points of the subspace A_{i} fixed. Then we consider the subgroup G of O(d) algebraically generated by all such groups G_{i} : 1 ≤ i ≤ k, which is certainly an embedded Lie subgroup of O(d).

Our aim in this work is to formulate precise statements about the aforementioned transitivity of the action of G. Having collected some necessary notions in section 2, the general results are formulated in section 3. According to our Conjecture 1, transitivity of G is equivalent to the hyperbolic ergodicity of the flow.

Conjecture 1. For every cylindric billiard flow (M, {S^{t}}, μ) the transitivity of the action of G on the velocity sphere 𝕊^{d−1} is equivalent to the hyperbolic ergodicity (or, equivalently, to the hyperbolicity and Bernoulli property) of the billiard map.

The main result of section 3 is

Theorem 3.6. The irreducibility of the G-action on ℝ^{d} implies the transitivity on 𝕊^{d−1}.

This theorem has the following important corollary:

Corollary A. The transitivity of the G-action on 𝕊^{d−1} is a necessary condition for the ergodicity of the cylindric billiard.

Though this corollary provides the necessity of the transitivity property, it seems to be extremely hard to establish the sufficiency of this condition. Indeed, for the “simple” subclass of cylindric billiards: hard ball systems, this is not known, in general. In section 4 of the present work, we will be able to show that for hard ball systems, transitivity of G holds for any number N of particles, in any dimension ν, and for arbitrary masses m_{1}, …, m_{N} and radii r of the particles. As a contrast, we note that in our recent, quite involved paper [S-Sz(1998)], we were only able to prove hyperbolicity — and still not ergodicity — for masses m_{1}, …, m_{N} and radii r of the particles not belonging to a countable union of exceptional analytic submanifolds of parameters. In the light of this last result, the merit of the transitivity statement of section 4 mentioned above is that it holds for arbitrary geometric parameters of the system — without any exceptional set. Motivated by the importance of the concept of transitivity, in section 3, we also introduce related notions: the tightness and the orthogonal non-splitting property of G, the first of them being stronger than, while the second one equivalent to transitivity. Furthermore, Proposition 3.18 of Section 3 gives a new, we think beatiful and quite surprising characterization of sufficiency.

2. Prerequisites

As to the basic notions about semi-dispersive billiards we refer to the paper [K-S-Sz (1990)]. For convenience and brevity, we will throughout use the concepts and notations of Sections 2 and 3 of that paper. Here we only summarize some further notions from [K-S-Sz (1991)] , [K-S-Sz (1992)] , and [Sim(1992)]. These are either new or their exposition is simpler than that given in the original work.

An often used abbreviation is the shorthand S^{[a,b]}x for the trajectory segment {S^{t}x : a ≤ t ≤ b}. The natural projections from M onto its factor spaces are denoted, as usual, by π : M → Q and p : M → S^{d−1} or, sometimes, we simply write π(x) = Q(x) = Q and p(x) = V (x) = V for x = (Q, V) ∈ M. Any t ∈ [a, b] with S^{t}x ∈ ∂M is called a collision moment or collision time. Denote ∂Q = ∪_{i=1}^{k}∂C_{i}, where ∂C_{i} are the smooth components of the boundary. Since we want to consider typical situations, we will always assume that at every point q ∈ ∂Q of the boundary of the configuration space Q the spherical angle subtended by the compact domain Q is strictly positive.

As pointed out in previous works on billiards, the dynamics can only be defined for trajectories where the moments of collisions do not accumulate in any finite time interval (cf. Condition 2.1 of [K-S-Sz(1990)]). An important consequence of Theorem 5.3 of [V(1979)] is that — for semi-dispersive billiards we are considering — there are no trajectories at all with a finite accumulation point of collision moments.

As a result, for an arbitrary non-singular orbit segment S^{[a,b]}x of the cylindric billiard flow, there is a uniquely defined sequence a ≤ τ_{1} < τ_{2} < ⋯ < τ_{m} ≤ b (m ≥ 0) of collision times and a uniquely defined sequence σ_{1}, σ_{2}, …, σ_{m} (1 ≤ σ_{i} ≤ k) of labels of cylinders with the properties that

(i) for every t ∈ [a, b], S^{t}x ∈ ∂M if and only if t = τ_{i} with some i = 1, …, m, and

(ii) π (S^{τ_{i}}x) ∈ ∂C_{σ_{i}}, i = 1, …, m.

The sequence Σ := Σ(S^{[a,b]}x) := (σ_{1}, σ_{2}, …, σ_{m}) is called the symbolic collision sequence of the trajectory segment S^{[a,b]}x.

As well known, billiards are dynamical systems with singularities. A collision at a point x ∈ ∂M such that in π(x), at least two smooth pieces of ∂Q meet, is called a multiple collision. A collision is called tangential if x ∈ ∂M and p(x) ∈ T_{π(x)}∂Q, i. e. p(x) is tangential to ∂Q at the point of reflection. We shall use the collection SR^{+} of all singular reflections:

Definition 2.1. The set SR^{+} is the collection of all phase points x ∈ ∂M for which the reflection, occurring at x, is singular (tangential or multiple) and, in the case of a multiple collision, x is supplied with the outgoing velocity V^{+}.

It is not hard to see that SR^{+} is a compact cell-complex in M and dim(SR^{+}) = dim M − 2 = 2d − 3.

Neutral Subspaces and Sufficiency

Consider a non–singular trajectory segment S^{[a,b]}x. Suppose that a and b are not moments of collision. Before defining the neutral linear space of this trajectory segment, we note that the tangent space of the configuration space Q at interior points can be identified with the common linear space ℝ^{d}.

Definition 2.1. The neutral space N_{0}(S^{[a,b]}x) of the trajectory segment S^{[a,b]}x at time zero (a < 0 < b) is defined by the following formula:

N_{0}(S^{[a,b]}x) = {W ∈ ℝ^{d} : ∃(δ > 0) s. t. ∀α ∈ (−δ, δ)
p (S^{a} (Q(x) + αW, V (x))) = p(S^{a}x) &
p (S^{b} (Q(x) + αW, V (x))) = p(S^{b}x)}.

It is known (see (3) in Section 3 of [S-Ch (1987)]) that N_{0}(S^{[a,b]}x) is a linear subspace of ℝ^{d} indeed, and V (x) ∈ N_{0}(S^{[a,b]}x). The neutral space N_{t}(S^{[a,b]}x) of the segment S^{[a,b]}x at time t ∈ [a, b] is defined as follows:

(2.2)  N_{t}(S^{[a,b]}x) = N_{0} (S^{[a−t,b−t]}(S^{t}x)) .

It is clear that the neutral space N_{t}(S^{[a,b]}x) can be identified canonically with N_{0}(S^{[a,b]}x) by the usual identification of the tangent spaces of Q along the trajectory S^{(−∞,∞)}x (see, for instance, Section 2 of [K-S-Sz (1990)] ). As we will see in Section 3, the neutral subspace is the orthocomplement of the positive subspace of the second fundamental form of the image of a parallel beam of light, see the proof of Proposition 3.18 later.

It is now time to bring up the basic notion of sufficiency of a trajectory (segment). This is the utmost important necessary condition for the proof of the Theorem on Local Ergodicity for Semi–Dispersing billiards, see Condition (ii) of Theorem 3.6 and Definition 2.12 in [K-S-Sz (1990)] .

Definition 2.3.

(1) The non–singular trajectory segment S^{[a,b]}x (a and b are supposed not to be moments of collision) is said to be sufficient if and only if the dimension of N_{t}(S^{[a,b]}x) (t ∈ [a, b]) is minimal, i.e. dim N_{t}(S^{[a,b]}x) = 1.

(2) The trajectory segment S^{[a,b]}x containing exactly one singularity is said to be sufficient if and only if both branches of this trajectory segment are sufficient.

For the notion of trajectory branches see, for example, the end of Section 2 in [Sim(1992)-I].

Definition 2.4. The phase point x ∈ M with at most one singularity is said to be sufficient if and only if its whole trajectory S^{(−∞,∞)}x is sufficient which means, by definition, that some of its bounded segments S^{[a,b]}x is sufficient.

In the case of an orbit S^{(−∞,∞)}x with exactly one singularity, sufficiency requires that both branches of S^{(−∞,∞)}x be sufficient.

3. Characterization of the Action of G

As has been said in the introduction, we consider cylindric billiards with the configuration space Q = 𝕋^{d} \(C_{1} ∪ ⋯ ∪ C_{k}), where 𝕋^{d} = ℝ^{d}/ℤ^{d} (d ≥ 2) is the unit torus, the generator space of the cylinder C_{i} is A_{i} ⊂ ℝ^{d}, L_{i} = A_{i}^{⊥}, and dim L_{i} ≥ 2. Recall that the linear subspace A_{i} ⊂ ℝ^{d} is a lattice subspace of ℝ^{d}, so that it defines a subtorus T_{i} = A_{i}/ℤ^{d} of 𝕋^{d}.

For a cylinder C_{i} (i = 1, …, k) consider the group G_{i} consisting of all orthogonal transformations U of ℝ^{d} that leave the points of the subspace A_{i} fixed. Since the initial studies of the close relationship between the ergodicity of cylindric billiards and the rotation groups determined by the generator spaces (cf. Lemma 4.4 and Sublemma 4.5 in [S-Sz (1994)] and the role of these lemmas in the proof of Main Lemma 4.1 there) it has become apparent that the transitivity of the action of the algebraic generate of these G_{i}’s on the sphere 𝕊^{d−1} is vital for proving ergodicity. We investigate the subgroup G of O(d) algebraically generated by all such groups G_{i}, which is certainly an embedded Lie subgroup of O(d). Observe that every orbit GV (V ∈ 𝕊^{d−1}) of the action of G on the unit sphere 𝕊^{d−1} is certainly an embedded, smooth submanifold of 𝕊^{d−1}, for it is naturally diffeomorphic to the homogeneous space that can be obtained by taking G modulo the isotropy subgroup of V.

Being aware of our general, three-step strategy of proving ergodicity for semidispersive billiards, it is clear that without the mentioned transitivity there would not be a great chance to prove the hoped ergodicity for such billiards. Later on, in Theorem 3.6, we will see that without the transitivity of the G-action actually there must exist some very simple first integrals of the flow, thus hindering the ergodicity. At this point, however, it is already easy to see that without that transitivity the flow (M, {S^{t}}, μ) simply can not be hyperbolic. Indeed, hyperbolicity just means that for μ−almost every point x ∈ M the relevant Lyapunov exponents of the system are nonzero. On the other hand, if for an x ∈ M, the relevant exponents do not vanish, then the point is necessarily sufficient. By the following lemma, however, the existence of just one sufficient point already implies the transitivity of the action of G.

Denote by δ the maximum dimension of the orbits of the action of G on the unit sphere 𝕊^{d−1}.

Lemma 3.1. The action of G on 𝕊^{d−1} is transitive if and only if δ = d − 1, i. e. there is an open orbit.

Proof. Transitivity obviously implies δ = d − 1.

Suppose that the orbit GV of some V ∈ 𝕊^{d−1} is an open subset of the unit sphere 𝕊^{d−1}. Then there exists an ϵ_{0} > 0 such that B(V; ϵ_{0}) ⊂ GV. Since G ⊂ O(d), we have that B(gV; ϵ_{0}) ⊂ GV for all g ∈ G. Using the connectedness of 𝕊^{d−1}, this is only possible if GV = 𝕊^{d−1}. □

First we put forward a simple observation regarding the (possibly existing) dense orbits of the G-action.

Lemma 3.2. Suppose that the orbit GV ⊂ 𝕊^{d−1} is dense for some V ∈ 𝕊^{d−1}. Then every orbit of the G-action is also dense.

Proof. Denote by \overline{G} ⊂ O(d) the closure of the group G in the orthogonal group O(d). It is obvious that the closure \overline{GV} of the orbit GV is precisely the orbit \overline{G}V of the \overline{G}-action on 𝕊^{d−1}. Thus, \overline{GV} = 𝕊^{d−1} means that the \overline{G}-action is transitive, hence \overline{G}V′ = \overline{GV′} = 𝕊^{d−1} for every V′ ∈ 𝕊^{d−1}. □

Let us observe that the existence of a single dense orbit implies the irreducibility of the G-action on the space ℝ^{d}, for every orbit GV′ (V′ ∈ 𝕊^{d−1}) is then dense in 𝕊^{d−1}. Thus, irreducibility of the G-action is a necessary condition for the transitivity.

Now we present another necessary condition for transitivity:

Definition of Orthogonal Non-Splitting Property, ONSP. We say that the system of subspaces L_{1}, …, L_{k} has the Orthogonal Non-Splitting Property (ONSP) iff there is no orthogonal splitting ℝ^{d} = B_{1} ⊕ B_{2} with dim B_{i} > 0 and with the property that for every i = 1, …, k either L_{i} ⊂ B_{1} or L_{i} ⊂ B_{2}.

Lemma 3.3. The irreducibility of the G-action on ℝ^{d} implies the ONSP.

Proof. If ONSP fails to hold, then the subspaces B_{i} in the splitting described above (with the property that for every i = 1, …, k either L_{i} ⊂ B_{1} or L_{i} ⊂ B_{2}) are G-invariant.

We note that in this case, in addition to the total kinetic energy \frac{1}{2}∥V ∥^{2}, the cylindric billiard clearly has further first integrals (invariant quantities), namely, the kinetic energies corresponding to the invariant norm square ∥P_{B_{i}}(V)∥^{2} (i = 1, 2) of the orthogonal projection of the velocity V on the subspace B_{i}. □

Lemma 3.4. The irreducibility of the action of G on ℝ^{d} is equivalent to the ONSP.

Proof. We have seen that the irreducibility implies the ONSP. Suppose now the reducibility. i. e. the existence of an orthogonal splitting ℝ^{d} = B_{1} ⊕ B_{2}, dim B_{i} > 0, and B_{i} is G-invariant. We show that ONSP is violated by the same splitting ℝ^{d} = B_{1} ⊕ B_{2}. Consider and fix an arbitrary index i, 1 ≤ i ≤ k.

Sublemma. If the linear subspace B ⊂ ℝ^{d} is G_{i}-invariant, then B has the orthogonal direct sum splitting B = B^{∗} ⊕ B^{∗∗}, where B^{∗∗} ⊂ A_{i} and B^{∗} = 0 or B^{∗} = L_{i}. (Recall that G_{i} ⊂ G consists of all orthogonal transformations U ∈ O(d) that leave all vectors of A_{i} fixed.)

Proof. If B is a subspace of A_{i} = L_{i}^{⊥}, then we are done: B^{∗∗} = B, B^{∗} = 0. Assume now that the orthogonal projection P_{L_{i}}(B) is nonzero. Select a vector b ∈ B with P_{L_{i}}(b) ≠ 0. Since the tangent space T_{b} (G_{i}b) is equal to the orthocomplement L_{i} ⊖ P_{L_{i}}(b) = H_{i}, by the invariance of B we have that H_{i} ⊂ B. Since G_{i} acts transitively on the unit sphere of L_{i}, it follows that L_{i} ⊂ B. Take B^{∗} = L_{i}, B^{∗∗} = B ⊖ L_{i}. □

Consider now the splittings B_{1} = B_{1}^{∗} ⊕ B_{1}^{∗∗}, B_{2} = B_{2}^{∗} ⊕ B_{2}^{∗∗}. Then we have that either B_{1}^{∗} = L_{i} or B_{2}^{∗} = L_{i}, for B_{1} ⊕ B_{2} ⊄ A_{i}. This shows that either L_{i} ⊂ B_{1} or L_{i} ⊂ B_{2}, i = 1, …, k. □

Remark 3.5. Each of the above properties obviously implies that ⋂_{i=1}^{k} A_{i} = 0.

It is now a bit surprising fact that the irreducibility of the G-action (or, equivalently, the ONSP) in turn implies the transitivity of the action on 𝕊^{d−1}, thus giving us an easily checkable necessary and sufficient condition for transitivity. The equivalence of transitivity and irreducibility is, indeed, unexpected to some extent because of the following fact:

Consider the case when the dimension of every space L_{i} is one. (Of course, this situation is not in the scope of our study.) Then the action of the group generated by the reflections across the hyperplanes A_{i} = L_{i}^{⊥} (i = 1, …, k) can very well produce dense (but, of course, countable) orbits on the sphere 𝕊^{d−1}, thus making the representation irreducible without transitivity on 𝕊^{d−1}. This somehow shows the essential difference between the cases dim L_{i} > 1 and dim L_{i} = 1.

Theorem 3.6. The irreducibility of the G-action on ℝ^{d} implies the transitivity on 𝕊^{d−1}, i. e. these properties are actually equivalent.

Corollary A. The transitivity of the G-action on 𝕊^{d−1} is a necessary condition for the ergodicity of the cylindric billiard.

Proof. Without transitivity there must exist a non-trivial orthogonal splitting

ℝ^{d} = B_{1} ⊕ B_{2}, and the partial kinetic energies \frac{1}{2} ∥P_{B_{i}}(V)∥^{2} are first integrals of the motion. □

Corollary B. The Orbit Structure. The natural representation of the group G in ℝ^{d} is always completely reducible, i. e. the space ℝ^{d} splits into the orthogonal direct sum ℝ^{d} = ⨁_{j=1}^{r} B_{j}, where each subspace B_{j} is G-invariant, and the restriction of the representation to B_{j} is irreducible. Note that the above splitting is also unique, as long as the subspaces L_{i} span the whole space ℝ^{d}. Two subspaces L_{i} and L_{j} belong to the same component B_{s} (s = 1, …, r) if and only if these subspaces can be connected by a finite chain of spaces L_{i(1)}, L_{i(2)}, …, L_{i(p)} so that the consecutive elements in this chain are not orthogonal to each other. It follows from Theorem 3.6 that every orbit of the G-action on 𝕊^{d−1} has the following form:

{V ∈ 𝕊^{d−1}| ∥P_{B_{j}}(V)∥^{2} = 2E_{j}, j = 1, …, r} .

Here the quantity E_{j} is the flow invariant partial kinetic energy in the direction of the subspace B_{j}, j = 1, …, r. The orbit is uniquely determined by the vector (E_{1}, E_{2}, …, E_{r}) fulfilling E_{j} ≥ 0 and ∑_{j=1}^{r} E_{j} = 1/2. Thus every orbit is compact, for it is the product of spheres. The typical orbit is the topological product of (ν_{j}−1)- dimensional spheres, j = 1, …, r, where ν_{j} = dim B_{j}.

Proof of Theorem 3.6. The proof will be split into several lemmas, some of them being just very simple observations.

Lemma 3.7. Let us introduce the following equivalence relation ∼_{i} (i = 1, …, k) between vectors of ℝ^{d}:

(3.8)

V_{1} ∼_{i} V_{2} if and only if
V_{1} − P_{i}V_{1} = V_{2} − P_{i}V_{2} and ∥P_{i}V_{1}∥ = ∥P_{i}V_{2}∥.

(Recall that P_{i} denotes the orthogonal projection onto L_{i}.) Suppose that some subgroups (not necessarily Lie subgroups) H_{i} ⊂ G_{i} = O(L_{i}) are given with the property that H_{i} acts transitively on the unit sphere of L_{i}, i = 1, …, k.

Then the algebraic generate H = ⟨H_{1}, …, H_{k}⟩ of the groups H_{i} acts transitively on 𝕊^{d−1} if and only if the transitive hull of the equivalence relations ∼_{i} on 𝕊^{d−1} (i = 1, …, k) is the trivial equivalence relation on 𝕊^{d−1} making every pair of velocities equivalent.

Proof. The assertion of the lemma immediately follows from the definitions. □

Based on the previous lemma, it will be convenient to say that the system of subspaces {L_{1}, …, L_{k}} of ℝ^{d} (dim L_{i} ≥ 2, i = 1, …, k) is transitive if and only if the transitive hull of the relations ∼_{i} (i = 1, …, k) associated with the subspaces L_{1}, …, L_{k} is the trivial relation 𝕊 × 𝕊 on the unit sphere 𝕊 of the linear span L_{1} + L_{2} + ⋯ + L_{k} of these subspaces.

Lemma 3.9. If the systems of linear subspaces {L_{1}, L_{2}, …, L_{p}} and

{L_{1} + L_{2} + ⋯ + L_{p}, L_{p+1}, …, L_{k}}

are both transitive (1 ≤ p ≤ k), then the system {L_{1}, …, L_{k}} is transitive, as well.

Proof. The lemma is an immediate consequence of Lemma 3.7. □

The next proposition deals with the transitivity of two subspaces {L_{1}, L_{2}} of ℝ^{d}.

Proposition 3.10. The system of two subspaces {L_{1}, L_{2}} of ℝ^{d} (dim L_{i} ≥ 2, i = 1, 2) is transitive if and only if L_{1} and L_{2} are not orthogonal to each other.

Proof. Non-orthogonality is, of course, necessary for transitivity. Assume now that L_{1} ⊥̸ L_{2}, and show that the system {L_{1}, L_{2}} is transitive. In order to simplify the notations, we assume that L_{1} + L_{2} = ℝ^{d}.

Case I. L_{1} ∩ L_{2} ≠ 0. Select a vector V ∈ 𝕊^{d−1} such that V ∉ (L_{1} ∩ L_{2})^{⊥}. It is easy to see that the tangent space T_{V} G_{i}V of the orbit G_{i}V at V (i = 1, 2) is the linear space V^{⊥} ∩ L_{i} where, as usual, G_{i} denotes the group of all orthogonal transformations of ℝ^{d} that keep the vectors of L_{i}^{⊥} = A_{i} fixed. Elementary computation with the dimensions shows that

dim [(V^{⊥} ∩ L_{1}) + (V^{⊥} ∩ L_{2})] = dim L_{1} − 1 + dim L_{2} − 1 − (dim L_{1} ∩ L_{2} − 1) = dim(L_{1} + L_{2}) − 1 = d − 1.

Due to Lemma 3.1, the system {L_{1}, L_{2}} is indeed transitive in Case I.

Case II. L_{1} ∩ L_{2} = 0. In this case the above calculus of dimensions provides

dim [(V^{⊥} ∩ L_{1}) + (V^{⊥} ∩ L_{2})] = d − 2,

so we have by one less dimension than the needed d − 1. Interestingly enough, the missing one dimension of Case II and the trick to obtain transitivity of the action by using Lie brackets already appeared in [K-S-Sz(1991)]. Actually, we apply now the commutator method used in the proof of Sublemma 4.5 of [S-Sz(1994)]. Indeed, we select two-dimensional subspaces L̃_{i} ⊂ L_{i} (i = 1, 2) so that L̃_{1} and L̃_{2} are still not orthogonal to each other, then we consider the linear span H = L̃_{1} + L̃_{2}. In the spirit of Lemma 3.9, in order to prove the transitivity of the system {L_{1}, L_{2}} it is enough to show that

(A) the system {L̃_{1}, L̃_{2}} is transitive, and

(B) the system {L_{1}, L_{2}, H} is transitive.

By the result of Case I, both systems {L_{1} + H, L_{2}} and {L_{1}, H} are transitive and, therefore, according to Lemma 3.9, the statement of (B) is indeed true.

The only outstanding task in the proof of Proposition 3.10 is to show (A).

In order to simplify the notations, we assume that L̃_{1} + L̃_{2} = ℝ^{4}. We have arrived at the set-up of Sublemma 4.5 of [S-Sz(1994)], except that now we do not have the bit stronger assumption L̃_{1} ∩ (L̃_{2})^{⊥} = 0 but, rather, we only have that L̃_{1} ⊥̸ L̃_{2}. This means that the cosine

(3.11)  b = min {∥P_{2}(x)∥| x ∈ L̃_{1}, ||x|| = 1}

of the maximum angle between a unit vector of L̃_{1} and the plane L̃_{2} (see (4.6) in [S-Sz(1994)]) may be zero, whereas the cosine

(3.12)  a = max {∥P_{2}(x)∥| x ∈ L̃_{1}, ||x|| = 1}

of the minimum such angle (see (4.5) in [S-Sz(1994)]) must be strictly between zero and one, thanks to the conditions L̃_{1} ∩ L̃_{2} = 0 and L̃_{1} ⊥̸ L̃_{2}.

However, the whole machinery of computing the commutator [X_{1}, X_{2}] of the infinitesimal generators X_{1}, X_{2} of the one-parameter rotation groups G_{1} = O \binom{}{L̃_{1}},

G_{2} = O \binom{}{L̃_{2}} (presented in the proof of Sublemma 4.5 of [S-Sz(1994)]) still works, and we obtain the matrix expansion

(3.13)  [X_{1}, X_{2}] = ( \begin{matrix}−ab & 0 & −b & 0 \\ 0 & −ab & 0 & −a \\ b & 0 & ab & 0 \\ 0 & a & 0 & ab \end{matrix} )

for the commutator [X_{1}, X_{2}] written in an appropriate basis {e_{1}, f_{1}, e_{2}, f_{2}} of ℝ^{4}. Here e_{1} ∈ L̃_{1} is a unit vector (out of the two possible, antipodal unit vectors) for which the maximum value a of ∥P_{2}(x)∥ is attained, e_{2} = P_{2}(e_{1})/a ∈ L̃_{2}, whereas f_{1} ∈ L̃_{1} ⊖ e_{1} is a unit vector (out of the two possible, antipodal ones) for which the minimum value b of ∥P_{2}(x)∥ is attained, and, finally, f_{2} ∈ L̃_{2} ⊖ e_{2} is a unit vector making the angle arccos(b) with f_{1}. Note that ⟨e_{1}, e_{2}⟩ = a, ⟨f_{1}, f_{2}⟩ = b, and ⟨e_{i}, f_{j}⟩ = 0, i, j = 1, 2. Let us compute now the action of the infinitesimal generators X_{1}, X_{2}, and [X_{1}, X_{2}] on the nonzero vector f_{1}+f_{2}, that is, multiply this vector from the left by the matrices of the above infinitesimal generators. Formula (3.13) and the definition of the infintesimal generators X_{i} of the one-parameter rotation groups G_{i} = O(L̃_{i}) yield the following results:

[X_{1}, X_{2}] · (f_{1} + f_{2}) = a(b + 1)(f_{2} − f_{1}),
X_{1} · (f_{1} + f_{2}) = X_{1} · (P_{1}(f_{1} + f_{2})) = ±(b + 1)e_{1},

and, similarly,

X_{2} · (f_{1} + f_{2}) = ±(b + 1)e_{2}.

The resulting three vectors are apparently linearly independent and, therefore, the G-orbit of the vector f_{1} + f_{2} (in the sphere of radius ∥f_{1} + f_{2}∥) is open. By Lemma 3.1, this implies the transitivity of the G-action on the unit sphere 𝕊^{3}. This completes the proof of Proposition 3.10. □

Finishing the proof of Theorem 3.6. By using induction on the number k of the subspaces L_{1}, L_{2}, …, L_{k} (dim L_{i} ≥ 2), we prove that the system {L_{1}, …, L_{k}} is transitive, as long as it has the ONSP (see the definition preceding Lemma 3.3) in the linear span L_{1} + L_{2} + ⋯ + L_{k}.

Indeed, the statement is obviously true for k = 1. Let k > 1, and assume that the theorem has been proven for all smaller values of k. By the ONSP of the system {L_{1}, …, L_{k}}, one can find two subspaces among L_{1}, …, L_{k}, say L_{1} and L_{2}, such that L_{1} ⊥̸ L_{2}. According to Proposition 3.10, the system {L_{1}, L_{2}} is transitive. The ONSP of the collection {L_{1}, …, L_{k}} immediately implies the ONSP for the system {L_{1} + L_{2}, L_{3}, …, L_{k}}. By the induction hypothesis, the latter system is transitive. Finally, Lemma 3.9 and the mentioned transitivity of {L_{1}, L_{2}} yield the transitivity of {L_{1}, …, L_{k}}. Theorem 3.6 is now proved. □

Remark 3.14. The validity of the following assertion easily follows from the proof of Lemma 3.7, actually from the characterization of the transitivity in terms of the equivalence relations ∼_{i}, i = 1, …, k:

If the system {L_{1}, L_{2}, …, L_{k}} (dim L_{i} ≥ 2, L_{1}+⋯+L_{k} = ℝ^{d}) is transitive, then even the connected component G^{0} of the unity in the Lie group G acts transitively on the unit sphere 𝕊^{d−1}. Note that G^{0} is the group algebraically generated by the connected components SO(L_{i}) of the groups G_{i} = O(L_{i}), i = 1, …, k.

It is worth mentioning the following, easily checkable sufficient condition for transitivity:

Definition of tightness. We say that the system of subspaces L_{1}, …, L_{k} is tight if and only if there exists a system of d − 1 linearly independent vectors e_{1} ∈ L_{i(1)}, e_{2} ∈ L_{i(2)}, …, e_{d−1} ∈ L_{i(d−1)} such that the linear span span {e_{1}, …, e_{d−1}} of these vectors does not contain any of the subspaces L_{i(1)}, …, L_{i(d−1)}.

As a matter of fact, as we will see in the proof of the forthcoming lemma, under the condition of tightness, transitivity can be obtained directly by merely taking the linear span of the Lie algebras of G_{i} instead of the entire generated Lie algebra.

Lemma 3.15. The tightness of the system L_{1}, …, L_{k} implies the transitivity of the action of G on 𝕊^{d−1}.

Proof. Select a system of linearly independent vectors e_{1}, …, e_{d−1} featuring the definition of tightness. Let V ∈ 𝕊^{d−1} be orthogonal to all vectors e_{1}, …, e_{d−1}. Consider one of these vectors, say, e_{1}. Let us denote by P_{i(1)} the orthogonal projection onto the subspace L_{i(1)}. The projection P_{i(1)}V is nonzero, for L_{i(1)} ⊄ span {e_{1}, …, e_{d−1}}. Since the tangent space T_{V} (G_{i(1)}V ) of the orbit G_{i(1)}V ⊂ GV at V is the orthocomplement L_{i(1)} ⊖ span {P_{i(1)}V } of the nonzero vector P_{i(1)}V in L_{i(1)}, we have that e_{1} ∈ T_{V} (G_{i(1)}V ) ⊂ T_{V} (GV). By the same argument, e_{j} ∈ T_{V} (GV) for j = 1, …, d − 1 and, therefore, dim (GV) = d − 1. □

Basic examples related to the notion of tightness are given in Remarks 3.24 and 3.26. The first of them shows that transitivity does not imply tightness, whereas the second one provides an example of a tight family.

Characterization of the Positive Subspace of the Second Fundamental Form

We define the “parallel beam of light” B (formerly called an orthogonal manifold) around the phase point x_{0} = (Q_{0}, V_{0}) ∈ M as follows:

(3.16)  B = {x = (Q, V_{0}) ∈ M : Q − Q_{0} ⊥ V_{0} and ||Q − Q_{0}|| < ε_{0}}

with a fixed and sufficiently small number ε_{0} > 0. There are two interpretations of the manifold B:

(a) a (d − 1)-dimensional submanifold of the phase space M, or

(b) a codimension-one flat submanifold π(B) of the configuration space Q supplied with a field of unit normal vectors, where π : M → Q is the natural projection.

We shall use these two interpretations alternately: when dealing with the second fundamental form we use (b), however, when defining the image S^{t}(B) of B under the flow, we use the first interpretation. Consider a non-singular trajectory segment S^{[0,T]}x_{0} and denote by W_{+} = W_{+} (S^{[0,T]}x_{0}) the positive subspace of the positive semi-definite, symmetric, second fundamental form W of S^{T} (B) at the point S^{T} x_{0} = (Q′_{0}, V′_{0}). Since W acts in the orthocomplement (V′_{0})^{⊥} of V′_{0} in ℝ^{d}, W_{+} is a subspace of (V′_{0})^{⊥}.

With the first collision with the cylinder C_{σ_{1}} we associate a (ν_{1} − 1)-dimensional real projective space P_{1} ≅ ℙ^{ν_{1}−1}(ℝ) of all orthogonal reflections of the space ℝ^{d} across all possible hyperplanes H that contain the generator space A_{σ_{1}} of the cylinder C_{σ_{1}}. Plainly, such reflections (or hyperplanes H) can be characterized uniquely by the orthocomplement line H^{⊥} ⊂ L_{σ_{1}}, so the collection P_{1} of all such reflections is naturally diffeomorphic to the real projective space ℙ^{ν_{1}−1}(ℝ), where ν_{1} = dim L_{σ_{1}}.

Similarly, other real projective spaces P_{i} ≅ ℙ^{ν_{i}−1}(ℝ) are attached to the symbolic collisions σ_{i}, i = 1, …, m. By using these definitions, we obtain a mapping

(3.17)  Φ_{Σ} = Φ : 𝕊^{d−1} × ∏_{i=1}^{m} P_{i} → 𝕊^{d−1}

which assignes to every (m + 1)-tuple

(V; h_{1}, h_{2}, …, h_{m}) ∈ 𝕊^{d−1} × ∏_{i=1}^{m} P_{i}

the image V h_{1}h_{2}·. . .·h_{m} = V′ of V ∈ 𝕊^{d−1} under the composite action h_{1}h_{2}·. . .·h_{m}. (Here, by convention, h_{1} is applied first.) Plainly V′_{0} = Φ(V_{0}; g_{1}, g_{2}, …, g_{m}), where g_{i} denotes the orthogonal reflection that causes the abrupt change of the velocity at the i-th collision σ_{i} of the given orbit segment S^{[0,T]}x_{0}. The space 𝕊^{d−1} × ∏\begin{matrix}m \\ i=1 \end{matrix} P_{i} will often be called the phase space of the virtual dynamics, or the phase space of the velocity process.

Now we can consider the partial derivative \frac{∂Φ}{∂P̃} (V; h_{1}, h_{2}, …, h_{m}) of Φ with

respect to the factor P̃ = ∏\begin{matrix}m \\ i=1 \end{matrix} P_{i}: It is a linear mapping from the tangent space Th⃗ \binom{}{P̃} into the tangent space T_{V′}𝕊^{d−1}, where V′ = Φ(V; h_{1}, h_{2}, …, h_{m}). The next result characterizes the positive subspace W_{+} as the range of the above mentioned partial derivative:

Proposition 3.18. Using the definitions and notations from above,

W_{+} (S^{[0,T]}x_{0}) = Ran (\frac{∂Φ}{∂P̃} (V_{0}; g_{1}, g_{2}, …, g_{m})) .

Proof. The left-hand-side is obviously a subspace of the one on the right. Conversely, suppose a vector

Y = (y_{1}, y_{2}, …, y_{N}) ∈ (V′_{0})^{⊥}

is orthogonal to W_{+}. We will show that Y is orthogonal to

Ran (\frac{∂Φ}{∂P̃} (V_{0}; g_{1}, g_{2}, …, g_{m})) ,

as well. Because of

Ran (\frac{∂Φ}{∂P̃} (V_{0}; g_{1}, g_{2}, …, g_{m})) =

L \{Ran (\frac{∂Φ}{∂P_{i}} (V_{0}; g_{1}, g_{2}, …, g_{m})) : i = 1, …, m\}

it is sufficient to show that for every integer i (1 ≤ i ≤ m)

Y ⊥ Ran (\frac{∂Φ}{∂P_{i}} (V_{0}; g_{1}, g_{2}, …, g_{m})) := R_{i}.

For the sake of simplifying the notations we suppose that σ_{i} = 1. Then

R_{i} = {zg_{i+1} … g_{m} : z ∈ L_{1}, z ⊥ V_{i}^{+}} ,

where V_{i}^{+} is the velocity right after the i-th collision with the cylinder C_{1}. By the assumed neutrality of Y with respect to S^{[0,T]}x_{0} (which says that the vector Y g_{m}^{−1} … g_{i+1}^{−1} belongs to the linear span of A_{1} = A_{σ_{i}} and V_{i}^{+}), we have that Y g_{m}^{−1} … g_{i+1}^{−1} ⊥ z for every vector z ∈ L_{1}, z ⊥ V_{i}^{+}. Thus, by the orthogonality of the mapping g_{i+1} … g_{m}, we have Y ⊥ R_{i}.

Hence Proposition 3.18 follows. □

Remark 3.19. The big advantage of the above lemma is that it gives us a new characterization of sufficiency in terms of the pure velocity process without the configuration history.

Proposition 3.18 pregnantly shows an intimate relationship between the sufficiency of a trajectory segment S^{[0,T]}x_{0} (i. e. when the left-hand-side in the statement of that lemma has the maximum dimension d − 1) and the transitivity of the action of G on the velocity sphere 𝕊^{d−1}. This circumstance and the results of the papers [S-Sz(1994)], [S-Sz(1995)], and [Sim(1998)] (Especially Lemma 4.4 and Sublemma 4.5 in [S-Sz (1994)], and the role of those lemmas in the proof of Main Lemma 4.1 in [S-Sz(1994)]) strongly suggest the following conjectures:

Conjecture 1. For every cylindric billiard flow (M, {S^{t}}, μ) the transitivity of the action of G on the velocity sphere 𝕊^{d−1} is equivalent to the hyperbolic ergodicity (or, equivalently, to the hyperbolicity and Bernoulli property; cf. [Ch-H(1996)] and [O-W(1996)]) of the billiard map.

Corollary 3.20. Dichotomy. If the above conjecture holds true, then for every cylindric billiard flow (M, {S^{t}}, μ) exactly one of the following two possibilities will occur:

(I) The system {L_{1}, …, L_{k}} of the orthocomplements of the generators has the ONSP in ℝ^{d}, and the billiard map is hyperbolic and enjoys the Bernoulli property;

(II) The system {L_{1}, …, L_{k}} has an orthogonal splitting ℝ^{d} = B_{1} ⊕ B_{2} (see the definition right before Lemma 3.3), and the partial kinetic energies ∥P_{B_{1}}(V)∥^{2} and ∥P_{B_{2}}(V)∥^{2} are trivial first integrals of the motion.

The above dichotomy shows that non-ergodicity can only be caused by the presence of some very simple invariant quantity, namely the kinetic energy of a “subsystem”.

Corollary 3.21. Another consequence of Conjecture 1 is that “the more cylinders the better”, i. e. (hyperbolic) ergodicity can not be spoilt by the removal of additional cylinders from the configuration space, that is, by adjoining more cylindric scatterers to the billiard.

Corollary 3.22. If Conjecture 1 is valid, then the phenomena of ergodicity and partial hyperbolicity (i. e. the case when there exist zero and nonzero Lyapunov exponents alike) can not coexist in cylindric billiards.

It is interesting to note that for billiards with two cylindric scatterers the recent generalization of the result of [S-Sz(1994)] by Péter Bálint [B(1998)] actually verifies Conjecture 1 for that case.

For completeness, we recite here the conjecture appeared in [Sz(1996)] which easily turns out to be a weakened version of the previous conjecture:

Conjecture 2. For every cylindric billiard flow (M, {S^{t}}, μ) the existence of a single sufficient trajectory is equivalent to the hyperbolic ergodicity (or, equivalently, to the hyperbolicity and Bernoulli property) of the billiard map.

Remark 3.23. By virtue of Proposition 3.18 it is obvious that the existence of a sufficient trajectory implies the transitivity of the G-action. Thus Conjecture 1 is formally stronger than Conjecture 2.

Remark 3.24. Transitivity does not imply tightness. Indeed, in the model of [S-Sz(1994)] we had d = 4, and ℝ^{4} was the linear direct sum of the two-dimensional subspaces L_{1} and L_{2}. Thus, for any system e_{1} ∈ L_{i(1)}, e_{2} ∈ L_{i(2)}, e_{3} ∈ L_{i(3)} of linearly independent vectors either L_{1} or L_{2} is a subspace of span {e_{1}, e_{2}, e_{3}}, and the system {L_{1}, L_{2}} is not tight, although, as has been shown in [S-Sz(1994)], the G-action is still transitive.

Remark 3.25. Orthogonal Cylindric Billiards. In his paper [Sz(1994)] the second author studied the so-called orthogonal cylindric billiards. i. e. the cylindric billiards for which each space L_{i} ⊂ ℝ^{d} is spanned by the coordinate axes belonging to the set K^{i} ⊂ {1, 2, …, d}. The sufficient and necessary condition of ergodicity found there was the following one: There is no splitting {1, 2, …, d} = B_{1} ∪ B_{2}, B_{j} ≠ ∅, B_{1} ∩ B_{2} = ∅, such that every set K^{i} is the subset of B_{1} or B_{2}. It is clear that this condition is equivalent to our ONSP.

An interesting, special family of the above orthogonal cylindric billiards is the one for which ⋂_{i=1}^{k} K^{i} ≠ ∅, ⋃_{i=1}^{k} K^{i} = {1, 2, …, d}. For simplicity assume that d ∈ ⋂_{i=1}^{k} K^{i}. Then we can select the standard coordinate unit vectors e_{1}, …, e_{d−1} (in the direction of the first, second, ..., (d−1)-th coordinate axes), and this system obviously fulfills all requirements in the definition of tightness. By the result of [Sz(1994)], the corresponding cylindric billiard map is hyperbolic and ergodic.

Remark 3.26. Consider the case when dim A_{j} = 1 for j = 1, …, k. Assume that not all lines A_{j} are parallel, for example, A_{1} is not parallel to A_{2}. Choose a linear basis e_{1}, …, e_{d} in ℝ^{d} so that {e_{1}, e_{3}, …, e_{d}} is a basis of L_{1} and {e_{2}, e_{3}, …, e_{d}} is a basis of L_{2}. (Thus, {e_{3}, …, e_{d}} is automatically a basis of L_{1} ∩ L_{2}.) Then the system {e_{1}, e_{2}, …, e_{d−1}} obviously fulfills all requirements in the definition of tightness. Thus, we obtained that any cylindric billiard with one-dimensional generators is necessarily tight, unless all generators are parallel. (In which case the G-action is clearly not transitive.)

For such models the proof of (hyperbolicity and) ergodicity can go ahead straightforwardly along the lines of the proof of ergodicity developed in the papers [K-S-Sz(1989)] and [S-Sz(1995)] for two cylinders. Thus, for the case of one-dimensional generator spaces, the methods of [K-S-Sz(1989)] and [S-Sz(1995)] actually prove Conjecture 1.

4. Hard Sphere Systems

Consider the system of N (≥ 2) hard spheres, labelled by 1, 2, …, N, with positive masses m_{1}, …, m_{N} (∑_{i=1}^{N} m_{i} = 1) moving uniformly and colliding elastically in the unit ν-torus (ν ≥ 2) 𝕋^{ν} = ℝ^{ν}/ℤ^{ν}. For simplicity we assume that these spheres have the common radius r > 0, so that even the interior of the configuration space is connected. We make the standard reductions I = ∑_{i=1}^{N} m_{i}v_{i} = 0, 2E = ∑_{i=1}^{N} m_{i}∥v_{i}∥^{2} = 1, where v_{i} = q̇_{i} is the velocity of the i-th sphere, while q_{i} ∈ 𝕋^{ν} is the position of its center. Corresponding to the reduction I = 0, we need to factorize the configuration space with respect to uniform translations as follows: the configuration (q_{1}, q_{2}, …, q_{N}) is considered to be equivalent to another configuration (q′_{1}, q′_{2}, …, q′_{N}) iff there is an element a ∈ 𝕋^{ν} such that q_{i} − q′_{i} = a for every index i. After this factorization the configuration space Q is still a torus (of dimension d = ν(N − 1)) from which we remove the convex sets

C_{i,j} = {(q_{1}, …, q_{N})| ∥q_{i} − q_{j}∥_{e} < 2r} ,

1 ≤ i < j ≤ N. Here ∥ . ∥_{e} denotes the original Euclidean metric of 𝕋^{ν} = ℝ^{ν}/ℤ^{ν}.

The relevant Riemannian metric in Q, however, is given by (the double of) the kinetic energy:

(4.1)  ∥ds∥^{2} = ∑_{i=1}^{N} m_{i} ∥dq_{i}∥_{e}^{2} .

With the above Riemannian metric ∥ . ∥ the configuration space Q is still a flat torus and — as it is easy to see — the removed scatterers C_{i,j} are cylinders with a subtorus T_{i,j} as the generator, and a ν-dimensional, solid disk D_{i,j} as the base. Let us understand in more detail the tangent space A_{i,j} of the generator subtorus T_{i,j} and its orthocomplement L_{i,j} = A_{i,j}^{⊥} with respect to the inner product given by the kinetic energy (4.1).

First of all, the common tangent space of Q at all of its points is the d-dimensional Euclidean space

(4.2)  Z = {z = (dq_{k})_{k=1}^{N}| ∑_{k=1}^{N} m_{k}dq_{k} = 0}

with the Euclidean structure (4.1). Secondly, the tangent space A_{i,j} ⊂ Z of T_{i,j} is easily seen to be

(4.3)  A_{i,j} = {z = (dq_{k})_{k=1}^{N}| ∑_{k=1}^{N} m_{k}dq_{k} = 0 and dq_{i} = dq_{j}} .

The orthocomplement L_{i,j} = A_{i,j}^{⊥} of A_{i,j} is

(4.4)  L_{i,j} = {z = (dq_{k})_{k=1}^{N}| dq_{k} = 0 for k ≠ i, j and m_{i}dq_{i} + m_{j}dq_{j} = 0} .

For the norm square ∥z∥^{2} = m_{i}∥dq_{i}∥_{e}^{2} + m_{j}∥dq_{j}∥_{e}^{2} of an element z ∈ L_{i,j} in (4.4) we immediately have

(4.5)  ∥z∥^{2} = \frac{m_{i}(m_{i} + m_{j})}{m_{j}}∥dq_{i}∥_{e}^{2} = \frac{m_{j}(m_{i} + m_{j})}{m_{i}}∥dq_{j}∥_{e}^{2}.

Therefore, the base D_{i,j} of the removed body C_{i,j} is

(4.6)  \begin{matrix}D_{i,j} = \{z = (dq_{k})_{k=1}^{N} ∈ L_{i,j}| ∥dq_{i} − dq_{j}∥_{e} < 2r\} \\ = \{z ∈ L_{i,j}| ∥z∥ < 2r · \sqrt{\frac{m_{i}m_{j}}{m_{i} + m_{j}}} \} . \end{matrix}

The (d − 1)-dimensional sphere 𝕊^{d−1} (on which the action of G is defined) is, of course, the unit sphere

(4.7)  𝕊^{d−1} = {z = (dq_{k})_{k=1}^{N} ∈ Z| ∑_{k=1}^{N} m_{k} ∥dq_{k}∥_{e}^{2} = 1}

of the space Z.

Remark 4.8. In Section 2 we assumed that the cylindric billiard has the following property:

At every point q ∈ ∂Q of the boundary of the configuration space Q the spherical angle subtended by the compact domain Q is strictly positive.

The reason for postulating this axiom was to ensure that in any trajectory the collision times can not accumulate in finite time. Unfortunately, this property does not hold for every hard sphere system, it only holds for typical values of the radius r. (Apart from countably many exceptional values.) However, even if this property fails to hold, the set of phase points

A_{∞}^{±} = {x ∈ M| S^{[0, b)}x contains infinitely many collisions with some ± b > 0}

has Liouville measure zero, which is a direct consequence of the invariance of the Liouville measure. Then we can discard the zero measure union of the above sets, and this minor modification of the phase space does not influence in any way the hyperbolicity status of the hard sphere system.

The result of this section is

Proposition 4.9. In the case of the hard sphere system (N; ν; r; m_{1}, m_{2}, …, m_{N}) the action of G on 𝕊^{d−1} is transitive.

Proof. By using (4.4), we will simply check the meaning of the orthogonality of two subspaces L_{i,j} and L_{k,l} (1 ≤ i < j ≤ N, 1 ≤ k < l ≤ N). One immediately sees that L_{i,j} ⊥ L_{k,l} if and only if the pairs {i, j} and {k, l} have no common element. By the connectedness of the full collision graph, the system of subspaces {L_{i,j}| 1 ≤ i < j ≤ N} has the ONSP. Therefore, according to Lemma 3.4 and Theorem 3.6, the G-action is transitive on 𝕊^{d−1}. This finishes the proof of the proposition. □

Remark 4.10. Let us call a transformation g ∈ G_{i,j} (with some pair (i, j), 1 ≤ i < j ≤ N) elementary. If, further, g is a reflection across a hyperplane H (containing the generator space A_{i,j}), then we call g an elementary reflection.

By the compactness of the sphere 𝕊^{d−1}, there exists an upper bound

C = C(N; ν; m_{1}, …, m_{N}) < ∞

such that for every pair of velocities V_{1}, V_{2} ∈ 𝕊^{d−1} the velocity V_{2} can be obtained from V_{1} by applying at most C elementary transformations or, equivalently, at most C elementary reflections. We note that every product gV, V ∈ 𝕊^{d−1}, g elementary, can also be obtained as g′V, where g′ is an elementary orthogonal reflection across a hyperplane.

Example 4.11. The upper bound C(N; ν; m_{1}, …, m_{N}) is far from uniform; it depends heavily on the masses m_{1}, m_{2}, …, m_{N}! Actually, the bound C may be forced to tend to infinity, as some ratios between masses blow up. A simple example showing this phenomenon is the following one:

Let m_{1} = m_{2} = M >> 1 (very big), m_{3} = m_{4} = ⋯ = m_{N} = 1, v_{1} = v_{2} = 0, N ≥ 4. By the theorem and the previous remark this velocity configuration can be transformed into (V, −V, 0, 0, …, 0) (where ∥V ∥^{2} = 1/(2M)) by applying at most C elementary transformations. We can, however, easily estimate (from above) the maximum amount of kinetic energy that can be conveyed from the subsystem {3, 4, …, N} to the system {1, 2}. Assume, for instance, that the first and third spheres collide. It is easy to see that the maximum amount of such an energy exchange occurs when

(a) the collision is a “head-on” collision, i. e. the two spheres move on a straight line so that (by identifying this line with ℝ by also suitably orienting it) the precollision velocities of the colliding spheres are v_{3}^{−} > v_{1}^{−} > 0;

(b) the velocities v_{1}^{−}, v_{3}^{−} have the maximum possible values. The simple reason for (b) to hold is that by increasing either v_{1}^{−} or v_{3}^{−} the amount of energy gained by the first particle during this collision also increases. Straightforward upper bounds for these positive velocities are v_{1}^{−} ≤ 1/\sqrt{M}, v_{3}^{−} ≤ 1. An elementary calculation shows that in the extreme case v_{1}^{−} = 1/\sqrt{M}, v_{3}^{−} = 1 the after-collision velocity v_{3}^{+} is equal to

\frac{2\sqrt{M} + 1 − M}{M + 1}

and, therefore, the maximum order of magnitude of the amount of kinetic energy conveyed from the subsystem {3, 4, …, N} to the subsystem {1, 2} (by a single collision) is not greater than 2/\sqrt{M}. Thus, in order for the subsystem \{3, …, N\} to lose its energy 1/2, there must be at least \sqrt{M}/4 collisions between the two subsystems of heavy and light particles, i. e. C(N; ν; M, M, 1, 1, …, 1) ≥ \sqrt{M}/4, M >> 1.

Remark 4.12. Arbitrary collision graphs. Suppose that not all cylinders C_{i,j} are removed from the original, ν(N − 1)-dimensional configuration torus, but, instead, only the cylinders whose pair of labels {i, j} belongs to the set of edges E(G) of a prescribed graph G of allowed collisions on the set of vertices {1, 2, …, N}. Then the proof of Proposition 4.9 yields that the G-action corresponding to the arising cylindric billiard is transitive if and only if the graph G is connected on the whole vertex set {1, 2, …, N}. Hyperbolic ergodicity, that is, actually Conjecture 1 was proved to be true in [S-Sz(1995)] for such classes of generalized hard sphere systems in the case ν ≥ 4, provided that the connected graph G is either the simple path of length N − 1, or the simple loop of length N. For such graphs of allowed collision, in the case of ν = 3, the complete hyperbolicity of the flow is proved there, too.

Finally, by taking into account (4.3), observe the interesting fact that, in the realm of such generalized hard sphere systems, the condition ⋂_{(i,j)∈E(G)} A_{i,j} = 0 is also equivalent to the connectedness of the graph G. Thus, at least for these generalized hard sphere systems, the transitivity and the otherwise much weaker property ⋂_{i,j} A_{i,j} = 0 are equivalent.

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