Introducing symplectic billiards
Peter Albers Serge Tabachnikov



Height transitions, shape evolution, and coarsening of equilibrating quantum nanoislands
    Mikhail Khenner
    December 30, 2023
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In this article we introduce a simple dynamical system called symplectic billiards. As opposed to usual/Birkhoff billiards, where length is the generating function, for symplectic billiards symplectic area is the generating function. We explore basic properties and exhibit several similarities, but also differences of symplectic billiards to Birkhoff billiards.




§ INTRODUCTION
 

Birkhoff billiard describes the motion of a free particle in a domain: when the particle hits the boundary, it reflects elastically so that the tangential component of its velocity remains the same and the normal component changes the sign. In the plane, this is the familiar law of geometric optics: the angle of incidence equals the angle of reflection, see Figure <ref>.

 

This reflection law has a variational formulation: if the points x and z are fixed, the position of the point y on the billiard curve is determined by the condition that the length |xy|+|yz| is extremal. In particular, periodic billiard trajectories are inscribed polygons of extremal perimeter length.

Another well-studied dynamical system is outer billiard, one such transformation in the exterior of a convex curve also depicted in Figure <ref>. Outer billiard admits a variational formulation as well: periodic outer billiard orbits are circumscribed polygons of extremal area. 

It is natural to consider two other planar billiards: inner with area, and outer with the length as the generating functions. The variational formulation of the reflection depicted in Figure <ref> on the right is as follows: if the points x and z are fixed, the position of the point y on the billiard curve is determined by the condition that the area of the triangle xyz is extremal. It would be natural to call this area billiard [A competing name would be affine billiard since this system commutes with affine transformations of the plane.]; due to higher-dimensional considerations, we prefer the term symplectic billiard.  

Analytically, the generating function of the symplectic billiard map is ω(x,y), where ω is the area form, as opposed to |xy|, the generating function of the usual billiard map. 

The aim of this paper is to introduce symplectic billiards and to make the first steps in their study. 
We believe that a system that has such a simple and natural definition should have interesting properties; in particular, one wants  to learn which familiar properties of the usual inner (Birkhoff) and outer billiards are specific to them, and which extend to the symplectic billiards and, perhaps, other similar systems. 

One can also define symplectic billiard in a convex domain with smooth boundary in a linear symplectic space (^2n,ω). Let M be its boundary. To define a reflection similarly to the planar case, one needs to choose a tangent direction at every point of M. This is canonically provided by the symplectic structure: the tangent line at x∈ M is the characteristic direction ω|_T_xM× T_xM, that is, the kernel of the restriction of the symplectic form ω on the tangent hyperplane T_x M. (A similar idea is used in the definition of the multi-dimensional outer billiard, see, e.g., the survey <cit.>.) The generating function of this map is again ω(x,y), where x,y ∈ M.

This paper consists of three parts, the two-dimensional smooth and polygonal symplectic billiards, and multi-dimensional ones. Let us describe our main results.

In the planar case, we show that symplectic billiard is a monotone, area preserving twist map. In Theorem <ref> we calculate the area of the phase space: it equals four times the area of the central symmetrization of the billiard table. 

Theorem <ref> is a version of the well known theorem by Mather <cit.>: if the billiard curve has a point of zero curvature, then the symplectic billiard possesses no caustics. Theorem <ref> is a KAM theory result, an analog of Lazutkin's theorem <cit.>: if the curve is sufficiently smooth and the curvature is everywhere positive, then the symplectic billiard has smooth caustics arbitrary close to the boundary. The coordinates in which the symplectic billiard map is a small perturbation of an integrable map are provided by the affine parameterization of the billiard curve. 

In Section <ref>, we apply the approach via exterior differential systems, introduced and implemented in the study of Birkhoff and outer billiards in <cit.>. It makes it possible to prove the existence of billiard tables that possess caustics consisting of periodic points. We remark that, for period 4, the billiard bounded by a Radon curve has this property. Another application of exterior differential systems is to small-period cases of the Ivrii conjecture which states that the set of periodic billiard orbits has zero measure (a weaker version says that this set has empty interior). Theorem <ref> asserts that  for planar symplectic billiards the set of 3-periodic points and that of 4-periodic points has empty interior.

In Section <ref>, we consider the area spectrum, that is, the set of areas of the polygons corresponding to the periodic orbits of the planar symplectic billiard. Let A_n be the maximal area of an n-gon inscribed in the billiard curve. The theory of interpolating Hamiltonians <cit.> implies an asymptotic series expansion of A_n in negative even powers of n whose first two terms are equal, up to numerical factors, to the area bounded by the billiard curve and its  affine length. Then the affine isoperimetric inequality implies that ellipses are uniquely determined by their area spectrum – Theorem <ref> (a similar result for outer billiards was obtained in <cit.>). 

We also show that symplectic billiards do not possess the finite blocking property (or, in different terminology, are insecure): for every pair of distinct points x,y on the boundary curve  and every finite set S inside γ, there exists a symplectic billiard trajectory from x to y that avoid S. See <cit.> for insecurity of Birkhoff billiards.

Section <ref> concerns polygonal symplectic billiards. Theorems <ref> and <ref> assert that if the polygon is affine-regular or is a trapezoid, then all orbits are periodic, with explicitly described periods. In Propositions <ref> and <ref>, we describe two types of periodic orbits: those that appear in 2-parameter families (similarly to the periodic orbits of polygonal outer billiards), and the isolated, stable ones that survive small perturbations of the polygon. We do not know whether every convex polygon carries a periodic symplectic billiard orbit (this question is also open for the usual polygonal billiards,  in contrast with outer polygonal billiards where such a result is known).

After giving a careful definition of multi-dimensional symplectic billiard, we describe, in Section <ref>, the case of ellipsoids. As one may expect, this is a completely integrable case. Namely, in Theorem <ref>, we relate the symplectic billiard in an ellipsoid to the usual billiard in a different ellipsoid: given a trajectory of the symplectic billiard, construct a new polygonal line by connecting every second consecutive impact points (i.e, always skip one impact point); then a linear transformation (that depends only on the original ellipsoid) takes this polygonal line to a Birkhoff billiard trajectory on another ellipsoid (that again depends only on the original ellipsoid.) This result is analogous to a theorem of Moser and Veselov <cit.> that relates the discrete Neumann system with billiards in ellipsoids. We also explicitly describe the symplectic billiard dynamics inside a round sphere (Proposition <ref>).

The last Section <ref> concerns periodic orbits in multi-dimensional symplectic billiards inside an arbitrary strictly convex closed smooth hypersurface. Theorem <ref> asserts that, for every k≥ 2, there exist k-periodic trajectory. This is a weak result and, in Theorem <ref>, we obtain a stronger  one for small periods: 
the number of 3-periodic, and of 4-periodic, symplectic billiard orbits in 2n-dimensional symplectic space is no less than 2n (the dihedral group D_k acts on k-periodic trajectories by cyclic permutation of the vertices and reversing their order; what one counts are these D_k-orbits). 

We conclude this introduction with a remark concerning the interplay between convexity and symplectic geometry. Even though convexity is not a symplectically invariant notion it has long been know that convex domains in symplectic space enjoy special rigidity properties. This culminated in Viterbo's conjecture <cit.> which asserts an inequality between symplectic capacities and volume for convex domains. It is well known that Viterbo's conjecture fails if the convexity assumption is dropped. And even though it has been proved in special cases in general, it is considered wide open. The fundamental nature of Viterbo's conjecture is demonstrated by the fact that it implies Mahler's conjecture from convex geometry <cit.>. Recently, a renewed interested in this interplay arose, in part due to related results about systolic inequalities, see e.g., <cit.>.

Of course, the definition of the symplectic billiard map crucially relies on the convexity of the domain. At this point it is not more than idle speculation that this fact is more than a coincidence.




Acknowledgments. We are grateful to V. Dragovic and B. Jovanovic for a discussion of periodic  billiard orbits in ellipsoids, to R. Montgomery for a discussion of the Lexell theorem,  and to R. Schwartz for writing a computer program for experiments with polygons and for his insight concerning the symplectic billiards in trapezoids. In the past, the second author had numerous discussions of the topic of four planar billiards with E. Gutkin; his untimely death made it impossible for him to participate in this project. S.T. also acknowledges stimulating discussions of these ideas with S. Troubetzkoy. 

The first author is grateful to the hospitality of the Pennsylvania State University. He was supported by SFB/TRR 191. The second author is grateful to the hospitality of the Heidelberg University. He was supported by the NSF grant  DMS-1510055.



§ SYMPLECTIC BILLIARD IN THE PLANE
 



 §.§ Symplectic billiard as an area preserving twist map
 


  §.§.§ A precise look at the definition
 
Let  be a smooth, strictly convex, closed, positively oriented curve, the boundary of our billiard table. 
Since  is strictly convex, for every point x∈ there exists a unique point x^*∈ with the property

    T_x=T_x^*⊂^2.

Clearly we have x^**=x. 


Denote by ν the outer normal of . Then

    T_x=T_y  ⟺  ω(ν_x,ν_y)=0.

Using the orientation of , we define

    :={(x,y)∈×| x < y < x^*}.

Note that this is actually antisymmetric in x and y:

    ={(x,y)∈×| y^* < x <y},

see Figure <ref>. Indeed,

    ={(x,y)∈×|ω(ν_x,ν_y)>0}.

We think of  as the (open, positive part of) phase space. The next lemma formalizes the definition of the symplectic billiard map.


Given (x,y)∈, there exists a unique point z∈ with

    z-x∈ T_y.

Moreover, the new pair (y,z) lies again in our phase space: (y,z)∈.


Since (x,y)∈, we know that T_x⋔ T_y. Using that  is convex, we obtain

    (x+T_y)∩={x,z}.

Moreover, z≠ x since otherwise T_y=T_z=T_x, which contradicts x<y<x^*. In other words, the line through x parallel to T_y intersects  in a new point z, see  Figure <ref>. Equation (<ref>) is equivalent to z-x∈ T_y.

To show that (y,z)∈, we observe that if y is close to x, then so is z, and therefore ω(ν_x,ν_y)>0 implies ω(ν_y,ν_z)>0. 

Now assume that x<y<x^* and ω(ν_y,ν_z)≤0. By continuity and moving y close to x, we can arrange ω(ν_y,ν_z)=0. But then equation (<ref>) implies that T_y=T_z, which implies y=x, a contradiction.

Thus the map

    Φ:→,   (x,y)↦(y,z),
 
with z∈ being the unique point satisfying z-x∈ T_y, is well-defined.


Of course the same dynamics/map is defined on the negative part of phase space {(x,y)∈γ×γ| x^* < y < x} simply by reversing the orientation of .



We extend Φ to the closure ={(x,y)∈γ×γ| x≤ y≤ x^*} by continuity. The first case is obvious

    lim_y→ xΦ(x,y)=(x,x),

that is, the map extends as the identity. In the other case, we claim

    lim_y→ x^*Φ(x,y)=(x^*,x).

This follows from the observation that, due to convexity, y↦ z(y) is monotone, and lim_y→ x^*T_y=T_x^*. 


The continuous extension Φ(x,x^*)=(x^*,x) is characterized by the 2-periodicity. That is,  Φ(x,y)=(y,x) is equivalent to y∈{x,x^*}.


One direction is exactly the continuous extension. Now assume that Φ(x,y)=(y,x). If (x,y)∈ then, by Lemma <ref>,

    (x+T_y)∩={x,y}

with x≠ y. This is clearly a contradiction since T_y∩={y}. Therefore, either (x,y)=(x,x) or (x,y)=(x,x^*).

 
The envelope of the 1-parameter family of chords x x^* is a caustic of our billiard. This envelope is called the centre symmetry set of , and it was studied in the framework of singularity theory <cit.>.



Now we identify a generating function for the symplectic billiard map Φ.


The function 

    S:→,   (x,y)↦ S(x,y):=ω(x,y)
 
is a generating function for Φ, that is

    Φ(x,y)=(y,z)   ⟺  d/dy[S(x,y)+S(y,z)]=0.




    d/dy[S(x,y)+S(y,z)]=0      ⟺  ω(x,v)+ω(v,z)=0  ∀ v∈ T_y
       ⟺  ω(x-z,v)=0  ∀ v∈ T_y
       (∗)⟺   x-z∈ T_y.

In (∗) we used x≠ z which is due to (x,y)∈, compare with Lemma <ref>.




The area of the triangle xyz on the right of Figure <ref> equals

    1/2[ ω(x,y) + ω(y,z) + ω(z,x) ],

which, ignoring the factor, differs from S(x,y)+S(y,z) by a function of x and z, having no effect on the partial derivative with respect to y.



The symplectic billiard map commutes with affine transformations of the plane. Obviously, circles are completely integrable: concentric circles are the caustics; by affine equivariance, ellipses are completely integrable as well.



  §.§.§ Invariant area form and the twist condition
 

Let (t) be a parameterization of the curve  with 0≤ x ≤ L. Denote by t^* the involution on the circle of parameters: the tangents at (t) and (t^*) are parallel. We have
S(t_1,t_2) = ω((t_1),(t_2)),
and Φ(t_1,t_2) = (t_2,t_3) if

    S_2 (t_1,t_2) + S_1(t_2,t_3) =0,

where S_1 and S_2 denote the first partial derivatives with respect to the first and second arguments.

Following the theory of twist maps (see, e.g., <cit.>), we introduce new variables 

    s_1 = - S_1(t_1,t_2), s_2 = S_2(t_1,t_2),

and a 2-form Ω = S_12(t_1,t_2) dt_1∧ dt_2 on S^1× S^1.

 
The variables (t,s) are coordinates on the  phase space , and Ω is an area form therein.
The map Φ is a monotone twist map, and the area form is Φ-invariant: Φ^*Ω = Ω.


Identify ^2 with .
Then

    -∂ s_1/∂ t_2(t_1,t_2)= S_12(t_1,t_2) = ω('(t_1),'(t_2)) = ω(-i'(t_1),-i'(t_2)) >0,

since -i '(t) is an outward normal vector to  at point (t) and, in , one has by definition ω(ν_ (t_1),ν_(t_2))>0. It follows that Ω is an area form on .
It also follows that the Jacobian of the map (t_1,t_2) ↦ (t_1,s_1) does not vanish, therefore (t_1,s_1) are coordinates.
The twist condition ∂ t_2/∂ s_1 < 0 follows as well: indeed, ∂ s_1 / ∂ t_2 = - S_12 < 0.

Finally, it follows that equation (<ref>) implies that the 2-form S_12 (t_1,t_2) dt_1∧ dt_2 is Φ-invariant, indeed take the exterior derivative of  (<ref>) and wedge multiply by d t_2.  This completes the proof.

Thus a variety of results about monotone twist maps apply to our symplectic billiard. In particular, we need to the following fact. For every period n ≥ 2 and any rotation number 1≤ k ≤ [n/2], the symplectic billiard has at least two distinct n-periodic orbits with rotation number k.




  §.§.§ Spherical and hyperbolic versions
 

Using the same generating function, the area of a triangle, one can define the symplectic billiard map in the  spherical and hyperbolic geometries. The definition is based on the Lexell theorem of spherical geometry, and its hyperbolic analog, that describes the locus of vertices of triangles with a given base and a fixed area, see <cit.>. 



Here is the formulation of Lexell's theorem. Let xyz be a spherical triangle, and let x^* and z^* be the points antipodal to x and z. Consider the arc of a spherical circle x^* y z^*. This arc is one half of the locus of points w such that the area of the spherical triangle xwz equals that of the triangle xyz, the other half being its reflection in the geodesic xz.

Consider the hyperbolic plane in the Poincaré disk model. Let now x^* and z^* be the inversions of points x and z in the unit circle. Then the above formulation holds without change (in hyperbolic terms, the arc x^* y z^* is an equidistant curve, a curve of constant curvature less than 1). 


This leads to the following definition of the symplectic billiard in the spherical and hyperbolic geometries. Let  be a closed convex curve, and let x, y ∈. Let δ be the circle through point x^* that is tangent to  at point y, and let δ^* be its image under the  antipodal involution, respectively, inversion in the absolute. Then  Φ(x,y) = (y,z),  where z is the intersection point of δ^*  with , different from x (assuming that such a point is unique, otherwise the map Φ becomes multi-valued). See Figure <ref>.



 §.§ Total phase space area
 

In this section we calculate the Ω-area of the phase space.  

Let D be the billiard table with the boundary . 
Parameterize  by the direction α of its tangent line. Choose an origin O inside D, and let p(α) be the support function, that is, the distance from O to the tangent line to  having direction α+π/2, see Figure <ref>.



The Cartesian coordinates of the point (α) are given by the formulas

    x(α)=p(α) cosα - p'(α) sinα, y(α)= p(α) sinα + p'(α) cosα.

The perimeter length of  and the area bounded by it are given, respectively, by the integrals

    ∫_0^2π p(α) dα   and  1/2∫_0^2π [p(α)+p”(α)] p(α) dα,

see, e.g., <cit.>. 

The phase space  consists of pairs α_1,α_2 with α_1 < α_2 < α_1+π. Let w(α)=p(α)+p(α+π) be the width of D in the direction α.

The  symmetrization D̅ of the domain D  is the Minkowski sum with the centrally symmetric domain, scaled by 1/2. The support function p̅(α) of D̅ equals [p(α)+p(α+π)]/2. 

 
The total Ω-area of the phase space  equals four times the area of the symmetrization D̅.


Since Ω = S_12 dα_1∧ dα_2, we need to calculate S_12 = ['(α_1),'(α_2)].

One has from (<ref>)

    '(α) = [p(α)+p”(α)] (-sinα,cosα),

hence

    ['(α_1),'(α_2)] = [p(α_1)+p”(α_1)] [p(α_2)+p”(α_2)] sin(α_2-α_1).

Therefore the phase area is

    ∫_0^2π∫_α_1^α_1+π [p(α_1)+p”(α_1)] [p(α_2)+p”(α_2)] sin(α_2-α_1) dα_2 dα_1.

Consider the inner integral:

    ∫_α_1^α_1+π p(α_2) sin(α_2-α_1) dα_2 + ∫_α_1^α_1+π p”(α_2) sin(α_2-α_1) dα_2 .

We use integration by parts twice on the second summand to compute:

    ∫_α_1^α_1+π p”(α_2)   sin(α_2-α_1) dα_2
       =p'(α_2) sin(α_2-α_1)|_α_1^α_1+π_=0-∫_α_1^α_1+π p'(α_2) cos(α_2-α_1) dα_2
       =-∫_α_1^α_1+π p'(α_2) cos(α_2-α_1) dα_2
       =- p(α_2) cos(α_2-α_1)|_α_1^α_1+π - ∫_α_1^α_1+π p(α_2) sin(α_2-α_1) dα_2
       = p(α_1) + p(α_1+π)- ∫_α_1^α_1+π p(α_2) sin(α_2-α_1) dα_2 .

Thus the inner integral (<ref>) evaluates to 
p(α_1) + p(α_1+π).

Now consider the outer integral and recall that the support function p̅(α)  of the symmetrization D̅ equals 12[p(α)+p(α+π)]. 

    ∫_0^2π   [p(α_1)+p”(α_1)] [p(α_1) + p(α_1+π)] dα_1 
       =1/2∫_0^2π [p(α_1)+p”(α_1) + p(α_1+π)+p”(α_1+π)] [p(α_1) + p(α_1+π)] dα_1 
        =2 ∫_0^2π [p̅(α) + p̅”(α)] p̅(α) dα
       = 4  area(D̅),

as claimed.

It is interesting to compare this with the usual billiards: in that case, the area of the phase space (with respect to the canonical area form on the space of oriented lines) equals twice the perimeter length of the boundary curve (see, e.g., <cit.>).



 §.§ Existence and non-existence of caustics
 



  §.§.§ Non-existence of caustics
 

The next result is a symplectic billiard version of Mather's theorem: If a smooth convex billiard curve has a point of zero curvature, then the (usual) billiard inside this curve has no caustics <cit.> or <cit.>. 

 
Let  be a smooth closed convex curve whose curvature vanishes at some point. Then the symplectic billiard in  has no caustics.


According to Birkhoff's theorem, an invariant curve of an area preserving twist map is a graph of a function; see, e.g., <cit.>. 

Assume that our billiard has a caustic. Then one has a 1-parameter family of chords x_1 x_2 of the curve , corresponding to the points of the invariant curve. The graph property implies that if x̅_1 x̅_2 is a nearby chord from the same family and x̅_1 has moved along  in the positive direction from x_1, then x̅_2 also has moved in the positive direction from x_2. It follows that the chords intersect inside the curve  and, as a consequence, the caustic, which is the envelope of the lines containing these chords, lies inside the billiard table.



Assume that a caustic exists, a chord x_1 x_2, tangent to the caustic, reflects to x_2 x_3, and the curvature at x_2 vanishes. Consider an infinitesimally close chord x̅_1 x̅_2, tangent to the same caustic. Since the curvature at x_2 vanishes, the tangent line at x̅_2 is, in the linear approximation, the same as the one at x_2. Therefore, in the same linear approximation, the line x̅_1 x̅_3 is parallel to x_1 x_3, hence these lines do not intersect inside the billiard table. This contradiction proves the non-existence of the caustic.

Alternatively, one may use Mather's analytic condition from <cit.>. This necessary condition for the existence of a caustic is S_22 (t_1,t_2) + S_11 (t_2,t_3) < 0.

Let t be an arc length parameter on . Since S(t_1,t_2) = ω((t_1),(t_2)), we have

    S_22 (t_1,t_2) = ω((t_1),”(t_2)) = k(t_2) ω((t_1),i '(t_2)),

and likewise for S_11 (t_2,t_3), 
where k is the curvature of .  If k(t_2)=0, then S_22 (t_1,t_2) + S_11 (t_2,t_3) =0, violating Mather's  criterion.

It would be interesting to find analogs of the results of Gutkin and Katok on the regions of the billiard table  free of caustics <cit.> and of Bialy  on the part of the phase space free of invariant curves <cit.>.



  §.§.§ Existence of caustics
 

The existence of caustics is provided by KAM theory, namely, Lazutkin's theorem <cit.>, applied to symplectic billiards.  Let us make a simplifying assumption that the billiard curve  is infinitely smooth (this can be replaced by sufficiently high finite smoothness) and has everywhere positive curvature. 

 
Arbitrary close to the curve , there exist smooth caustics for the symplectic billiard map; the union of these caustics has positive measure.


It will be convenient  to use an affine parameterization of ; let us recall the pertinent notions, e.g., <cit.>.

We first give a geometric definition of the affine length of the curve followed by a more computational one.

Given v∈ T_xγ, we consider a parametrization (t) with (0)=x and d/dt(0)=v.  Our aim is to define a cubic form B(x,v) on . For this we denote by A(,t) the area bounded by the curve and the segment (0) (). We keep t in the notation as a reminder for the parametrization, see below. This area is of third order in , hence 

    B(x,v):= lim_→ 0A(,t)/^3

is well-defined and, of course, independent of the parametrization (as long as (0)=x and d/dt(0)=v.) 

That B indeed is a cubic form is basically the chain rule. Indeed, replace v by av for some a≠0. If we choose a new parametrization (τ) with t=aτ then at t=τ=0

    d/dτ=d/dt dt/dτ=av.

We now can define two enclosed areas with respect to these two parametrizations: A(, t) and A(,τ). Of course, area is area, i.e.,

    A(,t)=A(a,τ),

and thus

    B(x,v)=lim_→ 0A(, t)/^3=lim_→ 0A(a,τ)/(a)^3=a^-3lim_→ 0A(a,τ)/^3=a^-3B(x,av).

This confirms that B is a cubic form. Therefore, B^1/3 is a 1-form, called an element of affine length. The integral of this form is called the affine length of the curve.

More conveniently for computations, a parameterization (t) is affine if ['(t),”(t)]=1 for all t. In this case, ”'(t) = -k(t) '(t), where the positive function k is called the affine curvature. The relation between the affine length parameter t and the Euclidean arc length parameter x is dt = κ^1/3 dx, where κ is the (Euclidean) curvature.

Consider an affine parameterization of the billiard curve. A chord (t_1) (t_2) is characterized by the numbers t_1 and :=t_2-t_1. We use (t,) as coordinates on the phase cylinder: t is a cyclic coordinate,  is non-negative and bounded above by some function of t (since t_1 ≤ t_2 ≤ t_1^*).

Let us describe the billiard map in these coordinates. Let Φ: (t-,) ↦ (t,δ), where δ(t,) is a function on the phase space. We claim that 

    δ(t,) =  + ^3 f(t,),

where f is a smooth function. 

To prove this claim, let δ = a_0 + a_1  + a_2 ^2 + a_3 ^3 + O(^4), where a_i are functions of t. Since the boundary =0 consists of fixed points of the billiard map, a_0=0. By definition of the billiard reflection,

    [(t+δ)-(t-),γ'(t)]=0,

where the brackets denote the determinant made by two vectors.

Expand (t+δ) and (t-) in Taylor series up to 4th derivative and substitute to (<ref>):

    [(δ+)' + (δ^2-^2/2)” + (δ^3+^3/6)”' + (δ^4-^4/24)^””,']=0,

where we suppress the argument t from (t) and its derivatives.

Since [',”]=1, the quadratic term in (<ref>) yields a_1=1. Since [',”']=0, the cubic term does not give any information. Since ”'=-k, one has ””=-k''-k”, and hence [””,']=k ≠ 0. Therefore the quartic term in (<ref>) yields a_2=0,  proving (<ref>). 

Formula (<ref>) implies that the billiard map is given by the formula:

    (t,) ↦ (t+,  + ^3 g(t,)),

where g is a smooth function. This is an area preserving map, a small perturbation of the integrable map (t,) ↦ (t+, ), satisfying the assumptions of  Lazutkin's theorem (Theorem 2 in <cit.>). Applying this theorem concludes the proof.

Figure <ref> shows a computer generated phase portrait of the symplectic billiard. It looks like a typical area preserving twist map.



 
A useful tool in the study of billiards is the string construction that recovers a billiard table from its caustic; it produces a 1-parameter family of tables. A similar, area, construction is known for outer billiards, see, e.g., <cit.>. In the context of symplectic billiards, we failed to discover a string construction.







 §.§ Periodic caustics, Radon curves,  and n=3,4 versions of the Ivrii conjecture
 



  §.§.§ Distribution on the space of polygons
 

In this section we apply the approach to periodic billiard trajectories via exterior differential systems, developed by Baryshnikov and Zharnitsky <cit.> and, independently, by Landsberg <cit.>, and applied to outer billiards in <cit.>; see also <cit.>.

Suppose that the symplectic billiard has an invariant curve consisting of n-periodic points. A periodic point is a polygon P=(z_1,z_2,…,z_n) inscribed into the billiard curve  and having an extremal area. These n-gons form a 1-parameter family of inscribed polygons, connecting  P=(z_1,z_2,…,z_n) to the polygon P'=(z_2,…,z_n,z_1) with cyclically permuted vertices. The area remains constant in this  1-parameter family of area-extremal polygons.

Assume that n≥ 3. One can reverse the situation: start with an n-gon P and try to construct a billiard curve  that has an invariant curve of n-periodic points including P. The main observation is that the directions of  at the vertices z_i, i=1,…,n are uniquely determined: they are the directions of the diagonals z_i+1 z_i-1 (here and elsewhere the indices are taken in the cyclic order, so that n+1=1).

Consider the 2n-dimensional space of n-gons in ^2, and let  𝒫_n be its open dense subset given by the condition that z_i+1-z_i-1 is not collinear with z_i+2-z_i (in particular, each vector z_i+1-z_i-1 is non-zero).
Restricting the motion of the i-th vertex z_i to the direction of the diagonal z_i+1 z_i-1 defines an n-dimensional distribution 𝒟 on 𝒫_n (its analog for the usual inner billiards was called in <cit.> the Birkhoff distribution). An invariant curve of n-periodic points of the symplectic  billiard gives rise to a curve in 𝒫_n, tangent to 𝒟. We call curves tangent to 𝒟 horizontal curves. We point out that, by the very definition of 𝒫_n, we only consider invariant curves consisting of non-degenerate polygons, i.e., polygons in 𝒫_n.

Let A: 𝒫_n → be the algebraic area of a polygon given by

    A = 1/2∑_i=1^n ω(z_i,z_i+1).


 
The distribution 𝒟 is tangent to the level hypersurfaces of the area function A. The distribution 𝒟 is totally non-integrable on these level hypersurfaces: the tangent space at every point is generated by the vectors fields tangent to 𝒟 and by their first commutators.



Let z_i=(p_i,q_i)∈^2n, i=1,…,n. Introduce the vector fields

    v_i = (p_i+1-p_i-1) ∂/∂ p_i + (q_i+1-q_i-1) ∂/∂ q_i,

where, as always, the indices are understood cyclically.
The fields v_1,…,v_n are linearly independent and they span 𝒟 at every point. 

Geometrically, it is obvious that the fields v_i preserve the area of a polygon. Analytically, 

    dA = 1/2∑_i=1^n (q_i+1-q_i-1) dp_i - (p_i+1-p_i-1) dq_i.

Let 

    θ_i = (q_i+1-q_i-1) dp_i - (p_i+1-p_i-1) dq_i,

so that dA = ∑θ_i. 
Then θ_i (v_j)=0 for all i,j, and it follows that dA vanishes on 𝒟.
The common kernel of the 1-forms θ_i is the distribution 𝒟.

Next, one has [v_i,v_j]=0 for |i-j|≥ 2, and 

    [v_i+1,v_i] = (p_i+2-p_i)∂/∂ p_i + (p_i+1-p_i-1)∂/∂ p_i+1 +
    (q_i+2-q_i)∂/∂ q_i + (q_i+1-q_i-1)∂/∂ q_i+1.

It follows that 

    θ_i([v_i,v_i+1]) = - θ_i+1 ([v_i,v_i+1]) = (z_i+1-z_i-1,z_i+2-z_i) ≠ 0,

where the last inequality is due to the definition of 𝒫_n. 

The distribution 𝒟 has codimension n-1 on a constant-area hypersurface, and it suffices to show that the rank of the matrix
θ_i ([v_j,v_j+1]), i=1,…,n-1, j=1,… n, is n-1. But it follows from (<ref>) that the square submatrix with i,j=1,…,n-1 is upper-triangular with non-zero diagonal entries. This concludes the proof.

As in <cit.>, one can draw two conclusions from Theorem <ref>. 

The first is that one has an extreme flexibility of deforming horizontal curves of the distribution 𝒟, and hence of  billiard curves, still keeping an invariant curve consisting of n-periodic points (the technical statement is that there is a Hilbert manifold worth of such curves, obtained by deforming a circle and depending on functional parameters). We do not dwell on these technical issues; see <cit.> for a detailed discussion in the usual billiard set-up or <cit.> for the case of outer billiards.  However, we present one concrete example for n=4.



  §.§.§ Radon curves
 

 A centrally symmetric convex closed planar curve  is called a Radon curve if it has the property depicted in Figure <ref>. 
 



Radon curves are unit circles of a particular kind of Minkowski (normed) planes, called Radon planes, that share many features with Euclidean planes. In particular, in a Radon plane, normality (also called the Birkhoff orthogonality) is a symmetric relation.
Radon curves have been extensively studied since their introduction by Radon in 1916; see <cit.> for a modern account. 

The relevance of Radon curves for us is that they have  invariant circles consisting of 4-periodic points: the parallelogram xyx^*y^* is a symplectic billiard orbit for every x. Conversely, if a centrally symmetric curve has an invariant circle consisting of 4-periodic points, then it is a Radon curve.

An obvious example of Radon curve is an ellipse. However Radon curves are very flexible, depending on a functional parameter. For instance, here is a construction of a C^1-smooth Radon curve, see <cit.>. 

Let a and b be two vectors with [a,b]=1. Connect a and b by a smooth convex curve C_1 that lies in the parallelogram spanned by a and b and is tangent to its sides at the end points. Parameterize C_1 by a parameter t∈[0,T] so that C_1(0)=a, C_1(T)=b, and [C_1(t),C_1'(t)]=1 for all t. The latter condition means that the rate of change of the sectorial area is constant. Differentiating, we obtain [C_1(t),C_1”(t)]=0, hence the acceleration of the curve C_1 is proportional to the position vector.

Since C'_1(0) is proportional to b, and [a,b]=1, we have C'_1(0)=b. For the same reason, C'_1(T)=-a. 
Now consider the curve C_2(t)=C'_1(t). This curve lies in the second quadrant and connects  b with -a. Furthermore, C_2'(0)=C_1”(0) is proportional to C_1(0)=a, and C_2'(T)=C_1”(T) is proportional to C_1(T)=b. Therefore the union of C_1, C_2 and their reflections in the origin is a C^1-smooth curve.

Since [C_1(t),C_2(t)]=1, one has [C'_1(t),C_2(t)]+[C_1(t),C'_2(t)]=0. Thus one summand vanishes if and only if so does the other. This implies  the Radon property. 

For example, one can combine ℓ_p- and ℓ_q-norms with 1<p,q < ∞ and 1/p+1/q=1, taking C_1 and C_2 to be the quarters of their respective unit circles, see Figure <ref>. It is worth mentioning that, as p→∞, q→ 1, the respective Radon curve tends to an affine-regular hexagon.


The above construction, in general, gives rise to C^1-smooth Radon curves. It can obviously adapted to produce C^k or C^∞-smooth examples.








  §.§.§ Scarcity of periodic points
 

The second consequence of Theorem <ref> is a version of the n=3 and n=4 cases of the Ivrii conjecture. This conjecture asserts that the set of periodic trajectories of the usual billiard has zero measure; a seemingly weaker (but, in fact, equivalent) version is that this set has  empty interior.  For period n=3, the Ivrii conjecture is proved, by a number of authors and in different ways, in  <cit.>; currently, the best known result is for n=4, see <cit.>. See also <cit.> for periods n=3,4,5,6 for outer billiards. 

 
The set of 3-periodic points of the planar symplectic billiard has empty interior. 
If  is strictly convex, then the set of 4-periodic points also has empty interior.



First we consider the case n=3.  The proof is essentially the same as in <cit.> and <cit.>.

Let M be the 5-dimensional manifold of triangles of unit area and 𝒟 be the above defined distribution on it. We point out that triangles of unit area lie automatically in the space 𝒫_3, i.e., M⊂𝒫_3, on which 𝒟 is defined. If the set of 3-periodic points of the symplectic billiards map contains an open set, then the respective triangles form a horizontal surface U ⊂ M (i.e., tangent to 𝒟.) Choosing local coordinates in this surface yields a pair of commuting  vector fields on U.

Since these fields are horizontal, they can be written in the form 
f_1 v_1+f_2v_2+f_3v_3  and g_1 v_1+g_2v_2+g_3v_3, where f_i,g_i are functions and v_i are as in (<ref>).
Without loss of generality, assume that f_1≠ 0 and g_2≠ 0. Taking linear combinations, we obtain vector fields w_1=v_1 + fv_3, w_2=v_2+gv_3 with the property that [w_1,w_2] is a linear combination of w_1 and w_2 with functional coefficients. In particular, [w_1,w_2] is horizontal.

Let d=[z_2-z_1,z_3-z_1] be twice the area of the triangle z_1z_2z_3.
One calculates

    [w_1,w_2] = [v_1,v_2] -f [v_2,v_3] - g [v_3,v_1] 𝒟,

and hence, using (<ref>),

    θ_1([w_1,w_2]) = d(1+g), θ_2([w_1,w_2]) = -d (1+f).

Since d 0, it follows that f=g=-1, and 

    [w_1,w_2] = [v_1,v_2] + [v_2,v_3] + [v_3,v_1] = v_1+v_2+v_3.
 
But this vector field is not a linear combination of w_1=v_1-v_3 and w_2=v_2-v_3. This contradiction concludes the proof of this case.

Now consider the case n=4. Let x_1 x_2 x_3 x_4 be a 4-periodic orbit in the curve . Then the tangents T_x_1 and T_x_3 are parallel to the diagonal x_2 x_4, and hence x_3=x_1^*. Likewise, x_4=x_2^*.

Let x̅_1 x̅_2 x̅_3 x̅_4 be another 4-periodic orbit, a perturbation of the first one. Assume that point x̅_1 has moved from point x_1 in the positive direction (with respect to the orientation of the curve ). We claim that then the other points x̅_i, i=2,3,4, have also moved in the positive direction. We shall refer to this property as the monotonicity condition.

To prove the monotonicity condition, note that since  is strictly convex, the relation x↦ x^* is an orientation preserving involution. Since x̅_3=x̅_1^*, this point has moved in the positive direction. Hence the segment x̅_1 x̅_3 has turned in the positive sense (compared to x_1 x_3). Since T_x̅_2 is parallel to (x̅_1) (x̅_3), point x̅_2 has moved in the positive direction as well, and so has x̅_4=x̅_2^*.

Now we argue similarly to the n=3 case. Let M be the manifold of quadrilaterals of constant area, and let 𝒟 be the respective 4-dimensional distribution. Again we claim that 4-periodic points of the billiards map give rise to polygons which automatically lie in 𝒫_4. Indeed, by the definition of the symplectic billiard map the characteristic directions of  at x_1 and x_3 are parallel to x_2x_4 and similarly the characteristic directions at x_2 and x_4 are parallel to x_1x_3. Now, if we assume that x_1x_3 and x_2x_4 are parallel we find find four points with the same characteristic direction which directly contradicts the strict convexity of  which allows for exactly two such point.

Therefore, we now assume that there exists a horizontal surface in M. Then one has two linearly independent commuting vector fields, ξ and η, that are lineal combinations of the vector fields v_1,…,v_4 with functional coefficients. 

Let θ_1,…,θ_4 be as above. It follows from (<ref>) that 

    θ_i([v_i,v_i+1]) = - θ_i+1 ([v_i,v_i+1]) = 1, i=1,…,4,

because the determinant involved is twice the oriented area of the quadrilateral.

Without loss of generality, assume that the coefficient of v_1 in ξ is non-zero. In the following we distinguish two cases. If the coefficient of v_2 or of v_4 in η is non-zero, one can replace ξ and η by their linear combinations so that, perhaps after reversing the order of indices, one has

    ξ = v_1 + f v_3 + g v_4, η = v_2 + f̅ v_3 + g̅ v_4,
 
and [ξ,η] is a linear combination of ξ and η with functional coefficients. We refer to this as case 1.

If the coefficients of v_2 and of v_4 in η vanish then one can replace ξ and η by their linear combinations so that ξ = v_1, η = v_3. This is case 2.

In case 1, 

    [ξ,η]= [v_1,v_2]-g̅ [v_4,v_1] - f [v_2,v_3] + (f g̅ - f̅ g) [v_3,v_4] 𝒟.

Evaluating θ_i, i=1,…,4 on this commutator and equating to zero yields

    f = g̅ = -1, f̅ g = 0.

Without loss of generality, assume that g = 0. Then ξ = v_1 - v_3 is a vector field tangent to the disc U consisting of 4-periodic orbits. The flow of this field moves points x_1 and x_3 in the opposite directions, contradicting the monotonicity condition.

Likewise, in case 2, ξ = v_1. The flow of this field moves point x_1 in the positive direction, but leaves the other points fixed, again contradicting the monotonicity condition. This completes the proof.

 
As follows from the analysis in section <ref>, the monotonicity condition does not hold for
4-periodic orbits in a square: in fact, when such an orbit is perturbed, the points x_1 and x_3 move in the opposite directions, and so do x_2 and x_4. In particular, this shows that strict convexity is necessary.





 §.§ Interpolating Hamiltonians and area spectrum
 



  §.§.§ Area spectrum of symplectic billiard
 



Consider the maximal action, that is, the maximal area of a simple n-gon, inscribed in the billiard curve . Let A_n be this area. We are interested in the asymptotics of A_n as n→∞.

The theory of interpolating Hamiltonians, applied to the symplectic billiard, implies that the symplectic billiard  map equals an integrable symplectic map, the time-one map of a Hamiltonian vector field, composed with a smooth symplectic map that fixes the boundary of the phase space point-wise to all orders, see <cit.> and <cit.>; see also <cit.> for an application to  outer billiards.
In particular, this theory provides an asymptotic expansion of A_n in negative even powers of n:

    A_n ∼ a_0 + a_1/n^2 + a_2/n^4 + a_3/n^6 + …
 
In our situation, the coefficient a_0 is, of course, the area of the billiard table. The next two coefficients, a_1 and a_2, were found in <cit.>: 

    a_1= L^3/12,   a_2=-L^4/240∫_0^L k(t) dt,

where t is the affine parameter on the billiard curve, L=∫_ dt is the total affine length, and k the affine curvature of . For comparison, for  outer billiards, that is for the circumscribed polygons of the least area, one has a_1 = L^3/24. 

In affine geometry one also has an isoperimetric inequality for all strictly convex closed curves, with equality only for ellipses, see <cit.>. It reads

    L^3 ≤ 8π^2 A,
 
where A is the area. We point out that it “goes in the wrong direction” if compared to the usual isoperimetric inequality. Similarly to <cit.>, one has the following immediate consequence.

 
The first two coefficients, a_0 and a_1, make it possible to recognize an ellipse: one always has the inequality

    3 a_1 ≤ 2π^2 a_0,

with equality if and only of  is an ellipse.


There is nothing to prove since by (<ref>) the affine isoperimetric inequality (<ref>) and (<ref>) are equivalent.

 
Of course, we can rephrase Theorem <ref> as “one can hear the shape of an ellipse". This leads to an interesting open question. Can one interpret the sequence a_0, a_1, a_2, … really as a spectrum? That is, is there a differential operator whose spectrum is this sequence? For usual billiards this is well-known, see for instance <cit.>.

Similarly it would be very interesting to determine the higher terms a_3, a_4, …, even in the case of ellipses, directly. In fact, for ellipses there is a little miracle that a_0 and a_1 determine all other terms. This is the affine isoperimetric inequality.





  §.§.§ Insecurity of symplectic billiards
   
 A classical billiard is called secure (or has the finite blocking property) if, for every two points A and B in the billiard table, there exists a finite set of points S, such that every billiard trajectory from A to B visits S (so the set S blocks A from B). For example, billiard in a square is secure.

It is proved in <cit.> that planar billiards with smooth boundary are not secure. In a nutshell, the argument is as follows. 

Let A and B be on the convex part of the boundary curve , and consider the shortest n-link billiard trajectory from A to B. For sufficiently large n, no points that are not on the boundary can block this trajectory. 

Using the theory of interpolating Hamiltonians, one shows that, modulo errors of order 1/n^2, the reflection points are regularly distributed on the arc AB with respect to the measure κ ^2/3 dx, where x is an arc length parameter, and κ is the curvature of . 

One can introduce a coordinate t so that dt = κ ^2/3 dx and normalize so that t ∈ [0,1] on the arc AB. If the reflection points were regularly distributed, and there were n reflections, then the reflection points would have coordinates m/n, 1≤ m < n. Then it is clear that for every finite set S ⊂ [0,1], there exists n such that S contains no fractions with denominator n: one can take n to be a prime number greater than all the denominators of the rational numbers contained in S.

The actual argument uses some number theory (Proposition 2 or the more general Theorem 3.2 of <cit.>) to deal with the errors of order 1/n^2 in the distribution of the reflection points. To recap, no finite set of points on the arc AB can block all billiard trajectories from A to B.

One can use the same approach to prove an analogous result for symplectic billiards that we formulate below without proof. The key observation is contained in <cit.>: with respect to the affine parameter t, the 
vertices of inscribed n-gons of maximal area, that is, the reflection points of the n-link symplectic billiard trajectory, are equidistributed  modulo errors of order 1/n^2. 

 
The symplectic billiard inside a smooth strictly convex curve is insecure. More precisely, for every pair of distinct points x,y on the boundary curve  and every finite set S⊂, there exists a symplectic billiard trajectory from x to y that avoids S.




§ POLYGONS
 

In this section we collect a few simple results on polygonal symplectic billiards. In our opinion, this subject deserves a thorough study.

We start with a remark concerning the definition: the symplectic billiard map is not defined for a chord xy if the sides of the polygon that contain these points are parallel. The map is also not defined if point x is a vertex of the polygon. Note that if the end points of a segment of a billiard trajectory are not on parallel sides of a polygon, then the same is true for the next segment of the trajectory. 

Let P be a convex polygon.
The phase space  of the symplectic billiard is the torus P× P, and it is naturally decomposed into rectangles, the products of pairs of sides of P (if P has pairs of parallel sides then the map is not defined on the respective rectangles). The symplectic billiard map Φ is a piecewise affine map of this phase space.



 §.§ Regular polygons
 

The case of regular polygons is very interesting in both `classical' cases, the inner and the outer billiards. In the  case of inner billiards, the Veech Dichotomy holds: for every direction in a regular polygon, the billiard flow is either periodic or uniquely ergodic, see, e.g., <cit.>, just like in the well known case of a square. In the case of outer billiards, (affine) regular polygons have an intricate fractal orbit structure (except for n=3, 4, 6 when all orbits are periodic) which is analyzed only for n=5,8, see <cit.>.

Let P be a regular n-gon whose sides are cyclically labeled 0,1,…, n-1. Assume that the initial segment of a billiard trajectory connects side 0 with side k; here 1≤ k ≤ [(n-1)/2]. We shall call k the rotation number of the trajectory. See Figures <ref> 
and <ref>. 





 
  =1.5ex

  * The rotation number of an orbit is well defined: each link of the orbit connects side i with side i+k.

  * Let 

    g(n,k) = n/ gcd(n,2k).

  =1ex

  * If g(n,k) is even, then the respective billiard orbits are 2g(n,k)-periodic.

  * If g(n,k) is odd, the orbits are 4g(n,k)-periodic.




Let the orbit be x_0,x_1,… with x_0  on the side labeled 0, and x_1 on the side labeled k. Due to the dihedral symmetry of the polygon, the projection along the kth side takes  the 0th side to the (2k)th side. Hence x_2 lies on the (2k)th side, and so on. It follows that the rotation number of the orbit is well defined. 

The segments connecting x_0 to x_2 to x_4, etc. are parallel to the sides of the polygon, and likewise for the segments connecting x_1 to x_3 to x_5, etc. One obtains two polygonal lines, even and odd. Due to the symmetry of a regular polygon P, both are usual billiard trajectories in P, see Figure <ref>. We refer to them as the even and odd (usual) billiard trajectories. 




Consider a particular case of such a billiard trajectory, the one whose initial segment connects the midpoint of side 0 to the midpoint of side 2k. This billiard trajectory is periodic with period g(n,k). A parallel trajectory is also periodic, with period g(n,k) if g(n,k) is even, and period 2g(n,k) if g(n,k) is odd. 

Next we observe that the even billiard trajectory depends only on the choice of the point x_0 (and the fixed number k), and the odd one only on the choice of the point x_1. This implies that, generically, these two billiard trajectories are different. Therefore the total number of vertices of the orbit x_0,x_1,… is 2g(n,k) if g(n,k) is even, and  4g(n,k) if g(n,k) is odd. 


Since all triangles are affine equivalent, all orbits in triangles are periodic, generically, of period 12.





 §.§ Trapezoids
 

Up to affine transformations, trapezoids form a 1-parameter family; it is convenient to normalize them so that they are isosceles. We assume that the lower horizontal side AB is greater than the upper one, CD. Define the modulus of a trapezoid ABCD to be

    [ |AB|/|AB|-|CD|].

The modulus is a positive integer. Let us call a trapezoid generic if |AB|/(|AB|-|CD|) is not an integer. For example, one may consider a triangle (with |CD|=0) as a trapezoid of modulus one, but it is not generic.

 
All billiard orbits in a trapezoid are periodic. If the modulus of a generic trapezoid is n, then there are three periods: 16n-4, 16n+4, and 16n+12.




Let x_0, x_1, x_2, … be a trajectory.  Similarly to the proof of Theorem <ref>, we consider the even and odd subsequences x_0, x_2, x_4, … and x_1, x_3, x_5, … separately.  

Let us define a transformation F of the (boundary of the) trapezoid as follows.
Let X be an interior point of a side of the trapezoid. Through X there pass two lines parallel to one of the sides of the trapezoid and intersecting its interior. Choose one of them and move the point X in this direction until it lands on the trapezoid  at point Y. Through Y there again pass two lines parallel to one of the sides of the trapezoid. One of them takes Y back to X; choose the other one and move the point Y to the new point Z, etc. We have described the map F: X → Y → Z →….



Since every trajectory necessarily visits all sides we take the starting point on the side BC and move horizontally left. 
After two moves, the point lands on the  horizontal side AB and makes an even number of `bouncing' moves between the horizontal sides, after which one more move in the direction AD takes the point to side BC. Thus we have a return map T to the side BC, see Figure <ref>.

There is a break point E on the side BC such that T(E)=C, see Figure <ref> on the right. For X ∈ CE, the    number of the bouncing moves between the horizontal sides equals 2(n-1), and for X ∈ BE, this number is 2n. 

Note that the map T consists of an odd number of moves, hence it reverses orientation.   In addition, T is a local isometry: the bouncing between the horizontal sides does not distort the length, and the two moves, from a side  AD to AB and from AD to BC, distort the length by reciprocal factors. It follows that T is a reflection in a point, namely, the mid-point of the segment BE or EC. 
Thus we find the period of point X under the map T: it equals 4n+2 if X ∈ CE, and 4n+6 if X ∈ BE. 

In what follows we shall call the polygonal lines connecting the consecutive images of points under the map F the guides. A guide that is an 4n+2-gon is called short, and a guide that is an 4n+6-gon is called long.

Now we can claim that every orbit of the symplectic billiard map Φ is periodic. Indeed, let 
x_0 x_1 be the initial segment of an orbit. As we have proved, the point x_0 determines a finite set of possible positions for even-numbered points x_i, and likewise, the point x_1 determines a finite set of possible odd-numbered points x_j. Therefore there are only finitely many possibilities for each segment x_i x_i+1, and 
hence the orbit is periodic.

Now we calculate the periods. Let x_0,x_1,… be an orbit. Consider the guides of points x_0 and x_1. Each one of them can be either short or long. Respectively, one needs to consider three cases: short-short, short-long, and long-long. The combinatorics of the orbits from each case are the same, but  differ from case to case. This is why one has three different periods.

Let us describe the short-short case in  detail. After that, we indicate how the other two cases differ. 



Consider Figure <ref> that depicts a trapezoid of modulus 2: one sees a 28-periodic orbit and the two short guides, each a decagon. Even for small n=2, this picture is already a mess. It is easier to analyze a particular case when the two guides coincide and contain the fixed point of the map T, see Figure <ref>. The effect of considering this special case is making the period four times as small (in other words, when we move the guides apart, the number of arrows quadruples).   



Let us analyze the combinatorics. 
The guide has 2n+1 vertices: two on the lateral sides, n on the side AB, and n-1 on the side CD (we use the notation in Figure <ref>). Let us label these points along the guide as shown in Figure <ref> (the odd-numbered points are on side AB and the even-numbered ones on side CD). Then the symplectic billiard orbit is 
4n-1-periodic and has the following code:

    (1 R 2 L 3 R 4 L … (2n-1) R L).

Quadrupling this number yields the short-short period 16n-4. 



The long-long case is similar to the short-short one. 

The short-long case is illustrated in Figure <ref>. Once again, we place the two guides in a special position and label their vertices as shown in the figure. This time the effect of the special position is in halving the number of arrows. The symbolic orbit in the case of n=2 is as follows:

    (L  I R I I L I I I R P 1 Q 2 P 3 Q 4 P 5 Q)

and, in general, the period consists of 1+ 2(2n-1) + 2(2n+1) +1 = 8n+2 symbols. Doubling this numbers gives the short-long period 16n+4.


It is an interesting problem to describe those polygons in which all symplectic billiard orbits are periodic. Another problem is to determine whether every convex polygon possesses a periodic symplectic billiard orbit.





 §.§ Stable and unstable periodic trajectories
 

It is known, and easy to prove, that an n-periodic point of a polygonal outer billiard transformation has a neighborhood consisting of periodic points of period n, if n is even, and period 2n, if n is odd, see for instance <cit.>. A similar property holds for odd-periodic orbits of the polygonal symplectic billiard. For the statement of the following Propositions we need a little preparation.

Let (x_0,…,x_n-1) be an n-periodic orbit. Let ℓ_i be the line through point x_i parallel to x_i-1 x_i+1. One cannot reconstruct the billiard polygon P from the periodic orbit, but one knows that the side of P through point x_i lies on the line ℓ_i.

Let (x̅_0 x̅_1), x̅_0 ∈ℓ_0, x̅_1 ∈ℓ_1, be the initial segment of a nearby billiard trajectory. As in the proof of Theorem <ref>, the evolutions of points x̅_0 and x̅_1 decouple: x̅_0 is projected to line ℓ_2 along ℓ_1, then to ℓ_4 along ℓ_3, etc., and likewise for x̅_1. Denote the projection of ℓ_i-1 to ℓ_i+1 along ℓ_i by π_i.

Let α_i+1/2 be the angle between ℓ_i and ℓ_i+1. Let dl_i  be the length element on the line ℓ_i. Then the distortion of length under the affine map π_i is given by the formula:

    dl_i+1/dl_i-1=sinα_i-1/2/sinα_i+1/2.

If n is odd, then the first time that point x̅_0 returns to line ℓ_0 is after n projections, that is, under the affine map 
π_n-1…π_3 π_1. This map is orientation reversing and it is an isometry because

    ∏_i=1^n sinα_i-1/2/sinα_i+1/2 =1.

Hence the second iteration of this map is the identity. 

A similar argument applies to x̅_1. Thus the orbits of both points, x̅_0 and x̅_1 consist of 2n points, and altogether, the billiard orbit is 4n-periodic. Moreover, since the map is the identity, an entire neighborhood of this orbit is periodic with the same period. This proves the following.

 
An n-periodic point in phase space with n being odd has an open neighborhood in phase spave which consists of 4n-periodic orbits.



If n is even, then the point x̅_0 returns to the line ℓ_0 after n/2 projections. The derivative of this affine map is

    ∏_i  odd mod nsinα_i-1/2/sinα_i+1/2.

If this product is not equal to 1 then the fixed point x_0 is isolated. A similar argument applies to point x̅_1, with the derivative given by the formula

    ∏_i  even mod nsinα_i-1/2/sinα_i+1/2,

i.e., if this product is not equal to 1 then x_1 is isolated.

In conclusion, if the derivatives of first return maps to ℓ_0, resp. to ℓ_1, at the fixed point are both different from 1 then the fixed points are hyperbolic. Hyperbolic fixed points do not vanish under small perturbations of the map, and hence of the polygon. This proves the following.

 
If n≥ 6 is even then, for a generic polygon, an n-periodic point is isolated and is stable in the sense that it does not disappear under a small perturbation of the polygon.




































§ SYMPLECTIC BILLIARD IN SYMPLECTIC SPACE
 



 §.§ Definition and continuous limit
 



  §.§.§ A precise look at the definition
 

We now extend the definition of the symplectic billiard map to linear symplectic space ^2n. 

Consider a smooth closed hypersurface M⊂^2n bounding a strictly convex domain. On M× M we consider the  function 

    S:M× M →, 
    (x_1,x_2) ↦ S(x_1,x_2):=ω(x_1,x_2).

In analogy to Section <ref>, we define the (open, positive part of) phase space as

    :={(x_1,x_2)∈ M× M|ω(ν_x_1,ν_x_2)>0}.

Here ν is again the outer normal. Since M is strictly convex, the Gauss map M∋ x↦ν_x∈ S^2n-1 is a bijection. Denote by R(x)=ω|_T_xM× T_xM the characteristic direction of M at the point x. 


The relation R(x_2)⊂ T_x_1M is equivalent to ω(ν_x_1,ν_x_2)=0. In particular, it is symmetric in x_1 and x_2:

    R(x_2)⊂ T_x_1M   ⟺   R(x_1)⊂ T_x_2M.



Since 
R(x_2)=·(Jν_x_2),
where J in the standard complex structure on ^2n≅^n, we observe

    R(x_2)⊂ T_x_1M   ⟺  ⟨ Jν_x_2,ν_x_1⟩=0   ⟺  ω(ν_x_2,ν_x_1)=0,

as claimed.


The sets ={(x_1,x_2)∈ M× M|ω(ν_x_1,ν_x_2)>0} and {(x_1,x_2)∈ M× M|ω(ν_x_1,ν_x_2)<0} are connected.


Consider the Gauss map M∋ x↦ν_x∈ S^2n-1. Then  is mapped to

    _0:={(a,b)∈ S^2n-1× S^2n-1|ω(a,b)>0}.

Clearly b≠± a. Therefore, the simple interpolation

    a_t:=a+t(b-a)/||a+t(b-a)||∈ S^2n-1,   t∈(0,1]

and 

    ω(a,a_t)=ω(a,a+t(b-a))/||a+t(b-a)||=tω(a,b)/||a+t(b-a)||

shows that we can move the second factor close to a. That is, we can move any point in  _0 into a tubular neighborhood of the diagonal which is connected. The same argument works for <. Since the Gauss map is a bijection the results follows.

We define the map Φ:→ geometrically similarly to the procedure from Section <ref>, see Figure <ref>.




Given (x_1,x_2)∈, there exists a unique point x_3∈ M with

    (x_1+R(x_2))∩ M ={x_1,x_3}.
 
Moreover, (x_2,x_3)∈ and

    x_3-x_1=tJν_x_2

with t>0.



Using the language of contact geometry one, of course, realizes that Jν is positively proportional to the Reeb vector field R⃗ of the standard contact form α:=-dr∘ J on M.  Here r denotes the radial coordinate on ^2n. In particular, R⃗ orients (or trivializes) the line field R and the definition of the positive part of the phase space corresponds to conforming to this orientation.
 


Since M is convex, (x_1+R(x_2))∩ M ={x_1,x_3} with possibly x_1=x_3. Since (x_1,x_2)∈, we know that R(x_2)⊄T_x_1M, and therefore, x_1≠ x_3. 

To show that (x_2,x_3)∈, we argue, as in the 2-dimensional case, by continuity. The statement is clearly true if x_2 is close to x_1. If ω(ν_x_2,ν_x_3)≤0 then, using the connectedness of {(x_1,x_2)∈ M× M|ω(ν_x_1,ν_x_2)<0}, we can move the point x_2 in order to achieve ω(ν_x_2,ν_x_3)=0. The latter is equivalent to R(x_2)⊂ T_x_3M which contradicts x_1≠ x_3.

To determine the sign in x_3-x_1=tJν_x_2 consider the set M_x^+:={y∈ M|ω(ν_x,ν_y)>0} and similarly M_x^-. Both are connected since they correspond to hemispheres under the Gauss map. We can rephrase the definition of  as 

    (x_1,x_2)∈  ⟺   x_1∈ M_x_2^-   ⟺   x_2∈ M_x_1^+,

and the above discussion as

    x_3∈ M_x_2^+ .
Since x_1,x_2 determine x_3 we write

    x_3-x_1=t(x_2)Jν_x_2,

where we think of x_1 as fixed and x_2 as a variable. 

First we claim that the sign of t(x_2) does not depend on x_2 as long as (x_1,x_2)∈, i.e., x_2∈ M_x_1^+. Assume that t(x_2) changes sign or vanishes. Since M_x_1^+ is connected, we can always reduce to the latter case, i.e., we find a point x_2 with t(x_2)=0. But this means x_3=x_1, which directly contradicts the very first assertion of this proof. 

Since the sign of t(x_2) is fixed, we now compute it at a convenient point. For that, let x_2 be the point where ν_x_2=aJν_x_1 with a>0. This point exists and is unique again by the Gauss map being a diffeomorphism. It follows that (x_1,x_2)∈ since

    ω(ν_x_1,ν_x_2)=ω(ν_x_1,aJν_x_1)= a·|ν_x_1|^2=a>0.

Finally we observe that

    x_3-x_1=t(x_2)Jν_x_2=at(x_2)JJν_x_1=-at(x_2)ν_x_1

which, again by convexity, forces t(x_2)>0, as ν is the outward normal.

By Lemma <ref>, we can again define a map

    Φ:→,(x_1,x_2)↦Φ(x_1,x_2):=(x_2,x_3)

by the above rule. Now it follows that S(x_1,x_2)=ω(x_1,x_2) is a generating function, indeed:

    (x_2,x_3)=Φ(x_1,x_2)  ⟺  ∂/∂ x_2(S(x_1,x_2)+S(x_2,x_3))=0,

by the latter we, of course, mean that 

    ω(x_1,v)+ω(v,x_3)=0  ∀ v∈ T_x_2M.

Since x_1≠ x_3, this is equivalent to the condition 
x_3-x_1∈ R(x_2),
as needed.

We can again extend the map Φ continuously to 

    :={(x_1,x_2)∈ M× M|ω(ν_x_1,ν_x_2)≥0}

 by

    Φ(x,x):=(x,x)
       ω(ν_x_1,ν_x_2)=0  ⇒  Φ(x_1,x_2):=(x_2,x_1).

Equation (<ref>) implies that the map Φ preserves the closed 2-form 

    Ω_(x_1,x_2):= ∂^2 S/∂ x_1∂ x_2(x_1,x_2) dx_1∧ dx_2∈Ω^2(M× M).

That is, for (u_i,v_i)∈ T_(x,y)M× M,

    Ω_(x,y)((u_1,v_1),(u_2,v_2))=ω(u_1,v_2)-ω(u_2,v_1).




For x,y∈ M, we have

    Φ(x,y)=(y,x)  ⟺  ω(ν_x,ν_y)=0   ⟺  Ω_(x,y) is degenerate.
 



For (u_1,v_1)≠0 we have

    Ω_(x,y)   ((u_1,v_1),(u_2,v_2))=0  ∀ (u_2,v_2)∈ T_(x,y)M× M
       ⟺  ω(u_2,v_1)=0 ∀ u_2∈ T_xM and ω(u_1,v_2)=0 ∀ v_2∈ T_yM
       ⟺   R(x)⊂ T_yM  or/and  R(y)⊂ T_xM
       ⟺  ω(ν_x,ν_y)=0
       ⟺  Φ(x,y)=(y,x),

as claimed. We recall from Lemma <ref> that the relation R(x)⊂ T_yM is symmetric in x and y. Therefore “or” and “and” in the third line are equivalent.


 

  * As in the planar case,t strict convexity gives rise to an involution M∋ z↦ z^*∈ M characterized by R(z)=R(z^*).

  * Similarly to polygons in the plane, one can define symplectic billiards in convex polyhedra in symplectic space. For example, it would be interesting to study a simplex in ^2.






  §.§.§ Continuous limit
 

Consider a (normal) billiard trajectory inside a convex hypersurface M that makes very small angles with the hypersurface (a grazing trajectory); as the angles tend to zero, one expects such a trajectory to have a geodesic on M as a limit. One
observes this at the level of generating functions: the chord length |xy|, x,y∈ M, becomes, in the limit y → x, the  function L(x,v) = |v| of the tangent vectors v ∈ T_x M. The extremals of the Lagrangian L are non-parameterized geodesics on M.

What happens in the continuous limit with the symplectic billiard? 

Geometrically, one expects the following. Let Z_0,Z_1,Z_2,Z_3,… be an orbit. As the points merge together, the direction of the segment Z_0 Z_2 tends to the characteristic direction R(Z_1), and the direction of Z_1 Z_3 to R(Z_2). Therefore, in the limit, the even-numbered points lie on a characteristic line, and so do the odd-numbered points; and in the limit, the two characteristic lines merge together. Note however that the direction Z_0 Z_1 is not necessarily characteristic, and  the limiting behavior of the directions Z_j Z_j+1 is not determined.

One observes the same phenomenon at the level of generating functions. The continuous limit of the generating function ω(x,y), x,y ∈ M, is the Lagrangian L(x,v)=ω(x,v) on the tangent bundle TM. The Euler-Lagrange equation with constraints reads

    d L_v/dt - L_x = λν,

where t is time, ν is a normal, and λ is a Lagrange multiplier. Since L(x,v)=⟨ Jx,v⟩, one has
L_v = Jx, L_x = -Jv, and the Euler-Lagrange equation reduces to v = μ(x,v) J(ν). This is the equation of non-parameterized characteristic line on M.

Note however that, due to the degeneracy of the Lagrangian L, we end up with a first order differential equation, rather than the second order one.

Perhaps a better way to take a continuous limit is to treat the even-numbered and odd-numbered vertices of a symplectic billiard trajectory separately: points Z_0,Z_2,… converge to one curve _0(t) ⊂ M, and points Z_1,Z_3,… to another curve _1(t)⊂ M. In this limit, one obtains the following relation between the curves:

    R(_1(t)) ∼_0'(t), R(_0(t)) ∼_1'(t),

where a∼ b means that the vectors a and b are proportional. Compare to the discussion for symplectic ellipsoids in Section <ref>.

The problem of finding pairs of curves on M satisfying (<ref>) is variational.

 
A pair of closed curves (_0(t),_1(t)) on M ⊂^n, satisfying (<ref>), is critical for the functional 

    L(_0,_1)= ∫ω(_0'(t),_1(t)) dt.

 

 
Integrating by parts, we see that L(_0,_1) = L (_1,_0). Consider a variation of _1 given by the vector field v(t) on M along _1. Having

    ∫ω(_0'(t),v(t)) dt =0

for all such v is equivalent to _0'(t) being symplectically orthogonal to T__1(t)M for all t, that is, _0'(t) ∼ R(_1(t)). Due to symmetry of the functional L, the same argument yields _1'(t) ∼ R(_0(t)).

It would be interesting to describe pairs of curves on M satisfying (<ref>). A particular solution is _0=_1, a characteristic curve. If M is a unit sphere, then a pair (_0,_1) of geodesic circles related by _1 = J(_0) are all other solutions.



 §.§ Ellipsoids
 



  §.§.§ Symplectic billiard and the usual billiard
 

It is well known that the billiard ball map in an ellipsoid is completely integrable, see, e.g., <cit.>. In this section we describe a close relation between the symplectic and the usual billiard in ellipsoids that, in particular, implies complete integrability of the former.

Consider an ellipsoid in ^2n with Darboux coordinates  x_1,…, x_n, y_1, …, y_n and the symplectic structure ω_0 = ∑ dx_j ∧ dy_j. Applying a linear symplectic transformation and homothety, we may assume that the ellipsoid is given by the equation

    x_1^2+y_1^2/a_1 + x_2^2+y_2^2/a_2 + … + x_n^2+y_n^2/a_n =1,     a_1,…, a_n>0.

The diagonal linear transformation 

    x_j ↦√(a_j) x_j, y_j ↦√(a_j) y_j, j=1,…,n,

 takes the ellipsoid to the unit sphere and transforms the symplectic form to ω = ∑ a_j dx_j ∧ dy_j. We shall consider the symplectic billiard inside the unit sphere S^2n-1 defined by this symplectic form. The characteristic direction at the point 
z∈ S^2n-1 is · R(z), where the complex linear operator R:^n →^n is diagonal with the entries ia_1^-1,…, i a_n^-1.

Consider the linear map R^-1: ^n →^n. It takes the unit sphere to the ellipsoid E given by the equation

    |w_1|^2/a_1^2 + |w_2|^2/a_2^2 + … + |w_n|^2/a_n^2 =1,

where w_1,…,w_n are complex coordinates in the target space.

 
Let (…, Z_0, Z_1, Z_2,…),  be a trajectory of the symplectic billiard map in the unit sphere with respect to the symplectic form ω = ∑ a_j dx_j ∧ dy_j. Then a sequence (…, R^-1(Z_0), R^-1 (Z_2), R^-1 (Z_4), …) is a billiard trajectory in E. Conversely, to a billiard trajectory (…, W_0, W_2, W_4, …) in E there corresponds a unique symplectic billiard trajectory (…, Z_0, Z_1, Z_2,…) in S^2n-1 with Z_0 = R(W_0), Z_2 = R(W_2), etc.


Consider the points Z_0, Z_2, Z_4 of a symplectic billiard trajectory. The points Z_1 and Z_3 are uniquely determined by the symplectic billiard reflection law:

    R(Z_1) = t_1 (Z_2-Z_0), R(Z_3) = t_3 (Z_4-Z_2), t_1 > 0, t_3 > 0.

Hence Z_1 = t_1 R^-1(Z_2-Z_0), and the normalization |Z_1|^2=1 uniquely determines t_1; likewise for t_3. Thus

    Z_1 = R^-1(Z_2-Z_0)/|R^-1(Z_2-Z_0)|,  Z_3 = R^-1(Z_4-Z_2)/|R^-1(Z_4-Z_2)|.

The symplectic billiard reflection law also implies that

    R(Z_2) = t (Z_3-Z_1) = t (R^-1(Z_4-Z_2)/|R^-1(Z_4-Z_2)| + R^-1(Z_0-Z_2)/|R^-1(Z_0-Z_2)|).

Set R^-1 (Z_j)=W_j, j=0,2,4, and rewrite the last equation as

    R^2 (W_2) =t (W_0 - W_2/|W_0 - W_2| + W_4 - W_2/|W_4 - W_2|).

Note that the vector R^2 (W_2) is normal to the ellipsoid E given by (<ref>). Therefore equation (<ref>) describes the billiard reflection in E at point W_2 that takes  W_0 W_2 to W_2 W_4, as claimed.

Conversely, given a segment of a billiard trajectory W_0,W_2,W_4 in E, one defines Z_j = R(W_j), j=0,2,4, 

    Z_1=W_2-W_0/|W_2-W_0|, Z_3=W_4-W_2/|W_4-W_2|.

Then equation (<ref>) implies that Z_0,…, Z_4 is a segment of a symplectic billiard trajectory. 



  §.§.§ Continuous version
 

Let us also present a continuous version of Theorem <ref>.

 
 Let A =  diag (a_1,a_2,…,a_n) be a diagonal matrix with real positive entries, and let
 x(t) be a characteristic curve on the ellipsoid ⟨ Ax,x⟩=1. The linear map A^-1/2: ^n →^n takes this curve to a geodesic curve on the ellipsoid ⟨ A^2 y,y⟩=1.


If y=A^-1/2x and ⟨ Ax,x⟩=1, then ⟨ A^2 y,y⟩=1. Note that Ax is a normal to the ellipsoid ⟨ Ax,x⟩=1 at the point x.
If x(t) is a characteristic, then x' = f JA x where f(t) is a non-vanishing function. The matricies A and J commute. Differentiate:

    x” =  f' JA x + f JA x' = f'/f x' - f^2 A^2 x,

hence

    y” = f'/f y' - f^2 A^2 y.

Since A^2 y is a normal to the ellipsoid ⟨ A^2 y,y⟩=1 at the point y, we conclude that y”∈ Span (y',ν_y), that is, y(t) is a geodesic.

One can reverse the argument:  start with a geodesic on the ellipsoid ⟨ A^2 y,y⟩=1 and construct a pair of curves on the ellipsoid ⟨ Ax,x⟩=1 in the relation (<ref>). We use the same notations as before.

 
Let y(t) be a geodesic on the ellipsoid ⟨ A^2 y,y⟩=1 satisfying 

    y”(t) = f'(t)/f(t) y'(t) +g(t) A^2 y(t),

where f and g are some functions with f(t)≠0 for all t. Set 

    x(t)=A^1/2 y(t), x̅(t)= c/f(t) J A^-1 x'(t).

Then, for a suitable choice of the constant c, the pair of curves (x,x̅) lie on the ellipsoid ⟨ Ax,x⟩=1 and satisfy (<ref>).


We know from the previous proof that x(t) lies on the ellipsoid ⟨ Ax,x⟩=1. Also

    x”(t) = f'(t)/f(t) x'(t) +g(t) A^2 x(t).
 
Let h(t)=⟨ x'(t),A^-1 x'(t)⟩. Then 

    h' = 2 ⟨ x”,A^-1 x'⟩ = 2 f'/f⟨ x',A^-1 x'⟩ + 2g ⟨ A^2 x,x'⟩ = 2  f'/f h,

where we used (<ref>) and the fact that Ax is orthogonal to x'. It follows that h=C f^2 where C is a constant. Then

    ⟨ A x̅, x̅⟩ = c^2/f^2(t)⟨ x'(t),A^-1 x'(t)⟩ = c^2 C,

hence, if c=C^-1/2, we have ⟨ A x̅, x̅⟩ =1.

It remains to check (<ref>). That R(x̅) ∼ x is clear, since R = JA. Finally,

    x̅' = -cf'/f^2 J A^-1 x' + c/f(t) J A^-1 x” = -cf'/f^2 J A^-1 x' + c/f J A^-1( f'/f x' +g A^2 x ) ∼ JAx,

as needed.

Proposition <ref> seems to indicate that the correct continuous limit of symplectic billiard is a pair of curves satisfying (<ref>), as discussed in Section <ref>.


The curve y̅ = A^-1/2x̅ is also a geodesic on the ellipsoid ⟨ A^2 y,y⟩=1. The  geodesics y and y̅ are related by the composition of J and the skew-hodograph transformation, see section 3 in <cit.>. 






  §.§.§ Symplectic billiards and the discrete Neumann system
 

The discrete Neumann system (Z_0,Z_1) ↦ (Z_1,Z_2) is a Lagrangian map on the Cartesian square of the unit sphere given by the equation

    Z_0 + Z_2 = λ A(Z_1),

where A is a self-adjoint linear map and λ is a factor  determined by the normalization |Z_2|=1, see <cit.>. Symplectic billiard inside the unit sphere is given by a similar equation

    Z_2 - Z_0 = t R(Z_1),

where R is an anti self-adjoint linear map and t is a suitable factor. 

Our Theorem <ref> is an analog of Theorem 6 in <cit.> that relates, in a similar way, the discrete Neumann system and the billiard inside an ellipsoid. Thus, similarly to the Neumann system, the symplectic billiard is “a square root" of the billiard system in an ellipsoid (cf. <cit.>). Note however that the latter ellipsoid is not generic: its axes are equal pairwise.



  §.§.§ Integrals
 

The symplectic billiard map possesses  a collection of particularly simple integrals described in the following proposition.
For a point Z_j ∈ S^2n-1⊂^2n, write Z_j=(x_1j,y_1j,…,x_nj,y_nj).

 
The following functions are integrals of the symplectic billiard map (Z_0,Z_1) ↦ (Z_1,Z_2):

    I_k (Z_0,Z_1) = x_k0 x_k1 + y_k0 y_k1, k=1,…,n,   and  J(Z_0,Z_1) = ⟨ R(Z_0), Z_1⟩.



One has

    Z_2 = Z_0 + 2 ⟨ R(Z_0),Z_1⟩/|R(Z_1)|^2 R(Z_1).

Substitute this Z_2 to J(Z_1,Z_2) and simplify to obtain J(Z_0,Z_1).

Likewise, to show that I_k is an integral it suffices to check that I_k(R(Z_1),Z_1) =0. This follows from the fact that R is a diagonal map of ^n that, up to a factor, multiplies each  coordinate by i. 

In particular, a trajectory (…, Z_0, Z_1, Z_2, …) of the symplectic billiard in a sphere is an equilateral polygonal line.

The integrals I_k occur due to the symmetry of the ellipsoid (or, equivalently, of the operator R): I_k corresponds to the rotational symmetry in kth coordinate complex line via E. Noether's theorem. 

 Let us explain the billiard origin of the integral J.

Consider the billiard system in an ellipsoid E given by the equation ⟨ Aw,w⟩ = 1. The  phase space of the billiard consists of the tangent vectors (w,v) with the foot point w∈ E and a unit inward vector v. It is known that the function ⟨ Aw,v⟩ is an integral of the billiard transformation, see, e.g., <cit.>.

Consider the pull-back of the integral ⟨ Aw,v⟩ under the map R^-1: ^n →^n. This is a function of (Z_0,Z_2), and since Z_2 is determined by Z_0 and Z_1 via the symplectic billiard reflection, this is also a function of (Z_0,Z_1), a phase point of the symplectic billiard.

 
This function equals ⟨ Z_0, R(Z_1)⟩.


 
One has R^-1 (Z_0) = w, that is, R(w) = Z_0. Likewise, R(v) is positive-proportional to Z_2-Z_0 which, by the symplectic billiard reflection law, is positive-proportional to R(Z_1). Since both v and Z_1 are unit, v=Z_1. Since A= -R^2 and R is anti self-adjoint, we have

    ⟨ Aw,v⟩ = ⟨ Rw,Rv⟩ = ⟨ Z_0,R(Z_1)⟩,

as claimed.



  §.§.§ Low-period orbits
  

Let us also mention a property of low-period orbits of the symplectic billiard in an ellipsoid. 

By a coordinate subspace in ^n we mean a subspace spanned by any number of the complex coordinate lines. As before, we consider symplectic billiard inside the unit sphere, with the characteristic vector  given by a diagonal complex linear operator R with the entries ia_1^-1,…, i a_n^-1. Assume that R is generic in the sense that a_1<a_2<… < a_n.


Let Z_1,…,Z_k be a k-periodic symplectic billiard orbit with k<2n. If k is even, then the orbit is contained in a coordinate subspace of dimension at most k, and if k is odd, in a coordinate subspace of dimension at most k-1.


Let L be the vector space spanned by Z_j, j=1,…,k. The law of the symplectic billiard reflection implies that R(Z_j) is proportional to Z_j+1-Z_j-1 for all j=1,…,k (the indices are understood cyclically mod k). Therefore L is an invariant subspace of the linear map R.

We claim that L is a coordinate subspace. Let _1,…,_n be the complex coordinate lines, and let π_j be the projection of ^n on _j. 

Consider the projections π_j(L). If π_j(L)=0 for some j, then we may ignore the jth coordinate in what follows. In other words, assume that π_j(L)≠ 0 for all j. The claim now is that L is the whole space ^n.

Let Z=(1,z_2, …,z_n) ∈ L. Then a_1^4m R^4m(Z) ∈ L. In the limit m→∞, we obtain the basic vector (1,0,…,0), hence _1 ⊂ L. Factorize by _1 and repeat the argument. It implies that _1 + _2 ⊂ L, and so on. Thus L = ^n, as claimed. 

To finish the proof, note that a coordinate subspace is always even-dimensional.

Proposition <ref> is in perfect agreement with Proposition  7.16 of <cit.>: “if the billiard trajectory within an ellipsoid in d-dimensional Euclidean space is periodic with period n ≤ d, then it is placed in one of the (n - 1)-dimensional planes of symmetry of the ellipsoid".



  §.§.§ Round sphere
 

The case of a_1=a_2=… =a_n=1, that is, the case when M=S^2n-1 is the unit sphere, is special. In this section, we describe this case in further detail. 

Let z_0 z_1 be the initial segment of a billiard trajectory, z_0,z_1 ∈ S^2n-1. The symplectic billiard map Φ is given by the formula

    z_2 = z_0 + 2 ω (z_0,z_1) J(z_1),

where J is multiplication by i. The quantity ω (z_0,z_1) is an integral: ω (z_0,z_1) = ω (z_1,z_2), and we denote it simply by ω.

In the phase space  we have 0 ≤ω, and also ω≤ 1 because M is the unit sphere. Set ω = sinα for 0≤α≤π/2,
and let 

    λ_1 = e^iα, λ_2 = - e^-iα.


The case ω=0 is special: this is the boundary of the phase space, and the orbits are 2-periodic.
The case ω=1 is also special. In this case, the orbit of Φ is 4-periodic. Indeed, if ω (z_0,z_1)=1 then z_1=J(z_0) and hence one obtains the sequence of points 

    z_0    ↦    z_1= J(z_0)    ↦    z_2= -z_0    ↦    z_3=- J(z_0)    ↦    z_4=z_0    ↦   …

The general case is described in the next proposition, where we assume that ω < 1.

 
One has

    z_n = λ_1^n-1 - λ_2^n-1/λ_1 - λ_2 z_0 + 
    λ_1^n - λ_2^n/λ_1 - λ_2 z_1.

The Φ-orbit of a point lies on the union of two circles. The orbit is periodic if α is π-rational and dense on the two circles otherwise. If α= 2π (p/q), where p/q is in the lowest terms, then the period equals q for even q, and 2q for odd q.


Equation (<ref>) is a second order linear recurrence with constant coefficients that generates the sequence z_0,z_1,z_2,… Its solution is a linear combination of two geometric progressions whose denominators are the roots of the characteristic equation λ^2 - 2iωλ -1=0. These roots are distinct, and they are given by the formula

    λ_1,2 = i ω±√(1-ω^2),

coinciding with  (<ref>). Choosing the coefficients of the two geometric progressions to satisfy the initial conditions, one obtains formula (<ref>). 

The map Φ: (z_0,z_1) ↦ (z_1,z_2) is a complex linear self-map of ^2n, it has the eigen-values λ_1 and λ_2, each with multiplicity n. Writing (z_0,z_1) as a column vector, Φ has the matrix

    (
    
    0   E
    
    E     2iω E
    ),

where each entry is an n× n block.

Decompose ^2n into the direct sum of the eigen-spaces. Specifically, 

    (
    
    a 
    
    b  
    ) = 
     1/λ_2-λ_1(
    
    b - λ_1 a 
    λ_2 b + a  
    ) +
     1/λ_1-λ_2(
    
    b - λ_2 a 
    λ_1 b + a  
    ),

where a and b are column vectors in ^n.

Writing a vector accordingly as (u,v), one has

    Φ: (u,v) ↦ (e^iα u , - e^-iα v).
 
The orbit of (u,v) lies on the union of two circles (e^it u ,  e^-it v) and (e^it u , - e^-it v), where t∈. The orbit is finite if α is π-rational, and dense on the two circles otherwise. Let

    α = 2πp/q,

where p and q are coprime. Assume that v≠ 0. If q is even then the orbit closes up after q iterations, but if q is odd, one needs twice as many, due to the alternating sign of the second component in (<ref>). 




 §.§ Periodic orbits
 

Let M ⊂^2n be a smooth, strictly convex, closed hypersurface. In this section we discuss periodic trajectories of the symplectic billiard map in M. Given a k-periodic trajectory, one can cyclically permute its vertices or reverse their order; accordingly, we  count the orbits of this dihedral group D_k action.



  §.§.§ Existence of periodic orbits of any period
 

 
For every k≥ 2, the symplectic billiard map has a k-periodic trajectory.


If k is not prime, we do not exclude the case that this trajectory may be multiple, that is, a lower-periodic trajectory, traversed several times.


A periodic trajectory is a critical point of the symplectic area function F(z_1,…,z_k)=∑_i=1^k ω(z_i,z_i+1) on k-gons inscribed in M. Due to compactness, this function attains maximum. Let us show that the respective critical point is a genuine periodic trajectory of the symplectic billiard map.

Let z_1,…,z_k be a critical polygon for F(z_1,…,z_k)=∑_i=1^k ω(z_i,z_i+1). Then
ω(z_i-1,v)+ω(v,z_i+1) =0  for all v∈ T_z_i M, i=1,…,n,
that is, ω(v,z_i+1-z_i-1) =0. 
If z_i+1≠ z_i-1 then z_i+1-z_i-1∈ R(z_i), as the symplectic billiard map requires.
 A problem arises if z_i+1 = z_i-1.

Let us show that if z_i+1 = z_i-1 then the value of F(z_1,…,z_k) is not maximal. Indeed, in this case the two terms, ω(z_i-1,z_i) and ω(z_i,z_i+1) cancel each other, and the function F is the symplectic area of the (k-2)-gon z_1,…,z_i-1,z_i+2,…,z_k. Thus it suffices to show that the maximum of the symplectic area F is attained on non-degenerate k-gons (and not on polygons with fewer sides).

To do this we show that, given an inscribed polygon P, one can add to it one vertex  (and thus two vertices) so that the symplectic area  increases. Indeed, let ac be a side of P. We want to find a point b∈ M so that ω (a,b) + ω(b,c) > ω (a,c). This is equivalent to saying that the symplectic area of the triangle abc is positive. To achieve this, take an affine symplectic plane through ac and choose point b appropriately on its  intersection curve with M.




  §.§.§ Periods three and four
 

The result of Theorem <ref> is quite weak: we believe, the actual number of periodic orbits is much larger (see, e.g., <cit.> for the usual multi-dimensional billiards). The next theorem concerns small periods.

 
For every M as above, the number of 3-periodic symplectic billiard trajectories is not less than 2n.  The same lower bound holds for the number of 4-periodic trajectories.


The case k=3 is contained in <cit.> where 3-periodic trajectories of the outer billiard are studied. The function whose critical points are these 3-periodic trajectories is the same: it is the symplectic area of an inscribed triangle. This is also clear geometrically: if ABC is a 3-periodic trajectory of the outer billiard, then the midpoints of the sides of the triangle ABC form a 3-periodic trajectory of the symplectic billiard. 

Let us consider the case k=4. 

Let z_1,z_2,z_3,z_4 be a 4-periodic orbit. Then z_3-z_1 ∈ R(z_2) and z_3-z_1 ∈ R(z_4), and likewise, z_4-z_2 ∈ R(z_3) and z_4-z_2 ∈ R(z_1). It follows that R(z_2)=R(z_4) and R(z_1) = R(z_3). Using strict convexity of M, we conclude that z_4=z_2^* and z_3=z_1^*, where the involution z ↦ z^* is as before: the tangent hyperplanes to M at z and z^* are parallel.

The (oriented) chords z^* z are called affine diameters of M. The observation made in the preceding paragraph suggests to consider the following function of a pair of oriented affine diameters

    ω(z_1-z_1^*,z_2-z_2^*) = ω(z_1,z_2) + ω(z_2,z_1^*) + ω (z_1^*,z_2^*) + ω (z_2^*,z_1).

We shall show that the critical points of this function are 4-periodic orbits of the symplectic billiard. We start with a technical statement.

 
The map φ: M → S^2n-1 to the unit sphere, given by the formula

    z ↦z-z^*/|z-z^*|,

is a diffeomorphism. 


One has the following characterization of affine diameters: a chord of a convex body is its affine diameter if and only if it is a longest chord of the body in a given direction, see <cit.>. Since M is strictly convex, for every direction v ∈ S^2n-1, there is a unique affine diameter z z^*. The map v ↦ z is inverse of the map φ, and this map is smooth. Thus the smooth map φ has a smooth inverse map, that is, φ is a diffeomorphism.












Next we consider critical points of the function (<ref>).


 
The critical points of the function ω(z_1-z_1^*,z_2-z_2^*) are 4-periodic orbits of the symplectic billiard.


Let z∈ M and let v∈ T_z M be a tangent vector. Denote by v^* ∈ T_z^* M the image of v under the differential of the involution z ↦ z^*.

Let (z_1,z_2) be a critical point of the function ω(z_1-z_1^*,z_2-z_2^*). Then, for every v∈ T_z_1 M, one has ω(v-v^*,z_2-z_2^*)=0. 

Note that ψ, composed with normalization to unit vectors, is the map φ of Lemma <ref>, that is, a diffeomorphism. Hence the map dψ(z)v=v-v^*:T_zM→^2n is an injection.



It follows that the vector z_2-z_2^* is symplectically orthogonal to the hyperplane T_z_1 M, that is, z_2-z_2^* ∈ R(z_1)=R(z_1^*). The same argument shows that z_1-z_1^* ∈ R(z_2)=R(z_2^*). Hence z_1 z_2 z_1^* z_2^* is an orbit of the symplectic billiard in M.


Let 𝒟 be the set of pairs of oriented affine diameters of M. One has an action of the group _4 on this set:

    (D_1,D_2) ↦ (D_2,-D_1) ↦ (-D_1,-D_2) ↦ (-D_2, D_1) ↦ (D_1,D_2).

Let F(D_1,D_2) be the function (<ref>). This function is _4-invariant. 

Let U ⊂𝒟 be the manifold with boundary given by the inequality F(D_1,D_2) ≥ for a sufficiently small generic positive . The gradient of the function F has the inward direction on the boundary of U, therefore the usual Morse-Lusternik-Schnirelman inequalities for the number of critical points apply. 

Note that if (D_1,D_2) ∈ U, then D_1 ≠± D_2, and the action of _4 on U is free. 
We need to describe  the topology of the quotient space U/_4. We claim that it is homotopically equivalent to the lens space L = S^2n-1/_4, where _4 acts on the unit sphere in ^n by
(z_1,…,z_n) ↦ (iz_1,…,iz_n).

Let V be the set of pairs of unit vectors (e_1,e_2) with ω(e_1,e_2) >0. Using Lemma <ref>, we normalize D_1 and D_2 to unit vectors. Thus, at the first step, we get a _4-equivariant retraction of U to V.

Next we want to retract V to the set of complex 2-frames (e_1,e_2) satisfying e_2=J e_1. To this end, consider the function ω(e_1,e_2) on V. 

We claim that the critical points of this function are complex  frames with e_2= J e_1. Indeed, let (e_1,e_2) be a critical point. Then, for every v ∈ T_e_1 S^2n-1, one has ω(v,e_2) =0. Hence e_2 = ± J e_1. The condition ω(e_1,e_2) > 0 excludes the minus sign. 

Thus the gradient of the function ω(e_1,e_2) retracts V   to the set of complex 2-frames. This set is S^2n-1 with the action of _4 given by J.

It remains to use the Lusternik-Schnirelman lower bound for the number of critical points given by the category of the lens space L. It is known that cat(L)=2n (Kransnoselski  <cit.>; alternatively, one can use the 2-fold covering ^2n-1→ L that implies
cat(L) ≥ cat (^2n-1) = 2n,
where the inequality follows from the homotopy lifting property, see <cit.>). 
This completes the proof of the theorem.

99

ABHS A. Abbondandolo, B. Bramham, U. Hryniewicz, P. Salomão. A systolic inequality for geodesic flows on the two-sphere. Math. Ann. 367 (2017), 701–753.

AKO S. Artstein-Avidan, R. Karasev, Y. Ostrover. From symplectic measurements to the Mahler conjecture. Duke Math. J. 163 (2014), 2003–2022.

BZ Yu. Baryshnikov, V. Zharnitsky. Sub-Riemannian geometry and periodic orbits in classical billiards. Math. Res. Lett. 13 (2006), 587–598.

Bia2 M. Bialy. Effective bounds in E. Hopf rigidity for billiards and geodesic flows. Comment. Math. Helv. 90 (2015), 139–153.

BKNZ V. Blumen, K. Y. Kim, J. Nance, V. Zharnitsky.
Three-period orbits in billiards on the surfaces of constant curvature.
Int. Math. Res. Not. IMRN 2012, no. 21, 5014–5024. 

DG T. Dauer, M. Gerber. Generic absence of finite blocking for interior points of Birkhoff billiards. Discrete Contin. Dyn. Syst. 36 (2016), 4871–4893.

DT F. Dogru, S. Tabachnikov. Dual billiards. Math. Intelligencer 27 (2005), no. 4, 18–25. 

DR V. Dragović, M. Radnović.  Poncelet porisms and beyond. 
Integrable billiards, hyperelliptic Jacobians and pencils of quadrics. Birkhäuser/Springer Basel AG, Basel, 2011.

FT M. Farber, S. Tabachnikov. Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards. Topology 41 (2002),  553–589. 



GT D. Genin, S. Tabachnikov. On configuration space of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. J. Modern Dynamics 1 (2007), 155–173.

GH P. Giblin, P. Holtom. The centre symmetry set. Geometry and topology of caustics–CAUSTICS '98 (Warsaw), 91–105, Banach Center Publ. 50, Warsaw, 1999. 

GlK A. Glutsyuk, Yu. Kudryashov. No planar billiard possesses an open set of quadrilateral trajectories.  J. Mod. Dyn. 6 (2012),  287–326. 

Go C. Golé. Symplectic twist maps.  Global variational techniques. World Scientific Publ. Co., Inc., River Edge, NJ, 2001.

Gug H. Guggenheimer. Differential geometry. McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963

GK E. Gutkin, A. Katok. Caustics for inner and outer billiards. Comm. Math. Phys. 173 (1995),  101–133. 

HS P. Hubert, T.  Schmidt. An introduction to Veech surfaces. Handbook of dynamical systems. Vol. 1B, 501–526, Elsevier B. V., Amsterdam, 2006. 

Ja I. M.James. On category, in the sense of Lusternik-Schnirelmann. Topology 17 (1978),  331–348. 

KZ V. Kaloshin, K. Zhang. Density of convex billiards with rational caustics. arXiv:1706.07968v1. 

KH A. Katok, B. Hasselblatt. Introduction to the modern theory of dynamical systems. Cambridge Univ. Press, Cambridge, 1995.

Kr M. Krasnoselski. On special coverings of a finite-dimensional sphere.
Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 961–964. 

La J. Landsberg. Exterior differential systems and billiards, Geometry, integrability and quantization, 35–54, Softex, Sofia, 2006.

Laz V. Lazutkin. The existence of caustics for a billiard problem in a convex domain.  Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 186–216. 

LSZZ A.-M. Li, U. Simon, G. Zhao, Z. Hu. Global affine differential geometry of hypersurfaces. Second revised and extended edition. De Gruyter, Berlin, 2015.

Lud M. Ludwig. Asymptotic approximation of convex curves. Arch. Math. (Basel) 63 (1994), 377–384.

Lut E. Lutwak. Selected affine isoperimetric inequalities. Handbook of convex geometry, Vol. A, B, 151–176, North-Holland, Amsterdam, 1993.

MaMa H. Maehara, H. Martini. On Lexell's theorem. Amer. Math. Monthly 124 (2017), 337–344. 

MS H. Martini, K. Swanepoel.  Antinorms and Radon curves.
Aequationes Math. 72 (2006), 110–138. 

MM S. Marvizi and R. Melrose. Spectral invariants of convex planar regions. J. Diff. Geom. 17 (1982), 475–502.

MT H. Masur, S. Tabachnikov. 
Rational billiards and flat structures. Handbook of dynamical systems, Vol. 1A, 1015–1089, North-Holland, Amsterdam, 2002. 

Ma J. Mather. Glancing billiards.  Ergodic Theory Dynam. Systems 2 (1982), 397–403. 

MF J. Mather, G. Forni, Giovanni. Action minimizing orbits in Hamiltonian systems. Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), 92–186, 
Lecture Notes Math., 1589, Springer, Berlin, 1994. 

McV D. McClure, R. Vitale. Polygonal approximation of plane convex bodies. 
J. Math. Anal. Appl. 51 (1975), 326–358. 

Me R. Melrose. Equivalence of glancing hypersurfaces. Invent. Math. 37 (1976), 165–192.

MV J. Moser, A. Veselov. Discrete versions of some classical integrable systems and factorization of matrix polynomials. Comm. Math. Phys. 139 (1991), 217–243.

PaSu A. Papadopoulos, W. Su. On hyperbolic analogues of some classical theorems in spherical geometry. arXiv:1409.4742v2. 

PS V. Petkov, L. Stoyanov. Geometry of reflecting rays and inverse spectral problems.  John Wiley & Sons, Ltd., Chichester, 1992.

Ry M. Rychlik. Periodic points of the billiard ball map in a convex domain. J. Diff. Geom. 30
(1989), 191–205.

Sa L. Santaló. Integral geometry and geometric probability. Addison-Wesley Publ. Co., Reading, Mass.-London-Amsterdam, 1976. 

Sc R. Schwartz. The octogonal PETs.  Amer. Math. Soc., Providence, RI, 2014. 

Si K. F. Siburg. The principle of least action in geometry and dynamics. Lecture Notes  Math., 1844. Springer-Verlag, Berlin, 2004.

So V. Soltan. Affine diameters of convex-bodies – a survey. Expo. Math. 23 (2005), 47–63. 

St L. Stojanov Note on the periodic points of the billiard. J. Diff. Geom. 34 (1991), 835–837.



Ta1 S. Tabachnikov. On the dual billiard problem. Adv. Math. 115 (1995),  221–249. 

Ta2 S. Tabachnikov. On three-periodic trajectories of multi-dimensional dual billiards. Algebr. Geom. Topol. 3 (2003), 993–1004.

Ta S. Tabachnikov. Geometry and billiards. Amer. Math. Soc., Providence, RI, 2005.

Ta3 S. Tabachnikov. Birkhoff billiards are insecure. Discrete Contin. Dyn. Syst. 23 (2009),  1035–1040. 

TZ A. Tumanov,  V. Zharnitsky. Periodic orbits in outer billiard. Int. Math. Res. Not. 2006, Art. ID 67089, 17 pp.

Tu A. Tumanov. Scarcity of periodic orbits in outer billiards. arXiv:1706.03882v1. 

Ve1 A. Veselov. Integrable systems with discrete time, and difference operators. Funct. Anal. Appl. 22 (1988),  83–93 .

Ve2 A. Veselov. Integrable mappings. Russian Math. Surveys 46 (1991), 1–51. 

Viterbo C. Viterbo. Metric and isoperimetric problems in symplectic geometry. J. Amer. Math. Soc. 13  (2000), 411–431.

Vo Ya. Vorobets. On the measure of the set of periodic points of a billiard. Math. Notes 55 (1994),
455–460.

Wo M. Wojtkowski. Two applications of Jacobi fields to the billiard ball problem. J. Diff. Geom. 40
(1994), 155–164.



