On the local Birkhoff Conjecture for convex billiards

On the local Birkhoff Conjecture for convex billiards

Vadim Kaloshin Department of Mathematics, University of Maryland, College Park, MD, USA, & ETH Zurich, Institute for Theoretical Studies, Zurich Switzerland vadim.kaloshin@gmail.com  and  Alfonso Sorrentino Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Rome, Italy. sorrentino@mat.uniroma2.it
July 17, 2019
Abstract.

The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in [3], where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.

Dedicated to the memory of our thesis advisor John N. Mather:

a great mathematician and a remarkable person

1. Introduction

A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary (which is assumed to have infinite mass). This simple model has been first proposed by G.D. Birkhoff as a mathematical playground where “the formal side, usually so formidable in dynamics, almost completely disappears and only the interesting qualitative questions need to be considered”, [7, pp. 155-156].

Since then billiards have captured much attention in many different contexts, becoming a very popular subject of investigation. Not only is their law of motion very physical and intuitive, but billiard-type dynamics is ubiquitous. Mathematically, they offer models in every subclass of dynamical systems (integrable, regular, chaotic, etc.); more importantly, techniques initially devised for billiards have often been applied and adapted to other systems, becoming standard tools and having ripple effects beyond the field.

Let us first recall some properties of the billiard map. We refer to [45, 48, 49] for a more comprehensive introduction to the study of billiards.

Let be a strictly convex domain in with boundary , with . The phase space of the billiard map consists of unit vectors whose foot points are on and which have inward directions. The billiard ball map takes to , where represents the point where the trajectory starting at with velocity hits the boundary again, and is the reflected velocity, according to the standard reflection law: angle of incidence is equal to the angle of reflection (figure 1).

Remark 1.

Observe that if is not convex, then the billiard map is not continuous; in this article we will be interested only in strictly convex domains (see Remark 4). Moreover, as pointed out by Halpern [19], if the boundary is not at least , then the flow might not be complete.

Let us introduce coordinates on . We suppose that is parametrized by arc-length and let denote such a parametrization, where denotes the length of . Let be the angle between and the positive tangent to at . Hence, can be identified with the annulus and the billiard map can be described as

Figure 1.

In particular can be extended to by fixing and for all . Let us denote by

the Euclidean distance between two points on . It is easy to prove that

(1)
Remark 2.

If we lift everything to the universal cover and introduce new coordinates , then the billiard map is a twist map with as generating function and it preserves the area form . See [45, 48, 49].

Despite the apparently simple (local) dynamics, the qualitative dynamical properties of billiard maps are extremely non-local. This global influence on the dynamics translates into several intriguing rigidity phenomena, which are at the basis of several unanswered questions and conjectures (see, for example, [3, 5, 12, 18, 21, 22, 41, 40, 42, 45, 46, 48, 49, 51]). Amongst many, in this article we will address the question of classifying integrable billiards, also known as Birkhoff conjecture. As an application of our main result, in subsection 1.2 we will also discuss certain spectral rigidity properties of ellipses.

1.1. Integrable billiards and Birkhoff conjecture

The easiest example of billiard is given by a billiard in a disc (for example of radius ). It is easy to check in this case that the angle of reflection remains constant at each reflection (see also [49, Chapter 2]). If we denote by the arc-length parameter (i.e., ) and by the angle of reflection, then the billiard map has a very simple form:

In particular, stays constant along the orbit and it represents an integral of motion for the map. Moreover, this billiard enjoys the peculiar property of having the phase space – which is topologically a cylinder – completely foliated by homotopically non-trivial invariant curves . These curves correspond to concentric circles of radii and are examples of what are called caustics, i.e., (smooth and convex) curves with the property that if a trajectory is tangent to one of them, then it will remain tangent after each reflection (see figure 2).

Figure 2. Billiard in a disc

A billiard in a disc is an example of an integrable billiard. There are different ways to define global/local integrability for billiards (the equivalence of these notions is an interesting problem itself):

  • either through the existence of an integral of motion, globally or locally near the boundary (in the circular case an integral of motion is given by ),

  • or through the existence of a (smooth) foliation of the whole phase space (or locally in a neighborhood of the boundary ), consisting of invariant curves of the billiard map; for example, in the circular case these are given by . This property translates (under suitable assumptions) into the existence of a (smooth) family of caustics, globally or locally near the boundary (in the circular case, the concentric circles of radii ).

In [5], Misha Bialy proved the following result concerning global integrability (see also [52]):

Theorem (Bialy). If the phase space of the billiard ball map is globally foliated by continuous invariant curves which are not null-homotopic, then it is a circular billiard.

However, while circular billiards are the only examples of global integrable billiards, integrability itself is still an intriguing open question. One could consider a billiard in an ellipse: this is in fact integrable (see Section 2). Yet, the dynamical picture is very distinct from the circular case: as it is showed in figure 3, each trajectory which does not pass through a focal point, is always tangent to precisely one confocal conic section, either a confocal ellipse or the two branches of a confocal hyperbola (see for example [49, Chapter 4]). Thus, the confocal ellipses inside an elliptical billiards are convex caustics, but they do not foliate the whole domain: the segment between the two foci is left out (describing the dynamics explicitly is much more complicated: see for example [50] and Section 2).

Figure 3. Billiard in an ellipse

Question (Birkhoff). Are there other examples of integrable billiards?

Remark 3.

Although some vague indications of this question can be found in [7], to the best of our knowledge, its first appearance as a conjecture was in a paper by Poritsky [41], where the author attributes it to Birkhoff himself111Poritsky was Birkhoff’s doctoral student and [41] was published several years after Birkhoff’s death.. Thereafter, references to this conjecture (either as Birkhoff conjecture or Birkhoff-Poritsky conjecture) repeatedly appeared in the literature: see, for example, Gutkin [18, Section 1], Moser [33, Appendix A], Tabachnikov [48, Section 2.4], etc.

Remark 4.

In [29] Mather proved the non-existence of caustics (hence, the non-integrability) if the curvature of the boundary vanishes at one point. This observation justifies the restriction of our attention to strictly convex domains.

Remark 5.

i) Interestingly, Treschev in [51] gives indication that there might exist analytic billiards, different from ellipses, for which the dynamics in a neighborhood of the elliptic period- orbit is conjugate to a rigid rotation. These billiards can be seen as an instance of local integrability; however, this regime is somehow complementary to the one conjectured by Birkhoff. Here one has local integrabilility in a neighborhood of an elliptic periodic orbit of period , while Birkhoff conjecture is related to integrability in a neighborhood of the boundary. This gives an indication that these two notions of integrability do differ.
ii) An algebraic version of this conjecture states that the only billiards admitting polynomial (in the velocity) integrals are circles and ellipses. For recent results in this direction, see [6].

Despite its long history and the amount of attention that this conjecture has captured, it remains still open. As far as our understanding of integrable billiards is concerned, the most important related results are the above–mentioned theorem by Bialy [5] (see also [52]), a result by Innami222We are grateful to M. Bialy for pointing out this reference. [23], in which he shows that the existence of caustics with rotation numbers accumulating to implies that the billiard must be an ellipse333This regime of integrability is somehow diametrically opposed to ours, since we are interested in integrability near the boundary of the billiard domain., a result by Delshams and Ramírez-Ros [11] in which they study entire perturbations of elliptic billiards and prove that any nontrivial symmetric perturbation of the elliptic billiard is not integrable, near homoclinic solutions, and a very recent result by Avila, De Simoi and Kaloshin [3] in which they show a perturbative version of this conjecture for ellipses of small eccentricity.

Let us introduce an important notion for this paper.

Definition 6.

(i) We say is an integrable rational caustic for the billiard map in , if the corresponding (non-contractible) invariant curve consists of periodic points; in particular, the corresponding rotation number is rational.
(ii) If the billiard map inside admits integrable rational caustics of rotation number for all , we say that is rationally integrable.

Remark 7.

A simple sufficient condition for rational integrability is the following (see [3, Lemma 1]). Let denote the union of all smooth convex caustics of the billiard in ; if the interior of contains caustics of rotation number for any , then is rationally integrable.

Our main result is the following.

Main Theorem (Local Birkhoff Conjecture).

Let be an ellipse of eccentricity and semi-focal distance ; let . For every , there exists such that the following holds: if is a rationally integrable -smooth domain so that is --close and --close to , then is an ellipse.

Remark 8.

One could replace the smallness condition in the -norm with a smallness condition with respect to the -topology (this can be showed by using interpolation inequalities and the convexity of the domains)444This remark was suggested to the authors by Camillo De Lellis..

Remark 9.

In [21] we prove a similar rigidity statement for a different type of rational integrability. Namely, we describe an algorithm to prove that for any given there exists such that every sufficiently smooth perturbation of , with , having integrable rational caustics of rotation numbers , for all must be an ellipse. This algorithm is conditional on checking the invertibility of finitely many explicit matrices, which we prove in the cases . Observe that the analysis in [21] only applies to ellipses of small eccentricity as in [3], since Taylor expansions with respect to are needed in order to get higher order (integrability) conditions.

One of the crucial ideas to extend the analysis beyond the almost circular case in [3], is to consider analytic extensions of the action-angle coordinates of the elliptic billiard (more specifically, of the boundary parametrizations induced by each integrable caustic) and to study their singularities (see Section 7). These functions can be explicited expressed in terms of elliptic integrals and Jacobi elliptic functions (see subsection 3.1). This analysis will be exploited to define a dynamically-adapted basis for , which will provide the main framework to carry out our analysis. See subsection 4.2 for a more detailed description of the scheme of the proof.

In addition to this, in Appendix F we propose a possible strategy to use the affine length shortening (ALS) flow (see, for instance, [43]) as a potential approach to prove the global Birkhoff conjecture. Our proposal is based on the fact that the ALS flow evolves any convex domain with smooth boundary into an ellipse in finite time.

1.2. Applications for spectral rigidity of ellipses

In this subsection we describe an interesting application of our Main Theorem to spectral rigidity properties of ellipses555This was suggested to the authors by Hamid Hezari..

Let be a smooth strictly convex (planar) domain. While the dependence of the dynamics on the geometry of the domain is well perceptible, an intriguing challenge is to understand to which extent dynamical information can be used to reconstruct the shape of the domain. A particular interesting problem in this direction is to unravel which information on the geometry of the billiard domain, the set of periodic orbits does encode. More ambitiously, one could wonder whether a complete knowledge of this set allows one to reconstruct the shape of the billiard and hence the whole of its dynamics. Several results in this direction (and in related ones) are contained, for instance, in [4, 12, 16, 20, 21, 27, 28, 39, 40, 45, 46, 53].

Let us start by introducing the Length Spectrum of a domain .

Definition 10 (Length Spectrum).

Given a domain , the length spectrum of is given by the set of lengths of its periodic orbits, counted with multiplicity:

where denotes the length of the boundary of .

A remarkable relation exists between the length spectrum of a billiard in a convex domain and the spectrum of the Laplace operator in with Dirichlet boundary conditions (similarly for Neumann boundary conditions):

(2)

From the physical point of view, the eigenvalues ’s are the eigenfrequencies of the membrane with a fixed boundary. K. Andersson and R. Melrose [2] proved the following relation between the Laplace spectrum and the length spectrum. Call the function

the wave trace. Then, the wave trace is a well-defined generalized function (distribution) of , smooth away from the length spectrum, namely,

(3)

So if belongs to the singular support of this distribution, then there exists either a closed billiard trajectory of length , or a closed geodesic of length in the boundary of the billiard table.

Generically, equality holds in (3). More precisely, if no two distinct orbits have the same length and the Poincaré map of any periodic orbit is non-degenerate, then the singular support of the wave trace coincides with (see e.g. [39]). This theorem implies that, at least for generic domains, one can recover the length spectrum from the Laplace one.

This relation between periodic orbits and spectral properties of the domain, immediately recalls a more famous spectral problem (probably the most famous): Can one hear the shape of a drum?, as formulated in a very suggestive way by Mark Kac [24] (although the problem had been already stated by Hermann Weyl). More precisely: is it possible to infer information about the shape of a drumhead (i.e., a domain) from the sound it makes (i.e., the list of basic harmonics/ eigenvalues of the Laplace operator with Dirichlet or Neumann boundary conditions)? This question has not been completely solved yet: there are several negative answers (for instance by Milnor [32] and Gordon, Webb, and Wolpert [14]), as well as some positive ones.

Hezari and Zelditch, going in the affirmative direction, proved in [20] that, given an ellipse , any one-parameter -deformation which preserves the Laplace spectrum (with respect to either Dirichlet or Neumann boundary conditions) and the symmetry group of the ellipse has to be flat (i.e., all derivatives have to vanish for ). Popov–Topalov [40] recently extended these results (see also [53]). Further historical remarks on the inverse spectral problem can be also found in [20]. In [35, 36, 37] Osgood, Phillips and Sarnak showed that isospectral sets are necessarily compact in the topology in the space of domains with boundary. In [44] Sarnak conjectures that the set of smooth convex domains isospectral to a given smooth convex domain is finite (for a partial progress on this question, see [12]).

One of the difficulties in working with the length spectrum is that all of these information on the periodic orbits come in a non-formatted way. For example, we lose track of the rotation number corresponding to each length. A way to overcome this difficulty is to “organize” this set of information in a more systematic way, for instance by associating to each length the corresponding rotation number. This new set is called the Marked Length Spectrum of and denoted by :

where denotes the rotation number of .666 In the case of negatively curved surfaces without boundary the marked length spectrum consists of pairs of homotopy classes and length of the shortest geodesic in that homotopy class. Guillemin and Kazhdan [17] proved local rigidity with respect to this marked length spectrum. Global version of this result was obtained by Otal [34] and Croke [9].

One could also refine this set of information by considering not the lengths of all orbits, but selecting some of them. More precisely, for each rotation number in lowest terms, one could consider the maximal length among those having rotation number . We call this map the Maximal Marked Length Spectrum of , namely given by:

Remark 11.

The maximal marked length spectrum is closely related to Mather’s minimal average action (or Mather’s -function) of the associated billiard map in the domain, as it was pointed out in [45]. Briefly speaking, this function – which can be defined for any exact area preserving twist map, not necessarily a billiard map – associates to any fixed rotation number (not only rational ones) the minimal average action of orbits with that rotation number (whose existence, inside a suitable interval, is ensured by the twist condition). These action-minimizing orbits are of particular interest from a dynamical point of view and play a key-role in what is nowadays called Aubry-Mather theory; we refer the reader to [4, 31, 45, 47] for a presentation of this topic.
In the case of billiard maps, since the action coincides (up to a negative sign) with the euclidean length of the segment joining two subsequent rebounds, we have that the minimal average action of periodic orbits can be expressed in terms of the maximal marked length spectrum; namely:

(4)

In particular, this object encodes many interesting dynamical information on the billiard map. For example, using the result in [30], one can deduce that is differentiable at if and only if there exists a rational caustic of rotation number . See [45] for a detailed presentation of this and many other properties.

Let us now address the following question.

Question. Let and be two strictly convex planar domains with smooth boundaries and assume that they have the same maximal marked Length spectrum, namely (or equivalently, ). Is it true that and are isometric?

Remark 12.

It is known that if has the same marked length spectrum of a disc, then it is indeed a disc; for a proof of this result, see for example [45, Corollary 3.2.17]. Another proof can be obtained by looking only at the Taylor coefficients of the -function at (which are related to the so-called Marvizi-Melrose invariants); it turns out that the first and the third order coefficients always satisfy an inequality, which becomes an equality if and only if the domain is a disc (see [27, Section 8] and [46, Corollary 1]).

It would be interesting to find a similar characterization for elliptic billiards, namely that the maximal marked length spectrum (resp., the -function) univocally determines ellipses amongst all possible Birkhoff billiards.

In [46, Proposition 1], by looking at the Taylor expansion of the -function at (actually, only at the first and third order coefficients), it was pointed out a much weaker result, namely that the isospectrality condition determines univocally a given ellipse within the family of ellipses (up to rigid motions, i.e., the composition of a translation and a rotation)).

From our Main Theorem, we can now deduce the following spectral rigidity results for ellipses.

Corollary 13 (Local length–spectral uniqueness of ellipses).

Let be a smooth strictly convex domain sufficiently close to an ellipse.

  • If has the same maximal marked length spectrum (or Mather’s -function) of an ellipse, then it is an ellipse.

  • If its Mather’s -function is differentiable at all rationals with , then is an ellipse.

Moreover, the following spectral rigidity result holds.

Corollary 14 (Spectral rigidity of ellipses).
  • Ellipses are (maximal) marked-length-spectrally rigid, meaning that if is a smooth deformation of an ellipse which keeps fixed the (maximal) marked length spectrum, then it consists of a rigid motion.

  • Ellipses are length-spectrally rigid, meaning that if is a smooth deformation of an ellipse which keeps fixed the length spectrum, then it consists of a rigid motion.

Proof.

(Corollary 13) Assertion i) follows from assertion ii), using (4) and recalling that the -function of an ellipse is differentiable in , since the corresponding billiard map is integrable. As for the proof of ii), it follows from the differentiability assumptions on and from what recalled at the end of Remark 11 (see also [30, 45]), that there exist integrable rational caustics for all rotation number for any . Hence our billiard is rationally integrable (see Definition 6). Applying the Main Theorem, since is close to an ellipse, then it must be an ellipse. ∎

Proof.

(Corollary 14) Assertion i) follows from Corollary 13 ii) and the fact that the function (equivalently, the maximal marked length spectrum) univocally determines a given ellipse within the family of ellipses (up to rigid motions); see [46, Proposition 1].
To prove assertion ii), one needs to use [45, Proposition 3.2.2], which shows that a iso-length spectral deformation is necessarily an iso-marked length spectral deformation. Then, the claim follows by applying i). ∎

1.3. Organization of the article

For the reader’s convenience, here follows a brief description of how the article is organized.

In Section 2 we describe our setting and introduce elliptic coordinates (see subsection 2.1), while in Section 3 we recall some definitions and some needed properties of elliptic integrals and elliptic functions (see subsection 3.1) and use them to provide a more precise description of the billiard dynamics inside an ellipse (see subsection 3.2).

In Section 4 we outiline the scheme of the proof of our Main Theorem, both for perturbations of circular billiards (see subsection 4.1) and for perturbations of general elliptic ones (see subsection 4.2); we refer to this latter subsection for a detailed description of the contents of Sections 38.

In order to make the presentation clearer and easier to follow, we deferred several proofs of technical claims and some complementary material to Appendices AE. Finally, in Appendix F we outline a possible strategy to approach the global Birkhoff conjecture, by means of the affine length shortening flow.

1.4. Acknowledgements.

VK acknowledges partial support of the NSF grant DMS-1402164 and the hospitality of the ETH Institute for Theoretical Studies and the support of Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation. AS acknowledges the partial support of the Italian MIUR research grant: PRIN- 2012-74FYK7 “Variational and perturbative aspects of nonlinear differential problems”. VK is grateful to Jacopo De Simoi and Guan Huang for useful discussions. AS would like to thank Pau Martin and Rafael Ramírez-Ros for useful discussions during his stay at UPC. The authors are also indebted to Hamid Hezari, whose valuable remarks led to Corollaries 13 and 14. Finally, the authors wish to express their sincere gratitude to a referee for really careful reading of the paper and many useful suggestions, which led to significant improvements of the exposition and clarity of the proof.

2. Notation and Setting

Let us consider the ellipse

centered at the origin and with semiaxes of lenghts, respectively, ; in particular denotes its eccentricity, given by and the semi-focal distance. Observe that when , then and degenerates to a -parameter family of circles centered at the origin.

The family of confocal elliptic caustics in is given by (see also figure 3):

(5)

Observe that the boundary itself corresponds to , while the limit case corresponds to the the two foci . Clearly, for we recover the family of concentric circles described in Figure 2.

2.1. Elliptic polar coordinates

A more convenient coordinate frame for addressing this question is provided by the so-called elliptic polar coordinates (or, simply, elliptic coordinates) , given by:

where represents the semi-focal distance (in the case , this parametrization degenerates to the usual polar coordinates). Observe that for each , the equation represents a confocal ellipse, while for each the equation corresponds to one of the two branches of a confocal hyperbola; these grid-lines are mutually orthogonal. Moreover, the degenerate cases and describe, respectively, the (cartesian) segment , and the (cartesian) half-lines , , and .

Therefore, in these elliptic polar coordinates becomes:

where (the dependence on is in the definition of the coordinate frame).

Let us denote by the set of ellipses in with circles being degenerate points. This is a -dimensional family of strictly convex curves parametrized, for example, by the cartesian coordinates of its centre , the semi-focal distance , the parameter corresponding to the eccentricity, and the angle between the major semiaxis and the -axis (notice that is not well defined for circles). More specifically, for each we associate the (parametrized) ellipse

(6)

In the following we will use the shorthand for . In particular, consists of a 1-parameter family of circles centered at the origin.

3. Action-angle coordinate of elliptic billiards

Here we define and study action-angle coordinates for elliptic billiards.

3.1. Elliptic integrals and Jacobi elliptic functions

Let us recall some basic definitions on elliptic integrals and elliptic functions that will be used in the following; we refer the reader, for instance, to [1] for a more comprehensive presentation.

Let . We define the following elliptic integrals.

  • Incomplete elliptic integral of the first kind:

    In particular, is called the modulus and the amplitude. Moreover, the quantity is often called the complementary modulus. Observe that for we have ; on the other hand, has a pole at .

  • Complete elliptic integral of the first kind:

Let us recall that an elliptic function is a doubly-periodic meromorphic function, i.e., it is periodic in two directions and hence, it is determined by its values on a fundamental parallelogram. Of course, a non-constant elliptic function cannot be holomorphic, as it would be a bounded entire function, and by Liouville’s theorem it would be constant. In particular, elliptic functions must have at least two poles in a fundamental parallelogram (counting multiplicities); it is easy to check, using the periodicity, that a contour integral around its boundary must vanish, implying that the residues of all simple poles must cancel out.

Jacobi Elliptic functions are obtained by inverting incomplete elliptic integrals of the first kind. More specifically, let

(7)

( is often called the argument). If and are related as above (we can also write , called the amplitude of ) then we define the Jacobi elliptic functions as:

Remark 15.

These two elliptic functions have periods (in the real direction) and (in the imaginary direction). Moreover, they have two simple poles: at , with residue, respectively, and , and at with residue, respectively, and .

3.2. Elliptic billiard dynamics and caustics

Now we want to provide a more precise description of the billiard dynamics in .

The following result has been proven in [8] (see also [10, Lemma 2.1]).

Proposition 16.

Let and let

Let us denote, in cartesian coordinates, . Then, for every the segment joining and is tangent to the caustic .

Observe that:

  • is a strictly increasing function of ; in particular as , while as . Observe that represents the eccentricity of the ellipse .

  • is also a strictly increasing function of ; in fact, is clearly strictly increasing in both and . Moreover, as , and as .

Remark 17.

Using elliptic polar coordinate, one can easily check that and therefore

which is exactly the eccentricity of the confocal ellipse of parameter .


Let us now consider the parametrization of the boundary induced by the dynamics on the caustic :

We define the rotation number associated to the caustic to be

(8)

In particular is strictly increasing as a function of and as , while as .

It is easy to deduce from the above expressions that, in elliptic coordinates , the boundary parametrization induced by the caustic is given by

(9)

More precisely, the orbit starting at and tangent to , hits the boundary at .

4. Outline of the proof

In this section we provide a description of the strategy that we will follow to prove our Main Theorem.

4.1. A scheme for proving Main Theorem for circular billiards.

For small eccentricities Main Theorem was proven in [3] and we now describe the proof therein. Let us start with the simplified setting of integrable infinitesimal deformations of a circle. This provides an insight into the strategy of the proof in the general case.

Let be a circle centered at the origin and radius . Let be a one-parameter family of deformations given in the polar coordinates by

Consider the Fourier expansion of :

Theorem 18 (Ramírez-Ros [42]).

If has an integrable rational caustic of rotation number , for any sufficiently small, then we have for any integer .

Let us now assume that the domains are -rationally integrable for all sufficiently small and ignore for a moment dependence of parametrisation: then the above theorem implies that for , i.e.,

where and are appropriately chosen phases.

Remark 19.

Observe that

  • corresponds to an homothety;

  • corresponds to a translation in the direction forming an angle with the polar axis ;

  • corresponds to a deformation of the circle into an ellipse of small eccentricity, whose major axis forms an angle with the polar axis.

This implies that, infinitesimally (as ), rationally integrable deformations of a circle are tangent to the -parameter family of ellipses.

Notice that, in the above strategy, one needs to take as . This means that we cannot take small, but only infinitesimal; hence one cannot use directly the above theorem to prove the main result. A more elaborate strategy is needed.

4.2. Our scheme of the proof of Main Theorem for elliptic billiards

One of the noteworthy contributions of this paper is the analysis of perturbations of ellipses of arbitrary eccentricity . Let us outline the main steps involved in the proof.

Let be an ellipse of eccentricity and semifocal distance , and let be the associated elliptic coordinates. Any domain close to can be written (in the elliptic coordinates associated to ) in the form

where is a smooth -periodic function (see also (11)). Recall that the ellipse admits all integrable rational caustics of rotation number for .

By analogy with [3] we proceed as follows:

Step 1 (Dynamical modes): In Section 5, we consider the one-parameter integrable deformation of an ellipse , given by the family of rationally integrable domains , whose boundaries are given, using the elliptic coordinates associated to , by

In Lemma 21 we show that if for all , has an integrable rational caustic of rotation number , with , then

(10)

where denotes the standard inner product in and are suitable dynamical modes, which can be explicitly defined using the action-angle coordinates; see (15). See also Remark 22 for a more quantitative version, that we need since we are interested in perturbations of ellipses and not necessarily deformations.

Step 2 (Elliptic motions): In Section 6 we consider infinitesimal deformations of ellipses by homotheties, translations, rotations and hyperbolic rotations (we call them elliptic motions since they preserve the class of ellipses) and derive their infinitesimal generators , , , and ; see (16)–(20). Moreover, in Proposition 23 we prove a certain approximation result for ellipses.

Step 3 (Basis property): In Section 7 we show that the collection of dynamical modes and elliptic motions form a basis of . In subsections 7.1 and 7.2 we will consider their complex extensions and study in details their singularities; this analysis will be important to deduce their linear independence (Proposition 28). Moreover, in Proposition 33 we show that they do generate the whole , hence they form a (non-orthogonal) basis.

Step 4 (Approximation): In Section 8 we prove an approximation lemma (Lemma 34) and use it to complete the proof of Main Theorem (see subsection 8.1), by means of an approximation procedure similar to the one in [3, Section 8].

5. Preservation of rational caustics

In this section we want to investigate perturbations of ellipses, for which the associated billiard map continues to admit rationally integrable caustics corresponding to some rational rotation numbers.

Let us consider an ellipse and let be an infinitesimal perturbation of the form

(11)

for . To simplify notation we write

which must be understood in the elliptic coordinates with semi-focal distance .

Let us denote and let be the generating function of the billiard map inside ; in particular,

(12)

where denotes the generating function of the billiard map inside and denotes a term bounded by times a factor depending on and . Notice that this formula makes sense only for infinitesimal perturbations.

Let us recall the following result (see [38, Corollary 9 and Proposition 11]).

Proposition 20.

Assume that the billiard map associated to has a rationally integrable caustic corresponding to rotation number, in lowest term, .
If we denote by the periodic orbit of the billiard map in with rotation number and starting at
(these orbits are all tangent to a caustic , for some , see (5)), then

(13)

where is a constant depending only on .

Let us consider rotation numbers , with , and denote by the value of corresponding to the caustic of rotation number . Similarly, denotes the associated modulus (see Proposition 16).

Therefore, with respect to the action-angle variables (9), we have that for any :

If denotes either and , the above equality implies that

which, using the expression in (9) is equivalent to

Consider now the change of coordinates

Then

and the above integral becomes

(14)

Define for each :

(15)

or equivalently in the complex form:

Lemma 21.

Assume that the billiard map in has rationally integrable caustics corresponding to rotation numbers for all . Then,

Moreover, if we denote , then:

where is a term whose absolute value is bounded by times a factor depending on and .

Remark 22.

It follows from [3, Lemma 13] that assuming we have

where is a term whose absolute value is bounded by times a factor depending on and -norm of .
In order to apply [3, Lemma 13] we need to translate notations: in [3, Section 4, pp. 7–8] action-angle variables are introduced and in [3, middle of page 16] is defined, which coincides with what we denote (compare with (9), where is such that , or with Appendix E). With this notation, the above integral is estimated as in [3, Lemma 13]. Notice also that the Lazutkin density in [3, (14) on page 14] coincides with our (37). Thus, integrating with respect to Lazutkin parametrization with Lazutkin density is the same as integrating with respect to .

Proof.

The first part follows from (14). As for the second part, observe that

In particular, recall that for all and that as . Hence, using the first statement of the proposition:

6. Elliptic Motions

We call translations, rotations, hyperbolic rotations, and homothety elliptic motions; indeed, all of these transformations keep the class of ellipses invariant.

In Appendix B, we show that infinitesimal perturbations of an ellipse by these motions, correspond to these functions (expressed in the elliptic coordinate frame with semi-focal distance ):

  • Translations:

    (16)
    (17)
  • Rotations:

    (18)
  • Homotheties:

    (19)
  • Hyperbolic rotations:

    (20)

We say that a strictly convex smooth domain is a deformation of an ellipse if there exist and a function

such that

By abusing notation, in the following we will write

(21)

We will need the following approximation result.

Proposition 23.

Let us consider the ellipse and let

where are assumed to be sufficiently small. Then, there exist a constant and an ellipse such that

The proof is presented in Appendix B.


7. An adapted basis for

In this section we want to determine a suitable basis of , where hereafter . This basis will be constructed by means of elliptic motions , see (16)–(20), and the functions defined in (15).

In order to prove their linear indepedence, we need to consider their analytic extension to and study their singularities.

7.1. Analyticity properties of and

Let us start by considering the complex extensions of the functions defined in (15):

(22)

where, to simplify the notation, we have denoted . In particular, represents the eccentricity of the caustic with rotation number ; moreover, for all ( denotes the eccentricity of the boundary), it is strictly decreasing in , and as . Denote and .

We are interested in the complex extensions of these functions and in their singularities.

Proposition 24.

For , the functions and have an holomorphic extension to the complex strip . This extension is maximal in the sense that these functions have singularities at (which are ramification singularities).

This proposition will be proven in Appendix C.1.

Remark 25.

Observe that is a strictly increasing function as a function of and as .
Moreover, since is a strictly increasing function of and as , then is a strictly decreasing function of and as .

7.2. Analyticity properties of , , ,