January 30, 1998

Hard Ball Systems Are Completely Hyperbolic

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. We consider the system of N (≥ 2) elastically colliding hard balls with masses m_{1}, …, m_{N}, radius r, moving uniformly in the flat torus 𝕋_{L}^{ν} = ℝ^{ν}/L · ℤ^{ν}, ν ≥ 2. It is proved here that the relevant Lyapunov exponents of the flow do not vanish for almost every (N + 1)-tuple (m_{1}, …, m_{N}; L) of the outer geometric parameters.

1. Introduction.

The proper mathematical formulation of Ludwig Boltzmann’s ergodic hypothesis, which has incited so much interest and discussion in the last hundred years, is still not clear. For systems of elastic hard balls on a torus, however, Yakov Sinai, in 1963, [Sin(1963)] gave a stronger, and at the same time mathematically rigorous version of Boltzmann’s hypothesis: the system of an arbitrarily fixed number N of identical elastic hard balls moving in the ν-torus 𝕋^{ν} = ℝ^{ν}/ℤ^{ν} (ν ≥ 2) is ergodic — of course, on the submanifold of the phase space specified by the trivial conservation laws. Boltzmann used his ergodic hypothesis when laying down the foundations of statistical physics, and its various forms are still intensively used in modern statistical physics. The importance of Sinai’s hypothesis for the theory of dynamical systems is stressed by the fact that the interaction of elastic hard balls defines the only physical system of an arbitrary number of particles in arbitrary dimension whose dynamical behaviour has been so far at least guessed — except for the completely integrable system of harmonic oscillators. (As to the history of Boltzmann’s hypothesis, see the recent work [Sz(1996)].)

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

Sinai’s hypothesis was partially based on the physical arguments of Krylov’s 1942 thesis (cf. [K(1979)] and its afterword written by Ya. G. Sinai, [Sin(1979)]), where Krylov discovered that hard ball collisions provide effects analogous to the hyperbolic behaviour of geodesic flows on compact manifolds of constant negative curvature, exploited so beautifully in the then recent works of Hedlund, [He(1939)] and Hopf, [Ho(1939)].

The aim of the present paper is to establish that hard ball systems are, indeed, fully hyperbolic, i. e. all relevant Lyapunov exponents of these systems are non-zero almost everywhere. Our claim holds for typical (N + 1)-tuples (m_{1}, …, m_{N}; L) ∈ ℝ_{+}^{N+1} of the outer geometric data of the system, that is — in contrast to earlier results — we do not require that particles have identical masses, though, on the other hand, we have to exclude a countable union of proper submanifolds of (N + 1)-tuples (m_{1}, …, m_{N}; L). (Which set is, in fact, very likely to be empty.)

Full hyperbolicity combined with Katok-Strelcyn theory (see [K-S(1986)]) immediately provides that the ergodic components of these systems are of positive measure. Consequently, there are at most countably many of them, and, moreover, on each of them the system is K-mixing. Our methods so far do not give the expected global ergodicity of the systems considered (an additional hypothesis to provide that is formulated in Section 6).

The equality of the radii of the balls is not essential, but, for simplicity, it will be assumed throughout. For certain values of the radii — just think of the case when they are large — the phase space of our system decomposes into a finite union of different connected components, and these connected parts certainly belong to different ergodic components. Now, according to the wisdom of the ergodic hypothesis, these connected components are expected to be just the ergodic components of the system, and on each of them the system should also possess the Kolmogorov-mixing property.

Let us first specify the model and formulate our result.

Assume that, in general, a system of N(≥ 2) balls, identified as 1, 2, …, N, of masses m_{1}, …, m_{N} and of radius r > 0 are given in 𝕋_{L}^{ν} = ℝ^{ν}/L · ℤ^{ν}, the νdimensional cubic torus with sides L, (ν ≥ 2). Denote the phase point of the ith ball by (q_{i}, v_{i}) ∈ 𝕋_{L}^{ν} × ℝ^{ν}. A priori, the configuration space Q̃ of the N balls is a subset of 𝕋_{L}^{N·ν}: from 𝕋_{L}^{N·ν} we cut out \binom{N}{2} cylindric scatterers:

C̃_{i,j} = {Q = (q_{1}, …, q_{N}) ∈ 𝕋_{L}^{N·ν} :| q_{i} − q_{j} |< 2r} ,  (1.1)

1 ≤ i < j ≤ N, or in other words Q̃ := 𝕋_{L}^{Nν} \ ⋃_{1≤i<j≤N} C̃_{i,j}. The energy H = \frac{1}{2} ∑_{1}^{N} m_{i}v_{i}^{2} and the total momentum P = ∑_{1}^{N} m_{i}v_{i} are first integrals of the motion. Thus, without loss of generality, we can assume that H = \frac{1}{2}, P = 0. (If P ≠ 0, then the system has an additional conditionally periodic or periodic motion.) Now, for these values of H and P, we define our dynamical system.

Remark. It is clear that not actual values of r and L, but merely their ratio L/r is elevant for our model. Therefore, throughout this paper we fix the value of r > 0, and will only consider (later in sections 3–4) the size L of the torus as a variable.

In earlier works (cf. [S-Sz(1995)-II] and the references therein), where the masses were identical, one could and — to obtain ergodicity — had to fix a center of mass. For different masses — as observed in [S-W(1989)] — this is generically not possible and we should follow a different approach.

The equivalence relation Ψ over Q̃, defined by Q ∼_{Ψ} Q^{∗} if and only if there exists an a ∈ 𝕋_{L}^{ν} such that for every i ∈ [1, N] q_{i}^{∗} = q_{i} + a, allows us to introduce Q := Q̃/Ψ (the equivalence relation means, in other words, that for the internal coordinates q_{i} − q_{j} = q_{i}^{∗} − q_{j}^{∗} holds for every i, j ∈ [1, N]). The set Q, a compact, flat Riemannian manifold with boundary will actually be the configuration space of our system, whereas its phase space will be the ellipsoid bundle M := Q × E, where E denotes the ellipsoid ∑_{i=1}^{N} m_{i}(dq_{i})^{2} = 1, ∑_{i=1}^{N} m_{i}dq_{i} = 0. Clearly, d := dim Q = Nν − ν, and dim E = d − 1. The well-known Liouville measure μ is invariant with respect to the evolution S^{ℝ} := {S^{t} : t ∈ ℝ} of our dynamical system defined by elastic collisions of the balls of masses m_{1}, …, m_{N} and their uniform free motion. Here we have two remarks:

(i) the collision laws for a pair of balls with different masses are well-known from mechanics and will also be reproduced in the equation (3.9);

(ii) the dynamics can, indeed, be defined for μ−a. e. phase point, see the

corresponding references in Section 2. The dynamical system (M, S^{ℝ}, μ)_{m⃗,L} is called the standard billiard ball system with the outer geometric parameters (m⃗; L) ∈ ℝ_{+}^{N+1}.

Denote by R_{0} := R_{0}(N, ν, L) the interval of those values of r > 0, for which the interior IntM = IntQ × E of the phase space of the standard billiard ball flow is connected.

The basic result of our paper is the following

Main Theorem. For N ≥ 2, ν ≥ 2 and r ∈ R_{0}, none of the relevant Lyapunov exponents of the standard billiard ball system (M, {S^{ℝ}}, μ)_{m⃗,L} vanishes — apart from a countable union of proper analytic submanifolds of the outer geometric parameters (m⃗; L) ∈ ℝ_{+}^{N+1}.

An interesting consequence of this theorem is the following

Corollary. For the good set of typical outer geometric parameters (m⃗; L) the ergodic components of the system have positive measure and on each of them the standard billiard ball flow has the K-property.

Note 1. In general, if r ∉ R_{0}, then IntM decomposes into a finite number of connected components. Our results extend to these cases, too.

Note 2. As it will be seen in Section 2 (see Lemma 2.1) our system is isomorphic to a semi-dispersing billiard, and, what is more, is a semi-dispersing billiard in a weakly generalized sense, and, consequently, we can use the theory of semidispersing billiards. As it has been proved in recent manuscripts by N. I. Chernov and C. Haskell, [C-H(1996)] on one hand, and by D. Ornstein and B. Weiss, [O-W(1996)] on the other hand, the K-mixing property of a semi-dispersing billiard flow on a positive ergodic component actually implies its Bernoulli property, as well.

Note 3. As to the basic results (and history) concerning the Boltzmann and Sinai hypotheses we refer to the recent survey [Sz(1996)].

The basic notion in the theory of semi-dispersing billiards is that of the sufficiency of a phase point or, equivalently, of its orbit. The conceptual importance of sufficiency can be explained as follows (for a technical introduction and our prerequisites, see Section 2): In a suitably small neighbourhood of a (typical) phase point of a dispersing billiard the system is hyperbolic, i. e. its relevant Lyapunov exponents are not zero. For a semi-dispersing billiard the same property holds for sufficient points only! Physically speaking, a phase point is sufficient if its trajectory encounters in its history all possible degrees of freedom of the system.

Our proof of the full hyperbolicity is the first major step in the basic strategy for establishing global ergodicity of semi-dispersing billiards as it was suggested in our series of works with A. Krámli (see the references), and, in fact, some elements of that approach are also used in this paper. In the sense of our strategy for proving global ergodicity, initiated in [K-S-Sz(1991)] and explained in the introductions of [K-S-Sz(1992)] and of [Sim(1992)-I] , there are two fundamental parts in the demonstration of hyperbolicity, once a combinatorial property, called richness of the symbolic collision sequence of a trajectory, had been suitably defined:

(1) The “geometric–algebraic considerations” on the codimension of the manifolds describing the non–sufficient trajectory segments with a combinatorially rich symbolic collision structure;

(2) Proof of the fact that the set of phase points with a combinatorially non–rich collision sequence has measure zero.

In contrast to earlier proofs of ergodic properties of hard ball systems our method for obtaining full hyperbolicity will not be inductive, more precisely, we do not use the hyperbolicity of smaller systems. This has become possible since our Theorem 5.1 (settling step (2) here) which is a variant of the so-called weak ball-avoiding theorems, can now be proven without any inductive assumption. Nevertheless, the proof of our crucial Key Lemma 4.1, which copes with part (1), is inductive and, indeed, one of the main reasons to use varying masses is just that by choosing the mass of one particle to be equal to zero permits us to use an inductive assumption on the smaller system (but about its algebraic behaviour, only). As just mentioned, to prove our theorem we have introduced varying masses, and can only claim hyperbolicity for typical (N + 1)-tuples (m_{1}, …, m_{N}; L) ∈ ℝ_{+}^{N+1} of outer geometric parameters. Furthermore, we will, in Section 3, complexify the dynamics. The complexification is required, on one hand, by the fact that our arguments in Section 4 use some algebraic tools that assume the ground field to be algebraically closed. Another advantage of the complexification is that, in the inductive derivation of Key Lemma 4.1, one does not have to worry about the sheer existence of orbit segments with a prescribed symbolic collision sequence: they do exist, thanks to the algebraic closedness of the complex field.

The paper is organized as follows: Section 2 is devoted to prerequisites. Section 3 then describes the complexified dynamics, while in Section 4 we establish Key Lemma 4.1 — both for the complexified and the real dynamics — settling, in particular, part (1) of the strategy. Section 5 provides the demonstration of our Main Theorem through the afore mentioned Theorem 5.1, and, finally, Section 6 contains some comments and remarks.

2. Prerequisites

Semi-dispersing billiards and hard ball systems

Our approach is based on a simple observation:

Lemma 2.1. The standard billiard ball flow with mass vector m⃗ ∈ ℝ_{+}^{N} is isomorphic to a semi-dispersing billiard.

Because of the importance of the statement we sketch the proof.

Proof. Introduce new coordinates in the phase space as follows: For (Q, V) ∈ M let q̂_{i} = \sqrt{m_{i}}q_{i}, v̂_{i} = \sqrt{m_{i}}v_{i}, where thus q̂_{i} ∈ \sqrt{m_{i}}𝕋_{L}^{ν} and ∑_{i=1}^{N} v̂_{i}^{2} = 1. Moreover, the vector V̂ = (v̂_{1}, …, v̂_{N}) necessarily belongs to the hyperplane ∑_{i=1}^{N} \sqrt{m_{i}}v̂_{i} = 0. Denote 𝕋_{m⃗,L} = ∏_{i=1}^{N}(\sqrt{m_{i}} 𝕋_{L}^{ν}) and

Ĉ_{i,j} = \{Q̂ = (q̂_{1}, …, q̂_{N}) ∈ 𝕋_{m⃗,L} : ∥\frac{q̂_{i}}{\sqrt{m_{i}}} − \frac{q̂_{j}}{\sqrt{m_{j}}} ∥ < 2r\}  (2.2)

and ˜ Q̂ = 𝕋_{m⃗,L}\∪_{1≤i<j≤N} Ĉ_{i,j}. The equivalence relation Ψ̂m⃗ over 𝕋_{m⃗,L} is as follows: Q̂ ∼Ψ̂m⃗ Q̂^{∗} if and only if there exists an a ∈ 𝕋_{L}^{ν} such that for every i ∈ [1, N] q̂_{i}^{∗} =

q̂_{i}+\sqrt{m_{i}}a allows us to define Q̂ = ˜ Q̂/Ψ̂m⃗. Let M̂_{m⃗,L} = Q̂×𝕊^{d−1} be the unit tangent bundle of Q̂, and denote by dμ̂ the probability measure const · dQ̂ · dV̂, where dV̂ is the surface measure on the (d − 1)−sphere 𝕊^{d−1}, and dQ̂ is the Lebesgue-measure on 𝕋_{m⃗,L}. Now the standard billiard ball system (M, S^{ℝ}, μ)_{m⃗,L} with mass vector m⃗ and the billiard system (M̂_{m⃗,L}, Ŝ^{ℝ}, μ̂) are isomorphic.

Indeed, we can reduce the question to the case of one-dimensional particles because for both models the velocity components perpendicular to the normal of impact remain unchanged. The claimed isomorphy for one-dimensional particles, however, is well-known, and for its simple proof we can refer, for instance, to Section 4 of Chapter 5 in [C-F-S(1981)]. Thus, the point is that in the isomorphic flow (M̂_{m⃗,L}, Ŝ^{ℝ}, μ̂) the velocity transformations at collisions become orthogonal

reflections across the tangent hyperplane of the boundary of Q̂, see also (3.9) for the mentioned velocity transformation.

The fact that the billiard system (M̂_{m⃗,L}, Ŝ^{ℝ}, μ̂) is semi-dispersing, is obvious, because the scattering bodies in Q̂ are cylinders built on ellipsoid bases (therefore this system is a cylindric billiard as introduced in [Sz(1993)]). □

For convenience and brevity, we will throughout use the concepts and notations, related to semi-dispersing billiards and hard ball systems, of the papers [K-S-Sz(1990)] and [Sim(1992)-I-II], respectively, and will only point out where and how different masses play a role.

Remark 2.3 By slightly generalizing the notion of semi-dispersing billiards with allowing Riemannian metrics different from the usual ones (see (2.2) of [K-S-Sz(1990)]), we could have immediately identified the standard billiard ball system (M, S^{ℝ}, μ)_{m⃗,L} with mass vector m⃗ with a (generalized) semi-dispersing billiard. (In the coordinates of the proof of Lemma 2.1, the Riemannian metric is

(dρ)^{2} = ∑_{i=1}^{N} (∥dq̂_{i}∥^{2} + ∥ds∥^{2})

where ∥ds∥^{2} is the square of the natural Riemannian metric on the unit sphere 𝕊^{d−1}.) Then the standard billiard ball system (M, S^{ℝ}, μ)_{m⃗,L} itself will be a (generalized) semi-dispersing billiard. The advantage is that — as it is easy to see — the results of [Ch-S(1987)] and [K-S-Sz(1990)] remain valid for this class, too. For simplifying our exposition therefore, we will omit the change of coordinates of Lemma 2.1 and will be using the notion of semi-dispersing billiards in this slightly more general sense. In this sense the model of the Introduction is a semi-dispersing billiard.

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^{N·ν−ν−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.

As it is 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-dispersing billiards! — there are no trajectories at all with a finite accumulation point of collision moments (see also [G(1981)] and [B-F-K(1995)]).

As a result, for an arbitrary non-singular orbit segment S^{[a,b]}x of the standard billiard ball flow, there is a uniquely defined maximal sequence a ≤ t_{1} < t_{2} < ⋯ < t_{n} ≤ b : n ≥ 0 of collision times and a uniquely defined sequence σ_{1} < σ_{2} < ⋯ < σ_{n} of “colliding pairs”, i. e. σ_{k} = {i_{k}, j_{k}} whenever Q(t_{k}) = π(S^{t_{k}}x) ∈ ∂C̃_{i_{k},j_{k}}. The sequence Σ := Σ(S^{[a,b]}x) := (σ_{1}, σ_{2}, …, σ_{n}) is called the symbolic collision sequence of the trajectory segment S^{[a,b]}x.

Definition 2.4. We say that the symbolic collision sequence Σ = (σ_{1}, …, σ_{n}) is connected if the collision graph of this sequence:

G_{Σ} := (V = {1, 2, …, N}, E_{Σ} := {{i_{k}, j_{k}} : where σ_{k} = {i_{k}, j_{k}}, 1 ≤ k ≤ n})

is connected.

Definition 2.5. We say that the symbolic collision sequence Σ = (σ_{1}, …, σ_{n}) is C-rich, with C being a natural number, if it can be decomposed into at least C consecutive, disjoint collision sequences in such a way that each of them is connected.

Neutral Subspaces, Advance 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

Z = {(w_{1}, w_{2}, …, w_{N}) ∈ (ℝ^{ν})^{N} : ∑_{i=1}^{N} m_{i}w_{i} = 0} .  (2.6)

Definition 2.7. 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 ∈ Z : ∃(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 Z 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:

N_{t}(S^{[a,b]}x) = N_{0} (S^{[a−t,b−t]}(S^{t}x)) .  (2.8)

It is clear that the neutral space N_{t}(S^{[a,b]}x) can be canonically identified 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)] ).

Our next definition is that of the advance. Consider a non-singular orbit segment S^{[a,b]}x with symbolic collision sequence Σ = (σ_{1}, …, σ_{n}) (n ≥ 1) as at the beginning of the present section. For x = (Q, V) ∈ M and W ∈ Z, ∥W ∥ sufficiently small, denote T_{W} (Q, V) := (Q + W, V).

Definition 2.9. For any 1 ≤ k ≤ n and t ∈ [a, b], the advance

α(σ_{k}) : N_{t}(S^{[a,b]}x) → ℝ

is the unique linear extension of the linear functional defined in a sufficiently small neighbourhood of the origin of N_{t}(S^{[a,b]}x) in the following way:

α(σ_{k})(W) := t_{k}(x) − t_{k}(S^{−t}T_{W} S^{t}x).

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 fundamental theorem for semi–dispersing billiards, see Condition (ii) of Theorem 3.6 and Definition 2.12 in [K-S-Sz(1990)] .

Definition 2.10.

(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 2.11. 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.

Connecting Path Formula for particles with different masses

The Connecting Path Formula, abbreviated as CPF, was discovered for particles with identical masses in [Sim(1992)-II] . Its goal was to give an explicit description (a linear coordinatization) of the neutral linear space N_{0}(S^{[−T,0]}x_{0}) in the language of the “advances” of the occuring collisions by using, as coefficients, linear expressions of the (pre-collision and post-collision) velocity differences of the colliding paticles. Since it relied upon the conservation of the momentum, it has been natural to expect that the CPF can be generalized for particles with different masses as well. The case is, indeed, this, and next we give this generalization for particles with different masses. Since its structure is the same as that of the CPF for identical masses, our exposition follows closely the structure of [Sim(1992)-II] .

Consider a phase point x_{0} ∈ M whose trajectory segment S^{[−T,0]}x_{0} is not singular, T > 0. In the forthcoming discussion the phase point x_{0} and the positive number T will be fixed. All the velocities v_{i}(t) ∈ ℝ^{ν} i ∈ {1, 2, …, N}, −T ≤ t ≤ 0 appearing in the considerations are velocities of certain balls at specified moments t and always with the starting phase point x_{0}. (v_{i}(t) is the velocity of the i-th ball at time t.) We suppose that the moments 0 and −T are not moments of collision. We label the balls by the natural numbers 1, 2, …, N (so the set {1, 2, …, N} is always the vertex set of the collision graph) and we denote by e_{1}, e_{2}, …, e_{n} the collisions of the trajectory segment S^{[−T,0]}x_{0} (i.e. the edges of the collision graph) so that the time order of these collisions is just the opposite of the order given by the indices. A few more definitions and notations:

1. t_{i} = t(e_{i}) denotes the time of the collision e_{i}, so 0 > t_{1} > t_{2} > ⋯ > t_{n} > −T.

2. If t ∈ ℝ is not a moment of collision (−T ≤ t ≤ 0), then

Δq_{i}(t) : N_{0}(S^{[−T,0]}x_{0}) → ℝ^{ν}

is a linear mapping assigning to every element W ∈ N_{0}(S^{[−T,0]}x_{0}) the displacement of the i-th ball at time t, provided that the configuration displacement at time zero is given by W. Originally, this linear mapping is only defined for vectors W ∈ N_{0}(S^{[−T,0]}x_{0}) close enough to the origin, but it can be uniquely extended to the whole space N_{0}(S^{[−T,0]}x_{0}) by preserving linearity.

3. α(e_{i}) denotes the advance of the collision e_{i}, thus

α(e_{i}) : N_{0}(S^{[−T,0]}x_{0}) → ℝ

is a linear mapping (i = 1, 2, …, n).

4. The integers 1 = k(1) < k(2) < ⋯ < k(l_{0}) ≤ n are defined by the requirement that for every j (1 ≤ j ≤ l_{0}) the graph {e_{1}, e_{2}, …, e_{k(j)}} consists of N − j connected components (on the vertex set {1, 2, …, N}, as always!) while the graph {e_{1}, e_{2}, …, e_{k(j)−1}} consists of N − j + 1 connected components and, moreover, we require that the number of connected components of the whole graph {e_{1}, e_{2}, …, e_{n}} be equal to N − l_{0}. It is clear from this definition that the graph

T = {e_{k(1)}, e_{k(2)}, …, e_{k(l_{0})}}

does not contain any loop, especially l_{0} ≤ N − 1.

Here we make two remarks commenting the above notions.

Remark 2.12. We often do not indicate the variable W ∈ N_{0}(S^{[−T,0]}x_{0}) of the linear mappings Δq_{i}(t) and α(e_{i}), for we will not be dealing with specific neutral tangent vectors W but, instead, we think of W as a typical (running) element of N_{0}(S^{[−T,0]}x_{0}) and Δq_{i}(t), α(e_{i}) as linear mappings defined on N_{0}(S^{[−T,0]}x_{0}) in order to obtain an appropriate description of the neutral space N_{0}(S^{[−T,0]}x_{0}).

Remark 2.13. If W ∈ N_{0}(S^{[−T,0]}x_{0}) has the property Δq_{i}(0)[W] = λv_{i}(0) for some λ ∈ ℝ and for all i ∈ {1, 2, …, N} (here v_{i}(0) is the velocity of the i-th ball at time zero), then α(e_{k})[W] = λ for all k = 1, 2, …, n. This particular W corresponds to the direction of the flow. In the sequel we shall often refer to this remark.

Let us fix two distinct balls α, ω ∈ {1, 2, …, N} that are in the same connected component of the collision graph G_{n} = {e_{1}, e_{2}, …, e_{n}}. The CPF expresses the relative displacement Δq_{α}(0) − Δq_{ω}(0) in terms of the advances α(e_{i}) and the relative velocities occuring at these collisions e_{i}. In order to be able to formulate the CPF we need to define some graph-theoretic notions concerning the pair of vertices (α, ω).

Definition 2.14. Since the graph T = {e_{k(1)}, e_{k(2)}, …, e_{k(l_{0})}} contains no loop and the vertices α, ω belong to the same connected component of T , there is a unique path Π(α, ω) = {f_{1}, f_{2}, …, f_{h}} in the graph T connecting the vertices α and ω. The edges f_{i} ∈ T (i = 1, 2, …, h) are listed up successively along this path Π(α, ω) starting from α and ending at ω. The vertices of the path Π(α, ω) are denoted by α = B_{0}, B_{1}, B_{2}, …, B_{h} = ω indexed along this path going from α to ω, so the edge f_{i} connects the vertices B_{i−1} and B_{i} (i = 1, 2, …, h).

When trying to compute Δq_{α}(0) − Δq_{ω}(0) by using the advances α(e_{i}) and the relative velocities at these collisions, it turns out that not only the collisions f_{i} (i = 1, 2, …, h) make an impact on Δq_{α}(0) − Δq_{ω}(0), but some other adjacent edges too. This motivates the following definition:

Definition 2.15. Let i ∈ {1, 2, …, h − 1} be an integer. We define the set A_{i} of adjacent edges at the vertex B_{i} as follows:

A_{i} = {e_{j} : j ∈ {1, 2, …, n} & (t(e_{j}) − t(f_{i})) · (t(e_{j}) − t(f_{i+1})) < 0 &
B_{i} is a vertex of e_{j}}.

We adopt a similar definition to the sets A_{0}, A_{h} of adjacent edges at the vertices B_{0} and B_{h}, respectively:

Definition 2.16.

A_{0} = {e_{j} : 1 ≤ j ≤ n & t(e_{j}) > t(f_{1}) & B_{0} is a vertex of e_{j}};
A_{h} = {e_{j} : 1 ≤ j ≤ n & t(e_{j}) > t(f_{h}) & B_{h} is a vertex of e_{j}}.

We note that the sets A_{0}, A_{1}, …, A_{h} are not necessarily mutually disjoint.

Finally, we need to define the “contribution” of the collision e_{j} to Δq_{α}(0) − Δq_{ω}(0) which is composed from the relative velocities just before and after the moment t(e_{j}) of the collision e_{j}.

Definition 2.17. For i ∈ {1, 2, …, h} the “contribution” Γ(f_{i}) of the edge f_{i} ∈ Π(α, ω) is given by the formula

Γ(f_{i}) = \{ \begin{matrix}v_{B_{i−1}}^{−}(t(f_{i})) − v_{B_{i}}^{−}(t(f_{i})), \\ if t(f_{i−1}) < t(f_{i}) & t(f_{i+1}) < t(f_{i}); \\ v_{B_{i−1}}^{+}(t(f_{i})) − v_{B_{i}}^{+}(t(f_{i})), \\ if t(f_{i−1}) > t(f_{i}) & t(f_{i+1}) > t(f_{i}); \\ \frac{1}{m_{B_{i−1}} + m_{B_{i}}} [m_{B_{i−1}}(v_{B_{i−1}}^{−}(t(f_{i})) − v_{B_{i}}^{−}(t(f_{i}))) \\ +m_{B_{i}}(v_{B_{i−1}}^{+}(t(f_{i})) − v_{B_{i}}^{+}(t(f_{i})))], \\ if t(f_{i+1}) < t(f_{i}) < t(f_{i−1}) \\ \frac{1}{m_{B_{i−1}} + m_{B_{i}}} [m_{B_{i−1}}(v_{B_{i−1}}^{+}(t(f_{i})) − v_{B_{i}}^{+}(t(f_{i}))) \\ +m_{B_{i}}(v_{B_{i−1}}^{−}(t(f_{i})) − v_{B_{i}}^{−}(t(f_{i})))], \\ if t(f_{i−1}) < t(f_{i}) < t(f_{i+1}). \end{matrix}

Here v_{B_{i}}^{−}(t(f_{i})) denotes the velocity of the B_{i}-th particle just before the collision f_{i} (occuring at time t(f_{i})) and, similarly, v_{B_{i}}^{+}(t(f_{i})) is the velocity of the same particle just after the mentioned collision. We also note that, by convention, t(f_{0}) = 0 > t(f_{1}) and t(f_{h+1}) = 0 > t(f_{h}). Apparently, the time order plays an important role in this definition.

Definition 2.18. For i ∈ {0, 1, 2, …, h} the “contribution” Γ_{i}(e_{j}) of an edge e_{j} ∈ A_{i} is defined as follows:

\begin{aligned}Γ_{i}(e_{j}) = & sign(t(f_{i}) − t(f_{i+1}))\frac{m_{C}}{m_{B_{i}} + m_{C}} · \\ & · [(v_{B_{i}}^{+}(t(e_{j})) − v_{C}^{+}(t(e_{j}))) − (v_{B_{i}}^{−}(t(e_{j})) − v_{C}^{−}(t(e_{j})))] \end{aligned}

where C is the vertex of e_{j} different from B_{i}.

Here again we adopt the convention of t(f_{0}) = 0 > t(e_{j}) (e_{j} ∈ A_{0}) and t(f_{h+1}) = 0 > t(e_{j}) (e_{j} ∈ A_{h}). We note that, by the definition of the set A_{i}, exactly one of the two possibilities t(f_{i+1}) < t(e_{j}) < t(f_{i}) and t(f_{i}) < t(e_{j}) < t(f_{i+1}) occurs. The subscript i of Γ is only needed because an edge e_{j} ∈ A_{i_{1}} ∩ A_{i_{2}} (i_{1} < i_{2}) has two contributions at the vertices B_{i_{1}} and B_{i_{2}} which are just the endpoints of e_{j}.

We are now in the position of formulating the Connecting Path Formula:

Proposition 2.19. Using all definitions and notations above, the following sum is an expression for Δq_{α}(0) − Δq_{ω}(0) in terms of the advances and relative velocities of collisions:

Δq_{α}(0) − Δq_{ω}(0) = ∑_{i=1}^{h} α(f_{i})Γ(f_{i}) + ∑_{i=0}^{h} ∑_{e_{j}∈A_{i}} α(e_{j})Γ_{i}(e_{j}).

The proof of the proposition follows the proof of Simányi’s CPF (Lemma 2.9 [Sim(1992)-II] ) with the only difference that Lemma 2.8 of [Sim(1992)-II] is replaced here by the following

Lemma 2.20. If e is a collision at time t between the particles B and C, then

v_{B}^{+}(t) − v_{C}^{−}(t) = \frac{m_{C}}{m_{B} + m_{C}} (v_{B}^{+}(t) − v_{C}^{+}(t)) + \frac{m_{B}}{m_{B} + m_{C}} (v_{B}^{−}(t) − v_{C}^{−}(t))

and

v_{B}^{+}(t) − v_{B}^{−}(t) = \frac{m_{C}}{m_{B} + m_{C}} [(v_{B}^{+}(t) − v_{C}^{+}(t)) − (v_{B}^{−}(t) − v_{C}^{−}(t))]

Proof. The lemma is an easy consequence of the conservation of momentum for the collision e. □

Remark 2.21. In Section 3, we will complexify the system, and also allow complex masses, in particular. It is easy to see that the CPF of Proposition 2.19 and the whole discussion of this subsection will still hold for complex masses if we assume, in addition, that for every e = {B, C}, occuring in the symbolic collision sequence of S^{[−T,0]}x_{0}, m_{B} + m_{C} ≠ 0.

3. The Complexified Billiard Map Partial Linearity of the Dynamics

The aim of this section is to properly understand the algebraic relationship between the kinetic data of the billiard flow measured at different moments of time.

From now on we shall investigate orbit segments S^{[0,T]}x_{0} (T > 0) of the standard billiard ball flow (M̃, {S^{t}}, μ, m⃗, L). We note here that M̃ = Q̃ × E, where the

configuration space Q̃ was defined right after (1.1) and E is the velocity sphere

E = {(v_{1}, …, v_{N}) ∈ ℝ^{νN} | ∑_{i=1}^{N} m_{i}v_{i} = 0 & ∑_{i=1}^{N} m_{i}∥v_{i}∥^{2} = 1}

introduced in Section 1. Also note that in the geometric-algebraic considerations of the upcoming sections we do not use the equivalence relation Ψ of the introduction. Later on even the conditions ∑_{i=1}^{N} m_{i}v_{i} = 0 and ∑_{i=1}^{N} m_{i}∥v_{i}∥^{2} = 1 will be dropped, cf. Remark 3.14.

The symbolic collision sequence of S^{[0,T]}x_{0} is denoted by

Σ (S^{[0,T]}x_{0}) = (σ_{1}, …, σ_{n}).

The symbol v_{i}^{k} = v_{i}^{+}(t_{k}) ∈ ℝ^{ν} shall denote the velocity q̇_{i} of the i-th ball right after the k-th collision σ_{k} = {i_{k}, j_{k}} (1 ≤ i_{k} < j_{k} ≤ N) occuring at time t_{k}, k = 1, …, n. Of course, there is no need to deal with the velocities right before collisions, since v_{i}^{−}(t_{k}) = v_{i}^{+}(t_{k−1}) = v_{i}^{k−1}. As usual, q_{i}^{k} ∈ 𝕋_{L}^{ν} = ℝ^{ν}/L · ℤ^{ν} denotes the position of the center of the i-th ball at the moment t_{k} of the k-th collision.

Let us now fix the symbolic collision sequence Σ = (σ_{1}, …, σ_{n}) (n ≥ 1), and explore the algebraic relationship between the data

{q_{i}^{k−1}, v_{i}^{k−1}| i = 1, …, N}

and

{q_{i}^{k}, v_{i}^{k}| i = 1, …, N} ,

k = 1, …, n. By definition, the data {q_{i}^{0}, v_{i}^{0}| i = 1, …, N} correspond to the initial (non-collision) phase point x_{0}. We also set t_{0} = 0.

Since we would like to carry out arithmetic operations on these data, the periodic positions q_{i}^{k} ∈ 𝕋_{L}^{ν} are not suitable for this purpose. Therefore, instead of studying the genuine orbit segments S^{[0,T]}x_{0}, we will deal with their Euclidean liftings.

Proposition 3.1. Let S^{[0,T]}x_{0} = {q_{i}(t), v_{i}(t)| 0 ≤ t ≤ T } be an orbit segment as above, and assume that certain pre-images (Euclidean liftings) q̃_{i}(0) = q̃_{i}^{0} ∈ ℝ^{ν} of the positions q_{i}(0) ∈ 𝕋_{L}^{ν} = ℝ^{ν}/L · ℤ^{ν} are given. Then there is a uniquely defined, continuous, Euclidean lifting {q̃_{i}(t) ∈ ℝ^{ν}| 0 ≤ t ≤ T } of the given orbit segment that is an extension of the initial lifting {q̃_{i}^{0}| i = 1, …, N}. Moreover, for every collision σ_{k} there exists a uniquely defined integer vector a_{k} ∈ ℤ^{ν} – named the adjustment vector of σ_{k} – such that

∥q̃_{i_{k}}^{k} − q̃_{j_{k}}^{k} − L · a_{k}∥^{2} − 4r^{2} = 0.  (3.2)

We shall call the orbit segment ω̃ = {q̃_{i}^{k}, v_{i}^{k}| i = 1, …, N; k = 0, …, n} a lifted orbit segment with the system of adjustment vectors A = (a_{1}, …, a_{n}) ∈ ℤ^{nν}.

The proof of this proposition is straightforward.

The next result establishes the already mentioned polynomial relationships between the kinetic data {q̃_{i}^{k−1}, v_{i}^{k−1}| i = 1, …, N} and {q̃_{i}^{k}, v_{i}^{k}| i = 1, …, N}, k = 1, …, n.

Proposition 3.3. Using all notations and notions from above, one has the following polynomial relations between the kinetic data at σ_{k−1} and σ_{k}, k = 1, …, n. In order to simplify the notations, we write down these equations as if σ_{k} were {1, 2}:

v_{i}^{k} = v_{i}^{k−1}, i ∉ {1, 2},  (3.4)
m_{1}v_{1}^{k} + m_{2}v_{2}^{k} = m_{1}v_{1}^{k−1} + m_{2}v_{2}^{k−1}  (3.5)

(conservation of the momentum),

v_{1}^{k} − v_{2}^{k} = v_{1}^{k−1} − v_{2}^{k−1} − \frac{1}{2r^{2}} ⟨v_{1}^{k−1} − v_{2}^{k−1}; q̃_{1}^{k} − q̃_{2}^{k} − L · a_{k}⟩ · (q̃_{1}^{k} − q̃_{2}^{k} − L · a_{k})

(3.6)

(reflection of the relative velocity determined by the elastic collision),

q̃_{i}^{k} = q̃_{i}^{k−1} + τ_{k}v_{i}^{k−1}, i = 1, …, N,  (3.7)

where the time slot τ_{k} = t_{k} − t_{k−1} (t_{0} := 0) is determined by the quadratic equation

∥q̃_{1}^{k−1} − q̃_{2}^{k−1} + τ_{k} (v_{1}^{k−1} − v_{2}^{k−1}) − L · a_{k}∥^{2} = 4r^{2}.  (3.8)

The proof of this proposition is also obvious.

Remark. Observe that the new velocities v_{1}^{k} and v_{2}^{k} can be computed from (3.5)- (3.6) as follows:

\begin{aligned}v_{1}^{k} & = v_{1}^{k−1} − \frac{m_{2}}{2r^{2}(m_{1} + m_{2})} ⟨v_{1}^{k−1} − v_{2}^{k−1}; q̃_{1}^{k} − q̃_{2}^{k} − L · a_{k}⟩ · (q̃_{1}^{k} − q̃_{2}^{k} − L · a_{k}) , \\ v_{2}^{k} & = v_{2}^{k−1} + \frac{m_{1}}{2r^{2}(m_{1} + m_{2})} ⟨v_{1}^{k−1} − v_{2}^{k−1}; q̃_{1}^{k} − q̃_{2}^{k} − L · a_{k}⟩ · (q̃_{1}^{k} − q̃_{2}^{k} − L · a_{k}) . \end{aligned}

(3.9)

We also note that the equations above even extend analytically to the case when one of the two masses, say m_{1}, is equal to zero:

\begin{aligned}v_{1}^{k} & = v_{1}^{k−1} − \frac{1}{2r^{2}} ⟨v_{1}^{k−1} − v_{2}^{k−1}; q̃_{1}^{k} − q̃_{2}^{k} − L · a_{k}⟩ · (q̃_{1}^{k} − q̃_{2}^{k} − L · a_{k}) , \\ v_{2}^{k} & = v_{2}^{k−1}. \end{aligned} (3.10)

We call the attention of the reader to the fact that in our understanding the symbol ⟨ .; . ⟩ denotes the Euclidean inner product of ν-dimensional real vectors and ∥ . ∥ is the corresponding norm.

The Complexification of the Billiard Map

Given the pair (Σ, A) = (σ_{1}, …, σ_{n}; a_{1}, …, a_{n}), the equations (3.4)-(3.8) make it possible to iteratively compute the kinetic data {q̃_{i}^{k}, v_{i}^{k}| i = 1, …, N} by using the preceding data {q̃_{i}^{k−1}, v_{i}^{k−1}| i = 1, …, N}. Throughout these computations one only uses the field operations and square roots. (The latter one is merely used when computing τ_{k} as the root of the quadratic equation (3.8). Obviously, since – at the moment – we are dealing with genuine, real orbit segments, in this dynamically realistic situation the equations (3.8) have two distinct, positive real roots, and τ_{k} is the smaller one.) Therefore, the data {q̃_{i}^{k}, v_{i}^{k}| i = 1, …, N} can be expressed as certain algebraic functions of the initial data {q̃_{i}^{0}, v_{i}^{0}| i = 1, …, N}, and the arising algebraic functions merely contain field operations, square roots, the radius r of the balls, the size L > 0 of the torus and, finally, the masses m_{i} as constants.

Since these algebraic functions make full sense over the complex field ℂ and, after all, our proof of the Theorem requires the complexification, we are now going to complexify the whole system by considering the kinetic variables, the size L, and the masses as complex ones and retaining the polynomial equations (3.4)-(3.8). However, due to the ambiguity of selecting a root of (3.8) out of the two, it proves to be important to explore first the algebraic frame of the relations (3.4)-(3.8) connecting the studied variables. This is what we do now.

The Field Extension Associated With the Pair (Σ, A)

For avoiding misunderstanding we immediately stress that the field extensions 𝕂 = 𝕂 (Σ; A) to be defined below will also depend on a sequence τ⃗ = (τ_{0}, …, τ_{n−1}) of field elements to be introduced succesively in Definition 3.11. If we also want to emphasize the dependence of 𝕂 on τ⃗, then we will write 𝕂 = 𝕂 (Σ; A) = (Σ, A, τ⃗).

We are going to define the commutative function field 𝕂 = 𝕂 (Σ; A) generated by all functions

{(q̃_{i}^{k})_{j}, (v_{i}^{k})_{j}| i = 1, …, N; k = 0, …, n; j = 1, …, ν}

of the lifted orbit segments corresponding to the given parameters

(Σ, A) = (σ_{1}, …, σ_{n}; a_{1}, …, a_{n})

in such a way that the field 𝕂 (Σ; A) incorporates all algebraic relations among these variables that are consequences of the equations (3.4)-(3.8). (Here the subscript j denotes the j-th component of a ν-vector.) In our setup the ground field of allowed constants (coefficients) is, by definition, the complex field ℂ. The precise definition of 𝕂_{n} = 𝕂 (Σ; A) is

Definition 3.11. For n = 0 the field 𝕂_{0} = 𝕂(∅; ∅) is the transcendental extension ℂ(B) of the coefficient field ℂ by the algebraically independent formal variables

B = {(q̃_{i}^{0})_{j}, (v_{i}^{0})_{j}, m_{i}, L| i = 1, …, N; j = 1, …, ν} .

Suppose now that n > 0 and the commutative field 𝕂_{n−1} = 𝕂 (Σ′; A′) has already been defined, where Σ′ = (σ_{1}, …, σ_{n−1}), A′ = (a_{1}, …, a_{n−1}). Then we consider the quadratic equation b_{n}τ_{n}^{2}+c_{n}τ_{n}+d_{n} = 0 in (3.8) with k = n as a polynomial equation defining a new field element τ_{n} to be adjoined to the field 𝕂_{n−1} = 𝕂 (Σ′; A′). (Recall that the coefficients b_{n}, c_{n}, d_{n} come from the field 𝕂_{n−1}.)

There are two possibilities:

(i) The quadratic polynomial b_{n}x^{2} + c_{n}x + d_{n} is reducible over the field 𝕂_{n−1}. Then we take 𝕂_{n} = 𝕂_{n−1}.

(ii) The polynomial b_{n}x^{2} + c_{n}x + d_{n} is irreducible over the field 𝕂_{n−1}. Then we define 𝕂_{n} = 𝕂 (Σ; A) as the extension of 𝕂_{n−1} = 𝕂 (Σ′; A′) by the root τ_{n} of this irreducible polynomial.

Remark 3.12. The importance of the field 𝕂_{n} is underscored by the fact that this field encodes all algebraic relations among the kinetic data that follow from the polynomial equations (3.4)-(3.8). Furthermore, Proposition 3.3 gives us an iterative computation rule for successively obtaining the kinetic variables

{q̃_{i}^{k}, v_{i}^{k}| i = 1, …, N} ,

k = 0, …, n. The field 𝕂_{n} = 𝕂(Σ, A) is, after all, the algebraic frame of such computations. We note that – everywhere in this paper – the symbols ⟨ .; . ⟩ and ∥ . ∥^{2} do not refer to an Hermitian inner product but, rather, ⟨x; y⟩ = ∑_{j=1}^{ν} x_{j}y_{j} and ∥x∥^{2} = ∑_{j=1}^{ν} x_{j}^{2}, so that these expressions retain their polynomial form.

Remark 3.13. Later on it will be necessary to express each kinetic variable (q̃_{i}^{k})_{j} and (v_{i}^{k})_{j} as an algebraic function of the initial variables of B. This raises, however, an important question: Which one of the two roots τ_{k} of (3.8) should be considered during these computations? This is no problem if the polynomial b_{k}x^{2} + c_{k}x + d_{k}, defining τ_{k}, is irreducible over the field 𝕂_{k−1}, because the two roots of this polynomial are algebraically equivalent over 𝕂_{k−1}, and any of them can be chosen as τ_{k}. However, if the polynomial b_{k}x^{2} + c_{k}x + d_{k} is reducible over the field 𝕂_{k−1}, then it is necessary to make a decision and assign to one of the two roots (both in the field 𝕂_{k−1} now) the role of τ_{k}. This is what we do. The result is the field 𝕂_{n} = 𝕂(Σ; A) endowed with a distinguished n-tuple (τ_{1}, …, τ_{n}) of its elements. The field 𝕂_{n} with the distinguished n-tuple τ⃗ is denoted by 𝕂_{n} = 𝕂(Σ; A; τ⃗). Then the algebraic object 𝕂(Σ; A; τ⃗) completely controls the whole process of the iterative computation of the kinetic variables.

Note that in the genuine, real case the two roots of b_{k}x^{2} + c_{k}x + d_{k} are distinct, positive real numbers, and – by the nature of the billiard dynamics – the chosen value is always the smaller one.

Finally, we mention here a simple case when the polynomial b_{k}x^{2} + c_{k}x + d_{k} is reducible over the field 𝕂_{k−1}. Namely, this takes place whenever σ_{k−1} = σ_{k} and a_{k−1} = a_{k}. In that case the value τ_{k} ≡ 0 is clearly a solution of the above polynomial b_{k}x^{2} + c_{k}x + d_{k}, and, being so, that polynomial is reducible over the field 𝕂_{k−1}, see also 3.31–3.33.

Remark 3.14. The reader may wonder why we did not postulate the algebraic dependencies ∑_{i=1}^{N} m_{i}v_{i}^{k} = 0, ∑_{i=1}^{N} m_{i}∥v_{i}^{k}∥^{2} = 1? The answer is the following: The definition of the neutral linear space together with its characterization via the Connecting Path Formula (CPF, see Section 2 of this paper or Lemma 2.9 of [Sim(1992)-II]), i. e. the partial linearity of the billiard flow, are invariant under

(1) all uniform velocity translations (adding the same vector to all velocities) and

(2) time rescalings.

Remark 3.15. All fields occuring in this paper are only defined up to an isomorphism over the coefficient field ℂ. Therefore, the statements like “the field 𝕂_{2} is an extension of 𝕂_{1}” should be understood as follows: 𝕂_{1} is a subfield of 𝕂_{2} after the natural identification of the generating variables bearing the same name. For the necessary notions, properties, and results from the theory of field extensions and Galois theory, the reader is kindly recommended to look up, for instance, the book by I. Stewart, [St(1973)].

Remark 3.16. As said before (see Definition 3.11), the collection B of elements of the field 𝕂(Σ; A; τ⃗) is a base of transcendence in that field over the subfield ℂ, and the degree of the extension 𝕂(Σ; A; τ⃗) : ℂ(B) = 𝕂_{n} : 𝕂_{0} is a power of two. Moreover, each of the following sets is a generator for the field 𝕂_{n}:

(a) ℂ ∪ B ∪ {τ_{1}, …, τ_{n}};

(b) ℂ∪{m_{1}, …, m_{N}, L}∪{(q̃_{i}^{n})_{j}, (v_{i}^{n})_{j}| i = 1, …, N; j = 1, …, ν}∪{τ_{1}, …, τ_{n}};

(c) ℂ∪{m_{1}, …, m_{N}, L}∪{(q̃_{i}^{k})_{j}, (v_{i}^{k})_{j}| i = 1, …, N; j = 1, …, ν; k = 0, …, n}.

Remark 3.17. The above procedure of constructing field extensions is closely related to the classical theory of geometric constructions by a ruler and compass, and this is not for surprise: the billiard trajectory is constructed by intersecting a straight line with a sphere and then reflecting it across the tangent hyperplane of the sphere. This is a sort of classical geometric construction in ν dimensions. For the details see, for instance, §57 and §60 of [VDW (1955)], or Chapter 5 of [St(1973)].

Let us fix the pair (Σ; A) and the n-tuple τ⃗ = (τ_{1}, …, τ_{n}) of elements of the field 𝕂(Σ, A). We are now defining a (2ν + 1)N + 1-dimensional complex analytic manifold Ω̃ = Ω̃ (Σ, A, τ⃗) and certain holomorphic functions q̃_{i}^{k}, v_{i}^{k} : Ω̃ → ℂ^{ν}, m_{i} : Ω̃ → ℂ, L : Ω̃ → ℂ, τ_{k} : Ω̃ → ℂ, i = 1, …, N, k = 0, …, n, (τ_{0} is not defined). But first we introduce

Definition 3.18. Define the domain D = D (Σ, A, τ⃗) ⊂ ℂ^{(2ν+1)N+1} as the set of all complex (2ν + 1)N + 1-tuples

{(q̃_{i}^{0})_{j}, (v_{i}^{0})_{j}, m_{i}, L| i = 1, …, N; j = 1, …, ν}

for which

(a) the leading coefficient ∥v_{i_{k}}^{k−1} − v_{j_{k}}^{k−1}∥^{2} and the discriminant of the quadratic equation (3.8) is never zero, k = 1, …, n, (The latter condition is equivalent to

⟨v_{i_{k}}^{k−1} − v_{j_{k}}^{k−1}; q̃_{i_{k}}^{k} − q̃_{j_{k}}^{k} − L · a_{k}⟩ ≠ 0,

see also Remark 3.29), and

(b) m_{i_{k}} + m_{j_{k}} ≠ 0 for k = 1, …, n, provided that we carry out the iterative computation of the kinetic variables

{(q̃_{i}^{k})_{j}, (v_{i}^{k})_{j}| i = 1, …, N; j = 1, …, ν; k = 0, …, n}

as governed by the polynomial equations (3.4)-(3.8) and by the fixed selection of elements τ⃗.

Remark 1. When inverting the dynamics, a useful remark is that

∥v_{i_{k}}^{k−1} − v_{j_{k}}^{k−1}∥^{2} = ∥v_{i_{k}}^{k} − v_{j_{k}}^{k} ∥^{2} ,

and

⟨v_{i_{k}}^{k−1} − v_{j_{k}}^{k−1}; q̃_{i_{k}}^{k} − q̃_{j_{k}}^{k} − L · a_{k}⟩ = − ⟨v_{i_{k}}^{k} − v_{j_{k}}^{k}; q̃_{i_{k}}^{k} − q̃_{j_{k}}^{k} − L · a_{k}⟩ .

Remark 2. Note that we require the validity of (a) for any branch of taking the square roots when computing the τ_{k}’s with irreducible defining polynomials (3.8).

Lemma 3.19. The complement set ℂ^{(2ν+1)N+1} \ D is a proper, closed, algebraic subset of ℂ^{(2ν+1)N+1}, especially a finite union of complex analytic submanifolds with codimension at least one, so the open set D ⊂ ℂ^{(2ν+1)N+1} is connected and dense.

Proof. We have the polynomial equations (3.4)-(3.8) making it possible to set up an algorithm for iteratively computing the kinetic variables measured at different times. The point is not just this iterative algorithm, but also its invertibility (time reversibility). The inverse process has similar algebraic properties, for it just means time reversal. One can easily prove by an induction on the length n of Σ that the iteratively defined dynamics determines a several-to-several mapping with maximum rank between a nonempty Zariski open set of the kinetic variables with superscript 1 and a nonempty Zariski open set of the kinetic variables measured right after the last reflection (with superscript n). (A Zariski open set is the complement of a closed algebraic set defined as the simultaneous zero set of finitely many polynomials.) Then one can go ahead with the induction from n to n+1, because all possible obstructions to extending the process only occur on proper algebraic submanifolds of the Zariski open set of kinetic variables with superscript n.

Definition 3.20. We define Ω̃ (Σ, A, τ⃗) as the set of all complex (2ν(n + 1)N + N + n + 1)-tuples

\begin{matrix}ω = \\ ( (q̃_{i}^{k})_{j} , (v_{i}^{k})_{j}, m_{i}, L, τ_{l}| i = 1, …, N; k = 0, …, n; j = 1, …, ν; l = 1, …, n) \end{matrix}

for which these coordinates (1) are interrelated by the equations (3.4)–(3.8),

(2) respect the choices of τ_{k} prescribed by τ⃗ (cf 3.11, and also 3.13), and the vector

x⃗(ω) = ((q̃_{i}^{0})_{j} , (v_{i}^{0})_{j}, m_{i}, L| i = 1, …, N; j = 1, …, ν)

of initial data belongs to the set D (Σ, A, τ⃗) defined above.

It is clear now that q̃_{i}^{k}, v_{i}^{k} : Ω̃ → ℂ^{ν}, m_{i} : Ω̃ → ℂ, L : Ω̃ → ℂ, and τ_{k} : Ω̃ → ℂ (i = 1, …, N, k = 0, …, n) are holomorphic functions on Ω̃ = Ω̃ (Σ, A, τ⃗). The complex analytic manifold Ω̃ endowed with the above holomorphic functions can justifiably considered as the complexification of the (Σ, A, τ⃗)-iterated billiard map.

Remark 1. The careful reader has certainly done the following principal observation: the τ_{k}’s figure in two different roles. They first denote successively chosen field elements of the extensions introduced in Definition 3.11, and secondly they also denote complex-valued functions on Ω̃ (or multivalued complex functions on D, cf. Definitions 3.20 and 3.18).

Remark. It is very likely that the manifold Ω̃ is connected. Nevertheless, in our proof we do not need this connectedness, and are not going to pursue the goal of proving it.

The Complex Neutral Space N (ω)

Fix a base point ω ∈ Ω̃ (Σ, A, τ⃗). The tangent space T_{ω}Ω̃ of Ω̃ at ω consists of all complex vectors

x = (δq̃_{i}^{0}, δv_{i}^{0}, δm_{i}, δL | i = 1, …, N) ∈ ℂ^{(2ν+1)N+1},

thus T_{ω}Ω̃ can be naturally identified with the complex vector space ℂ^{(2ν+1)N+1}. Set

\begin{matrix}N (ω) = \{x = (δq̃_{i}^{0}, δv_{i}^{0}, δm_{i}, δL | i = 1, …, N) ∈ T_{ω}Ω̃| \\ D_{x}(v_{i}^{k}) = D_{x}(m_{i}) = D_{x}(L) = 0 for i = 1, …, N; k = 0, …, n\}, \end{matrix}  (3.21)

where D_{x}( .) denotes the directional differentiation in the direction of x. Clearly, the complex linear subspace N (ω) of ℂ^{(2ν+1)N+1} is the proper complex analogue of the neutral space of a genuine, real orbit segment.

Remark. Since δv_{i}^{0}, δm_{i}, and δL must be equal to zero for a neutral vector x ∈ N (ω), we shall simply write x = (δq̃_{1}^{0}, …, δq̃_{N}^{0}) instead of indicating the zero entries.

The proof of the following proposition is completely analogous to that one for the real case, and therefore we omit it.

Proposition 3.22. For every tangent vector x ∈ N (ω) (ω ∈ Ω̃ (Σ, A, τ⃗)) and 1 ≤ k ≤ n we have

D_{x} (q̃_{i_{k}}^{k−1} − q̃_{j_{k}}^{k−1}) = (α_{k}(x) − α_{k−1}(x)) · (v_{i_{k}}^{k−1} − v_{j_{k}}^{k−1}) ,  (3.23)
D_{x}(t_{k}) = −α_{k}(x),  (3.24)

and

D_{x} (τ_{k}) = D_{x}(t_{k} − t_{k−1}) = α_{k−1}(x) − α_{k}(x),  (3.25)

where, by definition, t_{0} = 0 = α_{0}(x). The functions α_{k} : N (ω) → ℂ (k = 1, …, n) are linear functionals. The name of the functional α_{k} is the advance of the k-th collision σ_{k} = {i_{k}, j_{k}}, see also Section 2.

This result pregnantly shows that, indeed, the vector space N (ω) describes the linearity of the (Σ, A, τ⃗)-iterated billiard map.

Remark 3.26. It is a matter of simple computation to convince ourselves that all assertions of Section 2 of the present paper and [Sim(1992)-II] pertaining to the neutral space N (ω) remain valid for the complexified dynamics. Here we only point out the two most important statements out of those results:

(1) The vector space N (ω) measures the ambiguity in determining the orbit segment ω purely by its velocity history {v_{i}^{k}(ω) : i = 1, …, N; k = 0, …, n} and the outer geometric data (m⃗, L). In other words, this means that if, locally in the phase space Ω̃ (in a small open set), two phase points ω_{1} and ω_{2} have the same velocities (v_{i}^{k}(ω_{1}) = v_{i}^{k}(ω_{2}), i = 1, …, N; k = 0, …, n) and geometric parameters (m⃗, L), then the initial data of ω_{1} and ω_{2} can only differ by a spatial translation by a vector from the space N (ω_{1}) = N (ω_{2}), and this statement is obviously reversible.

(2) The reflection laws for the neutral vectors are exactly the same as for velocities.

(3) The Connecting Path Formula (CPF, Lemma 2.9 of [Sim(1992)-II], or Proposition 2.19 in this paper) is applicable, and it enables us to compute the neutral space N (ω) via solving a homogeneous system of linear equations (now over the field ℂ)

∑_{k=1}^{n} α_{k}Γ_{ik} = 0, i = 1, …, n + P_{Σ} − N,  (3.27)

where each equation in (3.27) is a ν-dimensional complex vector equation, the coefficients Γ_{ik} = Γ_{ik}(ω) ∈ ℂ^{ν} are certain linear combinations of relative velocities v_{i}^{k}(ω)−v_{j}^{k}(ω) (the coefficients of those linear combinations are just fractional linear expressions of the masses, see definitions 2.17-2.18) and, finally, P_{Σ} denotes the number of connected components of the collision graph of Σ. For a bit more detailed explanation of (3.27), see the beginning part of Remark 4.6.

Remark 3.28. We note here that the ordering of the moments of collisions t(f_{i}) (which playes a significant role in definitions 2.17-2.18) is no longer meaningful over the unordered field ℂ. However, the use of the corresponding inequalities in 2.17-2.18 is purely technical/notational. Those inequalities only serve to introduce the combinatorial ordering of the collisions f_{i} (which are σ’s) given by the indexes of the symbolic collision sequence Σ = (σ_{1}, σ_{2}, …, σ_{n}).

Remark 3.29. As a straightforward consequence of the equation (3.6), we obtain that v_{i}^{k−1} = v_{i}^{k} for i = 1, …, N (i. e. the k-th reflection does not change the compound velocity) if and only if

⟨v_{i_{k}}^{k−1} − v_{j_{k}}^{k−1}; q̃_{i_{k}}^{k} − q̃_{j_{k}}^{k} − L · a_{k}⟩ = 0,

and this is just the case of a tangential collision. Easy calculation shows that the mentioned tangentiality occurs if and only if the two roots τ_{k} of (3.8) coincide.

It is obvious that dim_{ℂ}N (ω) is at least ν + 1 (as long as not all velocities are the same), because the flow direction and the uniform spatial translations are necessarily contained by N (ω).

Definition 3.30. (Definition of sufficiency). The orbit segment ω ∈ Ω̃ (Σ, A, τ⃗) is said to be sufficient if and only if dim_{ℂ}N (ω) = ν + 1.

Finally, the main result of the next section (Key Lemma 4.1) will use the following Definition 3.31. We say that the triple (Σ, A, τ⃗) has Property (A) iff the following assertion holds:

For every pair of indices 1 ≤ k < l ≤ n for which σ_{k} = σ_{l}, a_{k} = a_{l}, and σ_{j} ∩σ_{k} = ∅ for j = k + 1, …, l − 1, we have that τ_{l} ≢ − ∑_{j=k+1}^{l−1} τ_{j} in the algebraic selection τ⃗ of the time slots. (Note that in this case the quadratic polynomial b_{l}x^{2} + c_{l}x + d_{l} is reducible over the field 𝕂_{l−1} because the time slot τ_{l} ≡ − ∑_{j=k+1}^{l−1} τ_{j} ∈ 𝕂_{l−1} is automatically a solution of (3.8).)

Remark 3.32. It is obvious that the schemes (Σ, A, τ⃗) corresponding to genuine, real billiard trajectory segments enjoy Property (A). Indeed, in those cases even the equations σ_{k} = σ_{l}, a_{k} = a_{l}, and σ_{j} ∩ σ_{k} = ∅ (j = k + 1, …, l − 1) cannot hold simultaneously. Concerning the necessity for introducing Property (A), see Example I at the end of the next section.

Remark 3.33. For combinatorial schemes satisfying Property (A), the quadratic polynomial b_{l}x^{2} + c_{l}x + d_{l} is reducible over the field 𝕂_{l−1} (i. e. 𝕂_{l−1} = 𝕂_{l}) only in the case described above in 3.31. This will be a consequence of Main Lemma 4.21 (cf. Corollary 4.36).

4. The Notion of Richness. Richness Implies Sufficiency

The result of this section is

Key Lemma 4.1. There exists a positive number C(N) (depending merely on the number of balls N ≥ 2) with the following property: If a symbolic sequence Σ = (σ_{1}, …, σ_{n}) is C(N)-rich and the triple (Σ, A, τ⃗) has Property (A) (see Definition 3.31), then dim_{ℂ}N (ω) = ν + 1 (i. e. the trajectory is sufficient) for almost every ω ∈ Ω̃ (Σ, A, τ⃗). The real version of this result is also valid. (For the definition of a C-rich symbolic sequence, see 2.5.)

Remark 1. Plainly, the exceptional zero measure subset of Ω̃ (Σ, A, τ⃗) mentioned in this lemma must actually be a countable union of proper, analytic submanifolds.

As mentioned in the Introduction, the aim of sections 3 and 4, in particular that of Key Lemma 4.1, is to settle part (1) of the strategy detailed there. The methods are mainly algebraic, and for an easier separation of our dynamical vs. algebraic arguments, we are now going to describe what are the only properties of the dynamics (and nothing more) that we are going to use.

(1) The dynamical properties of an orbit segment S^{[0,T]}x_{0} with a symbolic collision sequence Σ are completely characterized by the equations (3.4)–(3.8) if we, in addition, specify τ_{k} as the smaller of the two, necessarily positive, roots of the equation (3.8). In the inductive steps of lemmas 4.9, 4.39, and 4.40 we also use the remark to Proposition 3.3 saying that the collision laws (3.9), themselves consequences of (3.5)–(3.6), extend analytically to the case of zero mass, cf. (3.10).

(2) Since our goal is to show that orbits with a rich collision structure are typically sufficient or, equivalently, their neutral subspaces have typically minimal possible dimension, we also need an algebraic characterization of the neutral subspace. This is attained through the CPF (see Proposition 2.19 and the system of linear equations (3.27)), and our arguments also use some simple inference about the CPF and the neutral subspaces from [Sim(1992)-II] .

(3) As we have seen in Section 3, the algebraic formalisms of points (1) and (2) can be complexified, and our forthcoming discussion is based upon this complexification.

Remark 2. In the course of proving the key lemma we will see that the thresholds C(N) (N ≥ 2) can actually be defined by the recursion C(2) = 1, C(N) = \frac{N}{2} ·

max\{C(N − 1), 3\} (N ≥ 3), that is, C(2) = 1, C(N) = \frac{3N!}{2^{N−1}} for N ≥ 3. We note that, as it has been proved in [K-S-Sz(1991)], for N = 3 the value C(3) = 2 already does the job.

Proof of Key Lemma 4.1. All the rest of this section is devoted to the inductive prof of the key lemma, and it will be split into several lemmas.

First of all, we prove the original, complex version of the key lemma by an induction on the number N. Once it is completed, we will make the necessary remark by showing that the validity of the key lemma carries over to the real case.

The Dichotomy. Algebraic Characterization of Sufficiency

Lemma 4.2. Consider the open, connected and dense subset D of ℂ^{(2ν+1)N+1} formed by all possible complex (2ν + 1)N + 1-tuples

x⃗(ω) = ((q̃_{i}^{0}(ω))_{j} , (v_{i}^{0}(ω))_{j} , m_{i}(ω), L(ω)| i = 1, …, N; j = 1, …, ν) , (ω ∈ Ω̃)

of initial data of the complex (Σ, A, τ⃗)-dynamics, see Definitions 3.18 and 3.20. We claim that there are finitely many complex polynomials P_{1}(x⃗), …, P_{s}(x⃗) (x⃗ ∈ ℂ^{(2ν+1)N+1}), canonically determined by the discrete parameters (Σ, A, τ⃗), such that for every x⃗ ∈ D (Σ, A, τ⃗) the following implication holds true:

(∃ ω ∈ Ω̃ (Σ, A, τ⃗) such that x⃗ = x⃗(ω) & dim_{ℂ}N (ω) > ν + 1)
⟹ P_{1}(x⃗) = P_{2}(x⃗) = ⋯ = P_{s}(x⃗) = 0.

(4.3)

Remark. The left-hand-side in the above equivalence precisely says that some branch of the multiple-valued (Σ, A, τ⃗)-dynamics with initial data x⃗ is not sufficient, see also Remark 2 to Definition 3.18.

Proof. This proof is a typical application of the Connecting Path Formula (see Proposition 2.19 here or Lemma 2.9 of [Sim(1992)-II]) and Proposition 3.4 of [Sim(1992)-II]. Namely, Proposition 3.4 of [Sim(1992)-II] asserts that

dim_{ℂ}N (ω) = ν · P_{Σ} + dim_{ℂ}{α_{1}, …, α_{n}},  (4.4)

where, as said before, P_{Σ} denotes the number of connected components of the collision graph of Σ, and {α_{1}, …, α_{n}} is a shorthand for the complex linear space of all possible n-tuples (α_{1}(x), …, α_{n}(x)) of advances, x ∈ N (ω).

Remark 4.5. It is worth noting here that our present formula (4.4) differs from its counterpart in Proposition 3.4 of [Sim(1992)-II] by an additional term ν. This is, however, due to the fact that in the present approach we no longer have the reduction equation ∑_{i=1}^{N} m_{i}δq̃_{i} = 0. It follows easily from (4.4) that the sufficiency is equivalent to dim_{ℂ}{α_{1}, …, α_{n}} = 1, i. e. sufficiency means that all advances are equal to the same functional. Note that dim_{ℂ}{α_{1}, …, α_{n}} = 1 obviously implies P_{Σ} = 1.

A simple, but important consequence of the Connecting Path Formula is that the linear space {α_{1}, …, α_{n}} is the solution set of a homogeneous system of linear equations (3.27).

Remark 4.6. Note that, when applying the mentioned lemma, the left-hand-side of the Connecting Path Formula has to be written as the relative velocity of the colliding balls multiplied by the advance of that collision. It follows immediately from the exposition of [Sim(1992)-II] that the equations of this sort (arising from all CPF’s) are the only constraints on the advance functionals. The reason why this is true is that the fulfillment of all CPF’s precisely means that the relative displacement (variation of position) of every pair of particles right before their new collision is parallel to the relative incoming velocity of these particles (see also (3.23)), and this fact guarantees that the variation of the newly formed relative outgoing velocity will also be zero, just as required in (3.21).

Also note that the number n + P_{Σ} − N of equations in (3.27) is equal to the number of all collisions σ_{k} = {i_{k}, j_{k}} for which i_{k} and j_{k} are in the same connected component of the collision graph of {σ_{1}, σ_{2}, …, σ_{k−1}}.

The determinant D(M) of every minor M of the coefficient matrix of (3.27) is a homogeneous velocity polynomial and, therefore, D(M) is a holomorphic function on Ω̃ (Σ, A, τ⃗). Thus, there are finitely many velocity polynomials (actually, determinants) R_{1}(ω), …, R_{s}(ω) such that their simultaneous vanishing is equivalent to the non-sufficiency of the orbit segment ω. By using the equations (3.4)-(3.8) (which define the complex dynamics recursively), each of these velocity polynomials R_{i}(ω) can be – essentially uniquely – written as an algebraic function f_{i} (x⃗(ω)) = f_{i}(x⃗) of the initial data x⃗, and these algebraic functions f_{i}(x⃗) only contain (finitely many) field operations and square roots. By using the canonical method of successive elimination of the square roots from the equation f_{i}(x⃗) = 0, one obtains a complex polynomial P_{i}(x⃗) such that

P_{i}(x⃗) = 0 ⟺ f_{i}(x⃗) = 0.

We emphasize here that

(1) we understand the relation f_{i}(x⃗) = 0 in such a way that f_{i}(x⃗) becomes zero on some branch of evaluating the multiple-valued algebraic function f_{i}(x⃗), cf. Remark 2 to Definition 3.18;

(2) the equivalence is claimed to only hold on D, cf. the formulation of Lemma 4.2.

The process of eliminating the square roots from an equation f(x⃗) = 0 (of the above type), however, requires a bit of clarification. The algebraic function f has a natural representing element f̂ in the field 𝕂_{n}, see Definition 3.11. Suppose that f̂ ∈ 𝕂_{k} but f̂ ∉ 𝕂_{k−1}. Then the field element f̂ ∈ 𝕂_{k} can be written uniquely in the form f̂ = a + b\sqrt{δ}, where a, b, δ ∈ 𝕂_{k−1}, b ≠ 0, \sqrt{δ} ∉ 𝕂_{k−1}, and

\begin{matrix}δ = ⟨v_{i_{k}}^{k−1} − v_{j_{k}}^{k−1}; q̃_{i_{k}}^{k−1} − q̃_{j_{k}}^{k−1} − L · a_{k}⟩^{2} − \\ ∥v_{i_{k}}^{k−1} − v_{j_{k}}^{k−1}∥^{2} · (∥q̃_{i_{k}}^{k−1} − q̃_{j_{k}}^{k−1} − L · a_{k}∥^{2} − 4r^{2}) \end{matrix}

is the discriminant of the quadratic equation (3.8) with the unknown τ_{k}.

In the first step of the successive elimination of the square roots from the equation f(x⃗) = 0, we switch from the field element f̂ = a + b\sqrt{δ} ∈ 𝕂_{k} (representing f) to the new field element

f̂′ = (a + b\sqrt{δ}) · (a − b\sqrt{δ}) = a^{2} − b^{2}δ ∈ 𝕂_{k−1},

which is a representing element of an algebraic function f′(x⃗) containing less square roots than f(x⃗). Corresponding to the switching f̂ ↦ f̂′, we also replace the original equation f(x⃗) = 0 by the new equation f′(x⃗) = 0. Then we keep going by continuing this process until we arrive at a field element

ĝ(x⃗) = \frac{P (x⃗)}{Q(x⃗)} ∈ 𝕂_{0} = ℂ(x⃗),

which is a complex rational function \frac{P (x⃗)}{Q(x⃗)} of the initial variables x⃗. Of course,

here we assume that the polynomials P (x⃗) and Q(x⃗) have no non-trivial common divisor.

Actually, as it directly follows from Proposition 3.3 and from the method of eliminating the square roots (presented above), in each denominator Q_{l}(x⃗) (1 ≤ l ≤ s) associated to the algebraic function f_{l}(x⃗) there are only factors of the type m_{i} + m_{j} or ∥v_{i}^{k} − v_{j}^{k}∥^{2}, 1 ≤ i < j ≤ N, 0 ≤ k ≤ n. Therefore, none of the denominators Q_{l}(x⃗) vanishes on D. This explains property (2) formulated above.

As it also follows from the method of eliminating the square roots and from the actual form of the equations (3.4-8), the coefficients of the polynomials P_{i}(x⃗) are real.

By using some simple facts from Galois theory, we can easily detect an intimate relationship between the rational function ĝ(x⃗) ∈ 𝕂_{0} just constructed and the product α of all conjugate elements of f̂ ∈ 𝕂_{n}. (The conjugates of f̂ (over the base field 𝕂_{0}) are sitting in the normal hull \overline{𝕂_{n}} : 𝕂_{0} of the field extension 𝕂_{n} : 𝕂_{0}. The element α ∈ 𝕂_{0} is obviously the constant term in the normalized minimal polynomial of f̂over the field 𝕂_{0}. For the elementary concepts and facts from Galois theory, the reader is kindly referred to the book [St(1973)].) Namely, by establishing a natural matching between all conjugates of a^{2} − b^{2}δ and the conjugates of the ordered pair (a + b\sqrt{δ}, a − b\sqrt{δ}), we can easily see that there exists an integer l such that ĝ = α^{2^{l}}. (In the case l < 0 this should be understood as α = (ĝ)^{2^{−l}}.) Especially, α and ĝ have the same irreducible factors both in the numerator and in the denominator. Another consequence is that the field element f̂ is zero if and only if P (x⃗) ≡ 0.

So far we have seen that for each velocity determinant R_{i}(ω) (i = 1, …, s) one can find a canonically determined polynomial P_{i}(x⃗) with the following property: For every vector x⃗ ∈ D (Σ, A, τ⃗) the equation P_{i}(x⃗) = 0 is equivalent to the equation f_{i}(x⃗) = 0 on some branch of evaluating the algebraic function f_{i}(x⃗). Hence Lemma 4.2 follows. □

Corollary 4.7 (Dichotomy Corollary). As another consequence of the above identity ĝ = α^{2^{l}}, we obtain that if P ≡ 0, then f(x⃗) = 0 for every x⃗ ∈ ℂ^{(2ν+1)N+1} and on every branch of evaluating the multiple-valued algebraic function f(x⃗). This also means that if all polynomials P_{1}, …, P_{s} in Lemma 4.2 happen to be identically zero, then no orbit segment ω ∈ Ω̃ (Σ, A, τ⃗) is sufficient.

Therefore, thanks to the algebraic feature of the dynamics, we have got a nice dichotomy here: Either every (Σ, A, τ⃗)-orbit segment is non-sufficient, or almost every such trajectory segment is sufficient. □

Remark. We note that the reason why we do not have an equivalence in (4.3) is that – for a typical x⃗ ∈ ℂ^{(2ν+1)N+1} – the branches, on which f_{i}(x⃗) = 0, may be different for different values of i.

According to the above lemma, in order to prove Key Lemma 4.1, it is enough to show that at least one of the polynomials P_{i}(x⃗) is nonzero. This is exactly what we are going to do by an induction on the number N ≥ 2.

Lemma 4.8. Pertaining to the polynomials P_{i}(x⃗) associated with R_{i}(ω) = f_{i} (x⃗(ω)) in Lemma 4.2, we assert the following:

If the algebraic function f_{i}(x⃗) is identically zero on a non-empty, open subset B of ℝ^{(2ν+1)N+1}, then f_{i} is algebraically trivial, i. e. f̂_{i} = 0 or, equivalently, P_{i}(x⃗) ≡ 0.

Proof. Denote the minimal polynomial of f̂_{i} over the base field 𝕂_{0} = ℂ(x⃗) by M(x) = ∑_{j=0}^{k} c_{j}x^{j}, c_{j} ∈ 𝕂_{0}. Suppose, on the contrary, that f_{i} yet vanishes on an open ball B of ℝ^{(2ν+1)N+1}. Since f_{i} fulfills the identity ∑_{j=0}^{k} c_{j}(f_{i})^{j} ≡ 0, the vanishing of f_{i} on the ball B implies that the element c_{0} of the rational function field 𝕂_{0} vanishes on B, as well. Therefore, c_{0} = 0 in 𝕂_{0}, so M(x) is divisible by x. The irreducibility of M yields M(x) = x, i. e. f̂_{i} = 0. Hence the lemma follows. □

The Substitution {m_{N} = 0}. Derived Schemes

The next lemma describes the (Σ, A, τ⃗)-orbit segments in the case when the Nth particle is “infinitely light” compared to the others, i. e. when m_{N} = 0 but m_{1} · . . . · m_{N−1} ≠ 0.

Lemma 4.9. Suppose that m_{N} = 0 and m_{1} · . . . · m_{N−1} ≠ 0. Assume further that there exist indices 1 ≤ p < q ≤ n such that N ∈ σ_{p}, N ∈ σ_{q} and N ∉ σ_{j} for j = p + 1, …, q − 1. Then an orbit segment ω ∈ Ω̃ (Σ, A, τ⃗) is sufficient (i. e. dim_{ℂ}N (ω) = ν + 1) if (1) the {1, 2, …, N − 1}-part

{q̃_{i}^{k}(ω), v_{i}^{k}(ω), m_{i}(ω), L(ω) | i = 1, …, N − 1; k ∈ I_{N}}

of the orbit segment is sufficient as an orbit segment of the particles 1, …, N − 1 and

(2) the relative velocities v_{N}^{p}(ω) − v_{i_{p}}^{p}(ω) and v_{N}^{q−1}(ω) − v_{i_{q}}^{q−1}(ω) are not parallel.

(Here, as usual, i_{p} (i_{q}) is the index of the ball colliding with the N-th particle at σ_{p} (σ_{q}), and the index set I_{N} ⊂ {0, 1, …, n} contains 0 and those indices i > 0 for which N ∉ σ_{i}.)

Remark. It is worth noting that, as it follows easily from the conditions, v_{N}^{p}(ω) = v_{N}^{q−1}(ω).

Proof. First of all, we observe that the motion of the N-th ball with zero mass has absolutely no effect on the evolution of the {1, …, N −1}-part of the trajectory segment, and this statement is also valid for the time evolution of the {1, …, N −1}- part {δq̃_{i}^{k}| i = 1, …, N − 1} of a neutral vector (δq̃_{1}^{0}, …, δq̃_{N}^{0}) ∈ N (ω), see also the equations (3.4), (3.7)–(3.8) and (3.10).

Consider now an arbitrary neutral vector δQ = (δq̃_{1}^{0}, …, δq̃_{N}^{0}) ∈ N (ω). According to (1) and the above principle, we can modify the neutral vector δQ by a scalar multiple of the flow direction and by a uniform spatial displacement in such a way that δq̃_{i}^{0} = 0 for i = 1, …, N − 1, and then δq̃_{i}^{k} = 0 remains true for the whole orbit segment, i = 1, …, N − 1, k = 0, …, n. The neutrality of δQ with respect to the entire orbit segment means, however, that

δq̃_{N}^{p} = α_{p} · (v_{N}^{p}(ω) − v_{i_{p}}^{p}(ω)) and δq̃_{N}^{q−1} = α_{q} · (v_{N}^{q−1}(ω) − v_{i_{q}}^{q−1}(ω)) . (4.10)

According to our hypothesis (2), the consequence of (4.10) and of the obvious equation δq̃_{N}^{p} = δq̃_{N}^{q−1} is that α_{p} = α_{q} = 0, and thus δQ = 0. Hence the lemma follows. □

The crucial part of the inductive proof of Key Lemma 4.1, i. e. the substitution m_{N} = 0 (and L = 0, too), requires some preparatory thoughts and lemmas. The formulations given for the substitution m_{N} = 0 can be easily adapted to the substitution L = 0. Assume that a combinatorial-algebraic scheme (Σ, A, τ⃗) is given for the N-ball system {1, 2, …, N} so that Property (A) holds (see Definition 3.31). In the proof of Key Lemma 4.1 we want to use some combinatorial-algebraic (N − 1)- schemes (Σ′, A′, τ′) that

(i) govern the time evolution of the {1, 2, …, N − 1}-part of some (Σ, A, τ⃗)-orbit segments with an infinitely light N-th ball, i. e. m_{N} = 0 and

(ii) enjoy Property (A).

In the construction of such (N − 1)-schemes (Σ′, A′, τ′), the so called derived schemes, we want to follow the guiding principles below, which also serve as the definition of the derived schemes:

Definition 4.11. We say that the (N − 1)-scheme (Σ′, A′, τ′) is a scheme derived from (Σ, A, τ⃗) by putting m_{N} = 0 if it is obtained as follows:

(1) First of all, we discard all symbols σ_{j}, a_{j}, and τ_{j} from (Σ, A, τ⃗) that correspond to a σ_{j} containing the label N. This also means that we retain the other symbols σ_{j}, a_{j} without change: they only get re-indexed, due to the dropping of the other symbols;

(2) As far as the selection of τ′ is concerned, we want to retain all algebraic relations among the variables that are encoded in the original scheme (Σ, A, τ⃗), i. e. all algebraic relations that follow from the scheme (Σ, A, τ⃗) and from m_{N} = 0;

(3) We want the derived scheme (Σ′, A′, τ′) to enjoy Property (A), see 3.31.

It is straightforward that, following just the instructions in (1)-(2) above, one can easily construct such schemes (Σ, A, τ⃗) which, perhaps, do not have Property (A).

Remark 4.12. Part (2) of the above definition means the following: Initially, the algebra of (Σ, A, τ⃗) consists

(1) of the variables

{(q̃_{i}^{k})_{j} , (v_{i}^{k})_{j}, m_{i}, L, τ_{l}| i = 1, …, N; k = 0, …, n; j = 1, …, ν; l = 1, …, n};

(2) of the equations (3.4-8), and finally

(3) of the choices of the signs of the square roots in the solutions of the quadratic equations (3.8) whenever the equations are reducible (cf. Remark 3.13).

To obtain the algebra of the derived scheme (Σ′, A′, τ′), we put first m_{N} = 0 and then τ′_{l} = ∑_{j=k+1}^{l} τ_{j} whenever for some 1 ≤ k < l ≤ n we have N ∉ σ_{k} ∪ σ_{l}, N ∈ ⋂_{j=k+1}^{l−1} σ_{j} (here the indexing of the time slots τ′_{l} of (Σ′, A′, τ′) is lacunary). After trivial substitutions, the quadratic equation for τ_{l} leads to the natural quadratic equation for τ′_{l}, and then, finally, we can cancel all the variables {(q̃_{N}^{k})_{j}, (v_{N}^{k})_{j}, m_{N}| 0 ≤ k ≤ n, 1 ≤ j ≤ ν}, and all the equations they figure in as well. The collection of the remaining variables and equations with the corresponding vector τ′ make (Σ′, A′, τ′).

Remark 4.13. Below we show the inheritance of the reducibility 𝕂_{l−1} = 𝕂_{l} ⟹ 𝕂′_{l−1} = 𝕂′_{l}. We note, however, that this result is not used in the proof of Key Lemma 4.1. We merely include it – along with its nice consequence: part (3) of Lemma 4.37 – in order to give the reader a broader view of the subject. The subsequent proof of Lemma 4.40 only uses the weaker part (2) of 4.37 which, in turn, does not need the above mentioned inheritance of reducibilty.

Another interpretation of part (2) of the above definition is the following one: Once we have a complex rational expression g(z_{1}, …, z_{p}) of arbitrary elements

z_{1}, …, z_{p} ∈ 𝕂′_{n} = 𝕂 (Σ′, A′, τ′)

so that the same expression of these variables is zero in the field 𝕂_{n} = 𝕂 (Σ, A, τ⃗), then it must also be zero in the field 𝕂′_{n}. An important case (when (2) has special significance) is when N ∉ σ_{k} ∪ σ_{l} (1 ≤ k < l ≤ n), N ∈ ⋂_{j=k+1}^{l−1} σ_{j}, and 𝕂_{l−1} = 𝕂_{l}. Then the time slot τ′_{l} = ∑_{j=k+1}^{l} τ_{j} (Here the indexing of the time slots τ′_{l} of (Σ′, A′, τ′) is lacunary.) should be expressed as a rational function of the earlier defined (Σ, A, τ⃗)-variables (the variables constituting the field 𝕂_{l−1}), just as it is instructed by the rational dependence of τ_{l} on the variables constituting the field 𝕂_{l−1}. Therefore, τ′_{l} = ∑_{j=k+1}^{l} τ_{j} is a rational expression

τ′_{l} ≡ g (x⃗; q̃_{N}^{0}, v_{N}^{0}, m_{N}; τ_{1}, …, τ_{l−1})  (4.14)

of the initial variables (x⃗; q̃_{N}^{0}, v_{N}^{0}, m_{N}) ∈ ℂ^{(2ν+1)N+1} and the time slots τ_{1}, …, τ_{l−1}. Especially, the identity (4.14) is also true if we impose additionally the side condition m_{N} = 0:

τ_{l}^{′0} ≡ g (x⃗; q̃_{N}^{0}, v_{N}^{0}, 0; τ_{1}^{0}, …, τ_{l−1}^{0}) .  (4.15)

However, the {1, …, N − 1}-dynamics evolves independently of the data of the N-th ball with zero mass m_{N}. This means that

g (x⃗; q̃_{N}^{0}(1), v_{N}^{0}(1), 0; τ_{1}^{0}(1), …, τ_{l−1}^{0}(1)) = g (x⃗; q̃_{N}^{0}(2), v_{N}^{0}(2), 0; τ_{1}^{0}(2), …, τ_{l−1}^{0}(2))

whenever all cumulative sums ∑_{j=k_{1}+1}^{k_{2}} τ_{j}^{0}(1) are equal to the corresponding cumulative sums ∑_{j=k_{1}+1}^{k_{2}} τ_{j}^{0}(2) for 1 ≤ k_{1} < k_{2} < l, N ∉ σ_{k_{1}} ∪ σ_{k_{2}}, N ∈ ⋂_{j=k_{1}+1}^{k_{2}−1} σ_{j}. This precisely means that the rational function g (x⃗; q̃_{N}^{0}, v_{N}^{0}, 0; τ_{1}^{0}, …, τ_{l−1}^{0})

(i) does not contain the variables (q̃_{N}^{0})_{β}, (v_{N}^{0})_{β} (1 ≤ β ≤ ν), and

(ii) only depends through the above cumulative sums ∑_{j=k_{1}+1}^{k_{2}} τ_{j}^{0} = τ_{k_{2}}^{′0}. (The indexing of τ_{j}^{′0} is still lacunary!) Thus, we get that g (x⃗; q̃_{N}^{0}, v_{N}^{0}, 0; τ_{1}^{0}, …, τ_{l−1}^{0}) is, in fact, a rational expression of x⃗ and the time slots τ_{j}^{′0} preceding τ_{l}^{′0}. This shows that τ_{l}^{′0} ∈ 𝕂′_{l−1}, i. e. 𝕂′_{l−1} = 𝕂′_{l} follows from the similar reducibility 𝕂_{l−1} = 𝕂_{l}.

The main point is the following one: Once τ_{l} is the root of a reducible polynomial, it is selected “artificially”, following the instructions encoded in the scheme τ⃗. Now we want to avoid the selection of the other root when defining the derived scheme (Σ′, A′, τ′)! This is why (2) is a requirement, and not a corollary, and this is why, in Remark 4.12, we use the wording “the time slot τ′_{l} should be expressed”.

The fact that the reducibility of the polynomial defining τ_{l} is inherited during a substitution m_{N} = 0 followed from a general principle: If some algebraic function in 𝕂_{l−1} (the discriminant of the quadratic polynomial defining τ_{l}) is the square of another such algebraic expression, then this square relation remains even “more true” if one annihilates a variable, i. e. considers that square relation on a smaller set. It is now the nature of our model that m_{N} = 0 immediately kills all variables involving the motion of the N-th particle, since the considered new discriminant of the quadratic equation defining τ′_{l} is obviously independent of those variables.

Remark 4.16. a. Note that the construction of such schemes (Σ′, A′, τ′), by just requiring (1)-(2), is far from unique: It is well possible that, originally, for some k we have |𝕂_{k} : 𝕂_{k−1}| = 2 (i. e. the polynomial (3.8) determining τ_{k} is irreducible), but after the substitution m_{N} = 0 the corresponding field extension collapses to a trivial, degree-one extension. (The paramount example for this phenomenon is just the set-up of the upcoming Main Lemma 4.21.) If this phenomenon of “reducible splitting” takes place, then we have to make our choice: Which one of the two (now distinguishable!) roots of the reducible (3.8) to include in τ′?

The requirement (A) for the derived scheme (Σ′, A′, τ′), however, ties our hands when selecting this root in the scenario of Main Lemma 4.21, and it actually implies the uniqueness of the derived scheme (Σ′, A′, τ′) (see Corollary 4.36).

The phenomenon of this reducible splitting can also be well represented by the following, simple paradigm: Let a ∈ ℂ be a complex parameter (playing the role of m_{N}). Consider the Riemann-leaf of the function y = f(z) = \sqrt{z^{2} + a}, i. e. the algebraic variety defined by the equation y^{2} − z^{2} − a = 0. This is irreducible (undecomposable) for a ≠ 0, and it splits into two components (y − z = 0 and y + z = 0) for a = 0. However, given a sequence a_{k} → 0 (a_{k} ≠ 0), every point (z, y) on the curve y^{2} − z^{2} = 0 turns out to be a limit of a sequence (z_{k}, y_{k}) with y_{k}^{2} − z_{k}^{2} − a_{k} = 0.

b. The paradigm just given also makes it possible to illustrate why (2) in Definition 4.11 is a requirement and not a corollary. Indeed, consider the field 𝕂_{0} = ℂ(z, m_{N}), i. e. 𝕂_{0} is the complex field extended by two transcendental elements z and m_{N}. Let 𝕂_{1} := 𝕂_{0}(τ_{1}) = 𝕂_{0}(\sqrt{z^{2} + 1 + m_{N}}) be the extension of 𝕂_{0} by the solution τ_{1} of the irreducible equation τ_{1}^{2} − z^{2} − 1 − m_{N} = 0. Let us now extend 𝕂_{1} by the root τ_{2} of the quadratic polynomial x^{2} − 5τ_{1}x + 6τ_{1}^{2}. This polynomial is reducible over the field 𝕂_{1}: its two roots are 2τ_{1} and 3τ_{1}. Now the extended field 𝕂_{2} will be just equal to 𝕂_{1}, and the selection of a designated root (as τ_{2}) of the above polynomial amounts to selecting either 2τ_{1} or 3τ_{1}. Observe that the root of the considered reducible polynomial could have been an arbitrary rational expression of the elements z, and τ_{1}. In a general reducible step, such a rational expression shows the unique way of computing the actual τ_{n} from the initial data and the τ_{i}’s with i = 1, …, n − 1. However, when assigning actual complex values to the field element τ_{1} = \sqrt{z^{2} + 1 + m_{N}}, we have two possiblities. Summing everything up, our selection does not mean giving values to these τ’s but, rather, telling if we take (formally) 2τ_{1} or 3τ_{1} as τ_{2}. (This selection determines the discrete algebraic structure, which becomes transparent if we think about repeated σ’s with identical adjustment vectors.) When defining the derived scheme this means that once we have chosen 3\sqrt{z^{2} + 1 + m_{N}} (instead of 2\sqrt{z^{2} + 1 + m_{N}}) we still want to choose 3\sqrt{z^{2} + 1} after the substitution m_{N} = 0, and do not want to suddenly switch to the other root 2\sqrt{z^{2} + 1}.

Transversalities of the Degeneracies

Definition-notation 4.17. Denote by Φ_{p} (1 ≤ p ≤ n) the degeneracy ∥v_{i_{p}}^{p} − v_{j_{p}}^{p} ∥^{2}

= 0 and by Ψ_{p} (1 ≤ p ≤ n) the degeneracy defined by ⟨v_{i_{p}}^{p} − v_{j_{p}}^{p}; q̃_{i_{p}}^{p} − q̃_{j_{p}}^{p} − L · a_{p}⟩ = 0. (These degeneracies are represented here with post-collision velocities, but in view of Remark 1 to Definition 3.18 they also have an analogous representation in terms of pre-collision velocities.)

Remark 4.18. a. In the following arguments we will use a slightly different working domain D = D_{p+1,n−1}, which is not exactly the one of Definition 3.18. Instead, in general, the working domain D_{p+1,n−1} describes the orbit segments that are non-degenerate from the (p + 1)-st collision up to the (n − 1)-st one (inclusive). Here p ≤ n − 1, and if p = n − 1, then, by definition, D_{p+1,n−1} = ℂ^{(2ν+1)N+1}. The reference time (the upper index of the initial data) is now p + 1, i. e. the outgoing kinetic variables are taken at σ_{p+1}.

b. Observe that if for some p < p′

σ_{p} = σ_{p′} and, for p < j < p′, σ_{j} ∩ σ_{p} = either σ_{p} or ∅,  (4.19)

then by Remark 1 to Definition 3.18 Φ_{p} = Φ_{p′}. Similarly, if for some p < p′

(σ_{p}, a_{p}) = (σ_{p′}, a_{p′}) and for p < j < p′ either (σ_{j}, a_{j}) = (σ_{p}, a_{p}) or σ_{j} ∩ σ_{p} = ∅,

(4.20)

then necessarily Ψ_{p} = Ψ_{p′}.

c. In the upcoming Main Lemma transversality statements for Φ_{p} (or Ψ_{p}) and Φ_{n} (or Ψ_{n}) will be claimed on D_{p+1,n−1}. Because of our previous observations, these assertions can be nonvacuous only if the symbolic pair (Σ, A) is minimal in the sense that

(1) if Φ_{p} is involved in the statement, then there is no p′ : p < p′ < n such that (4.19) holds (this requirement is called Φ-minimality from the left);

(2) if Ψ_{p} is involved in the statement, then there is no p′ : p < p′ < n such that

(4.20) holds (this requirement is called Ψ-minimality from the left); Analogous requirements can be formulated when Φ_{n} (Ψ_{n}) is involved, and will be called Φ ( or Ψ) -minimality from the right.

In order to show that there really exist derived schemes (that is, fulfilling not only (1)-(2), but also (3) of Definition 4.11), we need to prove

Main Lemma 4.21. Assume that (Σ, A, τ⃗) enjoys Property (A). We claim that

(I) The codimension-one submanifolds defined either by m_{i} = 0 (1 ≤ i ≤ N) or by L = 0 can not locally coincide with any degeneracy Φ_{p} or Ψ_{p}, 1 ≤ p ≤ n;

(II) On D_{p+1,n−1}, the degeneracy Φ_{p} can not locally coincide with the degeneracy Ψ_{n} (1 ≤ p ≤ n − 1) whenever ((σ_{p}, …, σ_{n}), (a_{p}, …, a_{n})) is Φ-minimal from the left and Ψ-minimal from the right;

(III) On D_{p+1,n−1}, the degeneracy Ψ_{p} can not locally coincide with the degeneracy Ψ_{n}, whenever ((σ_{p}, …, σ_{n}), (a_{p}, …, a_{n})) is Ψ-minimal from both sides;

(IV) The degeneracies m_{i} = 0 (1 ≤ i ≤ N), L = 0 intersect D_{p+1,n−1} (1 ≤ p ≤ n − 1); moreover the degeneracies Φ_{p}, Φ_{n}, Ψ_{p} and Ψ_{n} intersect D_{p+1,n−1} (1 ≤ p ≤ n − 1) whenever ((σ_{p}, …, σ_{n}), (a_{p}, …, a_{n})) is Φ-minimal from the left or from the right, or is Ψ-minimal from the left or from the right, respectively;

(V) If (Σ, A, τ⃗) is Ψ-minimal from the right, then there exists a (non-degenerate) orbit segment

ω = {q̃_{i}^{α}(ω), v_{i}^{α}(ω), m_{i}(ω), L(ω) | i = 1, …, N; α = 0, 1, …, n − 1}

∈ Ω̃ (σ_{1}, …, σ_{n−1}; a_{1}, …, a_{n−1}; τ_{1}, …, τ_{n−1}) = Ω̃_{n−1}

for which the discriminant

δ(ω) := ⟨v_{i_{n}}^{n−1}(ω) − v_{j_{n}}^{n−1}(ω); q̃_{i_{n}}^{n−1}(ω) − q̃_{j_{n}}^{n−1}(ω) − L · a_{n}⟩^{2}
− ∥v_{i_{n}}^{n−1}(ω) − v_{j_{n}}^{n−1}(ω)∥^{2} · (∥q̃_{i_{n}}^{n−1}(ω) − q̃_{j_{n}}^{n−1}(ω) − L · a_{n}∥^{2} − 4r^{2})

of (3.8) (with k = n) is equal to zero and m_{i}(ω) ≠ 0 (i = 1, …, N), L(ω) ≠ 0;

(VI) If 1 ≤ k < n, σ_{k} = σ_{n}, a_{k} = a_{n}, N ∉ σ_{n}, and

(i) for k < j < n either σ_{j} ∩ σ_{n} = ∅ or N ∈ σ_{j},

(ii) there exists a j_{0} with k < j_{0} < n and σ_{j_{0}} ∩ σ_{n} ≠ ∅,

then |𝕂_{n} : 𝕂_{n−1}| = 2 or, equivalently, the polynomial (3.8), defining τ_{n}, is irreducible over the field 𝕂_{n−1};

(VII) If 1 ≤ k < n, σ_{k} = σ_{n}, a_{k} ≠ a_{n}, and for every k < j < n we have σ_{j} ∩ σ_{n} = ∅, then again |𝕂_{n} : 𝕂_{n−1}| = 2.

(VIII) On D_{p+1,n−1}, the degeneracy Φ_{p} can not locally coincide with the degeneracy Φ_{n} (1 ≤ p ≤ n − 1) whenever ((σ_{p}, …, σ_{n}), (a_{p}, …, a_{n})) is Φ-minimal from both sides.

Remark 4.22. a. Assertion (VI) of Main Lemma 4.21 is just taylor suited to ensure that — whenever carrying out the substitution m_{N} = 0 — at every “critical” step with the possibility of selecting the wrong τ′_{n} (and thus violating Property (A)), we are not forced to do so, because at an irreducible τ_{n} either branches of the corresponding square root of the discriminant of (3.8) may be selected for continuing the dynamics. Quite similarly, assertion (VII) guarantees the same thing for the substitution L = 0.

b. The careful reader has certainly observed that statement (VIII) is separated from the quite similar assertions (II-III) of the Main Lemma. The reason is the followong: By a slight improvement of the inductive arguments verfiying the transitions N − 1 → N when establishing (II-III) and (VIII), one can strengthen these claims by only requiring in each of them minimality from the right. The advantage of these stronger statements is that then (VIII) is not needed in the inductive proofs of (II-IV). As a result, (V-VII) can be proved completely without (VIII).

Proof.

1. First we prove (I). For the sake of brevity, we prove (I) for the case m_{i} = 0. The handling of the submanifold L = 0 is completely analogous. Assume that there is a counterexample, and p (the time of degeneracy) has the smallest possible value. Push forward the manifold m_{i} = 0 up to the collision σ_{p−1} by the algebraic dynamics: the form of the equation m_{i} = 0 defining this manifold does not change. (This push-forward can be made without difficulty by the minimality of the index p.) In this way we can reduce the proof of (I) to the case p = 1. However, the statement is obviously true in that case. □

For later use we prove the following Corollary 4.23. If for the combinatorial scheme (Σ, A, τ⃗) and for some ω_{0} ∈ Ω̃_{n−1} we have δ(ω_{0}) = 0, then we can also find an ω^{∗} ∈ Ω̃_{n−1} such that still δ(ω^{∗}) = 0, none of the masses m_{1}(ω^{∗}), …, m_{N}(ω^{∗}) is zero and, moreover, L(ω^{∗}) ≠ 0.

Proof. Indeed, the conditions imply that for a codimension-one algebraic subset Z of D \binom{}{Ω̃_{n−1}} it is true that for every x ∈ Z there exists an ω ∈ Ω̃_{n−1} such that x(ω) = x and δ(ω) = 0. By (I), Z is not even locally identical with any submanifold {m_{i} = 0} (i = 1, …, N). Consequently, we can select a small perturbation x^{∗} ∈ Z of x_{0} = x(ω_{0}) and an ω^{∗} ∈ Ω̃_{n−1} such that x(ω^{∗}) = x^{∗}, δ(ω^{∗}) = 0, and none of the masses m_{1}(ω^{∗}), …, m_{N}(ω^{∗}) and neither L(ω^{∗}) vanishes. □

2. The implication (II)&(III)&(IV) ⟹ (V) is obvious since, by the argument used to prove (I) it makes no problem to avoid the additional degeneracies m_{i}+m_{j} = 0, 1 ≤ i < j ≤ N.

3. We prove next that the existence of an ω ∈ Ω̃_{n−1} with δ(ω) = 0 (guaranteed in (V)) implies |𝕂_{n} : 𝕂_{n−1}| = 2. This will immediately prove the implications (V) ⟹ (V I) and (V) ⟹ (V II). (It is easy to check that the conditions in (VI) or (VII) actually imply the precondition of (V).)

Let, therefore, ω ∈ Ω̃_{n−1} be an orbit segment with δ(ω) = 0. By re-labeling the balls, we can assume that σ_{n} = {1, 2}. In a small neighborhood U_{0} ⊂ Ω̃_{n−1} of ω the discriminant δ(ω′) from (V) (ω′ ∈ U_{0}) is an analytic function of ω′ or, equivalently, δ(ω′) is an analytic, algebraic function f(x⃗) of the initial variables x⃗ = x⃗(ω) ∈ ℂ^{(2ν+1)N+1}.

Consider now the “normal vector of collision”

w = w(ω) = q̃_{1}^{n}(ω) − q̃_{2}^{n}(ω) − L · a_{n} =
q̃_{1}^{n−1}(ω) − q̃_{2}^{n−1}(ω) − L · a_{n} + τ_{n}(ω) · [v_{1}^{n−1}(ω) − v_{2}^{n−1}(ω)] ,

where τ_{n}(ω) is just the double root of (3.8) with k = n. Execute now the small perturbations ω(ε) ∈ U_{0}, (ε ∈ ℂ, |ε| is small) as follows:

q̃_{1}^{n−1} (ω(ε)) = q̃_{1}^{n−1} (ω) + ε · w;
q̃_{i}^{n−1} (ω(ε)) = q̃_{i}^{n−1} (ω) , i = 2, …, N;
v_{i}^{n−1} (ω(ε)) = v_{i}^{n−1} (ω) , i = 1, …, N.

(Note: If 1 ∈ σ_{n−1}, then the configuration perturbation defined right above ought to be modified by an appropriate scalar multiple of the flow direction in order

to ensure ∥q̃_{i_{n−1}}^{n−1} − q̃_{j_{n−1}}^{n−1} − L · a_{n−1}∥^{2} = 4r^{2}.) Easy geometric consideration yields that

\frac{d}{dε}δ (ω(ε)) |_{ε=0} = \frac{d}{dε}f (x⃗ (ω(ε))) |_{ε=0} = −8r^{2} ∥v_{1}^{n−1}(ω) − v_{2}^{n−1}(ω)∥^{2} ≠ 0.

That is, the holomorphic function g(ε) := f (x⃗ (ω(ε))) of ε ∈ ℂ (|ε| is small) has the properties g(0) = 0, g′(0) ≠ 0.

Suppose now, indirectly, that the algebraic function f(x⃗) ∈ 𝕂_{n−1} is a square (h(x⃗))^{2} in 𝕂_{n−1}. Consider the function z(ε) := h (x⃗ (ω(ε))) of the single complex variable ε ∈ ℂ, |ε| small. It is clear that z(ε) is a meromorphic function of ε, being a rational function of the variables {q̃_{i}^{α}, v_{i}^{α}, m_{i}, L | i = 1, …, N; α < n}, and thus being the ratio of two holomorphic functions. This is the point where we use the indirect assumption of h(x⃗) ∈ 𝕂_{n−1}. However, the square [z(ε)]^{2} = g(ε) of the meromorphic function z(ε), obviously, can not have the properties g(0) = 0, g′(0) ≠ 0.

This proves that, indeed, |𝕂_{n} : 𝕂_{n−1}| = 2. □

So far we have proved that (II),(III) and (IV) together imply assertions (V-VII) of Main Lemma 4.21.

4. We prove now (II-IV) ((VIII) will be settled later in point 5.) Let us first observe that (II-IV) (and similarly (VIII)) are consequences of the somewhat weaker statements

(II^{∗}) On D_{2,n−1}, the degeneracy Φ_{1} can not locally coincide with the degeneracy Ψ_{n} whenever ((σ_{1}, …, σ_{n}), (a_{1}, …, a_{n})) is Φ-minimal from the left and Ψ-minimal from the right;

(III^{∗}) On D_{2,n−1}, the degeneracy Ψ_{1} can not locally coincide with the degeneracy Ψ_{n} whenever ((σ_{1}, …, σ_{n}), (a_{1}, …, a_{n})) is Ψ-minimal from both sides;

(IV^{∗}) The degeneracies m_{i} = 0 (1 ≤ i ≤ N), L = 0 intersect D_{2,n−1}; moreover, the degeneracies Φ_{1}, Φ_{n}, Ψ_{1} and Ψ_{n} intersect D_{2,n−1} whenever ((σ_{1}, …, σ_{n}), (a_{1}, …, a_{n})) is Φ-minimal from the left or from the right, or is Ψ-minimal from the left or from the right, respectively;

(VIII^{∗}) On D_{2,n−1}, the degeneracy Φ_{1} can not locally coincide with the degeneracy Φ_{n} whenever ((σ_{1}, …, σ_{n}), (a_{1}, …, a_{n})) is Φ-minimal from both sides.

Indeed, we only have to shift the index (time) of the initial variables from 2 to p+1 and use the observation: If the triple (Σ, A, τ⃗) has Property(A) and (Σ′, A′) = (σ_{p}, …, σ_{n}; a_{p}, …, a_{n}) (1 < p ≤ n) is an end segment of (Σ, A), then the Σ′- restriction of every (Σ, A, τ⃗)-orbit segment ω is automatically a (Σ′, A′, τ′)-orbit segment with some τ′, so that (Σ′, A′, τ′) enjoys Property (A). Observe, moreover, that if τ′_{j} : p < j ≤ n is reducible in the truncated dynamics (Σ′, A′, τ′), then τ_{j} is certainly reducible in the original dynamics (Σ, A, τ⃗). In other words, any possible additional irreducibility in the truncated dynamics permits a larger variety of orbit segments to study than just the restrictions of the orbit segments of (Σ, A, τ⃗) to the collisions σ_{p}, ..., σ_{n}. Therefore it is sufficient to establish the claimed transversalities for (Σ′, A′, τ′), and this is exactly the content of (II^{∗}-III^{∗}) and (VIII^{∗}).

We will establish now (II^{∗}-IV^{∗}) (and (VIII^{∗}) later in point 5) by using a double induction: we assume that the statements are true for any number N′ ≤ N of balls and arbitrary length n′ ≤ n such that the pair (n′, N′) is not identical with (n, N). More precisely, when establishing (IV^{∗}) for the pair (n, N), we will only use (IV^{∗}) for pairs (n′, N) with n′ < n. On the other hand, when demonstrating (II^{∗}-III^{∗}) (and (VIII^{∗})), the validity of (II^{∗}-IV^{∗}) (and also of (VIII^{∗})) for pairs (n′, N′) with n′ < n and N′ < N will be exploited. The proof is formulated in such a way that we take first the outer induction on n, and then the statements for this fixed n are shown by using an inner induction on N.

Observe that, for an arbitrarily fixed number N of balls, assertion (IV^{∗}) is certainly true for the value n once (II^{∗}-IV^{∗}) and (VIII^{∗}) are known to hold for smaller values of n. Indeed, for the degeneracies m_{i} = 0 (1 ≤ i ≤ N) and L = 0, the statement is obvious. Our inductive hypothesis (i. e. the validity of (II^{∗})–(III^{∗}) and (VIII^{∗}) for all smaller values of n) guarantees that the claimed intersections of the singularities with the working domain D_{2,n−1} are nonempty, as long as the appropriate minimality conditions of (IV^{∗}) hold.

Let us turn to the inductive proofs of (II^{∗})–(III^{∗}) (and to that of (VIII^{∗}) later in point 5).

The base of the induction on n is n = 2: This case is a straightforward inspection. Assume now that n ≥ 3, and that (II^{∗}-IV^{∗}) and (VIII^{∗}) have been proved for all

smaller values of n. The base of the induction on N is N = 2 or 3 or 4, depending on the form of (Σ, A), when establishing (II^{∗}) and (III^{∗}). In both proofs the inductive steps will first be presented, and afterwards the base cases will be treated.

By re-labeling the balls, we can achieve that σ_{n} = {1, 2}, σ_{1} = {i, j}, 1 ≤ i < j ≤ 4, and if j = 4, then i = 3. (“Densely packed” indices.)

4/a: Proof of (II^{∗}) by using the inductive assumption, i. e. by assuming that (II^{∗}-IV^{∗}) and (VIII^{∗}) are valid for pairs (n′, N′) satisfying n′ < n, N′ < N. We argue by contradiction. Suppose, therefore, that, on D_{2,n−1}, Φ_{1} locally coincides with the degeneracy Ψ_{n} defined by the equation δ(ω) = 0, i. e. their intersection contains a non-empty, smooth, codimension-one submanifold of D_{2,n−1}. We apply our favorite

Globalization/substitution argument: In the sense of Remark 4.18, our variables (∈ D_{2,n−1}) will now be represented at reference time 2, and the (algebraic) dynamics exists (strictly) between the collisions 1 and n. By using two quadratic equations:

(1) the one expressing that σ_{1} is a collision of two particular balls;

(2) the one defining our considered σ_{1}-degeneracy Φ_{1}, we can eliminate two variables, say (q̃_{i}^{2})_{1} and (v_{i}^{2})_{1}, by expressing them as some algebraic functions f_{1}(y⃗), f_{2}(y⃗) of the other variables

y⃗ = {(q̃_{α}^{2})_{β} , (v_{α}^{2})_{β} , m_{α}, L| α = 1, …, N; 1 ≤ β ≤ ν} \ {(q̃_{i}^{2})_{1} , (v_{i}^{2})_{1}}

measured at the collision σ_{2}. The kinetic variables of y⃗ measured at σ_{2} (i. e. with superscript 2) are now serving as the initial, or reference variables in the upcoming globalization/substitution argument. As usual, the algebraic functions f_{1}(y⃗) and f_{2}(y⃗) only contain field operations and square roots. Then, by using the recursive feature of the (Σ, A, τ⃗)-dynamics and the inductive assumption of 4.21, we can express the discriminant δ(ω) as another (also essentially unique) algebraic function g(y⃗) of the variables y⃗.

We argue by contradiction. Assume, therefore, that the degeneracies Φ_{1} and Ψ_{n} locally coincide in the working domain D_{2,n−1}. The local coincidence of these degeneracies precisely means that

(*) the algebraic expression g(y⃗) is identically zero on some non-empty, open set of vectors y⃗ ∈ ℂ^{(2ν+1)N−1}.

Then, according to Lemma 4.8, this provides, on D_{2,n−1}, the global containment: Φ_{1} ⊂ Ψ_{n}. However, this statement remains valid after the substitution m_{N} = 0, provided that it was already valid in the original set-up. We remind the reader that the almost everywhere (non-degenerate) definability of the algebraic dynamics subjected to the side condition m_{N} = 0 easily follows from (I).

Remark 4.24. Since the substitutions m_{N} = 0, L = 0 play a basic role in the forthcoming arguments, it is very important to note here that, thanks to the inductive hypothesis of 4.21, we can assume that Property (A) is inherited to the algebraic billiard trajectory segment subjected to the side conditions m_{N} = 0 and L = 0. Indeed, as we mentioned in Remark 4.22, parts (VI)–(VII) are just designed to ensure the inheritance of Property (A).

Let us return now to our proof of the inductive step of (II^{∗}). Assume, of course, the inductive hypothesis. We have, moreover, already assumed indirectly that Φ_{1} and Ψ_{n} coincide locally in D_{2,n−1}. By the inductive assumption Φ_{1} and Ψ_{n} intersect D_{2,n−1}, whereas by the indirect assumption and by the globalization argument

Φ_{1} ⊂ Ψ_{n}  (4.25)

holds on D_{2,n−1}. If, under the substitution m_{N} = 0 (say), the degeneracy Φ_{1} (and thus Ψ_{n}, too) intersects D′, the working domain (cf. Remark 4.20) for the derived scheme (Σ′, A′, τ′) of (Σ, A, τ⃗) corresponding to m_{N} = 0, then the contradiction to the inductive hypothesis (II^{∗}) is evident. For every k with σ′_{k} ∈ Σ′, denote, in general, by Φ′_{k} and Ψ′_{k} the projections of Φ_{k} and Ψ_{k} to the phase space of the N −1ball system corresponding to taking m_{N} = 0 and cancelling the data of the N-th ball (as many times before, the indexing in the derived schemes is again lacunary).

Consider now the case when, under the substitution m_{N} = 0, the degeneracy Φ′_{1} or Ψ′_{n} does not intersect D′. The relation Φ′_{1} ∩D′ = ∅ can only hold if the symbolic sequence Σ′ is not Φ-minimal from the left and, similarly, Ψ′_{n} ∩ D′ = ∅ can only happen if the symbolic sequence (Σ′, A′) is not Ψ-minimal from the right.

In any case, there exists a subsegment (Σ′′, A′′, τ′′) of (Σ′, A′, τ′), which is minimal from both sides and, moreover, by denoting by p′′ and n′′ the indices of its initial and final elements (in the original indexing of (Σ, A, τ⃗)), then, on one hand, σ′_{1} = σ′_{p′′}, and for every σ′_{j} ∈ Σ′ with 1 < j < p′′, σ′_{j} ∩ σ′_{1} = either σ′_{1} or ∅, and on the other hand, (σ′_{n}, a′_{n}) = (σ′_{n′′}, a′_{n′′}), and for every σ′_{j} ∈ Σ′ with n′′ < j < n either (σ′_{j}, a′_{j}) = (σ′_{n}, a′_{n}) or σ′_{j} ∩ σ′_{n} = ∅. Note that, by virtue of Remark 4.18/b, the just formulated conditions also imply Φ′_{1} = Φ′_{p′′} and Ψ′_{n′′} = Ψ′_{n}. Also, by the inductive assumption (IV^{∗}), Φ′_{p′′} ∩ D_{p′′+1,n′′−1} ≠ ∅ and Ψ′_{n′′} ∩ D_{p′′+1,n′′−1} ≠ ∅ and further, by (II^{∗}), they can not coincide locally in D_{p′′+1,n′′−1}.

When completing the argument we will be exploiting the following lemma, which is a limiting variant of the observation of point b. of Remark 4.18 and can be proved easily.

Lemma 4.26. Consider a symbolic collision sequence (Σ, A, τ⃗) which satisfies Property (A) and is Φ-minimal from the left and Ψ-minimal from the right. Suppose we set m_{N} = 0 and denote by (Σ′, A′, τ′) the corresponding derived scheme. We have two statements (whose duals for Φ-minimality from the right and Ψ-minimality from the left are also true):

(1) Assume that σ′_{1} = σ′_{p′} (= {1, 2}, say) 1 < p′ ≤ n, and for every σ′_{j} ∈ Σ′ with 1 < j < p′ one has σ′_{j} ∩ σ′_{1} = σ′_{1} or ∅ (lacunary indices!). Take a sequence ω_{k} ∈ Ω̃ (Σ; A; τ⃗) such that

(i) all kinetic data of ω_{k} measured right after σ_{1} have a limit,

(ii) lim_{k→∞} m_{N}(ω_{k}) = 0

(iii) lim_{k→∞} m_{i}(ω_{k}) ≠ 0 1 ≤ i ≤ N − 1

(iv) lim_{k→∞} m_{i}(ω_{k}) + m_{j}(ω_{k}) ≠ 0 1 ≤ i < j ≤ N − 1

(v) lim_{k→∞} ∥v_{1}^{1}(ω_{k}) − v_{2}^{1}(ω_{k})∥^{2} = 0.

We claim that lim_{k→∞} ∥v_{1}^{p′}(ω_{k}) − v_{2}^{p′}(ω_{k})∥^{2} = 0.

(2) Assume that (σ′_{n′}, a′_{n′}) = (σ′_{n}, a′_{n}) (σ′_{n} = {1, 2}, say) 1 ≤ n′ < n, and for every σ′_{j} ∈ Σ′ with n′ < j < n one either has that (σ′_{j}, a′_{j}) = (σ′_{n}, a′_{n}) or σ′_{j} ∩ σ′_{n} = ∅ is true.

Take a sequence ω_{k} ∈ Ω̃ (Σ; A; τ⃗) such that (i-iv) hold (with σ_{1} replaced by σ_{n} in (i)), and further

(v)′ lim_{k→∞} ⟨v_{1}^{n}(ω_{k}) − v_{2}^{n}(ω_{k}); q̃_{1}^{n}(ω_{k}) − q̃_{2}^{n}(ω_{k}) − L · a_{n}⟩ = 0

is also true.

We claim that

lim_{k→∞} ⟨v_{1}^{n′}(ω_{k}) − v_{2}^{n′}(ω_{k}); q̃_{1}^{n′}(ω_{k}) − q̃_{2}^{n′}(ω_{k}) − L · a_{n′}⟩ = 0.

Take now convergent sequences of points x⃗_{k} ∈ Φ_{1} ∩ D_{2,n−1} ⊂ Ψ_{n} ∩ D_{2,n−1} such that m_{N}(x⃗_{k}) → 0 (k → ∞). Then, according to Lemma 4.26, the projections x′ of the points x⃗ to the N − 1-ball phase space of the system necessarily fill up a nonempty open piece U′ of Φ′_{p′′} = Φ′_{1} on one hand, which necessarily also belongs to Ψ′_{n′′} = Ψ′_{n} on the other hand. Consequently, Φ′_{p′′} coincides locally with Ψ′_{n′′}, a contradiction. The conclusion is that if (II^{∗}) holds for the N − 1-ball system, then so it does for the N-ball system, too. This completes the proof of the inductive step N − 1 → N for statement (II^{∗}).

As mentioned before, the base of the induction on N is N = 2 or 3 or 4 depending on the form of (Σ, A), and we treat two cases separately. We remind the reader of our notational convention to use densely packed indices.

Case 4/a/1: σ_{1} ≠ σ_{n}. In this case the base of the induction is N = 3 or 4.

The globalization argument provides that the indirect assumption that Φ_{1} and Ψ_{n} coincide locally leads to the global containment Φ_{1} ⊂ Ψ_{n} on D_{2,n−1}. Then this ought to hold under the side conditions m_{3} = m_{4} = 0, too. But this is impossible for the following reason: In the case m_{3} = m_{4} = 0 the motion of the particles {3, 4} has absolutely no effect on the time evolution of the {1, 2} subsystem. However, the equation defining Φ_{1} is obviously sensitive of the kinetic data (position and velocity) of the third ball, while the equation δ(ω) = 0 is written fully in terms of the subsystem {1, 2}. The obtained contradiction proves the base case of the induction for (II^{∗}) in the case σ_{1} ≠ σ_{n}.

Case 4/a/2: σ_{1} = σ_{n}. Now the base of the induction is N = 2.

Similarly to the previous case, the globalization argument implies the containment Φ_{1} ⊂ Ψ_{n}. This is an assertion completely in terms of the now independently evolving two-particle system {1, 2}! We will show that in such a complexified, two-particle algebraic system the tangential degeneracy δ(ω) = 0 of the collision σ_{n} cannot completely contain the degeneracy ∥v_{1}^{2} −v_{2}^{2}∥^{2} = 0. (Now the σ’s only denote the collisions of the two-ball system {1, 2}.) The proof will be a simple induction on the positive integer n. For n = 2 the statement is obviously true. Assume now that n > 2, and the assertion has been proven for all smaller values of n. According to our inductive hypothesis, for a typical element ω of the σ_{n}-degeneracy δ(ω) = 0 we have that ∥v_{1}^{3}(ω) − v_{2}^{3}(ω)∥^{2} ≠ 0. On the other hand, the collision σ_{3} is known to preserve the quantity ∥v_{1} − v_{2}∥^{2} (the velocity reflection is complex orthogonal), and, therefore, we get that ∥v_{1}^{2}(ω) − v_{2}^{2}(ω)∥^{2} ≠ 0, as well.

The obtained contradiction finishes the discussion of 4/a.

4/b: Proof of (III^{∗}) by using the inductive hypothesis. Assume now, besides the inductive hypothesis, the precondition of (III^{∗}).

The proof of the inductive step N − 1 → N is absolutely analogous to that given in part 4/a: one has just to substitute Φ_{1} with Ψ_{1}. The verification of the basis of the induction will be split into three parts depending on the form of (Σ, A). In all cases we again assume indirectly that Ψ_{1} coincides locally with Ψ_{n} in D_{2,n−1}. By the globalization/substitution argument from the proof of (II^{∗}) we then obtain Ψ_{1} ⊂ Ψ_{n} to hold globally in D_{2,n−1} for the base cases, too. What we show is that this inclusion leads to a contradiction in all cases.

Case 4/b/1: σ_{1} ≠ σ_{n}. In this case the base of the induction is N = 3 or 4, and the impossibility of the containment Ψ_{1} ⊂ Ψ_{n} for N = 3 or 4 can be obtained in the same way as in 4/a/1.

Case 4/b/2: σ_{1} = σ_{n} and there is a k, 1 < k < n, such that σ_{k} ∩ σ_{n} ≠ ∅ and σ_{k} ≠ σ_{n}.

Now the base case will be N = 3 and, as explained above, we can assume the global containment Ψ_{1} ⊂ Ψ_{n} on D_{2,n−1}. By re-labeling the particles, we can assume that σ_{1} = σ_{n} = {1, 2} and σ_{k} = {2, 3}.

The global containment Ψ_{1} ⊂ Ψ_{n} will naturally be preserved under the substitution L = 0, too. This substitution raises the same questions as the substitution m_{N} = 0, and they can be handled analogously. For instance, it is easily possible that though (Σ, A, τ⃗) is minimal from both sides, the scheme (Σ′, A′, τ′) derived from it via the substitution L = 0 is not minimal. Our way of taking care about this possibility in the case of the substitution m_{N} = 0 can also be repeated here and, therefore, by our combined indirect and inductive arguments, we can again assume that the containment Ψ_{1} ⊂ Ψ_{n} now holds even in the 3-particle system {1, 2, 3} (having a time evolution with Property (A)). On the other hand, the fact that this 3-particle system {1, 2, 3} also has the property L = 0 amounts to a_{l} = 0 for l = 1, …, n. Obviously, we can assume that k = 2, i. e. σ_{2} = {2, 3}. Case 4/b/2 will be taken care of by our

Lemma 4.27. Suppose that the triple (Σ = (σ_{0}, …, σ_{n}), 0, τ⃗) enjoys Property (A), N = 3, σ_{0} = σ_{n} = {1, 2} and, say, σ_{1} = {2, 3}. Then Ψ_{0} can not be (globally) a subset of Ψ_{n} and, therefore, the tangential degeneracies at σ_{0} and σ_{n} (i. e. Ψ_{0} and Ψ_{n}) can not even locally coincide in D_{1,n−1}.

Proof. In order to simplify the notations we assume that r = 1/2, and σ_{n−1} ≠ {1, 2}. Denote by Σ^{∗} = (σ_{1}^{∗}, …, σ_{s}^{∗}) the symbolic sequence that can be obtained from our original sequence (σ_{0}, …, σ_{n}) by discarding all symbols {1, 2}. The truncated sequence Σ^{∗} consists of a pure block of n(1) {2, 3}-collisions first, then a pure block of n(2) {1, 3}-collisions etc., up to the closing pure block of n(k) collisions of type {(3 + (−1)^{k+1}) /2, 3}.

Consider first the following, essentially two-dimensional and completely real limiting set-up: m_{1} = m_{2} = 1, m_{3} = 0, q̃_{1}^{0} = 0, q̃_{2}^{0} = e_{1}, q̃_{3}^{0} = −e_{2}, v_{1}^{0} = v_{2}^{0} = 0, v_{3}^{0} = (1+ε^{2})^{−1/2}·(e_{1}+εe_{2}), where e_{1} = (1, 0, …, 0) ∈ ℝ^{ν}, e_{2} = (0, 1, 0, …, 0) ∈ ℝ^{ν}, and ε > 0, ε → 0 is a positive parameter of perturbation. By using Property (A) of (Σ, 0, τ⃗) and the entire inductive hypothesis of 4.21, we can now construct a (Σ, 0, τ⃗) orbit segment

ω = {q̃_{i}^{α}, v_{i}^{α} | i = 1, 2, 3; α = 0, …, n}

by extending the fixed initial values m_{1} = m_{2} = 1, m_{3} = 0, q̃_{1}^{0} = 0, q̃_{2}^{0} = e_{1}, q̃_{3}^{0} = −e_{2}, v_{1}^{0} = v_{2}^{0} = 0, v_{3}^{0} = (1 + ε^{2})^{−1/2} · (e_{1} + εe_{2}) in such a way that

(i) at every collision σ_{j} = {2, 3} (0 < j < n) always the smaller root τ_{j} is selected out of the two distinct, real roots;

and

(ii) at every collision σ_{j} = {1, 3} (0 < j < n) always the greater root τ_{j} is selected out of the two distinct, real roots.

We note here that, obviously, the above constructed orbit segment ω is a highly degenerate one because of the degenerate {1, 2}-collisions. However, we will transform ω into a nondegenerate orbit segment (except one tangential collision at σ_{0}) by the upcoming, final perturbation.

For j = 1, …, k we denote by ν(j) the total number of collisions σ_{i} (i. e. together with the {1, 2}-collisions) from σ_{1} up to the closing collision σ_{h}^{∗} (h = n(1)+⋯+n(j)) of the j-th Σ^{∗} block. We note that, due to the selection of the initial kinetic data, the {1, 2}-collisions have no effect on the motion of the third disk.

An easy geometric calculation shows that the asymptotic formulae

\begin{aligned}q̃_{3}^{ν(1)}(ε) & = e_{1} − e_{2} − (2n(1) − 1) · (2ε)^{1/2} · e_{1} + o (ε^{1/2}) , \\ v_{3}^{ν(1)}(ε) & = e_{1} − 2n(1)(2ε)^{1/2} · e_{2} + o (ε^{1/2}) \end{aligned}

hold. By similar arguments we obtain the asymptotics

\begin{aligned}q̃_{3}^{ν(2)}(ε) & = −e_{2} + (2n(2) − 1) · [4n(1) · (2ε)^{1/2}]^{1/2} · e_{1} + o (ε^{1/4}) , \\ v_{3}^{ν(2)}(ε) & = e_{1} + 2n(2) · [4n(1) · (2ε)^{1/2}]^{1/2} · e_{2} + o (ε^{1/4}) . \end{aligned}

Then a simple induction yields that, in general, the following asymptotic formulae hold for j = 0, 1, …, k:

\begin{aligned}q̃_{3}^{ν(j)}(ε) & = \frac{1 + (−1)^{j+1}}{2} · e_{1} − e_{2} + (−1)^{j}b_{j}ε^{2^{−j}} · e_{1} + o (ε^{2^{−j}}) , \\ v_{3}^{ν(j)}(ε) & = e_{1} + (−1)^{j}c_{j}ε^{2^{−j}} · e_{2} + o (ε^{2^{−j}}) , \end{aligned}  (4.28)

where the positive constants b_{j} and c_{j} can be computed by the recursion b_{0} = 0, c_{0} = 1, b_{j} = (2n(j) − 1) \sqrt{2c_{j−1}}, c_{j} = 2n(j)\sqrt{2c_{j−1}}, j = 1, …, k.

Let us now focus our attention to the almost vertical velocity changes

ΔV_{j}(ε) = v_{3}^{ν(j)}(ε) − v_{3}^{ν(j−1)}(ε)

and their horizontal components ⟨ΔV_{j}(ε); e_{1}⟩, j = 1, …, k. The asymptotic formulae (4.28) for the velocities immediately yield the expressions

\begin{matrix}ΔV_{j}(ε) = (−1)^{j}c_{j}ε^{2^{−j}} · e_{2} + o (ε^{2^{−j}}) , \\ ⟨ΔV_{j}(ε); e_{1}⟩ = −\frac{1}{2}c_{j}^{2}ε^{2^{1−j}} + o (ε^{2^{1−j}}) , \end{matrix}  (4.29)

j = 1, …, k. Here we used the fact that the norm ||v_{3}^{i}(ε)|| is always 1.

From the asymptotic formulae (4.29) we conclude that the order of magnitude of the horizontal velocity change is the highest one in the last pure block of collisions B_{k} = (σ_{s−n(k)+1}^{∗}, …, σ_{s}^{∗}). (Actually, s − n(k) = ∑_{j=1}^{k−1} n(j).) These collisions take

place between the third disk and the disk with the label i(k) = (3 + (−1)^{k+1}) /2. However, after a perturbation (to be carried out a bit later) m_{3} ≈ 0, m_{i} ∈ ℝ_{+} (i = 1, 2, 3), only a certain part of the horizontal momentum given by the third disk to the i(k)-disk (during the collisions of B_{k}) will be retained by the disk i(k): the rest will be transported to the other heavy disk with label 3−i(k) by the almost tangential collisions {1, 2} (taking place in between the collisions of B_{k}) with almost horizontal normals of impact. This justifies the following definition:

H :={q ∈ ℤ| 1 ≤ q ≤ n(k) and the # of the
{1, 2}-collisions between σ_{s−n(k)+q}^{∗} and σ_{s}^{∗} is even},
\overline{H} := \{1, 2, …, n(k)\} \ H.

By using the method of the proof of the second line of (4.29) we see that the order of magnitude of the horizontal velocity change at the q-th collision of B_{k} (1 ≤ q ≤ n(k)) is

4(2q − 1)c_{k−1}ε^{2^{1−k}} + o (ε^{2^{1−k}}) .

Note that the cumulative horizontal velocity change caused by the collisions

σ_{s−n(k)+1}^{∗}, …, σ_{s−n(k)+q}^{∗}

has the asymptotic formula 4q^{2}c_{k−1}ε^{2^{1−k}}, cf. how one can deduce the formula c_{k} = 2n(k)\sqrt{2c_{k−1}}.

There are now two possibilities:

Case A. ∑_{q∈H}(2q − 1) ≠ \frac{1}{2} [n(k)]^{2}.

(Note that ∑_{q∈H∪\overline{H}}(2q − 1) = [n(k)]^{2}.)

In this case the small perturbation

0 < m_{3}(ε) << ε, m_{3}(ε) is very small, depending on ε,

v_{1}^{0}(ε) = −v_{2}^{0}(ε) = η(ε) · e_{2}, 0 < η(ε) = η(m_{3}(ε)) << m_{3}(ε) η(ε) is very small compared to m_{3}(ε)

(4.30)

(still m_{1} = m_{2} = 1) serves our purpose: the horizontal component of the relative velocity right before the collision σ_{n} = {1, 2} will be of the order of magnitude

4m_{3}c_{k−1} [∑_{q∈H}(2q − 1) − ∑_{q∈\overline{H}}(2q − 1)] ε^{2^{1−k}} ≠ 0.  (4.31)

We note that the asymptotic formulae (4.28-29) will not be destroyed as long as the quantities m_{3}(ε) and η(ε) << m_{3}(ε) are small enough depending on ε. Also note that the collision σ_{n} (as well as all other collisions {1, 2} amongst σ_{0}, …, σ_{n}) will be an asymptotically tangential collision with asymptotically horizontal normal of impact. These facts immediately follow from the asymptotic formulae (4.29). Hence, in Case A we managed to perturb out of the degeneracy Ψ_{n} from within Ψ_{0}.

Case B. ∑_{q∈H}(2q − 1) = ∑_{q∈\overline{H}}(2q − 1).

In this case the equation m_{1} = m_{2} does not suffice: we need to use the possibility of selecting different “heavy” masses. First we select and fix the positive values m_{1} ≠ m_{2}, then we carry out the perturbation (4.30) and compute the orders of magnitude of the horizontal components V_{1}^{h}, V_{2}^{h} of the velocities of the first and second disks measured right after the collision σ_{n−1} = σ_{s}^{∗}. By applying the collision laws for horizontal “head-on” collisions with masses m_{1} and m_{2} we easily obtain the asymptotics

\begin{aligned}V_{1}^{h} & = 2c_{k−1} [n(k)]^{2} [\frac{2m_{1}m_{3}}{m_{2}(m_{1} + m_{2})} + \frac{m_{3}}{m_{1}} − \frac{m_{3}}{m_{2}} ] · ε^{2^{1−k}} + o (ε^{2^{1−k}}) , \\ V_{2}^{h} & = 4c_{k−1} [n(k)]^{2} \frac{m_{1}m_{3}}{m_{2}(m_{1} + m_{2})} · ε^{2^{1−k}} + o (ε^{2^{1−k}}) . \end{aligned} (4.32)

These asymptotics are different, thanks to the inequality m_{1} ≠ m_{2}. Therefore, we again managed to perturb out of the degeneracy Ψ_{n} from within Ψ_{0} by using the facts that all collisions {1, 2} amongst σ_{0}, …, σ_{n} are asymptotically tangential collisions with asymptotically horizontal normals of impact.

Finally, by using the method of obtaining different asymptotics (4.31-32) for the horizontal components of v_{1}^{n−1} and v_{2}^{n−1}, we can also achieve different asymptotics for the horizontal components of v_{1} and v_{2} right before every collision σ_{i} = {1, 2} (i > 0) by appropriately selecting the perturbation. In that way all collisions σ_{1}, …, σ_{n} become non-singular, and all kinetic data of the constructed orbit segment will be real.

This finishes the proof of the Lemma 4.27. □

Our last outstanding task in proving (I-VII) is the handling of

Case 4/b/3: σ_{1} = σ_{n} (= {1, 2}), and for every k ∈ {1, …, n} it is true that either σ_{k} = σ_{n} or σ_{k} ∩ σ_{n} = ∅.

In this case not all adjustment vectors a_{k} with σ_{k} = σ_{n} (p ≤ k ≤ n) are the same.

The base case now is N = 2, and according to the indirect assumption we have a global containment of the tangential degeneracies: Ψ_{1} ⊂ Ψ_{n} in the 2-ball system {1, 2}. The discussion of Case 4/b/3 will be finished as soon as we prove our

Lemma 4.33. Suppose that N = 2, the triple (Σ, A, τ⃗) enjoys Property (A), and not all adjustment vectors a_{1}, …, a_{n} of A are the same. Then Ψ_{1} and Ψ_{n} can not locally coincide in D_{2,n−1}.

Proof.

1^{o}. We will construct a one-parameter family

ω(s) ∈ Ω̃ (Σ, A, τ⃗) = Ω̃, (s ∈ I ⊂ ℝ),

(I is an open and bounded interval of ℝ) of (Σ, A, τ⃗)-orbit segments in such a way that this bunch of trajectories will explicitly show that Ψ_{1} and Ψ_{n} are not the same. In fact, we will see that Ψ_{1} and Ψ_{n} intersect each other transversally at a large collection of points of intersection.

2^{o}. Throughout the construction of the family {ω(s)| s ∈ I} we shall follow these guiding principles:

(a) The dimension ν in this construction will be two: The transversality of Ψ_{1} and Ψ_{n} in a two-dimensional section q̃_{i}^{t}(s) ∈ ℂ^{2} ⊂ ℂ^{ν}, v_{i}^{t}(s) ∈ ℂ^{2} ⊂ ℂ^{ν} clearly implies the transversality in the general set-up ν ≥ 2;

(b) All data q̃_{i}^{t}(s) ∈ ℝ^{2}, v_{i}^{t}(s) ∈ ℝ^{2}, m_{1}(s) = m_{2}(s) = 1, r = 1/2, L(s) = L >> 1 will be real, the side length L of the 2-torus being a very large, fixed positive constant L;

(c) Due to the translation invariance of the system, we will only deal with the relative motion

q^{t}(s) := q̃_{1}^{t}(s) − q̃_{2}^{t}(s) ∈ ℝ^{2},
v^{t}(s) := v_{1}^{t}(s) − v_{2}^{t}(s) ∈ ℝ^{2}, ∥v^{t}(s)∥^{2} = 1,

(s ∈ I);

(d) In virtue of the condition ∑_{i=1}^{n} ||a_{i} − a_{1}||^{2} > 0, by cutting off a suitable starting segment of Σ (if necessary), we can assume that 0 = a_{1} ≠ a_{2};

(e) The construction of the family ω(s) will be such that for every k = 2, 3, …, n the polynomial (3.8) has two distinct, nonnegative real roots, and we will always take the greater one as τ_{k} out of these two roots. The inductive assumption on the irreducibility of these polynomials (3.8) over the field 𝕂_{k−1} (They are, in fact, irreducible, unless they have to be reducible, because of the repetitive adjustment vectors. This is part of the inductive hypothesis of 4.21.) and Property (A) of (Σ, A, τ⃗) together ensure that the time evolution of ω(s) will follow the algebraic pattern encoded in (Σ, A, τ⃗);

(f)

q^{0}(s) := (cos(ϕ_{0} + s), sin(ϕ_{0} + s)) + (1 − s) · (− sin(ϕ_{0} + s), cos(ϕ_{0} + s)) ,

v^{0}(s) := (− sin(ϕ_{0} + s), cos(ϕ_{0} + s)) ,

where the ray {q^{0}(0) + tv^{0}(0)| t ≥ 0} passes through the point L · a_{2}.

The above constructed family {q^{0}(s), v^{0}(s)| |s| < ϵ_{0}, s ∈ ℝ} is a so called convex family of rays, or convex wave front, see, for example, [B(1979)] or [W(1986)]. This means that

⟨\frac{d}{ds}q^{0}(s); v^{0}(s)⟩ = 0, ⟨\frac{d}{ds}q^{0}(s); \frac{d}{ds}v^{0}(s)⟩ > 0.

The time evolution of such diverging families of trajectories is pretty well discussed in the literature, see [W(1986)]. It is shown in that paper (see Section 3, especially Theorem 3 there) that our two-dimensional billiard with inner reflections at the circles

C_{a} := {q ∈ ℝ^{2}| d(q, L · a) = 1} , (a ∈ ℤ^{2})

has the so called convex scattering property, i. e. every such convex family of rays (with not too big curvature, see [W(1986)]) focuses after any inner reflection at C_{a} not later than when it reaches the midpoint of the chord of C_{a} along which it is traveling after the considered reflection.

3^{o}. Plainly, the trajectory ω(s) intersects the circle C_{0} tangentially in negative time τ_{1}(s) = s − 1, that is, the convex family of rays {q^{0}(s), v^{0}(s)| |s| < ϵ_{0}} is the “synchronized” family of trajectories belonging to Ψ_{1}. We recall that r = 1/2 and a_{1} = 0 ∈ ℤ^{2}.

4^{o}. The inner reflections at the circles C_{a_{2}}, …, C_{a_{n}} (yet to be constructed) will be in good harmony with the principle 2/e above.

5^{o}. If L is large enough, then there is a small, open subinterval I_{2} ⊂ (−ϵ_{0}, ϵ_{0}) (even the closure of I_{2} will be a subset of (−ϵ_{0}, ϵ_{0})) such that the rays {q^{0}(s) + tv^{0}(s)| t ≥ 0} intersect the circle C_{a_{2}} in two distinct points for s ∈ I_{2}, while these rays are tangent to C_{a_{2}} for s ∈ ∂I_{2}. For s ∈ I_{2} we elongate the above ray up to the second point of intersection with the circle C_{a_{2}}, and this way obtain the orbit segment up to the first collision with that circle.

6^{o}. Let a_{2} = a_{3} = ⋯ = a_{k(2)} (2 ≤ k(2) ≤ n), where either a_{k(2)+1} ≠ a_{k(2)} or k(2) = n. Suppose that k(2) is still less than n. By inspecting the geometry of the sequence of inner reflections σ_{2}, …, σ_{k(2)} at the circle C_{a_{2}} it is easy to see that the outgoing velocities v^{k(2)}(s) (s ∈ I_{2}) sweep out the entire space 𝕊^{1} of velocity directions. Therefore, there is an even smaller interval I_{3}, Cl(I_{3}) ⊂ I_{2}, such that the rays {q^{k(2)}(s) + tv^{k(2)}(s)| t ≥ 0} intersect the circle C_{a_{k(2)+1}} in two distinct points for s ∈ I_{3}, while these rays are tangent to C_{a_{k(2)+1}} for s ∈ ∂I_{3}. (Here q^{k(2)}(s) and v^{k(2)}(s) denote the position and the outgoing velocity at the reflection σ_{k(2)} taking place at time ∑_{i=1}^{k(2)} τ_{i}.)

For s ∈ I_{3} we again elongate the above ray up to the second point of intersection with the circle C_{a_{k(2)+1}}, and this way obtain the orbit segment up to the first collision with that circle.

7^{o}. Now we keep going with the iterative construction of the orbit segments ω(s) ∈ Ω̃ (Σ, A, τ⃗) described in 7 above, by also constructing the nested sequence of open intervals I_{2} ⊃ I_{3} ⊃ ⋯ ⊃ I_{l} := I, where k(l) = n. The two orbit segments {ω(s)| s ∈ ∂I} are now obviously in the transversal intersection of Ψ_{1} and Ψ_{n}. Thus the Lemma is proved. □

So far we have settled statements (I-VII) of the Main Lemma.

5. Our only remaining task in the proof of the Main Lemma is to verify statement (VIII). Similarly to (II-III) it is again sufficient to demonstrate the slightly weaker assertion (VIII^{∗}) formulated in point 4.

Assume now, besides the inductive hypothesis, the precondition of (VIII^{∗}). We again use double induction in the same way as it was explained before 4/a. Also, the proof of the inductive step N − 1 → N is absolutely analogous to that given in part 4/a: one has just to substitute Ψ_{n} with Φ_{n}. The verification of the basis of the induction will be split into two parts depending on the form of Σ. In all cases we again assume indirectly that Φ_{1} coincides locally with Φ_{n}. By the globalization/substitution argument from the proof of (II^{∗}) we then obtain Φ_{1} ⊂ Φ_{n} to hold globally for the base cases, too. What we show is that this inclusion leads to a contradiction in both cases.

Case 5/1: σ_{1} ≠ σ_{n}. In this case the base of the induction is N = 3 or 4, and the impossibility of the containment Φ_{1} ⊂ Φ_{n} for N = 3 or 4 can be obtained in the same way as in the proof of (II^{∗}).

Case 5/2: σ_{1} = σ_{n} and there is a k, 1 < k < n, such that σ_{k} ∩ σ_{n} ≠ ∅ and σ_{k} ≠ σ_{n}.

Now the base case will be N = 3, and as explained above we can derive the global containment Φ_{1} ⊂ Φ_{n}. By re-labeling the particles, we can assume that σ_{1} = σ_{n} = {1, 2} and σ_{k} = {2, 3}.

The global containment Φ_{1} ⊂ Φ_{n} will naturally be preserved under the substitution L = 0, too. The assumed containment Φ_{1} ⊂ Φ_{n} now implies that even in the 3-particle system {1, 2, 3} (having a time evolution with Property (A)) one has a global containment of the Φ- degeneracies at σ_{1} and σ_{n}. The fact that this 3-particle system {1, 2, 3} also has the property L = 0 amounts to a_{l} = 0 for l = 1, …, n.

Obviously, we can assume that k = 2, i. e. σ_{2} = {2, 3}. Case 5/2 will be taken care of by the following analogue of our Lemma 4.27:

Lemma 4.34. Suppose that the triple (Σ = (σ_{0}, …, σ_{n}), 0, τ⃗) enjoys Property (A), N = 3, σ_{0} = σ_{n} = {1, 2} and, say, σ_{1} = {2, 3}. Then Φ_{0} can not be (globally) a subset of Φ_{n} and, therefore, the Φ-degeneracies at σ_{0} and σ_{n} (i. e. Φ_{0} and Φ_{n}) can not even locally coincide in D_{1,n−1}.

Proof. The construction used in the proof of Lemma 4.27 also provides our recent claim. Indeed, let us set in that construction η(ε) = 0. Then, because of v_{1}^{0}(ε) = v_{2}^{0}(ε) = 0, the initial perturbation lies in Φ_{1}. At the same time, in view of (4.31-32), for the horizontal components V_{1}^{h}, V_{2}^{h} of the velocities of the first and second discs measured right after the collision σ_{n−1} = σ_{s}^{∗} we have ∥V_{1}^{h} − V_{2}^{h}∥^{2} ≠ 0 showing that the perturbation necessarily leads out of Φ_{n}. Thus the globalization/substitution argument provides the claimed transversality of Φ_{1} and of Φ_{n}, the statment of Lemma 4.34. □. This finishes the proof of Main Lemma 4.21, as well. □

Corollary 4.35. For every triple (Σ, A, τ⃗) with Property (A) and N ≥ 3, there exists at least one derived scheme (Σ′, A′, τ′). □

Corollary 4.36. For combinatorial schemes satisfying Property (A), the quadratic polynomial b_{l}x^{2} + c_{l}x + d_{l} is reducible over the field 𝕂_{l−1} (i. e. 𝕂_{l−1} = 𝕂_{l}) only in the case described above in 3.31.

Proof. Unless we are considering the case described in Remark 3.31, statement (V) of Main Lemma 4.21 gives the existence of an ω ∈ Ω̃_{n−1} with δ(ω) = 0. But then part 3 of the proof of the main lemma provides |𝕂_{n} : 𝕂_{n−1}| = 2. □

Proof of the Key Lemma. Final Induction

The next lemma establishes the link between the orbit segments of (Σ, A, τ⃗) and (Σ′, A′, τ′). It is a direct consequence of Remark 4.13 (actually, of the fact that the reducibility 𝕂_{l−1} = 𝕂_{l} is inherited to the (Σ′, A′, τ′)-dynamics during the substitution m_{N} = 0) and part (2) of the definition of the derived schemes.

Lemma 4.37. Fix a combinatorial scheme (Σ, A, τ⃗) satisfying Property (A) and an arbitrary scheme (Σ′, A′, τ′) derived from it. Then, for almost every choice of the vector of data (q̃_{N}^{0}, v_{N}^{0}) ∈ ℂ^{ν} × ℂ^{ν}

(1) The subset {(x′, q̃_{N}^{0}, v_{N}^{0}, m_{N} = 0)| x′ ∈ ℂ^{h}} ∩ D (Σ, A, τ⃗) is open and dense in the submanifold {(x′, q̃_{N}^{0}, v_{N}^{0}, m_{N} = 0)| x′ ∈ ℂ^{h}} (here h = (2ν + 1)(N − 1) + 1);

(2) For a nonempty, open set of orbit segments ω′ ∈ Ω̃ (Σ′, A′, τ′) there exists a trajectory segment ω ∈ Ω̃ (Σ, A, τ⃗) for which m_{N}(ω) = 0, trunc(ω) = ω′, and (q̃_{N}^{0}(ω), v_{N}^{0}(ω)) = (q̃_{N}^{0}, v_{N}^{0});

(3) For an open and dense set of orbit segments ω′ ∈ Ω̃ (Σ′, A′, τ′) there exists a trajectory segment ω ∈ Ω̃ (Σ, A, τ⃗) for which m_{N}(ω) = 0, trunc(ω) = ω′, and (q̃_{N}^{0}(ω), v_{N}^{0}(ω)) = (q̃_{N}^{0}, v_{N}^{0}).

(Here “trunc” denotes the cancellation of all kinetic data of the N-th ball with zero mass.)

Proof. Statement (1) is an obvious consequence of Main Lemma 4.21/(I) and the Fubini theorem. The second assertion follows immediately from the definition of the derived schemes, see also Remark 4.12. For demonstrating (3), we select and fix a typical pair (q̃_{N}^{0}, v_{N}^{0}) according to (1) of Lemma 4.37. Intersect the open and dense subset of ℂ^{h} – featuring in (1) – with the open and dense domain D (Σ′, A′, τ′) and obtain the open and dense intersection D′ ⊂ ℂ^{h}. Suppose now that ω′ ∈ Ω̃ (Σ′, A′, τ′), and the vector x′ = x(ω′) of the initial variables of ω′ belongs to D′. (This is an open and dense condition on ω′.) Launch the (Σ, A, τ⃗)-dynamics with the initial data (x′, q̃_{N}^{0}, v_{N}^{0}, m_{N} = 0) ∈ D (Σ, A, τ⃗) and try to continue the (Σ, A, τ⃗)-dynamics by also preserving the time evolution of the given ω′ ∈ Ω̃ (Σ′, A′, τ′). Remark 4.13 (more precisely, the inheritance of the reducibility proved there) and the definition of the derived schemes (especially (2) thereof) ensure that the continuation of the (Σ, A, τ⃗)-dynamics is, indeed, possible by also preserving the given time evolution of ω′. Hence Lemma 4.37 follows. □

The next lemma will use

Definition 4.38. Suppose that two indices 1 ≤ p < q ≤ n and two labels of balls i, j ∈ {1, …, N} are given with the additional requirement that if i = j, then i ∈ ⋃_{l=p+1}^{q−1} σ_{l}. Following the proof of Lemma 4.2, denote by Q_{1}(x⃗), Q_{2}(x⃗), …, Q_{ν}(x⃗) (x⃗ ∈ ℂ^{(2ν+1)N+1}) the polynomials with the property that for every vector of initial data x⃗ ∈ D (Σ, A, τ⃗) and for every k, k = 1, …, ν, the following equivalence holds true:

(∃ ω ∈ Ω̃ such that x⃗(ω) = x⃗ & (v_{i}^{p}(ω))_{k} = (v_{j}^{q−1}(ω))_{k})
⟺ Q_{k}(x⃗) = 0.

Lemma 4.39. Assume that the combinatorial-algebraic scheme (Σ, A, τ⃗) has Property (A), see Definition 3.31, and use the assumptions and notations of the above definition.

We claim that at least one of the polynomials Q_{1}, …, Q_{ν} is nonzero, i. e. v_{i}^{p}(ω) ≠ v_{j}^{q−1}(ω) for almost every ω ∈ Ω̃ (Σ, A, τ⃗), see also the Corollary 4.7.

Proof. Induction on the number N ≥ 2.

1. Base of the induction, N = 2: First of all, by performing the substitution L = 0, we can annihilate all adjustment vectors, see the proofs of parts 4/a–4/b in Main Lemma 4.21, in accordance with statement (VI) of that lemma. Then an elementary inspection shows that for any selection of positive masses (m_{1}, m_{2}), indeed, v_{i}^{p}(ω) ≠ v_{j}^{q−1}(ω) almost surely in the section Ω̃ (Σ, A, τ⃗, m⃗) of Ω̃ (Σ, A, τ⃗) corresponding to the selected masses, since any trajectory segment of a two-particle system with positive masses and A = 0 has a very nice, totally real (and essentially two-dimensional) representation in the relative coordinates of the particles: the consecutive, elastic bounces of a point particle moving uniformly inside a ball of radius 2r. Therefore, the statement of the lemma is true for N = 2.

We note here that there are merely four points in the entire proof of Key Lemma 4.1 where we need to explicitly use Property (A): The case N = 2, i = j in the proof of this lemma, the Lemmas 4.27, 4.33 and 4.34. Whenever a collision of type {1, 2} is repeated an even number of times with the same adjustment vector and zero time slots, the resulting velocity change is identically zero, because an even power of the same velocity reflection is applied. (See Example I at the end of this section.)

2. Assume that N ≥ 3 and the lemma has been proved for all smaller numbers of balls. By re-labeling the particles, if necessary, we can achieve that

(i) N ≠ i, N ≠ j and

(ii) if i = j, then the ball i has at least one collision between σ_{p} and σ_{q} with a particle different from N.

For the fixed combinatorial scheme (Σ, A, τ⃗), select a scheme (Σ′, A′, τ′) derived from it. By applying the inductive hypothesis to (Σ′, A′, τ′), we get that for almost every ω′ ∈ Ω̃ (Σ′, A′, τ′) the inequality v_{i}^{p}(ω′) ≠ v_{j}^{q−1}(ω′) holds. Moreover, by virtue of Lemma 4.37/(1), for almost every (q̃_{N}^{0}, v_{N}^{0}) ∈ ℂ^{ν} ×ℂ^{ν} one has (x′, q̃_{N}^{0}, v_{N}^{0}, m_{N} = 0) ∈ D (Σ, A, τ⃗) for a dense, open subset of x′ ∈ ℂ^{h}. Let us fix one such pair (q̃_{N}^{0}, v_{N}^{0}). By Lemma 4.37/(2), for a non-empty, open set of ω′ ∈ Ω̃ (Σ′, A′, τ′) it is true that (x(ω′), q̃_{N}^{0}, v_{N}^{0}, m_{N} = 0) ∈ D (Σ, A, τ⃗), v_{i}^{p}(ω′) ≠ v_{j}^{q−1}(ω′), and there exists an ω ∈ Ω̃ (Σ, A, τ⃗) with m_{N}(ω) = 0 and trunc(ω) = ω′.

The assertion of the lemma now follows for the N-ball systems since the bare existence of a single orbit segment with the above non-equality implies that not all polynomials Q_{l}(x⃗) are zero. □

Lemma 4.40. Assume that the combinatorial scheme (Σ, A, τ⃗) satisfying Property (A) is given for the N (N ≥ 3) ball system {1, 2, …, N} and, furthermore, there are two integers 1 ≤ p < q ≤ n such that N ∈ σ_{p} ∩ σ_{q}, N ∉ σ_{j} for p < j < q, and if σ_{p} = σ_{q} then we require additionally that there exists an index j, p < j < q, such that σ_{p} ∩ σ_{j} ≠ ∅. Denote by (Σ′, A′, τ′) an arbitrary derived scheme of (Σ, A, τ⃗). Denote by P_{1}(x⃗), …, P_{s}(x⃗) the canonically determined complex polynomials, the simultaneous vanishing of which is the consequence of non-sufficiency of the (Σ, A, τ⃗) orbit segments in the sense of Lemma 4.2. We claim that if all these polynomials are zero, then all the analogous polynomials associated with the derived scheme (Σ′, A′, τ′) are also zero.

In other words, if all (Σ, A, τ⃗)-orbit segments are non-sufficient, then the same is true for all (Σ′, A′, τ′)-orbit segments, see also the corollary at the end of the proof of Lemma 4.2.

Proof. Assume that the assertion is false for the (Σ′, A′, τ′)-orbit segments, that is, there exists at least one sufficient (Σ′, A′, τ′)-orbit segment or, equivalently, by Lemma 4.8, almost all (Σ′, A′, τ′)-orbit segments are sufficient.

By putting together Lemmas 4.37/(1-2), 4.39 and 4.8, we obtain a sufficient (Σ, A, τ⃗)-orbit. Indeed, for typical elements x′ ∈ D (Σ′, A′, τ′) we have that

for all ω′ ∈ Ω̃ (Σ′, A′, τ′) with x(ω′) = x′
ω′ is sufficient and v_{i_{p}}^{p}(ω′) ≠ v_{i_{q}}^{q−1}(ω′).

(4.41)

By Lemma 4.37/(1) for almost every (q̃_{N}^{0}, v_{N}^{0}) ∈ ℂ^{ν} × ℂ^{ν} it is true that for a dense open subset of x′ ∈ ℂ^{h}

(x′, q̃_{N}^{0}, v_{N}^{0}, m_{N} = 0) ∈ D (Σ, A, τ⃗) .  (4.42)

Thus, for almost every (q̃_{N}^{0}, v_{N}^{0}) ∈ ℂ^{ν} ×ℂ^{ν} we have the following: For a dense, open subset of x′ ∈ ℂ^{h} (4.41) and (4.42) hold. Further the velocity v_{N}^{q−1}(ω) = v_{N}^{p}(ω) can be chosen in such a typical manner that it be not contained by the complex line

v_{i_{p}}^{p}(ω′) + λ · (v_{i_{q}}^{q−1}(ω′) − v_{i_{p}}^{p}(ω′)) (λ ∈ ℂ),

so we can achieve that condition (2) of Lemma 4.8 be satisfied. (The possibility of the desired choice for v_{N}^{p}(ω) = v_{N}^{q−1}(ω) is easily seen if we represent our variables at the pth collison, cf. the proof of Lemma 3.19.)

By Lemma 4.37/(1-2), all this implies that for a typical (q̃_{N}^{0}, v_{N}^{0}) ∈ ℂ^{ν} × ℂ^{ν} there exists a nonempty, open set G of orbit segments ω′ ∈ Ω̃ (Σ′, A′, τ′), with (x(ω′), q̃_{N}^{0}, v_{N}^{0}, m_{N} = 0) ∈ D (Σ, A, τ⃗), such that for every ω′ ∈ G there exists an ω ∈ Ω̃ (Σ, A, τ⃗) satisfying m_{N}(ω) = 0, trunc(ω) = ω′, (q̃_{N}^{0}(ω), v_{N}^{0}(ω)) = (q̃_{N}^{0}, v_{N}^{0}), and, according to Lemma 4.8, ω is sufficient. Hence Lemma 4.40 follows. □

The next, purely combinatorial lemma is the last ingredient of the inductive proof of the key lemma.

Lemma 4.43. Define the sequence of positive numbers C(N) recursively by taking C(2) = 1 and C(N) = \frac{N}{2} · max \{C(N − 1); 3\} for N ≥ 3. Let N ≥ 3, and suppose that the symbolic collision sequence Σ = (σ_{1}, …, σ_{n}) for N particles is C(N)-rich. Then we can find a particle, say the one with label N, and two indices 1 ≤ p < q ≤ n such that

(i) N ∈ σ_{p} ∩ σ_{q},

(ii) N ∉ ⋃_{j=p+1}^{q−1} σ_{j},

(iii) σ_{p} = σ_{q} ⟹ ∃j p < j < q & σ_{p} ∩ σ_{j} ≠ ∅, and

(iv) Σ′ is C(N − 1)-rich on the vertex set {1, …, N − 1}.

Here, just as in the case of derived schemes, we denote by Σ′ the symbolic sequence that can be obtained from Σ by discarding all edges containing N.

Proof. The hypothesis on Σ implies that there exist subsequences Σ_{1}, …, Σ_{r} of Σ with the following properties:

(1) For 1 ≤ i < j ≤ r every collision of Σ_{i} precedes every collision of Σ_{j},

(2) the graph of Σ_{i} (1 ≤ i ≤ r) is a tree (a connected graph without loop) on the vertex set {1, …, N}, and

(3) r ≥ C(N).

Since every tree contains at least two vertices with degree one and C(N) = \frac{N}{2} · max \{C(N − 1); 3\} = \frac{N}{2} · b (b is a shorthand for max \{C(N − 1); 3\}) we have that there is a vertex, say the one labelled by N, such that N is a degree-one vertex of Σ_{i(1)}, …, Σ_{i(t)}, where 1 ≤ i(1) < ⋯ < i(t) ≤ r and t ≥ b.

Now t ≥ C(N − 1) and, therefore, (iv) obviously holds.

Let σ_{p′} the edge of Σ_{i(1)} that contains N and, similarly, let σ_{q′} be the edge of Σ_{i(t)} containing the vertex N. Then the fact t ≥ 3 ensures that the following properties hold:

(i)’ N ∈ σ_{p′} ∩ σ_{q′},

(iii)’ σ_{p′} = σ_{q′} ⟹ ∃j p′ < j < q′ & σ_{p′} ∩ σ_{j} ≠ ∅, σ_{j} ≠ σ_{p′}.

Let σ_{p}, σ_{q} (1 ≤ p < q ≤ n) be a pair of edges σ_{p′}, σ_{q′} (1 ≤ p′ < q′ ≤ n) fulfilling (i)’ and (iii)’ and having the minimum possible value of q′−p′. Elementary inspection shows that then (ii) must also hold for σ_{p}, σ_{q}. Lemma 4.43 is now proved. □

We are now able to prove the key lemma by an induction on the number N ≥ 2. Indeed, for N = 2 there are no non-sufficient trajectories with initial data in D (Σ, A, τ⃗) and, therefore, the assertion of the key lemma is obviously true.

The inductive step (N − 1) → N can be obtained by simply putting together lemmas 4.2, 4.8, 4.40 and 4.43.

The proof of the key lemma is now complete in the complex case. □

Finally, since a nonzero polynomial P_{i}(x⃗) (1 ≤ i ≤ s) takes nonzero values almost everywhere on the real space ℝ^{(2ν+1)N+1}, we immediately obtain the validity of the real version of Key Lemma 4.1. □

We conclude this paragraph by presenting to the reader two interesting examples. The first of them shows the absolute necessity of imposing Property (A) on the discrete structure (Σ, A, τ⃗), see Definition 3.31, while the second one sheds light on the limitations of our algebraic approach developed in sections 3–4.

Examples

Example I. Suppose that the collision graph of Σ is a tree (a connected graph without loop) with the possiblity of repeated edges. (This allows an arbirarily large number of consecutive, connected collision subgraphs.) Assume further that the combinatorial-algebraic scheme (Σ, A, τ⃗) is given in such a way that it has the following property:

There exists a sequence 1 = k_{1} < k_{2} < ⋯ < k_{r} < k_{r+1} = n + 1 such that

(1) k_{j+1} − k_{j} is an even number for j = 1, …, r, and

(2) σ_{p} = σ_{k_{j}}, a_{p} = a_{k_{j}}, and τ_{p} = 0 for k_{j} < p < k_{j+1}, j = 1, …, r.

Plainly, the orthogonal velocity reflections corresponding to the collisions

σ_{k_{j}}, σ_{k_{j}+1}, …, σ_{k_{j+1}−1}

are the same (j = 1, …, r) and, therefore, their product is the identity operator. Thus v_{i}^{0}(ω) = v_{i}^{k_{j}−1}(ω) for all i = 1, …, N, j = 1, …, r + 1, ω ∈ Ω̃ (Σ, A, τ⃗). It is an easy exercise then to show that these orbit segments ω are very far from being sufficient. Actually, dim_{ℂ}{α_{1}, …, α_{n}} = N − 1 and dim_{ℂ}N (ω) = ν + N − 1, see also (4.4).

Example II. Suppose now that the C(N)-rich symbolic sequence Σ only contains the collisions of type {i, i + 1} for i = 1, …, N − 1, i. e. the collision graph is the simple path of length N with the allowance of repeated edges. Consider the set X of orbit segments ω ∈ Ω̃ (Σ, A, τ⃗) with the side conditions m_{1} = m_{3} = 0. (Of course, m_{2} ≠ 0.) It is easy to see that the orbit segments from X are never sufficient: the vectors

(δq̃_{1}^{0}, …, δq̃_{N}^{0}) = (λ · (v_{1}^{0} − v_{2}^{0}), 0, …, 0)

(0 ≠ λ ∈ ℂ) are nontrivial neutral vectors. The interesting feature of this example is that the set X has two codimensions in Ω̃ (Σ, A, τ⃗), and the codimension of X decreases to one at the crucial step of the induction when we perform the substitution m_{1} = 0. (In earlier notations m_{N} = 0.)

5. Proof of the Main Theorem

Richness will throughout be understood in the sense of Defintion 2.5 by choosing C := C(N) as prescribed by Key Lemma 4.1 (see the Remark 2 after the lemma).

According to the results of [V(1979)], [G(1981)] and [B-F-K(1995)], in a semidispersing billiard system there are no orbits with a finite accumulation point of collision moments. Thus, it is sufficient to concentrate our attention to the set M^{0} of orbits containing no singular collision, since it is well-known and easy to see that phase points with a singular orbit form a countable union of proper submanifolds.

Theorem 5.1. Consider a system of N (≥ 3) particles on the L-torus 𝕋_{L}^{ν} (ν ≥ 2) satisfying r ∈ R_{0}. Let P = {P_{1}, P_{2}} be a given, two–class partition of the N particles, where, for simplicity, P_{1} = {1, …, n} and P_{2} = {n + 1, …, N} (n < N − 1). Then the closed set

F_{+} = {x ∈ M : S^{[0,∞)}x is partitioned by P }

has measure zero.

This statement would, in fact, allow to fix C(N) (the number of consecutive, connected collision graphs required for sufficiency) arbitrarily large.

Proof. Our reasoning is reminiscent to the proofs of Theorem 5.1 of [Sim(1992)-I] and of Proposition 2.8 of [S-Sz(1995)] irrespective of the fact that the notion of center of mass has lost its sense now. The two cases min{n, N − n} ≥ 2 and min{n, N −n} = 1 can be treated similarly, and thus we only consider the first one.

Every point x ∈ M can be characterized by the following coordinates in an essentially unique way:

(1) π_{P_{1}}(x) = x_{1} ∈ M_{1},  (2) π_{P_{2}}(x) = x_{2} ∈ M_{2},
(3) C_{1}(x) = q̃_{n+1} − q̃_{1} ∈ ℝ^{ν},  (4) \frac{I_{1}(x)}{∥I_{1}(x)∥} ∈ 𝕊^{ν−1},
(5) ∥I_{1}(x)∥ ∈ ℝ_{+},  (6) E_{1}(x) = \frac{1}{2} ∑_{i=1}^{n} m_{i}v_{i}^{2}(x) ∈ ℝ_{+}.

where I_{1}(x) = \frac{∑_{i=1}^{n} m_{i}v_{i}(x)}{∑_{i=1}^{n} m_{i}}. (Non-uniqueness only arises in choosing C_{1}(x) as

an arbitrary representant of q_{n+1} − q_{1} ∈ 𝕋_{L}^{ν}.) In what follows the six coordinates corresponding to the characterization given before will, in general, be denoted by b_{1}, …, b_{6}, thus it will always be assumed that b_{1} ∈ M_{1}, b_{2} ∈ M_{2}, b_{3} ∈ ℝ^{ν}, b_{4} ∈ 𝕊^{ν−1}, b_{5}, b_{6} ∈ ℝ_{+}. The relation μ(F_{+}) = 0 will certainly follow if we show that for almost every such choices of the b_{i}’s

μ_{b_{1},b_{2},b_{3},b_{5},b_{6}}(F_{+}(b_{1}, b_{2}, b_{3}, b_{5}, b_{6})) = 0,  (5.2)

where

F_{+}(b_{1}, b_{2}, b_{3}, b_{5}, b_{6}) = {y_{4} ∈ 𝕊^{ν−1}: (b_{1}, b_{2}, b_{3}, y_{4}, b_{5}, b_{6}) ∈ F_{+}},

and μ_{b_{1},b_{2},b_{3},b_{5},b_{6}} denotes the conditional measure of μ under the conditions corresponding to fixing the values of b_{1}, b_{2}, b_{3}, b_{5}, b_{6}. (This conditional measure is equivalent to the Lebesgue measure on its support.)

The relation x ∈ F_{+} is equivalent to saying that for every pair i ∈ P_{1}, j ∈ P_{2} and every t ≥ 0

ϱ (q_{i}^{t}(x) − q_{j}^{t}(x), L · ℤ^{ν}) ≥ 2r,  (5.3)

where ϱ(., .) denotes the euclidean distance. For simplicity, fix i = 1 and j = n + 1.

Now we will be considering the subdynamics corresponding to our two-class partition (cf. the “Subsystems, decompositions” part of Section 2 in the paper [Sim(1992)-I]), and will denote them, for simplicity, by S_{1} and S_{2}, respectively (their phase spaces are M_{1} and M_{2}, of course).

It is worthwile to note here that — according to the construction, given in the introduction, of the configuration space of our hard ball system with an arbitrary mass vector — we can not talk about the absolute configuration vector of a particle but rather about the relative position q_{i}^{t}(x) − q_{j}^{t}(x) for any pair of particles, only. Also, it makes sense to talk about the time-displacement q_{i}^{t}(x) − q_{i}^{0}(x) for any 1 ≤ i ≤ N, t ∈ ℝ and x ∈ M since

q_{i}^{t}(x) − q_{i}^{0}(x) = ∫_{0}^{t} v_{i}^{s}(x)ds (mod L · ℤ^{ν}).

Our remarks naturally apply to the subdynamics S_{1} and S_{2} as well.

Now, assuming for simplicity 1 < n < N − 1, a standard calculation, which is given for instance in the proof of Lemma 5.8 of [Sim(1992)-I], yields that

q_{n+1}^{t}(x)−q_{1}^{t}(x) = ∫_{0}^{βt} ṽ_{n+1}^{s}(x_{2})ds−∫_{0}^{αt} ṽ_{1}^{s}(x_{1})ds+C_{1}(x)+Ĩt (mod L·ℤ^{ν}) (5.4)

where ṽ_{1}^{s}(x_{1}) and ṽ_{n+1}^{s}(x_{2}) denote the time evolution of the corresponding velocity vectors under the subdynamics S_{1} and S_{2}, respectively and Ĩ is the relative velocity of the “baricenters” of the second and first subsystems (this term appears since in M_{1} and M_{2} the momenta of the subsytems are scaled to be equal to 0; actually,

Ĩ = \frac{−M_{1} − M_{2}}{M_{2}} I_{1} = \frac{−M_{1} − M_{2}}{M_{2}} ∥I_{1}∥\frac{I_{1}}{∥I_{1}∥}  (5.5)

where

M_{1} = ∑_{i=1}^{n} m_{i}  and  M_{2} = ∑_{i=n+1}^{N} m_{i}

are the masses of the subsystems) and, finally,

α = \sqrt{2E_{1}(x) − M_{1}∥I_{1}(x)∥^{2}},

β = \sqrt{1 − 2E_{1}(x) − \frac{M_{1}^{2}}{M_{2}} ∥I_{1}(x)∥^{2}}

are the corresponding time scalings.

Our task is to show that the event

∀t ≥ 0 ϱ (λt\frac{I_{1}}{∥I_{1}∥} + f(t), L · ℤ^{ν}) ≥ 2r  (5.6)

has measure zero for every fixed b_{1}, b_{2}, b_{3}, b_{5}, b_{6} where λ = (M_{2})^{−1}(−M_{1}−M_{2})∥I_{1}∥ and

f(t) : ℝ → ℝ^{ν}

is an arbitrary fixed function such that f(t) = f(t, x_{1}, x_{2}, C_{1}, ∥I_{1}∥, E_{1}). Actually, by (5.4),

f(t) = ∫_{0}^{βt} ṽ_{n+1}^{s}(x_{2})ds − ∫_{0}^{αt} ṽ_{1}^{s}(x_{1})ds + C_{1}(x).

In (5.6), the canonical meaning of zero measure is that in \frac{I_{1}}{∥I_{1}∥}.

Denote by L_{2r,L} the lattice of balls of radius 2r centered at points of L · ℤ^{ν}. Our proof of Theorem 5.1 will be based on the following well-known elementary lemma.

Lemma 5.7. Fix a vector n⃗ ∈ 𝕊^{ν−1} for which at least one ratio of coordinates is irrational. Consider arbitrary hyperplanes H perpendicular to n⃗, and denote by B_{R}(z) the (ν − 1)-dimensional ball of radius R in H centered at z ∈ H. Then, for a suitable γ(n⃗) > 0

lim inf_{R→∞} inf_{H} inf_{z∈H} meas (B_{R}(z) ∩ L_{2r,L}) ; meas (B_{R}(z)) ≥ γ(n⃗).  (5.8)

Proof. The lemma easily follows from the fact that the set B_{R}(z) modulo L · ℤ^{ν} is “uniformly dense” in the torus 𝕋_{L}^{ν}, i. e. it is an ϵ(R)-dense set in 𝕋_{L}^{ν}, where ϵ(R) → 0 as R → ∞. This statement is, however, an easy consequence of the wellknown density of the hyperplane H (mod L · ℤ^{ν}) in the torus 𝕋_{L}^{ν} = ℝ^{ν}/L · ℤ^{ν} and the compactness of 𝕋_{L}^{ν}. Indeed, since being an ϵ-dense set is a translation invariant property, the positioning of the foot point z is irrelevant. For a given positive ϵ, we choose first a finite ϵ/2-dense set F ⊂ 𝕋^{ν}, and then for every point y ∈ F we select a radius R_{y} > 0 with the property that B_{R_{y}}(z) is closer to y than ϵ/2. The maximum value R(ϵ) of the arising finitely many radii R_{y} (y ∈ F) provides now an ϵ-dense set B_{R(ϵ)}(z). Now select and fix ϵ = r. Observe that at least a fixed , positive percentage of the ball B_{R(r)}(z) belongs to L_{2r,L}. However, for every large enough R, also a fixed, positive percentage of the ball B_{R}(z) can be covered by disjoint balls B_{R(r)}(z_{i}). Hence Lemma 5.7 follows. □

Assume that the statement (5.6) is not true, i. e. the measure of the subset K of 𝕊^{ν−1} described by (5.6) is positive. Select then and fix a Lebesgue density point n⃗ of K with the property that at least one ratio of the components of n⃗ is irrational. Denote by G_{ε} ⊂ 𝕊^{ν−1} the ball of radius ε around n⃗. By (5.8) we can choose R_{0} so large that for R ≥ R_{0}

inf_{H} inf_{z∈H} meas (B_{R}(z) ∩ L_{2r,L}) ; meas (B_{R}(z)) ≥ \frac{γ(n⃗)}{2}.

The set λtG_{ε} can be arbitrarily well approximated by a ball of radius λtε = R (R is fixed, R ≥ R_{0}) in the hyperplane orthogonal to n⃗ through the point λtn⃗ + f(t) if only t is sufficiently large. Consequently, if R ≥ R_{0}, then by choosing t sufficiently large and at the same time putting ε = (λt)^{−1}R, we have

meas ((λtG_{ε} + f(t)) ∩ L_{2r,L}) ; meas ((λtG_{ε} + f(t)) ≥ \frac{γ(n⃗)}{4}.

But this inequality contradicts to the fact that n⃗ was chosen as a Lebesgue density point of the subset K ⊂ 𝕊^{ν−1}. Hence Theorem 5.1 follows. □

Return now to the demonstration of the Main Theorem. Restrict our attention to the subset M^{0}. From Theorem 5.1 it follows that almost every point is C(N)-rich whatever the value of C(N) may be. Let us fix C(N) according to the requirement of Key Lemma 4.1. Denote by R̂ ⊂ M^{0} the subset of C(N)-rich phase points. By the real version of Key Lemma 4.1, apart from a countable union of proper analytic submanifolds of the outer geometric parameters (m⃗, L), for every (Σ, A, τ⃗) satisfying Property (A), almost every non-singular orbit segment with the combinatorial scheme (Σ, A, τ⃗) is sufficient. Since the number of combinatorial schemes is countable, and the combinatorial schemes of actual (real) hard ball trajectories necessarily satisfy Property (A), we can conclude that, indeed, almost every orbit in R̂ is sufficient.

At this point we will use the following simple consequence of Poincaré recurrence and the ergodic theorem.

Lemma 5.9. (see [S-Ch(1987)]) If S^{[a,b]}x is a non-singular orbit segment and it is sufficient, then, in a suitable open neighbourhood of x, all relevant Lyapunov exponents are non-zero almost everywhere.

Adding Lemma 5.9 to what has been said before, one also obtains that for almost every phase point in R̂, the relevant Lyapunov exponents of the flow are non-zero. Thus our Main Theorem is proved. □

Proof of the Corollary. The positivity of the ergodic components is a consequence of the Katok-Strelcyn theory (cf. [K-S(1986)]), where the underlying assumption is that the relevant Lyapunov-exponents of the system do not vanish.

(We note that, in fact, the Katok-Strelcyn theory is a generalization of Pesin’s theory (see [P(1977)] for hyperbolic systems with singularities, also containing billiards isomorphic to hard ball systems). □

Finally, it is worthwile to note here that, in fact, on any positive ergodic component (i. e. on any ergodic component of positive measure) the Bernoulli property of the standard billiard flow (M, {S^{t}}, μ)_{m⃗,L} follows from its hyperbolicity and weak mixing, as it has been recently shown by Chernov–Haskell and Ornstein–Weiss, cf. [C-H(1996)] and [O-W(1996)]. For a brief summary of how the K-property follows from ergodicity and hyperbolicity, see – for instance – Section 7 of [Sim(1992)-I] , or [K-S(1986)].

6. Concluding remarks

1. Had we been able to strengthen the statement of Key Lemma 4.1 from “for almost every ω ∈ Ω̃ (Σ, A, τ⃗)”, i. e. from “for every ω ∈ Ω̃ (Σ, A, τ⃗) apart from a closed algebraic subset of codimension one” to a similar statement with an at least two-codimensional algebraic subset, we would have obtained the global ergodicity of the system. As shown by our earlier manuscript [S-Sz(1996)], a crucial step in getting to codimension two in the Key Lemma would be the strengthening of the last claim of Lemma 4.13 to stating v_{i}^{p}(ω) ≠ v_{j}^{q−1}(ω) apart from a subset of codimension two. Both lemmas would be implied by the following analogous algebraic properties: all common divisors of the polynomials

(1) P_{1}, …, P_{s} (in case of Key Lemma 4.1);

(2) Q_{1}, …, Q_{ν} ( in case of Lemma 4.13) are purely homogeneous polynomials of the masses (Example II at the end of section 4 illustrastes the essential difficulty arising in the inductive proof of this claim). In this way one can formulate a hypothetic statement about the Boltzmann-Sinai ergodic hypothesis: if this stronger form of Key Lemma 4.1 is true, then, in the Main Theorem, we can also claim the K-property instead of just full hyperbolicity of the system.

2. The system of N = 2 hard discs (i. e. ν = 2) with arbitrary masses m_{1}, m_{2} was studied by Simányi and Wojtkowski [S-W(1989)]. They were considering the same dynamics as here, but in the larger phase space Q̃ × S^{1}. By using the fact that this system is an isometric 𝕋^{2}-extension of the standard billiard ball flow (M, S^{ℝ}, μ)_{(m_{1},m_{2})}, which is a planar dispersing billiard, and was known to possess the Bernoulli property since [G-O(1974)], they showed that this extension is also a Bernoulli flow as long as the ratio m_{1}/m_{2} is irrational. If one thoroughly studies the steps of that proof, he realizes that, once the difficulties outlined in the above remark have been resolved, the methods of the present paper and [S-W(1989)] together with the results of [C-H(1996)] and [O-W(1996)] prove even the Bernoulli property of the flow on the extended phase space Q̃ × E (without the factorization Ψ, see the introduction), provided that not all ratios m_{i}/m_{j} are rational. (The latter condition means precisely that the “center of mass” is defined modulo a dense subgroup of 𝕋^{ν}.)

3. In [Sz(1994)], the second author obtained necessary and sufficient conditions for the K-property of the so-called “orthogonal cylindric billiards”. The subclass of cylindric billiards, introduced in [Sz(1993)], within the family of semi-dispersing billiards is of particular interest since it is this class for which the formulation of general — and at the same time in some sense constructively verifiable — necessary and sufficient conditions of the K-property seems possible. (We have seen in the proof of Lemma 2.1 that the system of N balls with an arbitrary mass vector is also isomorphic to a cylindric billiard.) Indeed, such a condition is found and shown to be necessary for the systems of N hard balls of identical masses in a forthcoming paper of the present authors [S-Sz(1997)], which is based upon the ideas from [S-Sz(1995-I)]. One apparent difficulty in establishing the sufficiency of this condition is the fact that so far no characterization of the neutral subspace of a trajectory segment of a general cylindric billiard is known (in the case of hard ball systems such a characterization was obtained by using the CPF, whose derivation, on the other hand, used the conservation of momentum, a property present in hard ball systems but absent in general cylindric billiards).

Acknowledgement. The authors are deeply indebted to N. Chernov for the large number of very valuable remarks and corrections he made during a careful reading of the manuscript. Special thanks are due to János Kollár who called our attention to the advantage of the complexification of the dynamics which turned out to be very important in sections 3-4. Also, he made useful remarks concerning some elementary facts from algebraic geometry. The authors express their sincere gratitude to A. Katok, C. Liverani, Ya. Pesin and L. Vaserstein for their helpful remarks on earlier versions of the paper.

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