A locally integrable multi-dimensional billiard system

We consider a multi-dimensional billiard system in an (n+1)-dimensional Euclidean space, the direct product of the"horizontal"hyperplane and the"vertical"line. The hypersurface that determines the system is assumed to be smooth and symmetric in all coordinate hyperplanes. Hence there exists a periodic orbit $\gamma$ of period 2 moving along the"vertical"coordinate axis. The question we ask is as follows. Is it possible to choose such a system to have the dynamics locally (near $\gamma$) conjugated to the dynamics of a linear map? Since the problem is local, the billiard hypersurface can be determined as the graphs of the functions $\pm f$, where $f$ is even and defined in a neighborhood of the origin on the"horizontal"coordinate hyperplane. We prove that $f$ exists as a formal Taylor series in the non-resonant case and give numerical evidence for convergence of the series.


Introduction
Let D ⊂ E be a domain with the smooth boundary S = ∂D in the Euclidean space E = R n+1 , n ≥ 1. Assuming that the closureD is compact, we define the billiard system in D as follows. A particle moves with a unit constant velocity inside D. The reflection from the boundary is elastic.
The word elastic means a well-known relation between the velocities of the particle before and after the impact ("the angle of the incidence equals the angle of the reflection"), but we will use an equivalent variational equation. Consider the generating function (the discrete Lagrangian) L : P → R, P = S × S, L(a, b) = |a − b|.
Then a, b, c ∈ S are 3 consecutive impact points iff (1.1) Billiard systems were introduced by Birkhoff [3] and since that time occupied a noticeable part of the dynamics. Many results and references on the subject, as a rule in the case n = 1, can be found in [11,12]. Now we define the billiard map β : P → P. A pair of consecutive impact points (a, b) is transformed to (b, c) = β(a, b) such that a, b, c satisfy (1.1). This means that β determines a discrete Lagrangian system (a general definition of a discrete Lagrangian system is contained in [5]). In particular, β is symplectic.
In this paper we consider the case when the domain D is symmetric in all coordinate hyperplanes. We assume that E splits into the direct product R n ×R, where the subspaces R n × {0} ⊂ E and {0} × R ⊂ E are called horizontal and vertical respectively. The map β has the periodic orbit γ of period 2 lying on the vertical coordinate axis. We are interested in the local dynamics near γ.
Consider the ball B = {x ∈ R n : |x| < d}. We determine S locally by the graphs where f : B → R is a negative even function. (1.2) Our question is as follows.
Q. Is it possible to choose f so that the corresponding billiard map β is locally (near γ) conjugated to a linear map?
Before a discussion of the motivations and results, we reformulate this question in a technically more convenient form.
Let I : E → E, I 2 = id be the symmetry in the horizontal hyperplane. Then I can be naturally extended to an involution of the phase space P: (a, b) → (Ia, Ib). Slightly abusing the notation, we denote this involution also as I. We define the quotient spacê P = P/I and the natural projection pr : P →P.
The maps I, β : P → P commute. Therefore there exists a unique mapβ :P →P such thatβ • pr = β.
The quotient mapβ is more convenient for the local study because γ projects to a fixed pointγ ofβ. Hence we can propose another (equivalent) version of the question Q.
Q. Is it possible to choose f so that the corresponding quotient billiard mapβ is locally (nearγ) conjugated to a linear map?
This linear map ρ is symplectic and coincides with the linearization ofβ atγ. Let λ ±1 1 , . . . , λ ±1 n be the eigenvalues of ρ. In this paper we consider only the most interesting situation whenγ is linearly stable. Then the eigenvalues lie on the unit circle. Explicit conditions of linear stability of a billiard orbit of period two (in general, not necessarily symmetric case) are well-known in the case n = 1 (see for example, [11]), some geometric interpretations of these conditions are given in [8]. In the case n = 2 such stability conditions can be found in [9].
If the eigenvalues form a resonant set i.e., λ k 1 1 · · · λ kn n = 1 for some nonzero vector k = (k 1 , . . . , k n ) ∈ Z n , then there is no hope to have a positive answer toQ (see Section 4). In the nonresonant case we show (Theorem 1) that f can be obtained as a formal Taylor series. The most intriguing question on the convergence of this series remains open. But numerical analysis makes reasonable the following Conjecture 1 If the set λ 1 , . . . , λ n is nonresonant and moreover, satisfies good Diophantine properties, the series, presenting f , is locally convergent. The same is true for the series, presenting the conjugacy map.
Consider the nonresonant case. If Conjecture 1 is true, we obtain a local real-analytic billiard system with linear quasi-periodic dynamics. In this case f turns out to be convex. This locally defined convex function f can be smoothly continued up to a function F such that graphs of ±F determine a smooth closed hypersurface S ⊂ E, the boundary of a convex domain D ⊂ E. Hence we obtain a globally defined billiard system such that the corresponding phase space P contains an open domain D ⊂ P filled by quasiperiodic motions with the same set of frequencies. In particular, the only periodic orbit in D is γ.
Billiard system in the domain D is locally integrable. The problem of integrability of billiard systems is widely discussed. According to the Birkhoff conjecture any domain on a plane bounded by a smooth closed curve generates an integrable billiard system iff this curve is an ellipse. Partial results confirming this conjecture are contained in [4,2,1], (see also [7], where an analog of the Birkhoff conjecture for outer billiard systems is proven). Usually the billiard integrability is discussed in the context of the existence of a first integral polynomial in the momenta, see a recent survey and a collection of new results in [10]. Local first integrals which should exist for the billiard system in D are real-analytic, but certainly not polynomial in the momenta.
Because of a strong degeneracy of the billiard dynamics in D we expect that the spectrum of the Laplace operator in D (say, with the Dirichlet boundary conditions) can be very special. Thereby an interesting question appears on possible values of the quantity meas D/ meas P. Numeric computations in the case n = 1 show that this ratio can exceed 50%.
Now we present some discussion of numeric results for n = 1 and n = 2.
The case n = 1. This situation is studied in [13] and [14]. We consider the normalization f (0) = −1/2. Taking λ = λ 1 = e iα , where α/π is an irrational number, we compute the sequence of coefficients f 2j , where Now we present some conjectures motivated by results of numeric computations.  This function is not defined for rational α/π, but looks smooth. Probably, this function is Whitney smooth, [15].
3. Independently of α numerically σ = −3/2. If this really the case, we have: This would imply that series (1.3) converges on the boundary of the convergence disk |x| ≤ x * = b −1/2 ∞ and has at the points x = ±x * singularities of type (x * ∓ x), in particular, the tangent line to the graph at these points is vertical.
This means that the graph of f can be probably continued through the points (±x * , f (±x * ) up to a longer real-analytic curve.
The case n = 2. Putting f (0) = −1/2, we compute the Taylor coefficients a j 1 j 2 (j 1 , j 2 are even), where The coefficients a 0k and a k0 can be computed from the case n = 1 because sections of the billiard domain by the vertical planes x 1 = 0 and x 2 = 0 give solutions of the problem with n = 1.
The further plan of the paper is as follows. In Section 2 we obtain equations (the conjugacy equations) from which the function f and the conjugacy map can be computed. We discuss basic symmetries of this equation and some properties of its (formal) solutions in Section 3. A further analysis of the solutions is contained in Section 4, where we prove a theorem about the existence of a formal solution in the non-resonant case. Other symmetries of the formal solution are discussed in Section 5. Finally we present another form of the conjugacy equation which contains the unknown functions polynomially. We used this equation in the numeric analysis of the Taylor series for f and the conjugacy map.

Conjugacy map
We consider billiard trajectories which hit S − and S + alternatively. If a and b are two consequtive impact points of a trajectory then In both cases Here the passage from the points a, b ∈ S to their projections to B corresponds to the passage from β to the quotient mapβ. Note that in these coordinateŝ Hence a, b, c are 3 consequtive points on a trajectory 1 of β iff a, b, c are 3 consequtive points on a trajectory ofβ iff (1.1) holds or equivalently, Given a collection of complex numbers consider the linear symplectic map The billiard map is locally conjugated with ρ iff there exists a diffeomorphism X : commutes. The map X has no relation with the complex structure on C n . Hence we have to use (z,z) = (z 1 , . . . , z n ,z 1 , . . . ,z n ) as coordinates on C n . We put X(z,z) = (a, b) = (χ − (z,z), χ(z,z)).
Let β(a, b) = (b, c) and a = χ − (z,z). Then and a, b, c are connected by (2.2). This implies and where ∂ 1 and ∂ 2 are defined by In principle, the functions f, χ can be computed from (2.5). But (2.5) turns out to be overdetermined w.r.t. f . This produces some difficulties in the proof of the existence of a solution. To avoid these difficulties, below we replace (2.5) by an equivalent system, free of such problems.

Averaging
We define the action of the torus T n = R n /(2πZ n ) on C n by the equation For any function g : U → C defined on a set U ⊂ C n , invariant with respect to the action of T n , we put ρ * α g = g • ρ α and define the average For any g = k ′ ,k ′′ ∈Z n + g k ′ k ′′ z k ′z k ′′ we have the identities Averaging can be applied to differential forms as well. In particular, we have the identity d g = dg for any function g : U → C. (2.6)

The forms ν,ν
Consider the forms µ,μ on B × B: Then dµ = −dμ is the standard symplectic structure for the billiard system, see for example [5]. We define two 1-forms ν = X * µ,ν = X * μ on U ⊂ C n . In more detail, It remains to use identity (2.6).

Main equation
Instead of (2.5) or (2.7) we consider an equivalent system [ν + ρ * ν] = 0, (2.8), or, in an explicit form 3 For any j = 1, . . . , n consider the maps Any map κ j is the symmetry in the j-th coordinate hyperplane and κ is the central symmetry w.r.t. the origin. Given the frequency vector (2.3) we search for a solution (f, χ) of (2.9),(2.10) satisfying the following properties.
It remains to use (3.1).
(b) Consider in (2.10) the homogeneous form of degree two in z,z: By using the notation after simple transformations we obtain: where e j ∈ Z n + is the j-th unit vector: its j-th components equals 1 while all others vanish. This means that for any l = 1, . . . , n 2 − λ l − λ −1 l + 8f 0 F e j |c jl | 2 = 0.
Since λ j = λ ±1 k for j = k, we see that for any l only one of coefficients c jl may be nonzero, the one, corresponding to j such that 2 − λ l − λ −1 l + 8f 0 F j = 0. By (4) for any j such j = j(l) exists. Without loss of generality we have: j(l) = l. Hence, Direct computation shows that by equations (3.4) the homogeneous forms of degree 2 in (2.9) vanish: We can assume that the coefficients c jj are real and positive. Indeed, by using the map s (3.2) s(z,z) = (|c 11 |c −1 11 z 1 , . . . , |c nn |c −1 nn z n , |c 11 |c −1 11z 1 , . . . , |c nn |c −1 nnz n ) for the gauge transformation (f, χ) → (f, χ•s), we obtain: (c) We define the maps (complex conjugacy of one coordinate) κ 1 , . . . , κ n : C n → C n , κ j (z) = w, w j =z j , w k = z k for any k = j.
Let ρ j : C n → C n be the map where (κ j λ) l is the l-th coordinate of the vector κ j λ. We have the identity
(d) We define the maps T jl : C n → C n which exchange the coordinates z j and z l in any vector z ∈ C n . Let ρ jl : C n → C n be the map z → w = ρ jl (z), w k = (T λ) k z k , k = 1, . . . , n.
We have the identity ρ • T jl = T jl • ρ jl .

Formal solution
Theorem 1 Suppose that the frequency vector λ (2.3) is nonresonant. Then for any f 0 < 0 and a 1 , . . . , a n > 0 system (2.9),(2.10) (1)-(4) has a formal solution given by power series Proof of Theorem 1. According to (4.1)-(4.2) we are looking for f even in x and χ odd in z,z. Hence even in z,z homogeneous forms in (2.9),(2.10) will vanish. The forms of degree 2 in (2.9) and (2.10) have been analyzed in item (b), Section 3. In this way we obtain (4.2).
Taking in (2.9),(2.10) the homogeneous form of degree 2k, we obtain: where the forms R

By (3.5) the second fraction in the brackets [ ] vanishes. Therefore
Consider equation (4.4). The first term in the left-hand side equals The second term equals 8f 0 n j=1 F e j a j (z j +z j ) (χ j l−e j l + χ j l l−e j )z lzl .
Hence the first two terms in the left-hand side of (4.4) cancel.
The third term equals Hence the coefficients F 2s are uniquely computed from (4.4). Now turn to equation (4.3). The first term in the left-hand side equals The condition l ′ − l ′′ = ±e j appears as a result of application of the operation [ ]. It implies that the coefficients λ j + λ −1 j − λ l ′ −l ′′ − λ l ′′ −l ′ do not vanish. The second term in the left-hand side has been already computed from equation (4.4). Hence the coefficients χ j l ′ l ′′ , l ′ − l ′′ = ±e j are computed uniquely from (4.3).
The coefficients χ j l−e j l and χ j l l−e j =χ j l−e j l can be chosen arbitrarily due to the gauge symmetry.

Other symmetries
When the existence theorem (Theorem 1) is proven, we can return to equation (2.5) which is equivalent to system (2.9),(2.10) but looks somewhat simpler. Now we can ignore overdeterminacy of (2.5) w.r.t. f . In an explicit form (2.5) looks as follows: Direct computation shows that, analogously to (4.3), f (2k) and χ (2k−1) satisfy the equations where the functions P (2k) j are polynomials w.r.t. coefficients of f (2m) and χ (2m−1) with m < k.
Proposition 5.1 (1) If χ j are chosen satisfying (4.2) then χ j are odd in z j ,z j and even in z l ,z l for all l = j χ j = −χ j • κ j = χ j • κ l for all l = j. (2) If the coefficients χ j l−e j , l =χ j l, l−e j (arbitrary due to the gauge symmetry) are chosen real, then all Taylor coefficients of χ are real.
(2) If the homogeneous forms χ (2m−1) with m < k satisfy (2.11) and coefficients of the forms χ (2m−1) are real, the functions P (2k) j are also real polynomials in z andz with real coefficients.
The case λ 1 = . . . = λ n = −1 is resonant and one should not expect a positive answer to questionQ. Although in this case χ can not be found, the function f can be computed explicitly. Indeed, in this case In this case in equations (5.1) we can regard χ j as independent variables. These equations are equivalent to This formal computation shows that, in a certain sense, any sphere with the center at the origin is a limit solution of our problem when all λ j tend to −1. Numeric computations confirm this statement.

Another version of the main equation
In this section we present another version of the equations from which the functions f and χ can be computed. For n = 1 these equations are the same as in [13,14]. We used these equations for n = 2 in numeric computations since they contain the unknown functions polynomially.

Dynamical equations
Let a, b, c ∈ E be three successive points of a billiard trajectory: . We refer to the map (a, b) →β(a, b) = (b, c) as the billiard map.

Conjugacy equation
QuestionQ is equivalent to the following one. Does the following conjugacy equation have a solution (f, X), where f satisfies (1.2) and X = (χ − , χ) : U → B 2 is a diffeomorphism of a neighborhood U ⊂ R 2n to its image. Equations (6.3), (2.4), and (6.2) imply For n = 1 equation (6.4) takes the form This equation is obtained in [13]. Its analysis is contained in [13,14]. If n = 2, (6.4) takes the form which implies 2 equations: Our numeric results in the case n = 2 presented in Section 1 are based on the analysis of system (6.6),(6.7).