Alexey Glutsyuk
email: aglutsyu@ens-lyon.fr

On 4-reflective complex analytic planar billiards

Keywords: billiards, analytic distributions, Birkhoff distributions and periodic orbits, 4, 2 Background material: Phaffian systems and involutivity, 4, 3 Case of three-dimensional distribution, 5 Applications to real pseudo-billiards 5, 1 Classification 3 Application 1: Tabachnikov's Commuting Billiard Conjecture 5, 4 Application 2: planar Plakhov's Invisibility Conjecture

 says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study its complex analytic version for quadrilateral orbits in two dimensions, with reflections from holomorphic curves. We present the complete classification of 4reflective complex analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the previous author's result [7] classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar C 4 -smooth pseudo-billiards; solutions of Tabachnikov's Commuting Billiard Conjecture and the 4-reflective case of Plakhov's Invisibility Conjecture (both in two dimensions; the boundary is required to be piecewise C 4 -smooth). We provide a survey and a small technical result concerning higher number of complex reflections.

Introduction

The famous V.Ya.Ivrii's conjecture [START_REF] Ivrii | The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary[END_REF] says that in every billiard with infinitely-smooth boundary in a Euclidean space of any dimension the set of periodic orbits has measure zero. As it was shown by V.Ya.Ivrii [START_REF] Ivrii | The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary[END_REF], his conjecture implies the famous H.Weyl's conjecture on the two-term asymptotics of the spectrum of Laplacian [START_REF] Weyl | Über die asymptotische verteilung der eigenwerte[END_REF]. A brief historical survey of both conjectures with references is presented in [START_REF] Glutsyuk | On quadrilateral orbits in planar billiards[END_REF][START_REF] Glutsyuk | No planar billiard possesses an open set of quadrilateral trajectories[END_REF].

For the proof of Ivrii's conjecture it suffices to show that for every k ∈ N the set ot k-periodic orbits has measure zero. For k = 3 this was proved in [START_REF] Baryshnikov | Billiards and nonholonomic distributions[END_REF][START_REF] Rychlik | Periodic points of the billiard ball map in a convex domain[END_REF][START_REF] Stojanov | Note on the periodic points of the billiard[END_REF][START_REF] Vorobets | On the measure of the set of periodic points of a billiard[END_REF][START_REF] Wojtkowski | Two applications of Jacobi fields to the billiard ball problem[END_REF]. For k = 4 in dimension two this was proved in [START_REF] Glutsyuk | On quadrilateral orbits in planar billiards[END_REF][START_REF] Glutsyuk | No planar billiard possesses an open set of quadrilateral trajectories[END_REF].

Remark 1.1 Ivrii's conjecture is open already for billiards with piecewiseanalytic boundaries, and we believe that this is its principal case. In the latter case Ivrii's conjecture is equivalent to the statement saying that for every k ∈ N the set of k-periodic orbits has empty interior.

Besides the traditional real billiards, where each ray hitting the boundary is reflected back to the same side, it is interesting to study the so-called pseudo-billiards (introduced in Section 5), where some reflections change the side. Pseudo-billiards naturally arise, e.g., in the invisibility theory. One can ask the following question analogous to Ivrii's Conjecture: classify those pseudo-billiards that have an open (positive measure) set of periodic orbits. This question is closely related, e.g., to Plakhov's Invisibility Conjecture [17, conjecture 8.2] and Tabachnikov's Commuting Billiard Conjecture [25, p. 58], which is related to the famous Birkhoff Conjecture on integrable billiards [24, p. 95].

It appears that planar Ivrii's conjecture and all its analogues for pseudobilliards have the same complexification stated and partially studied in [START_REF] Glutsyuk | On quadrilateral orbits in complex algebraic planar billiards[END_REF][START_REF] Glutsyuk | On odd-periodic orbits in complex planar billiards[END_REF] and recalled below. This is the problem to classify all the so-called kreflective complex planar analytic billiards: those collections of k complex analytic curves in CP 2 for which the corresponding billiard has an open set of k-periodic orbits. Its studying presents a unifying approach to the original Ivrii's conjecture and all its above-mentioned analogues altogether.

In the present paper we solve the complex classification problem completely for k = 4 (Theorem 1.7, the main result). As an application, we provide the complete classification of germs of C 4 -smooth real planar pseudobilliards having open set of quadrilateral orbits (Subsections 5.1, 5.2). As applications of the latter result, we give solutions of Tabachnikov's Commuting Billiard Conjecture and the 4-reflective Plakhov's Invisibility Conjecture, both in two-dimensional piecewise C 4 -smooth case (Subsections 5. 3, 5.4).

The classification of analytic 4-reflective germs of pseudo-billiards follows almost immediately from the main complex result. The proof of the classification of smooth pseudo-billiards is also done by complex methods and is based on the theory of Cartan's prolongations of Pfaffian systems.

It is analogous to Yu.G.Kudryashov's arguments from [10, section 2] reducing the piecewise C 4 -smooth case of 4-reflective Ivrii's Conjecture to the piecewise-analytic case.

The state of art of the classification problem of k-reflective complex billiards in the general case is presented in Section 6 together with small new technical results (Theorem 6.7 and Corollary 6.8).

Basic definitions and statement of main result are given below.

Main result: classification of 4-reflective complex analytic planar billiards

To recall the complexified Ivrii's conjecture and state the main result, let us recall some basic definitions contained in [7, section 1]. The complex plane C 2 with affine coordinates (z 1 , z 2 ) is equipped with the complexified Euclidean metric. It is the standard complex-bilinear quadratic form dz 2 1 + dz 2 2 . This defines the notion of symmetry with respect to a complex line, reflections with respect to complex lines and more generally, reflections of complex lines with respect to complex analytic (algebraic) curves. The symmetry is defined by the same formula, as in the real case. More details concerning the complex reflection law are given in Subsection 2.3.

Remark 1.2

The geometry of the complexified Euclidean metric is somewhat similar to that of its another real form: the pseudo-Euclidean metric. Billiards in pseudo-Euclidean spaces were studied, e.g., in [START_REF] Dragovic | Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics[END_REF][START_REF] Khesin | Pseudo-Riemannian geodesics and billiards[END_REF]. Definition 1.3 A complex projective line l ⊂ CP 2 ⊃ C 2 is isotropic, if either it coincides with the infinity line, or the complexified Euclidean quadratic form vanishes on l. Or equivalently, a line is isotropic, if it passes through some of two points with homogeneous coordinates (1 : ±i : 0): the so-called isotropic points at infinity (also known as cyclic (or circular) points).

Convention 1.4 Everywhere below by an irreducible analytic curve in CP n we mean a non-constant CP n -valued holomorphic function on a connected Riemann surface. Definition 1.5 [7, definition 1.3] A complex analytic (algebraic) planar billiard is a finite collection of irreducible complex analytic (algebraic) curves a 1 , . . . , a k ⊂ CP 2 that are not isotropic lines; set a k+1 = a 1 , a 0 = a k . A kperiodic billiard orbit is a collection of points A j ∈ a j , A k+1 = A 1 , A 0 = A k , such that for every j = 1, . . . , k one has A j+1 = A j , the tangent line T A j a j is not isotropic and the complex lines A j-1 A j and A j A j+1 are symmetric with respect to the line T A j a j and are distinct from it. (Properly saying, we have to take points A j together with prescribed branches of curves a j at A j : this specifies the line T A j a j in unique way, if A j is a self-intersection point of the curve a j .) Definition 1.6 [7, definition 1.4] A complex analytic (algebraic) billiard a 1 , . . . , a k is k-reflective, if it has an open set of k-periodic orbits. In more detail, this means that there exists an open set of pairs (A 1 , A 2 ) ∈ a 1 × a 2 extendable to k-periodic orbits A 1 . . . A k . (Then the latter property automatically holds for every other pair of neighbor mirrors a j , a j+1 .)

Problem (Complexified version of Ivrii's conjecture). Classify all the k-reflective complex analytic (algebraic) billiards. Theorem 1.7 A complex planar analytic billiard a, b, c, d is 4-reflective, if and only if it has one of the three following types:

1) one of the mirrors, say a is a line, c = a, the curves b and d are symmetric with respect to the line a and distinct from it, see Section 5, Fig. 7; 2) the mirrors are distinct lines through the same point O ∈ CP 2 , the pair of lines (a, b) is transformed to (d, c) by complex rotation around O, i.e., a complex isometry C 2 → C 2 fixing O with unit Jacobian, see Section 5, Fig. 8;

3) a = c, b = d, and they are distinct confocal conics, see Section 5, Fig. 9101112.

Remark 1.8 Theorem 1.7 in the algebraic case is given by [7, theorem 1.11], which implies the 4-reflectivity of billiards of types 2) and 3). The proof of 4-reflectivity of billiards of type 1) repeats the proof in the algebraic case, see [7, example 1.7].

The plan of the proof of Theorem 1.7

Theorem 1.7 is obviously implied by the two following theorems. Theorem 1.9 Every 4-reflective complex planar analytic billiard with at least one algebraic mirror has one of the above types 1)-3).

Theorem 1.10 Let in a complex planar analytic 4-reflective billiard no mirror be a line. Then all the mirrors are algebraic curves. Theorems 1.9 and 1.10 are proved in Subsection 3.4 and Section 4 respectively.

Remark 1.11 Theorem 1.7 is local and can be stated for a germ of 4reflective analytic billiard: a collection of irreducible germs of analytic curves (a, A), (b, B), (c, C), (d, D) in CP 2 such that the quadrilateral ABCD lies in an open set of quadrilateral orbits of the corresponding billiard.

For the proof of Theorem 1.7 we study the maximal analytic extensions of the mirrors. These are analytic curves parametrized by abstract connected Riemann surfaces, which we will denote by â, b, ĉ, d. The latter are called the maximal normalizations, see the corresponding background material in Subsection 2.2. We represent the open set of quadrilateral orbits as a subset in â × b × ĉ × d and will denote it by U 0 . Its closure

U = U 0 ⊂ â × b × ĉ × d
in the usual topology is an analytic subset with only two-dimensional irreducible components. It will be called the 4-reflective set, see [7, definition 2.13 and proposition 2.14]. The complement U \ U 0 consists of the so-called degenerate quadrilateral orbits: quadrilaterals ABCD satisfying the reflection law that have either a pair of coinciding neighbor vertices, or a pair of coinciding adjacent edges, e.g., an edge tangent to a mirror through an adjacent vertex, or an isotropic tangency vertex. One of the main ideas of the proof of Theorem 1.7 is similar to that from [START_REF] Glutsyuk | On quadrilateral orbits in planar billiards[END_REF][START_REF] Glutsyuk | No planar billiard possesses an open set of quadrilateral trajectories[END_REF][START_REF] Glutsyuk | On quadrilateral orbits in complex algebraic planar billiards[END_REF]: to study the degenerate orbit set U \ U 0 . This idea itself together with basic algebraic geometry allowed to treat the algebraic case in [START_REF] Glutsyuk | On quadrilateral orbits in complex algebraic planar billiards[END_REF]. One of the key facts used in the proof was properness (and hence, epimorphicity) of the projection U → â × b to the position of two neighbor vertices. In the algebraic case the properness is automatic (follows from Remmert's Proper Mapping Theorem [11, p.34]), but in the general analytic case under consideration it isn't. We prove that the above projection is indeed proper in the analytic case. The most part of the proof of Theorem 1.7, and in particular, the proof of properness are based on studying restricted versions of Birkhoff distribution, which was introduced in [START_REF] Baryshnikov | Billiards and nonholonomic distributions[END_REF]. All the Birkhoff distributions are briefly described below; more details are given in Subsection 2.7. Definition 1.12 Let M be an n-dimensional (real or complex) analytic manifold. Let D be a d-dimensional analytic distribution on M , i.e., D(x) ⊂ T x M is a d-dimensional subspace for every x ∈ M and the map x → D(x) is analytic. Let l ≤ d. An l-dimensional surface S ⊂ M is said to be an integral surface for the distribution D, if T x S ⊂ D(x) for every x ∈ S.

Consider the projectivization of the tangent bundle T CP 2 :

P = P(T CP 2 ).
It is the space of pairs (A, L): A ∈ CP 2 , L ⊂ T A CP 2 is a one-dimensional subspace. The space P is three-dimensional and it carries the standard two-dimensional contact distribution H: the plane H(A, L) ⊂ T (A,L) P is the preimage of the line L ⊂ T A CP 2 under the derivative of the bundle projection P → CP 2 . The product P k carries the product distribution H k . Let R 0,k ⊂ P k denote the subset of points ((A 1 , L 1 ), . . . , (A k , L k )) such that for every j one has A j ∈ C 2 ⊂ CP 2 , A j±1 = A j , the lines A j A j-1 , A j A j+1 are symmetric with respect to the line L j , and the three latter lines are distinct and non-isotropic. The above product distribution induces the socalled Birkhoff distribution D k on R 0,k , see [START_REF] Baryshnikov | Billiards and nonholonomic distributions[END_REF]. It is well-known [START_REF] Baryshnikov | Billiards and nonholonomic distributions[END_REF] that for every analytic billiard a 1 , . . . , a k the natural lifting to P k of any analytic family of its k-periodic orbits A 1 . . . A k with L j = T A j a j lies in R 0,k and is tangent to Birkhoff distribution. In particular, if the billiard is k-reflective, then the lifting to R 0,k of an open set of its k-periodic orbits is an integral surface of Birkhoff distribution.

We will study the following restricted versions D a and D ab of Birkhoff distribution that correspond respectively to 4-reflective billiards a, b, c, d with one given mirror a (or two given mirrors a and b). The products â × P 3 and â × b × P 2 admit natural inclusions to P 4 induced by parametrizations â → a, b → b. Let M a ⊂ â × P 3 , M ab ⊂ â × b × P 2 denote the closures of the corresponding pullbacks of the set R 0,4 . The distributions D a , D ab are the pullbacks of the Birkhoff distribution D 4 on R 0,4 . They are 3-and 2-dimensional singular analytic distributions on M a and M ab in the sense of Subsection 2.6. For every billiard as above the natural lifting to â × P 3 (â × b × P 2 ) of any open set of its quadrilateral orbits lies in M a (M ab ) and is an integral surface of the corresponding distribution D a (respectively, D ab ).

The proof of Theorem 1.7 is split into the following steps.

Step 1. Case of two neighbor algebraic mirrors. In this case it is easy to show that all the mirrors are algebraic (Proposition 2.1 in Subsection 2.1). This together with [7, theorem 1.11] implies that the billiard under question is of one of the types 1)-3), see Remark 1.8.

From now on we consider that no two neighbor mirrors are algebraic.

Step 2. Preparatory description of the complement U \U 0 . In Subsection 2.4 we study degenerate quadrilaterals ABCD ∈ U \ U 0 with a pair of coinciding neighbor vertices, say A = D, analogously to the arguments from [10, p.320]. Under mild additional assumptions, in particular, B, C = A = D, we show that the other mirrors b and c are special curves called triangular spirals centered at A. Namely, they are phase curves of algebraic line fields on CP2 : the so-called triangular line fields centered at A introduced in the same subsection (Proposition 2.19). One of the key arguments used in the proof of Theorem 1.7 is Proposition 2.21, which says that every triangular spiral with at least two distinct centers is algebraic, provided that the line through the centers is not isotropic. In Subsections 2.3 and 2.5 we recall the results of [7, subsections 2.1, 2.2] on partial description of degenerate quadrilaterals in U \ U 0 with either an isotropic tangency vertex, or an edge tangent to a mirror through an adjacent vertex.

Step 3. Properness of the projection U → â × b (Section 3, Corollary 3.4). To prove it, we study the Birkhoff distribution D ab and prove its nonintegrability in Subsections 3.1-3.3. Moreover, we show that the closure in M ab of the union of its integral surfaces (if any) is a two-dimensional analytic subset in M ab (Lemma 3.1 and Corollary 3.2.) In the proof of the latter statement and in what follows we use Proposition 2.38 from Subsection 2.6. It deals with an m-dimensional singular analytic distribution, a given union of m-dimensional integral surfaces and the minimal analytic set M containing the latter union. Proposition 2.38 states that the restriction to M of the distribution is m-dimensional and integrable. Proposition 2.38 is a key tool for the whole paper.

The set U is identified with either the above two-dimensional analytic subset in M ab , or a smaller analytic subset. This together with Proper Mapping Theorem implies properness of the projection U → â × b (Corollary 3.4). The proof of Lemma 3.1 is done by contradiction. The contrary would imply the existence of at least three-dimensional invariant irreducible analytic subset M ⊂ M ab where the distribution D ab is integrable. Then a complement M 0 ⊂ M to a smaller analytic subset is saturated by open sets of quadrilateral orbits of 4-reflective billiards a, b, c, d with variable mirrors c = c(x) and d = d(x), x ∈ M 0 . We treat separately two cases:

-some of the projections of the set M to the space of triples (A(x), B(x), G(x)), G = C, D, is not bimeromorphic.

-both latter projections are bimeromorphic.

The first case will be treated in Subsection 3.2. We show that there exist x, y ∈ M 0 projected to the same vertices A, B, D = D 0 but with distinct tangent lines T D 0 d(x) = T D 0 d(y), D 0 being not a cusp 1 of the curves d(x) and d(y), the projection to (A, B, D) being a local submersion at x, y. We then deduce that the billiard d(y), d(x), c(x), c(y) is 4-reflective (as in [7, proof of lemma 3.1]), and the mirror c(x) is a triangular spiral with center D 0 (Proposition 2.19, Step 2). Then we slightly deform y with fixed vertices A and B to a point y so that the corresponding mirror d(y ) intersects d(x) at a point D 1 = D 0 . We get analogously that the curve c(x) is a triangular spiral with two distinct centers D 0 and D 1 . This implies that c(x) is algebraic (Proposition 2.21, Step 2). Similarly, we show that c(y) is algebraic, fixing y and deforming x. Hence, the mirror d(x) of the 4-reflective billiard d(y), d(x), c(x), c(y) is algebraic, as are c(x) and c(y) (Proposition 2.1, Step 1). Similarly, a and b are algebraic, as are c(x) and d(x). The contradiction thus obtained implies that the first case is impossible.

In the second case we show (in Subsection 3.3) that for an open set of points x ∈ M 0 the mirrors c(x) and d(x) are lines. Hence, the curves a and b are algebraic, by Step 1, -a contradiction. Finally, none of the above cases is possible. The contradiction thus obtained will prove Lemma 3.1.

Step 4. Case of one algebraic mirror, say a: proof of Theorem 1.9 (Subsection 3.4). Properness of the projection U → â × b (Step 3) implies properness of the projection U → b (algebraicity). Therefore, the preimage in U of every point B ∈ b is a compact holomorphic curve. This immediately implies that the mirror c is algebraic and there are two possibilities:

-either all the mirrors are algebraic, and we are done; -or the projection of the above preimage to the position of the point D is constant for every B.

In the latter case we show that a = c is a line and the mirrors b, d = a are symmetric with respect to it: the billiard has type 1). This will prove Theorem 1.9.

From now on we consider that no mirror is algebraic. We show that this case is impossible. This will prove Theorem 1.10 and hence, Theorem 1.7.

Step 5. Case of intersected mirrors, say a and b intersect at a point A. Under the additional assumption that A is regular and not an isotropic tangency point for both a and b we show that a = c (Corollary 3.5 proved in Subsection 3.5). The set U contains a non-empty at most one-dimensional compact analytic subset of quadrilaterals AACD (properness of projection,

Step 3). For the proof of Corollary 3.5 we show in Subsection 3.5 that this is a discrete subset in U consisting of quadrilaterals with all the vertices coinciding with A. Indeed, otherwise, if the above subset were one-dimensional, this would immediately imply that some of the mirrors c or d is algebraic, -a contradiction.

Step 6. Proof of Theorem 1.10. To do this, we study the three-dimensional Birkhoff distribution D a on the 6-dimensional analytic set M a . We fix a connected component of the open set of quadrilateral orbits of the billiard a, b, c, d. It is an integral surface of the distribution D a , which we will denote S. We consider the minimal analytic subset M ⊂ M a containing S, which is irreducible (easy to show), and study the restriction D M to M of the distribution D a . We treat two separate cases: 1) dimD M = 2 (Subsection 4.1); 2) dimD M = 3 (Subsection 4.3). In the two-dimensional Case 1) we show that there exists an open subset V ⊂ M saturated by integral surfaces of the distribution D M that correspond to 4-reflective billiards a, b(x), c(x), d(x) with b(x) intersecting a (easily follows from transcendence of the curve a). We then deduce that either the mirror b(x) is a line for all x ∈ V (and hence, for all x regular for both M and D M ), or the mirror c(x) coincides with a for all x as above. This basically follows from Corollary 3.5, Step 5. The first subcase is impossible, since then the mirror b of the initial transcendental billiard would be a line, -a contradiction. In the second subcase the projection ν C (M ) of the whole variety M to the position of the vertex C lies in a. For a generic A ∈ â we consider its preimage W A ⊂ M under the projection ν a : M → â, which is a projective algebraic variety. It follows that the projection ν C (W A ) lies in a transcendental curve a, while it should be an algebraic subset in CP 2 (Remmert's Proper Mapping and Chow's Theorems). Hence, ν C (W A ) is discrete. On the other hand, it cannot be discrete, whenever b is neither a line, nor a conic, by [7, proposition 2.32]. The contradiction thus obtained shows that Case 1) is impossible. The three-dimensional Case 2) is treated analogously, but it is more technical. The existence of two-dimensional integral surfaces as above of a three-dimensional distribution D M is not automatic. Its proof is based on Cartan-Kuranishi-Rashevskii involutivity theory of Pfaffian systems. The corresponding background material is recalled in Subsection 4.2. Proof By symmetry, it suffices to prove algebraicity of the mirror c. Fix a quadrilateral orbit A 0 B 0 C 0 D 0 . Consider the family of quadrilateral orbits ABCD with fixed D = D 0 . They are locally parametrized by the line l = AD, which lies in the space CP 1 of lines through D. The point A depends algebraically on l, since a is algebraic. Similarly, the line AB, and hence, the point B depend algebraically on l, since a, b are algebraic and AB is symmetric to l with respect to the line T A a. Analogously, the line BC, which is symmetric to AB with respect to the line T B b, depends algebraically on l. The line DC also depends algebraically on l, being the reflected image of the line l with respect to the fixed line T D d. Finally, the variable intersection point C = BC ∩ DC should also depend algebraically on l. Hence, c is algebraic. The proposition is proved. 2

Preliminaries

Case of two neighbor algebraic mirrors

Maximal analytic extension

s 2 ∈ S 2 to CP n , f j : S j → CP n , j = 1, 2, f 1 (s 1 ) = f 2 (s 2 ). We say that f 1 ≤ f 2 , if there exists a holomorphic mapping h : S 1 → S 2 , h(s 1 ) = s 2 , such that f 1 = f 2 • h.
This defines a partial order on the set of classes of Riemann surface mappings to CP n up to conformal reparametrization respecting base points.

The following proposition is classical, see the proof, e.g., in [START_REF] Glutsyuk | On odd-periodic orbits in complex planar billiards[END_REF].

Proposition 2.3 [8, proposition 2]. Every irreducible germ of analytic curve in CP n has maximal analytic extension. In more detail, let (a, A) ⊂ CP n be an irreducible germ of analytic curve. There exists an abstract connected Riemann surface â with base point  ∈ â (which we will call the maximal normalization of the germ a) and a holomorphic mapping π a : â → CP n , π a ( Â) = A with the following properties:

-the image of germ at  of the mapping π a is contained in a; -π a is the maximal mapping with the above property in the sense of Definition 2.2.

Moreover, the mapping π a is unique up to composition with conformal isomorphism of Riemann surfaces respecting base points.

Corollary 2.4 Let M be a complex manifold, and let f : M → CP n be a non-constant holomorphic mapping. Let U ⊂ M be an irreducible analytic subset, and let the restriction f | U have rank one on an open subset. Let

x ∈ U be a regular point, A = f (x); then the image of the germ f : (U, x) → CP n is an irreducible germ (a, A) of analytic curve. Let π a : â → a be its maximal normalization. Let Û be the normalization of the analytic set U (see [5, p.78]), π U : Û → U be the natural projection (which is invertible on the regular part of U ). Then there exists a unique holomorphic lifting

F : Û → â such that f • π U = π a • F .
Proof For every point y ∈ Û and any point z ∈ Û close enough to y there exists an analytic curve in Û through y and z. This together with Proposition 2.3 (applied to the latter curves) and Hartogs' and Osgood's Theorems imply the corollary. 2

Complex reflection law

The material presented in this subsection is contained in [7, subsection 2.1].

We fix an Euclidean metric on R 2 and consider its complexification: the complex-bilinear quadratic form dz 2 1 + dz 2 2 on the complex affine plane

C 2 ⊂ CP 2 . We denote the infinity line in CP 2 by C ∞ = CP 2 \ C 2 .
Definition 2.5 The symmetry C 2 → C 2 with respect to a non-isotropic complex line L ⊂ CP 2 is the unique non-trivial complex-isometric involution fixing the points of the line L. It extends to a projective transformation of the ambient plane CP 2 . For every x ∈ L it acts on the space CP 1 of lines through x, and this action is called symmetry at x. If L is an isotropic line through a finite point x, then a pair of lines through x is called symmetric with respect to L, if it is a limit of symmetric pairs of lines with respect to non-isotropic lines converging to L. Lemma 2.6 [7, lemma 2.3] Let L be an isotropic line through a finite point x. A pair of lines (L 1 , L 2 ) through x is symmetric with respect to L, if and only if some of them coincides with L.

Convention 2.7 For every irreducible analytic curve a ⊂ CP 2 and a point A ∈ â the local branch a A of the curve a at A is the germ of curve π a : (â, A) → CP 2 , which is contained in a. By T A a we denote the tangent line to the local branch a A at π a (A). Sometimes we idendify a point (subset) in a with its preimage in the normalization â and denote both subsets by the same symbol. In particular, given a subset in CP 2 , say a line l, we set â ∩ l = π -1 a (a ∩ l) ⊂ â. If a, b ⊂ CP 2 are two irreducible analytic curves, and A ∈ â, B ∈ b, π a (A) = π b (B), then for simplicity we write A = B and the line π a (A)π b (B) will be referred to, as AB. Definition 2.8 A triple of points BAD ∈ (CP 2 ) 3 satisfies the complex reflection law with respect to a given line L through A, if one of the following statements holds:

-either B, D = A, the line L is non-isotropic and the lines AB, AD are symmetric with respect to L;

-or B, D = A, the line L is isotropic and some of the lines AB, AD coincides with L;

-or A coincides with some of the points B or D.

Definition 2.9 Let a 1 , . . . , a k ⊂ CP 2 be an analytic (algebraic) billiard, and let â1 , . . . , âk be the maximal normalizations of its mirrors. Let P k ⊂ â1 × • • • × âk denote the subset corresponding to k-periodic billiard orbits.

The set P k is contained in the subset Q k ⊂ â1 × • • • × âk of (not necessarily periodic) k-orbits: the k-gons A 1 . . . A k such that for every 2 ≤ j ≤ k -1 one has A j = A j±1 , the line T A j a j is not isotropic and the lines A j A j-1 , A j A j+1 are symmetric with respect to it and distinct from it. Let U 0 = Int(P k ) denote the interior of the subset

P k ⊂ Q k . Set U = U 0 ⊂ â1 × .
. . âk : the closure is taken in the usual product topology.

The set U will be called the k-reflective set. Addendum. For every k-reflective billiard the latter projections U → âj × âj+1 are local biholomorphisms on the set of those k-periodic orbits whose vertices are not cusps of the corresponding mirrors.

The addendum follows from definition.

Triangular algebraic line fields and spirals

Here we deal with a 4-reflective complex analytic billiard a, b, c, d whose 4-reflective set U contains a quadrilateral ABCD with coinciding vertices A = D. We show (Proposition 2.19) that under mild genericity assumptions (implying, e.g., that ABCD is not a single-point quadrilateral) either the mirrors b and c are conics, or they are so-called triangular spirals centered at A: phase curves of algebraic line fields invariant under the rotations around A. We show (Proposition 2.21) that every triangular spiral with two distinct centers is algebraic.

To define triangular spirals and state the above-mentioned results, we introduce yet another restricted Birkhoff distribution on the space of "framed triangles with fixed vertex". Let us fix a point A ∈ C 2 (take it as the origin) and a non-trivial complex isometry H ∈ SO(2, C) \ Id fixing A. Recall that P = P(T CP 2 ), and H is the standard contact plane field on P, see Subsection 1.2. Namely, for every x = (B, L) ∈ P, where B ∈ CP 2 , L ⊂ T B CP 2 is a one-dimensional subspace, the plane H(x) ⊂ T x P is the preimage of the line L under the differential of the bundle projection P → CP 2 . Consider the product P 2 equipped with the four-dimensional algebraic distribution H 2 = H ⊕ H. Let T A,H ⊂ CP 2 × CP 2 denote the subset of pairs (B, C) such that B, C = A, B = C, the lines AB, AC are distinct, nonisotropic and AC = H(AB). Let M 0 A,H ⊂ P 2 denote the subset of those pairs ((B, L B ), (C, L C )), for which (B, C) ∈ T A,H , the lines L B , L C are nonisotropic, the lines AB, BC are symmetric with respect to the line L B ; AC, BC are symmetric with respect to the line

L C ; AB = L B , AC = L C . Set M A,H = M 0
A,H ⊂ P 2 : the closure in the usual topology. This is a three-dimensional projective algebraic variety, and M 0 A,H ⊂ M A,H is its Zariski open and dense subset.

Proposition 2.11 The variety M 0

A,H is smooth and transversal to the distribution H 2 .

Proof The smoothness is obvious. The restriction ν : M 0 A,H → T A,H of the bundle projection P 2 → (CP 2 ) 2 is a local diffeomorphism, by construction. For every Corollary 2.12 For all x ∈ M 0 A,H the intersections

x = ((B, L B ), (C, L C )) ∈ M 0 A,H the subspace H 2 (x) ⊂ T x (P 2 ) is the preimage of the direct sum L B ⊕L C ⊂ T (B,C) (CP 2
D A,H (x) = H 2 (x) ∩ T x M 0 A,H
are one-dimensional and form an algebraic line field on M 0 A,H .

Proof The transversal variety M 0 A,H and distribution H 2 in the ambient six-dimensional space P 2 have dimensions 3 and 4 respectively. Hence, the intersections of their tangent spaces are one-dimensional. The algebraicity of the line field D A,H is obvious.

2

The next proposition shows that the line field D A,H has an algebraic first integral: an appropriate holomorphic branch of squared perimeter of triangle. To define the latter branch, let us introduce the following definition. Proof The contrary to the discreteness of preimages of the projection, say ν B would imply constance of the perimeter on a one-parameter family of triangles ABC with fixed vertices A and B, fixed line AC = H(AB) = L and variable C ∈ L. This is obviously impossible. The algebraicity and invariance of the projected line field obviously follow from the algebraicity and invariance of the surface S p and submersivity. 2

Definition 2.18 A triangular spiral centered at A is a complex orbit of a triangular line field centered at A, see Fig. 2a).

spirals, or conics

Proof There exists an irreducible germ Γ ⊂ U of analytic curve at ABCD parametrized by local small complex parameter t and consisting of quadrilaterals AB t C t D t with fixed vertex A: AB 0 C 0 D 0 = ABCD. Let us fix it. One has D t ≡ D = A. This follows from the fact that D t is found as a point of intersection of the curve d with the line L t symmetric to AB t with respect to the tangent line T A a. Indeed, the line L 0 is transverse to T D d, as is AC, by assumption and since L 0 and AC are symmetric with respect to the line T D d. Therefore, the intersection point D t ∈ L t ∩ d identically coincides with D = D 0 . Let H denote the composition of symmetries with respect to the tangent lines, first T A a, then T D d. Thus, H ∈ SO(2, C) fixes A and H(AB t ) = AC t for every AB t C t D ∈ Γ. We take A as the origin.

Case 1): AB = AC. Then B = C and the germ Γ is embedded into

M 0 A,H via the mapping t → ((B t , T Bt b), (C t , T Ct c))
, by construction and non-isotropicity condition. Its image is a phase curve of the line field D A,H , analogously to discussions in [START_REF] Baryshnikov | Billiards and nonholonomic distributions[END_REF] and [10, p.320]. This together with Proposition 2.17 implies that the projection Γ → CP 2 to the position of each one of the vertices B and C sends Γ to a triangular spiral centered at A, see Fig. 2a). Hence, b and c are triangular spirals.

Case 2): AB = AC. Then H = ±Id and AB t ≡ AC t . Note that at least one of vertices, either B t , or C t varies, since Γ is a curve. To treat the case under consideration, we use the following remark.

Remark 2.20 There exist no k-reflective analytic planar billiard such that some its two neighbor mirrors coincide with the same line: such a billiard would have no k-periodic orbits in the sense of Definition 1.5 (cf. [7, proof of corollary 2.19]). Subcase 2a): B t ≡ C t ≡ const. This implies that b = c and the line AB t is tangent to b at variable point B t , as in loc. cit. Therefore, b = c = AB, which is impossible by the above remark. Hence, this subcase is impossible.

Subcase 2b): B t ≡ C t . Without loss of generality we consider that B = C. Thus, for every t small enough the points A, B t and C t are distinct and lie on the same line. Note that T B b, T C c = AB = AC, by the condition of the proposition. Hence, T Bt b, T Ct c ⊥ AB t for all t. This implies that B t , C t ≡ const and b, c are complex circles centered at A, see Fig. 2b). This proves Proposition 2.19. 2

Proposition 2.21 Let a planar analytic curve be a triangular spiral with respect to two distinct centers, and let the line through them be non-isotropic.

Then it is algebraic.

Proof A triangular spiral is a phase curve of a triangular algebraic line field. The latter field is invariant under complex rotations: the isometries fixing the center of the spiral with unit Jacobian. Suppose the contrary: the spiral under consideration is not algebraic. Then the corresponding line field is uniquely defined: two algebraic line fields coinciding on a non-algebraic curve (which is Zariski dense) coincide everywhere. Thus, the latter line field is invariant under complex rotations around two distinct centers, and the line through the centers is not isotropic. The latter rotations generate the whole group of complex isometries of C 2 with unit Jacobian. Thus, the line field is invariant under all of them, which is impossible. The contradiction thus obtained proves the proposition. 2

Tangencies in k-reflective billiards

Here we recall the results of [7, subsection 2.2].

We deal with k-reflective analytic planar billiards a 1 , . . . , a k in CP 2 . Let U ⊂ â1 × • • • × âk be the k-reflective set. The results of loc.cit. presented below concern degenerate quadrilaterals in U \U 0 : limits A 1 . . . A k of k-periodic orbits such that for a certain j with a j being not a line the tangent line T A j a j and the adjacent edges A j±1 A j collide to the same non-isotropic limit. Then the limit vertex A j will be called a tangency vertex. Proposition 2.23 shows that the latter cannot happen to be the only degeneracy of the limit k-gon. Its Corollary 2.26 presented at the end of the subsection concerns the case, when k = 4. It says that if the tangency vertex is distinct from its neighbor limit vertices, then its opposite vertex should be either also a tangency vertex, or a cusp with a non-isotropic tangent line. Proposition 2.25 extends Proposition 2.23 to the case, when some subsequent mirrors coincide and the corresponding subsequent vertices of a limiting orbit collide. Definition 2.22 A point of a planar irreducible analytic curve is marked, if it is either a cusp, or an isotropic tangency point. Given a parametrized curve π a : â → a, a point A ∈ â is marked, if it corresponds to a marked point of the local branch a A , see Convention 2.7.

Proposition 2.23 [7, proposition 2.16] Let a 1 , . . . , a k and U be as above. Then U contains no k-gon A 1 . . . A k with the following properties:

-each pair of neighbor vertices correspond to distinct points, and no vertex is a marked point;

-there exists a unique s ∈ {1, . . . , k} such that the line A s A s+1 is tangent to the curve a s at A s , and the latter curve is not a line, see Fig. 3.

Remark 2.24

A real version of Proposition 2.23 is contained in [START_REF] Glutsyuk | No planar billiard possesses an open set of quadrilateral trajectories[END_REF] (lemma 56, p.315 for k = 4, and its generalization (lemma 67, p.322) for higher k).

Proposition 2.25 [7, proposition 2.18] Let a 1 , . . . , a k and U be as at the beginning of the subsection. Then U contains no k-gon A 1 . . . A k with the following properties:

1) each its vertex is not a marked point of the corresponding mirror;

2) there exist s, r ∈ {1, . . . , k}, s < r such that a

= a s = a s+1 = • • • = a r , A s = A s+1 = • • • = A r ,
and a is not a line;

3) For every j / ∈ R = {s, . . . , r} one has A j = A j±1 and the line A j-1 A j is not tangent to a j at A j , see Fig. 4.

k-2 A a a 1 k k A 1 k-1 A k-1 a k-2 a A Figure 3: Impossible degeneracy of simple tangency: s = k. a A =...=A A s k 1 a 1 a =...=a s k A 2 a 2 A s-1 s-1

Singular analytic distributions

Here we recall definitions and properties of singular analytic distributions. The author believes that the material of the present subsection is known to specialists, but he did not find it in literature. Definition 2.27 Let W be a complex manifold, n = dimW , Σ ⊂ W be a nowhere dense closed subset, m < n. Let D be an analytic field of codimension m vector subspaces D(y) ⊂ T y W , y ∈ W \ Σ. We say that D is a singular analytic distribution of codimension m (dimension n -m) with the singular set Σ = Sing(D), if it extends analytically to no point in Σ and each x ∈ W has a neighborhood U where there exists a finite collection Ω of holomorphic 1-forms such that D(y) = {v ∈ T y W | Ω(v) = 0} for every y ∈ U \ Σ. (This generalizes the definition of a codimension one singular holomorphic foliation [4, p.11]. A similar definition in smooth case can be found in [13, p.8].) Remark 2.28 Every k-dimensional singular analytic distribution on a complex manifold W is defined by a meromorphic 3 section of the Grassmanian k-subspace bundle Gr k (T W ) of the tangent bundle T W , and vice versa: each meromorphic section defines a k-dimensional singular analytic distribution. Its singular set is an analytic subset in W of codimension at least two, being the indeterminacy locus of a meromorphic section of a bundle with compact fibers.

Example 2.29 Let M be a complex analytic manifold, N ⊂ M be a connected complex submanifold, D be a (regular) analytic distribution on M . The intersection D| N (x) = T x N ∩ D(x) with x ∈ N has constant and minimal dimension on an open and dense subset N 0 ⊂ N . The subspaces D| N (x) ⊂ T x N form a singular analytic distribution D| N on N that is called the restriction to N of the distribution D. Its singular set is contained in the complement N \ N 0 : the set of those points x, where the above dimension is not minimal. The restriction to N of a singular analytic distribution D on M with N ⊂ Sing(D) is defined analogously; it is also a singular analytic distribution on N whose singular set is contained in the union of the intersection Sing(D) ∩ N and the set of those points x ∈ N , where the above dimension dim(D| N (x)) is not minimal.

Example 2.30 Let D be a singular analytic distribution on a complex manifold W . Let M be another connected complex manifold, and let φ : M → W be a non-constant holomorphic mapping such that φ(M ) ⊂ Sing(D). For every x ∈ M set

φ * D(x) = (dφ(x)) -1 (D(φ(x)) ∩ dφ(x)(T x M )) ⊂ T x M.
The subspaces φ * D(x) form a singular analytic distribution on M called the pullback distribution. In the case, when φ is an immersion on an open and dense subset, the dimension of the distribution φ * D equals the minimal dimension of the above intersection.

Convention 2.31 Let W be a complex manifold, M ⊂ W be an analytic subset. Everywhere below for simplicity we say that a subset N ⊂ M is analytic, if it is an analytic subset of the ambient manifold W . reg is the union of the set M sing and the set of those points x ∈ M reg where the distribution D| M does not extend analytically. The distribution D| M is also called the restriction to M of the distribution D. The restriction of a singular analytic distribution D| M to an irreducible analytic subset V ⊂ M , V ⊂ Sing(D| M ) is a singular analytic distribution on V defined analogously: it coincides with D| V . In the case, when the analytic set M is a union of several irreducible components, the restriction of the distribution D to each component will be referred to, as a singular distribution on M (which may have different dimensions on different components).

Example 2.33 The Birkhoff distribution D k introduced at the end of Section 1 extends to a singular analytic distribution on the closure R 0,k ⊂ P k . Definition 2.34 An integral l-surface of a singular analytic distribution D on an analytic variety5 M is a holomorphic connected l-dimensional surface S ⊂ M lying outside the singular set of D such that T x S ⊂ D(x) for every x ∈ S. An m-dimensional singular analytic distribution is integrable, if there exists an integral m-surface through each its regular point.

Remark 2.35

The singular set of a singular analytic distribution is always an analytic subset in the ambient variety (see Convention 2.31), as in Remark 2.28. In general an integral surface is not an analytic set. Indeed, a generic linear vector field on CP 2 has transcendental orbits. Hence, they are not analytic subsets in CP 2 , by Chow's Theorem [11, p.167]. Definition 2.36 Let M be an irreducible analytic subset in a complex manifold V . A p-dimensional intrinsic singular analytic distribution on M is a meromorphic section D : M → Gr p (T V )| M of the Grassmanian bundle (see Footnote 3) such that D(x) ⊂ T x M for regular points x ∈ M where D is holomorphic.

Remark 2.37 Each singular analytic distribution is an intrinsic one. Conversely, each intrinsic singular analytic distribution is transformed to a usual singular analytic distribution by a bimeromorphic mapping. Namely, consider the bundle projection π : Gr p (T V ) → V and the graph Γ of the section D (which is an analytic subset in Gr p (T V )). Let F be the tautological distribution on Gr p (T V ): for every λ ∈ Gr p (T V ) the subspace F(λ) ⊂ T λ Gr p (T V ) is the preimage of the subspace λ ⊂ T π(λ) V under the differential dπ(λ). The restriction F| Γ is a singular holomorphic distribution transformed to D by the bimeromorphic projection π : Γ → M . Proposition 2.38 Let an m-dimensional (intrinsic) singular analytic distribution D on an analytic subset N in a complex manifold have at least one m-dimensional integral surface. Given an arbitrary union S of mdimensional integral surfaces, let M ⊂ N denote the minimal analytic subset in N containing S. Then the restriction D| M is an integrable m-dimensional singular analytic distribution.

Proof It suffices to prove the proposition for a true singular analytic distribution, locally defined as kernel field of a system of holomorphic 1-forms (Remark 2.37). The set {x ∈ M 0 reg | D(x) ⊂ T x M } coincides with all of M 0 reg , since it contains S ∩ M 0 reg and its closure is an analytic subset in M . Similarly, the set of those points in M 0 reg where the distribution D| M satisfies the Frobenius integrability condition coincides with all of M 0 reg , since it contains S and its closure is an analytic subset in M . Thus, D| M is an m-dimensional integrable distribution. The proposition is proved. 2

Birkhoff distributions and periodic orbits

Here we recall the definition and basic properties of Birkhoff distribution and its restricted versions. Consider the space P = P(T CP 2 ), which consists of pairs (A, L), A ∈ CP 2 , L being a one-dimensional subspace in T A CP 2 . Its natural projection to CP 2 will be denoted by Π. The standard contact structure is the two-dimensional analytic distribution H on P given by the dΠ-pullbacks of the lines L:

H(A, L) = (dΠ(A, L)) -1 (L) ⊂ T (A,L) P.
The distribution

H k = ⊕ k j=1 H is the 2k-dimensional product distribution on P k . Recall that R 0,k ⊂ P k is the subset of k-tuples ((A 1 , L 1 ), . . . , (A k , L k )) such that for every j = 1, . . . , k one has A j ∈ C 2 = CP 2 \ C ∞ , A j = A j±1
, the lines A j A j-1 , A j A j+1 are symmetric with respect to the line L j , and the three latter lines are distinct and non-isotropic. This is a 2kdimensional smooth quasiprojective variety. The Birkhoff distribution D k is the restriction to R 0,k of the product distribution H k :

D k (x) = T x R 0,k ∩ H k (x) for every x ∈ R 0,k . (2.1)
This is a k-dimensional analytic distribution. It is the complexification of the real Birkhoff distribution introduced in [START_REF] Baryshnikov | Billiards and nonholonomic distributions[END_REF]. For every two irreducible analytic curves a, b ⊂ CP 2 with maximal normalizations π a : â → CP 2 , π b : b → CP 2 we will denote

P a = â × P 3 ; P ab = â × b × P 2 .
We consider the natural embeddings η a : P a → P 4 , η ab : P ab → P 4 :

η a (A, (B, L B ), (C, L C ), (D, L D )) = ((π a (A), T A a), (B, L B ), (C, L C ), (D, L D )), η ab (A, B, (C, L C ), (D, L D )) = ((π a (A), T A a), (π b (B), T B b), (C, L C ), (D, L D )).
Remark 2.39 The critical points of the mappings η a (η ab ) are contained in the sets Cusp a ⊂ P a , Cusp ab ⊂ P ab of those points for which A (A or B) is a cusp of the corresponding curve (see Footnote 1 in Section 1). The mappings η a and η ab are immersions outside the sets Cusp a and Cusp ab .

Consider the subsets

M 0 a = η -1 a (R 0,4 ) \ Cusp a ⊂ P a , M 0 ab = η -1 ab (R 0,4 ) \ Cusp ab ⊂ P ab , (2.2)
M a = M 0 a ⊂ P a ; M ab = M 0 ab ⊂ P ab : the closures are taken in the usual topology. The subsets M a ⊂ P a and M ab ⊂ P ab are obviously analytic. The restricted (pullback) Birkhoff distributions D a on M 0 a and D ab on M 0 ab respectively are the pullbacks of the Birkhoff distribution D 4 :

D a (x) = (dη a (x)) -1 (D 4 (η a (x)) ∩ dη a (x)(T x M 0 a )) ⊂ T x M 0 a , x ∈ M 0 a ; (2.3) D ab (x) = (dη ab (x)) -1 (D 4 (η ab (x)) ∩ dη ab (x)(T x M 0 ab )) ⊂ T x M 0 ab , x ∈ M 0 ab .
(2.4) They extend to singular analytic distributions on M a and M ab respectively in the sense of Subsection 2.6. For example, D a is the restriction to M a of the distribution T â ⊕ H 3 on P a = â × P 3 . One has

dimM a = 6, dimD a = 3; dimM ab = 4, dimD ab = 2. Definition 2.40 (complexification of [10, definition 14]) Let k ∈ N. A k- gon A 1 . . . A k ∈ (CP 2 ) k is said to be non-degenerate, if for every j = 1, . . . , k (we set A k+1 = A 1 , A 0 = A k ) one has A j ∈ C 2 = CP 2 \ C ∞ , A j+1 = A j , A j-1 A j = A j A j+1
and the line A j A j+1 is not isotropic. We will call the complex lines A j A j±1 the edges adjacent to A j .

Remark 2.41

The above sets R 0,k , M 0 a , M 0 ab are projected to the sets of non-degenerate k-gons (quadrilaterals). A periodic billiard orbit in the sense of Definition 1.5 is non-degenerate, provided that its vertices lie in C 2 , its edges are non-isotropic and every two adjacent edges are distinct. The kreflective set U of a k-reflective billiard contains an open and dense subset U 1 ⊂ U of those non-degenerate orbits whose vertices are not marked points (see Definition 2.22) of the corresponding mirrors. The proposition is the direct complexification of an analogous result from [START_REF] Baryshnikov | Billiards and nonholonomic distributions[END_REF] and Yu.G.Kudryashov's lemmas [10, section 2, lemmas [START_REF] Plakhov | Exterior billiards. Systems with impacts outside bounded domains[END_REF][START_REF] Plakhov | Invisibility in billiards[END_REF].

Everywhere below for every

x ∈ M 0 ab ⊂ P ab = â × b × P 2 we denote l a = l a (x) = A(x)D(x), l b = l b (x) = B(x)C(x).
Remark 2.44 The lines l a and l b depend only on (A, B) = (A(x), B(x)): these are the lines symmetric to AB with respect to the lines T A a and T B b respectively. Sometimes we will write l a = l a (A, B), l b = l b (A, B).

For every x ∈ M 0 ab (x ∈ M 0 a ) the corresponding lines L G , G = (B, )C, D, will be denoted by L G (x). The projections to the positions of vertices will be denoted by ν a : P a → â, ν ab :

P ab → â × b, ν G : P a , P ab → CP 2 , x → G(x) for G = B, C, D (respectively, G = C, D).
Remark 2.45 The above projections ν a and ν ab are proper and epimorphic. The corresponding preimages of points are compact and naturally identified with projective algebraic varieties.

Proposition 2.46

The planes of the Birkhoff distribution D ab on M 0 ab are non-trivial, and hence, so is each its integral surface.

Proof Suppose the contrary: there exists an x ∈ M 0 ab such that the differential of some of the above projections, say ν a vanishes identically on D ab (x). We consider only the case of projection ν a : the cases of other projections are treated analogously. Then the kernel

K(x) = Ker(dν b | D ab (x) ) ⊂ D ab (x)
ĝ : Ŝ → ĝ, g = c, d: ν G • π S = π g • ν ĝ on Ŝ (Corollary 2.4
). This yields an "inverse" mapping At the end of the section we prove Theorem 1.9 and Corollary 3.5. Both of them will be used further on in the proof of Theorem 1.10.

Ψ -1 = (ν ab • π S ) × ν ĉ × ν d : Ŝ → â × b × ĉ × d. Its image is contained in U ,
Plan of the proof of Lemma 3.1. Recall that a and b are not both lines. In the case, when they are both algebraic curves, there exist at most unique analytic curves c and d such that the billiard a, b, c, d is 4-reflective, and if they exist, they are algebraic (Proposition 2.1 and Theorem 1.7 in the algebraic case, see Remark 1.8). Thus, the only integral surfaces of the distribution D ab are given by the open set of its quadrilateral orbits, by Propositions 2.43 and 2.46. Moreover, the latter orbit set is a Zariski open dense subset in a projective algebraic surface. This immediately implies the statement of Lemma 3.1. Everywhere below we consider that some of the curves a or b is transcendental and prove the lemma by contradiction. Suppose the contrary to Lemma 3.1: there exists a three-or four-dimensional irreducible analytic subset M ⊂ P ab contained in M ab such that M ∩ M 0 ab = ∅ and the restriction D M to M 0 = M ∩ M 0 ab of the distribution D ab is two-dimensional and integrable. (In the second case M = M ab .) The complement

Σ 0 = M \ M 0 = M \ M 0 ab ⊂ M (3.1)
is an analytic subset of positive codimension in M , and M 0 is dense in M . Thus, every x ∈ M 0 where D M is regular is contained in an integral surface, and the latter is formed by quadrilateral orbits of a 4-reflective billiard a, b, c(x), d(x) (Propositions 2.43 and 2.46). We show that there exists an x ∈ M 0 such that the corresponding mirrors c(x), d(x) are algebraic. This together with Proposition 2.1 implies that a and b are algebraic. The contradiction thus obtained will prove Lemma 3.1.

For the proof of Lemma 3.1 we study the projections of the set M to the positions of three vertices (A, B, D): set

ν ab,D : x → (A(x), B(x), D(x)); M D = ν ab,D (M ) ⊂ â × b × CP 2 .
Analogous projections and spaces are defined with D replaced by C.

Case of a non-bimeromorphic projection

Here we prove Lemma 3.1 in the case, when some of the projections ν ab,C , ν ab,D , say ν ab,D is not bimeromorphic. Its proof is based on the following proposition. Thus, the 4-reflective billiard a, b, c(x), d(x) has two neighbor algebraic mirrors c(x) and d(x). Hence, a and b are also algebraic, by Proposition 2.1. This contradicts the non-algebraicity assumption and proves Lemma 3.1.

Case of bimeromorphic projections

Here we prove Lemma 3.1 in the case, when both projections ν ab,C , ν ab,D are bimeromorphic. To do this, we show that for every x ∈ M \ Σ both mirrors d(x) and c(x) are lines. We then get a contradiction as above.

It suffices to prove the above statement for the mirrors d(x) only. We first show (the next proposition) that the lines L D (x) ⊂ T D(x) CP 2 locally depend only on D(x) and form a holomorphic line field. Afterwards we show (Proposition 3.12) that the restriction of the latter line field to each projective line is tangent to a pencil of lines through the same point. This easily implies that its integral curves are lines (Proposition 3.13).

There exists an open and dense subset of points x ∈ M \ Σ where the germ of projection 12 for an open set of projective lines, e.g., for all the lines l a (A, B), (A, B) ∈ Reg. Suppose the contrary: for some (A, B) ∈ Reg the restriction to l a = l a (A, B) of the line field Λ D is not tangent to a pencil of lines, i.e., the curve Γ = Γ A,B has degree at least two. Note that the family of lines

λ(D) = DT A,B (D), D ∈ l a = l a (A, B) is a rational curve λ ⊂ CP 2 * parametrized by D ∈ l a .
Claim 1. The curve λ has degree at least three. Proof Let Q denote the intersection point of the infinity line C ∞ ⊂ CP 2 with non-isotropic line l a ; thus, Q = I 1,2 . Let us take a point P ∈ C ∞ \ {Q, I 1 , I 2 } and a finite line l through P . There are two symmetry lines of the pair of lines l a , l. They intersect the infinity line at distinct points E 1 , E 2 ∈ C ∞ depending only on P : E j = E j (P ). For a generic P ∈ C ∞ for every j = 1, 2 there are at least two distinct points D j , D j ∈ l a such that the projective lines Λ D (D j ), Λ D (D j ) pass through E j = E j (P ), since degΓ ≥ 2. Either the four points D 1 , D 1 , D 2 , D 2 are distinct, or at most two of them coincide: the latter happens exactly in the case, when, Λ D

(Q) = C ∞ ⊃ {E 1 , E 2 }; then D 1 = D 2 = Q.
This implies that there are always at least three distinct points D 1 , D 2 , D 3 ∈ l a such that each line Λ D (D i ), i = 1, 2, 3, passes through some of the points E j ; thus, P ∈ λ(D i ). The corresponding points λ(D i ) ∈ CP 2 * of the curve λ lie in the projective line P * , by construction. This proves the claim. 2

Claim 2. The curve λ has degree at most two. Proof Let X denote the point of intersection of the lines l a and l b , Y = T A,B (X) ∈ l b . For every point C ∈ l b \{Y } the curve λ intersects the dual line C * in at most two points with multiplicity 1: at the points λ(T -1

A,B (C)) and may be λ(X) (if λ(X) = l b ). The fact that in the latter case the multiplicity of the intersection point λ(X) equals one follows from the assumption that C = Y : the lines λ(D) with D close to X asymptotically focus at Y = C and are transverse to l a . This proves the claim.

2

Claims 1 and 2 contradict each other. This proves Proposition 3.12. 2 Proposition 3.13 The phase curves of the line field Λ D form a pencil of projective lines through the same point.

Proof Suppose the contrary: some phase curve S is not a line. Consider a projective line L = Λ D (P ) tangent to S at some point P . Then for every Q, R ∈ L, Q = R, close to P and distinct from it the projective lines Λ D (Q) and Λ D (R) are distinct from the line L = Λ D (P ) and intersect each other at a point outside the line L. Therefore, the lines of the restriction to L of the field Λ D are not tangent to a pencil of lines through the same point. The contradiction thus obtained to Proposition 3.12 proves Proposition 3.13. 2

Proposition 3.13 applied to both Λ C and Λ D implies that for every x ∈ M \ Σ the mirrors c(x) and d(x) are lines. Hence, a and b are algebraic (Proposition 2.1), -a contradiction. Lemma 3.1 is proved.

Case of one algebraic mirror. Proof of Theorem 1.9

In the present subsection we consider that a, b, c, d is a 4-reflective analytic planar billiard, and the mirror a is algebraic. Without loss of generality we consider that the curves b and d are transcendental: in the opposite case Theorem 1.9 follows immediately from Proposition 2.1 and [7, theorem 1.11]. As it is shown below, Theorem 1.9 is implied by the following proposition. This implies that either a is a line and B, D B are symmetric with respect to a for every B ∈ b, or a is a conic with one-dimensional family of foci pairs (B, D B ), see [7, proposition 2.32]. The latter case being obviously impossible, the curves b, d are symmetric with respect to the line a. Applying the above argument to the algebraic mirror c instead of a, we get that B and D B are symmetric with respect to the line c. Thus, the above pairs (B, D B ) are symmetric with respect to both lines a and c, by construction. Therefore, a = c = b, d, and the billiard is of type 1) from Theorem 1.7. Theorem 1.9 is proved.

Intersected neighbor mirrors. Proof of Corollary 3.5

In the conditions of Corollary 3.5 no mirror is a line, by Theorem 1.9 and since a, b are not lines. Without loss of generality we consider that each mirror is transcendental, since otherwise, a = c and a = b, by Theorem 1.9. Let U be the 4-reflective set. Its projection to â× b is proper and epimorphic, by Corollary 3.4. The projection preimage in U of the pair (A, A) ∈ â × b is a non-empty compact analytic subset Γ ⊂ U of dimension at most one.

Case 1): dimΓ = 1. Then at least one of the curves ĉ, d, say ĉ is a compact Riemann surface, -a contradiction to our non-algebraicity assumption. Thus, this case is impossible.

Case 2): dimΓ = 0: Γ is a finite set.

(A k , B k ) → (A, A) lifts to a converging sequence A k B k C k D k → AACD in U . Let us take two sequences (A k j , B k j ) → (A, A), j = 1,

2, with lines

A k j B k j converging to different limits for j = 1, 2; this is possible, since a = b. We get two sequences of quadrilaterals A k j B k j C k j D k j converging to the same quadrilateral AACD. On the other hand, the lines B k j C k j symmetric to A k j B k j with respect to the tangent lines T 4 Algebraicity: proof of Theorem 1.10 Theorem 1.7, and thus, Theorem 1.10 are already proved in the case, when at least one mirror is algebraic (Theorem 1.9). Here we prove Theorem 1.10 in the general case by contradiction. Suppose the contrary: there exists a 4-reflective billiard a, b, c, d with no algebraic mirrors. We study Birkhoff distribution D a on the space M a and consider its non-trivial integral surface S formed by a connected open set of quadrilateral orbits of the billiard. Recall that M a ⊂ P a is a six-dimensional analytic subset, and D a is a singular three-dimensional distribution on M a , see Subsection 2.7. Set M = the minimal analytic subset in M a containing S. This is an irreducible analytic subset in P a , by definition, see Convention 2.31. The intersections

D M (x) = D a (x) ∩ T x M, x
being a smooth point of the variety M, induce a singular analytic distribution D M on M , for which S is an integral surface. This is either two-or three-dimensional distribution, since dimS = 2 and dimD a = 3. The cases, when dimD M = 2, 3, will be treated separately in Subsections 4.1 and 4.3 respectively.

The methods of proof in both cases are similar. We show that an open set of points x ∈ M lie in integral surfaces corresponding to 4-reflective billiards a, b(x), c(x), d(x) with regularly intersected mirrors a and b(x). Then either b(x) is a line, or c(x) = a, by Corollary 3.5. We then show that either ν C (M ) ⊂ a, or the mirror b of the initial billiard is a line. We get a contradiction in both subcases. In the case, when dimD M = 3, the proof uses Cartan-Kuranishi-Rashevskii involutivity theory of Pfaffian systems. The corresponding background material will be recalled in Subsection 4.2.

The proof of the existence of the above integral surfaces is based on the following key proposition and corollary. They deal with the natural projection ν a : P a → â and its restriction to M , which are proper and epimorphic. For every A ∈ â the projection preimage A ⊂ W A is algebraic, since it is analytic (as is M \ N ) and by Chow's Theorem. Therefore, the complement CP 2 \ν B (W N A ) is contained in an algebraic subset in CP 2 of positive codimension (Proposition 4.1 and Chevalley-Remmert and Chow's Theorems). Its π a -preimage is at most discrete, since a is nonalgebraic. This implies the statement of the corollary. 2

W A = ν -1 a (A) ∩ M is a projective algebraic set.

Case of two-dimensional distribution

In the present section we consider that dimD M = 2. Then the distribution D M is integrable, by Proposition 2.38. We keep the previous notations ν a , ν G , G = B, C, D for the projections to the positions of vertices. Let M 0 reg ⊂ M ∩ M 0 a denote the subset of points regular for both M and D M , cf. Definition 2.32. Set

M = {x ∈ M 0 reg | dν a (x), dν G (x) ≡ 0 on D M (x) for G = B, C, D}. (4.1)
The set M contains the integral surface S, since S is non-trivial. It is an open and dense subset in M , and its complement Σ = M \ M is analytic. For every

A ∈ â set W 0 A = W A ∩ M . Proposition 4.4
There exists an open subset V ⊂ M of those x for which the mirrors a and b(x) intersect at some point non-marked for both their local branches.

Proof

There exists an open dense subset â ⊂ â such that for every A ∈ â the intersection ν B (W 0 A ) ∩ a contains a regular disk α ⊂ a without marked points for α (Corollary 4.2). Fix α and an x 0 ∈ M ∩ ν -1 B (α). The point ν B (x 0 ) ∈ b(x 0 )∩α is a non-marked point of the corresponding local branches of the curves b(x 0 ) and α (Remark 4.3). The latter branches are distinct, by Corollary 3.5. Therefore, there exists a neighborhood V = V (x 0 ) ⊂ M such that for every x ∈ V a regular branch of the curve b(x) intersects α at a non-marked point for both curves (Remark 4.3 and analyticity of the foliation by integral surfaces). This proves the proposition.

2

In Proposition 4.4 for every x ∈ V either c(x) = a, or b(x) is a line, by Corollary 3.5. Hence some of the latter statements holds for all x ∈ M , by analyticity. The mirror b(x) cannot be a line for all x, since the mirror b of the initial billiard is not algebraic by assumption. Hence, ν C (M ) ⊂ a. Let us show that this is impossible. To do this, we use the next proposition. Proof If C ≡ const, then b would be either a line, or a conic, by [7, proposition 2.32], -a contradiction.

2

Suppose, by contradiction, that ν C (M ) ⊂ a. Fix an A ∈ â such that π a (A) is not an isotropic point at infinity and there exists a one-parametric family of quadrilateral orbits ABCD of the initial billiard a, b, c, d with the given vertex A. The subset ν C (W A ) ⊂ CP 2 is non-discrete, by Proposition 4.5. On the other hand, it is an algebraic subset in CP 2 (Remmert's Proper Mapping and Chow's Theorems). It lies in a transcendental curve a. Hence, it is discrete. The contradiction thus obtained proves Theorem 1.10.

Background material: Phaffian systems and involutivity

Everywhere below in the present subsection whenever the contrary is not specified, F is a k-dimensional analytic distribution on an analytic manifold M , F(x) ⊂ T x M are the corresponding subspaces. Definition 4.6 [21, p.290] Let k, l ∈ N, k ≥ l, and let F be as above. A Pfaffian system F k,l is the problem to find l-dimensional analytic integral surfaces of the k-dimensional distribution F. Definition 4.7 [21, p.298] An m-dimensional integral element of the distribution F is an m-dimensional vector subspace E m (x) ⊂ F(x) such that for every 1-form ω on the ambient manifold vanishing on the subspaces of the distribution F its differential dω vanishes on E m (x). Definition 4.8 [21, p.300] A Pfaffian system F k,l is in involution (or briefly, involutive), if for every x ∈ M , p < l each p-dimensional integral element in T x M is contained in some (p + 1)-dimensional integral element. Example 4.9 A tangent subspace to an integral surface is an integral subspace. Every Pfaffian system defined by a Frobenius integrable distribution is involutive.

Remark 4.10 For every p-dimensional integral element E p (x) the set of ambient (p + 1)-dimensional integral elements E p+1 (x) ⊃ E p (x) either is empty, or consists of a unique integral element, or is a projective space: a pencil of (p + 1)-dimensional subspaces through E p (x), which saturate a linear subspace containing E p (x), see [21, formula (58.13), p.299]. Definition 4.11 [21, p.306] Let F be an analytic distribution on a connected manifold. A p-dimensional integral element E p (x) is said to be nonsingular, if the space of ambient (p + 1)-dimensional integral elements E p+1 (x) ⊃ E p (x) has minimal dimension. Theorem 4.12 (a version of Cauchy-Kovalevskaya Theorem; implicitly contained in [21, section 60]). Let an analytic Pfaffian system F k,l on a manifold M be involutive. Let p ≤ l, Γ ⊂ M be a (p -1)-dimensional analytic integral surface such that all its tangent spaces be nonsingular (p -1)dimensional integral elements. Then for every x ∈ Γ there exists a germ of p-dimensional integral surface through x that contains the germ of Γ at x. Definition 4.13 [12, p.188]. A subset N of a complex manifold V is called analytically constructible, if each point of the manifold V has a neighborhood U such that the intersection N ∩ U is a finite union of subsets defined by finite systems of equations f j = 0 and inequalities g i = 0; f j and g i are holomorphic functions on U .

Recall that for a singular analytic distribution D M on an irreducible analytic subset M in a complex manifold V by M 0 reg ⊂ M we denote the open and dense subset of points regular both for M and D M ; the complement M \ M 0 reg ⊂ V is an analytic subset.

Proposition 4.14 Let D M be a singular analytic distribution on an irreducible analytic subset M in a complex manifold V . Its nonsingular onedimensional integral elements form an open dense subset in the projectivized bundle P(T M reg ) that is an analytically constructible subset in P(T V ).

Proof Fix a point x 0 ∈ M . Let f 1 , . . . , f r be holomorphic functions on a neighborhood

U = U (x 0 ) ⊂ V that define M : M ∩U = {f 1 = • • • = f r = 0}, T x M = Ker(df 1 (x), .
. . , df r (x)) for every x ∈ M reg ∩ U . Let ω 1 , . . . , ω s be holomorphic 1-forms on U that define a singular analytic distribution whose restriction to M ∩U coincides with D M . A non-zero vector v ∈ T V generates a non-singular integral element, if and only if its base point x = x(v) lies in M 0 reg , (df i )(v) = 0 for i = 1, . . . , r, ω j (v) = 0 for j = 1, . . . , s and the system of (r + s) 1-forms df i (x), i v (dω j )(x), i = 1, . . . , r, j = 1, . . . , s has maximal rank. This implies the statements of the proposition. 2

Remark 4.15 For every singular analytic distribution F on an analytic set M in a complex manifold V the set of its two-dimensional integral elements (integral planes) is an analytically constructible subset in Gr 2 (T V )| M . The proof of this statement is analogous to the above proof of Proposition 4.14. Therefore, its image under the projection to M is also analytically constructible (Chevalley-Remmert Theorem).

Proposition 4.16 Every three-dimensional singular analytic distribution F on an irreducible analytic variety M satisfies one of the two following incompatible statements: 1) either the corresponding Pfaffian system F 3,2 on M 0 reg is involutive; 2) or there exists a proper analytic subset Σ ⊂ M such that a) either for every x ∈ M \ Σ the space F(x) contains no integral plane; b) or for each x ∈ M \Σ the space F(x) contains a unique integral plane.

Proof If F(x) contains some two distinct integral planes P 1 , P 2 , then it contains a pencil of integral planes through the line L = P 1 ∩ P 2 , which saturate the whole three-dimensional space F(x), by Remark 4.10. Thus, each F(x) either is a union of integral planes, or contains a unique integral plane, or contains no integral planes. This together with Remark 4.15 and Chevalley-Remmert Theorem easily implies that M 0 reg is a disjoint union of three analytically constructible subsets of points x ∈ M where F(x) satisfy respectively one of the three latter statements. One of them contains a complement to a proper analytic subset Σ ⊂ M . Therefore, either F(x) is a union of integral planes for all x ∈ M \ Σ (and hence, for all x ∈ M 0 reg by analyticity) and the Pfaffian system F 3,2 is involutive, or some of statements a) or b) of Proposition 4.16 holds. This proves Proposition 4.16. 2

Case of three-dimensional distribution

Here we consider that dimD M = 3. The subcase, when the Pfaffian system (D M ) 3,2 is non-involutive, is reduced to the two-dimensional case by Proposition 4.16. Indeed, the image of the projection to M of the set of integral planes is analytically constructible, by Remark 4.15, and contains the integral surface S. Therefore, it contains an open dense complement to an analytic subset in M , since M is the minimal analytic set containing S.

Hence, if the system is not involutive, then there exists a proper analytic subset Σ ⊂ M such that for every x ∈ M \ Σ the space F(x) contains a unique integral plane (Proposition 4.16). The latter integral planes form a two-dimensional intrinsic singular analytic distribution on M , by Definition 2.36 and Remark 4.15. Then we apply the arguments from Subsection 4.1 to the latter two-dimensional distribution instead of D M . Thus, without loss of generality we consider that the Pfaffian system (D M ) 3,2 is involutive. Corollary 4.2 easily implies the next proposition and corollary, which state that there exist a connected open subset V ⊂ M 0 reg , a regular analytic hypersurface V a ⊂ V , ν B (V a ) ⊂ a, and an analytic field L of one-dimensional nonsingular integral elements in D M on V whose all complex orbits intersect V a transversely (plus a mild genericity condition (4.2)). The germ through every x ∈ V of complex orbit is included into an integral surface of the distribution D M , by Theorem 4.12. Condition (4.2) implies that the integral surface is non-trivial, and hence, corresponds to an open set of quadrilateral orbits of a 4-reflective billiard a, b(x), c(x), d(x). If x ∈ V a , then the mirrors a and b(x) intersect at ν B (x), and we deduce that either c(x) = a, or b(x) is a line (Corollary 3.5). This easily implies that either ν C (M ) ⊂ a, or the image under the projection ν B of every analytic curve tangent to D M is a line. We show that none of the latter cases is possible. The contradiction thus obtained will prove Theorem 1.10.

Let M ⊂ M 0 reg be the subset from (4.1) defined for our three-dimensional distribution D M . It is open, dense and the complement M \M is analytic, as at the same place. By definition, dν a , dν G ≡ 0 on D M (x) for every x ∈ M . Proposition 4.17 There exist an x ∈ M and a one-dimensional nonsingular integral element L x ⊂ D M (x) such that ν B (x) ∈ a and

(dν a (x))(L x ) = 0, (dν G (x))(L x ) = 0 for every G = B, C, D; (4.2) the line (dν B (x))(L x ) is transverse to T ν B (x) a. (4.3)
Proof Let Q denote the set of one-dimensional nonsingular integral elements L x ⊂ D M (x), x ∈ M , satisfying (4.2). Let Q ⊂ M denote its projection to M . The sets Q and Q are analytically constructible subsets in P(T P a ) and M . They are open and dense subsets in P(D M ) and M respectively. The two latter statements follow from definition, Proposition 4.14 and Chevalley-Remmert Theorem. The intersection ν B (Q) ∩ a contains a regularly embedded disk α ⊂ a without isotropic tangent lines, and

V a = ν -1 B (α) ∩ Q is a hypersurface (Corollary 4.2). Claim. D M (x) ⊂ T x V a for an open dense set of points x ∈ V a .
Proof Suppose the contrary: each germ of integral curve (and hence, surface) of the distribution D M through each point in V a lies in V a . Fix an x ∈ V a , a nonsingular integral element L x ⊂ D M (x) satisfying (4.2), a germ of integral curve Γ tangent to L x and a germ of integral surface Ŝ containing Γ (given by Theorem 4.12). The surface Ŝ lies in V a , is non-trivial by (4.2), and hence, represents an open set of quadrilateral orbits of a 4-reflective billiard with two coinciding non-linear mirrors a = b(x). But the latter billiard cannot exist by Corollary 3.5. This proves the claim.

2

For every x ∈ V a such that D M (x) ⊂ T x V a a generic one-dimensional integral element L x ⊂ D M (x) is nonsingular and satisfies conditions (4.2), (4.3). This proves the proposition. 2

Corollary 4. [START_REF] Plakhov | Invisibility in billiards[END_REF] There exist an open subset V ⊂ M , a regularly embedded disk α ⊂ a without isotropic tangent lines, an analytic hypersurface V a ⊂ V with ν B (V a ) ⊂ α and an analytic line field L on V contained in D M and transverse to V a such that the lines of the field L are nonsingular integral elements satisfying inequalities (4.2) and each its complex orbit in V intersects V a .

The corollary follows immediately from the proposition and openness of the set of nonsingular integral elements satisfying (4.2). Proposition 4.19 Let V , V a and L be as in the above corollary. Then there are two possible cases:

Case 1): ν C (V ) ⊂ a; Case 2): the projection ν B sends complex orbits of the field L to lines.

Proof For every x ∈ V the germ of the orbit of the field L through x lies in a germ of integral surface of the distribution D M (Theorem 4.12). The latter surface is non-trivial by the inequalities from (4.2), and hence, is given by an open set of quadrilateral orbits of a 4-reflective billiard a, b(x), c(x), d(x). If x ∈ V a , then the mirrors a and b(x) intersect at the point B(x) = ν B (x), and the latter is not marked for their corresponding local branches. This follows from construction and the inequalities from (4.2). Hence, for every x ∈ V a either c(x) = a, or b(x) is a line (Corollary 3.5). Each orbit of the field L intersects V a , by assumption. Hence, either the projection ν C sends it to a, or the projection ν B sends it to a line. One of the two latter statements holds for all the orbits, by analyticity. This proves the proposition. 2

Now for the proof of Theorem 1.10 it suffices to show that none of the cases from the above proposition is possible.

Case 1): ν C (V ) ⊂ a. Then ν C (M ) ⊂ a, and we get a contradiction, as at the end of Subsection 4.1. Hence, Case 1) is impossible.

Case 2): ν C (M ) ⊂ a. Let V and L be the same, as in the above corollary. Let us deform L. The set of line fields L satisfying the conditions of the corollary is open in the space of line fields contained in the distribution D M . This together with the corollary implies that for every line field L contained in D M the projection ν B sends each its complex orbit to a line. This is equivalent to say that each analytic curve in M 0 reg tangent to D M is sent to a line by ν B . In particular, this holds for every one-parametric family of quadrilateral orbits lifted to M of the initial billiard a, b, c, d with variable B ∈ b. This implies that the curve b is a line. The contradiction thus obtained proves Theorem 1.10. The proof of Theorem 1.7 is complete.

Applications to real pseudo-billiards

In Subsection 5.1 we introduce and classify the germs of 4-reflective C 4smooth real planar pseudo-billiards. The proof of the classification Theorem 5.6 is presented in the same subsection (analytic case) and in Subsection 5.2 (smooth case). In the same Subsection 5.2 we prove Theorem 5.8 showing that there are no C 4 -smooth pseudo-billiards with only two skew reflection laws at some neighbor mirrors and a positive measure set of 4-periodic orbits. In Subsections 5.3, 5.4 we present applications of Theorems 5.6 and 5.8 respectively to Tabachnikov's Commuting Billiard Conjecture and 4-reflective Plakhov's Invisibility Conjecture.

Classification of real planar 4-reflective pseudo-billiards

Here by real smooth (analytic) curve in R 2 or RP 2 we mean the image of either R, or S 1 under a locally non-constant smooth (analytic) mapping to R 2 (respectively, RP 2 ). A smooth (analytic) germ of curve is given by a smooth (analytic) germ of immersion (R, 0) → R 2 ((R, 0) → RP 2 ). Definition 5.1 [7, remark 1.6] Let a line L ⊂ R 2 and a triple of points A, B, C ∈ R 2 be such that B ∈ L, A, C / ∈ L and the lines AB, BC are symmetric with respect to the line L. We say that the triple A, B, C and the line L satisfy the usual reflection law, if the points A and C lie on the same side from the line L. Otherwise, if they are on different sides from the line L, we say that the skew reflection law is satisfied.

Example 5.2 In every planar billiard orbit each triple of consequent vertices satisfies the usual reflection law with respect to the tangent line to the boundary of the billiard at the middle vertex. Definition 5.3 (cf. [7, definition 6.1]) A real planar pseudo-billiard is a collection of k curves a 1 , . . . , a k ⊂ R 2 called mirrors with a prescribed reflection law on each curve a j : either usual, or skew. Its k-periodic orbit is a k-gon A 1 . . . A k , A j ∈ a j , such that for every j = 1, . . . , k one has A j = A j±1 , A j A j±1 = T A j a j and the lines A j A j-1 , A j A j+1 are symmetric with respect to the tangent line T A j a j so that the triple A j-1 , A j , A j+1 and the line T A j a j satisfy the reflection law corresponding to a j . Here we set a k+1 = a 1 , A k+1 = A 1 , a 0 = a k , A 0 = A k . A real pseudo-billiard is called (piecewise) analytic/smooth, if so are its curves. A germ of real pseudo-billiard is a collection of k germs of curves (a j , A j ) with prescribed reflection laws for which the marked k-gon A 1 . . . A k is a k-periodic orbit. A germ of pseudo-billiard is called k-reflective, if the set of its k-periodic orbits has non-empty interior: contains a two-parameter family including A 1 . . . A k . A (germ of) pseudobilliard is called measure k-reflective, if the set of its k-periodic orbits has positive two-dimensional Lebesgue measure. This means that in the set of k-orbits A 1 . . . A k that satisfy the corresponding reflection laws at A j for all j = 1, k the set of k-periodic ones (i.e., those satisfying the reflection laws at A 1 , A k ) has positive two-dimensional Lebesgue measure. Remark 5.4 A k-reflective pseudo-billiard is automatically measure k-reflective. In the analytic case measure k-reflectivity is equivalent to k-reflectivity. The interior points of the set of k-periodic orbits will be called k-reflective orbits, cf. definition 6.1 in loc. cit. The complexification of each k-reflective planar analytic pseudo-billiard is a k-reflective complex billiard. In every 4-reflective orbit the reflection law at each pair of opposite vertices is the same; it is skew for at least one opposite vertex pair.

Addendum 1. In every pseudo-billiard of type 1) from Theorem 5.6 each quadrilateral orbit ABCD has the same type, as at Fig. 7. It is symmetric with respect to the line a, and the reflection law at A, C is skew. The reflection law at B, D is either usual at both, or skew at both.

Addendum 2 [7, addendums 2, 3 to theorem 6.3]. In pseudo-billiards of types 2), 3) the 4-reflective orbits have the same types, as at Fig. 89101112.

Remark 5.7 The main result of paper [START_REF] Glutsyuk | No planar billiard possesses an open set of quadrilateral trajectories[END_REF] (theorem 2) concerns usual real planar billiards with piecewise C 4 boundary; the reflection law is usual. It implies that the quadrilateral orbit set has empty interior. This statement also follows from the last statement of Theorem 5.6. Here we prove Theorem 5.6 for analytic pseudo-billiards. The general smooth case will be treated in the next subsection. We also prove the following theorem there, which will be applied to Plakhov's Invisibility Conjecture. Proof Reflections from curves act on the space of oriented lines in R 2 , which is a two-dimensional oriented manifold. Skew reflections change the orientation, and the usual ones don't. Therefore, a composition of odd number of skew reflections and a number of usual ones (in any order) is a local diffeomorphism changing the orientation. Hence, it cannot be equal to the identity on a set of positive measure. This proves the proposition.

2

The proposition implies that the only possible reflection law configurations for a potential measure 4-reflective C 4 pseudo-billiard are the following:

1) all the reflection laws are usual; 2) the reflection laws are skew only at some pair of neighbor mirrors; 3) all the reflection laws are skew; 4) the reflection laws are skew only at some pair of opposite mirrors.

Remark 5.10 Configuration 1) is already forbidden: the proof of its impossibility is implicitly contained in [START_REF] Glutsyuk | No planar billiard possesses an open set of quadrilateral trajectories[END_REF]. The proof of Theorem 5.6 given below does not use this result. Configuration 2) is already forbidden in the analytic case, by the last statement of Theorem 5.6 proved in this case.

A k+1 = A 1 , A 0 = A k ) one has A j+1 = A j and A j-1 A j = A j A j+1 .
The subset of degenerate k-gons in R 2k = (R 2 ) k will be denoted by Σ = Σ k .

Remark 5.12 A real non-degenerate k-gon is a complex non-degenerate k-gon in the sense of Definition 2.40.

To each configuration 1)-4) we put into correspondence a distribution D 4 α on R 8 \ Σ constructed below: the Birkhoff distribution corresponding to the chosen reflection laws. To define it, set

Ψ k = {±1} k , Ψ ± k = {α = (α 1 , . . . , α k ) ∈ Ψ k | j α j = ±1}.
For every k ∈ N, α ∈ Ψ k and every non-degenerate k-gon A 1 . . . A k in R 2 we consider the following collection of lines L A j = L A j (α) through A j :

-the line L A j is the exterior bisector of the angle ∠A j-1 A j A j+1 if α j = 1; -the line L A j is its interior bisector if α j = -1.

We will briefly call the configuration ((A 1 , L A 1 ), . . . , (A k , L A k )) α-framed. We identify each line L A j with the corresponding one-dimensional subspace in T A j R 2 . For every

x = A 1 . . . A k ∈ R 2k \ Σ set D k α (x) = ⊕ k j=1 L A j (α) ⊂ T x R 2k .
The latter subspaces define an analytic distribution D k α on R 2k \ Σ. The above reflection law configurations 1)-4) correspond to D 4 α with α ∈ Ψ + 4 . We show that all D k α with α ∈ Ψ + 4 naturally embed to the complex Birkhoff distribution D k on one and the same irreducible component R + 0,k of the variety R 0,k ⊂ P k = (P(T CP 2 )) k (the next proposition). We study the two-dimensional Pfaffian problem for the complex distribution D 4 on R + 0,4 : to find its two-dimensional integral surfaces. We study its first and second Cartan jet prolongations. (Basic theory of Cartan prolongations may be found in [START_REF] Cartan | Les Systèmes Différentiels Extérieurs et Leurs Applications Géométriques[END_REF][START_REF] Kuranishi | Cartan's prolongation theorem of exterior differential systems[END_REF][START_REF] Rashevskii | Geometrical theory of partial differential equations[END_REF]; the background material used in the proofs will be recalled below.) We show (the next two lemmas) that each second prolongation is a two-dimensional distribution outside a union of two subvarieties Λ 0 , Λ 1 . The latter union consists of the quadrilaterals symmetric with respect to some of the diagonals

A 1 A 3 or A 2 A 4 with L A 1 = L A 3 = A 1 A 3 (respectively, L A 2 = L A 4 = A 2 A 4 ).
We then deduce that each C 4 -smooth non-trivial integral surface of the distribution D 4 α should be either analytic or contained in the above set of symmetric quadrilaterals. Therefore, the pseudo-billiard under question should have one of the types 1)-3), by the analytic case of Theorem 5.6 proved in the previous subsection. This will prove Theorem 5.6 in the C 4 case. For the proof of Theorem 5.8 we show that the existence of a measure 4-reflective pseudo-billiard with reflection law configuration 2) implies the existence of a 4-reflective analytic one. This will be deduced from the above-mentioned lemmas and Kudryashov's results from [START_REF] Glutsyuk | No planar billiard possesses an open set of quadrilateral trajectories[END_REF]. But 4-reflective analytic pseudo-billiards with reflection law configuration 2) are forbidden by the last statement of Theorem 5.6, analytic case. The contradiction thus obtained will prove Theorem 5.8. Now let us pass to the proofs. Consider the k-dimensional complex Birkhoff distribution D k on the 2kdimensional smooth quasiprojective variety R 0,k ⊂ P k = (P(T CP 2 )) k introduced at the beginning of Subsection 2.7. For every α ∈ Ψ k there is a natural embedding

j α : R 2k \ Σ → R 0,k : j α (A 1 . . . A k ) = ((A 1 , L A 1 (α)), . . . , (A k , L A k (α))).
One has (j α ) * D k α ⊂ D k , and the latter image is a field of totally real subspaces in the corresponding spaces of the complex distribution D k . Proposition 5.13 For every k ≥ 3 the variety R 0,k consists of two irreducible components R ± 0,k . Each R ± 0,k contains the union of all the images j α (R 2k \ Σ) with α ∈ Ψ ± k . The component R + 0,k consists exactly of those configurations ((A 1 , L A 1 ), . . . , (A k , L A k )) for which the complex lengths |A j A j+1 |, j = 1, . . . , k can be simultaneously normalized so that for every j the lengths |A j-1 A j |, |A j A j+1 | are L A j -concordant, see Definition 2.13. In what follows the restriction of the distribution D 4 to the component R + 0,4 will be denoted D 4 + . Consider the complex Pfaffian system D 4,2 + : the problem to find two-dimensional integral surfaces of the distribution D 4 + . For every x ∈ R + 0,4 let l 1 (x), . . . , l 4 (x) denote the corresponding concordant collection of lengths l j = |A j A j+1 |. The lengths being defined up to simultaneous change of sign, their ratios are single-valued holomorphic functions on R

+ 0,4 . Set Λ = {l 1 l 3 = l 2 l 4 } ⊂ R + 0,4 , Λ 0 = {x ∈ R + 0,4 | A 1 (x)A 2 (x)A 3 (x)A 4 (x) is symmetric with respect to the line A 1 (x)A 3 (x) = L A 1 (x) = L A 3 (x)}, Λ 1 = {x ∈ R + 0,4 | A 1 (x)A 2 (x)A 3 (x)A 4 (x) is symmetric with respect to the line A 2 (x)A 4 (x) = L A 2 (x) = L A 4 (x)}. Remark 5.15 One has Λ 0 , Λ 1 ⊂ Λ. Indeed, if x ∈ Λ 0 , then l 4 (x) = -l 1 (x), l 2 (x) = -l 3 (x)
, by symmetry and length concordance (Definition 2.13). Hence, l 1 (x)l 3 (x) = l 2 (x)l 4 (x) and x ∈ Λ. For every x ∈ Λ 0 the lines L A 2 (x), L A 4 (x) are symmetric with respect to the line L A 1 (x), since the symmetry respects concordance of lengths and transforms the L A 2 (x)-concordant lengths l 1 (x), l 2 (x) to the L A 4 (x)-concordant lengths -l 4 (x), -l 3 (x). Similar statement holds for x ∈ Λ 1 and L A 1 (x), L A 3 (x).

Recall that an integral plane E (see Definition 4.7) of the distribution D 4 + is non-trivial, if for every j the restriction dA j | E is not identically zero. Let I 2 = I 2 (D denote the standard projection. The subspaces

F 3 (x, E) = (dπ gr ) -1 (E) ⊂ T (x,E) I 0 2 form a (singular) analytic distribution F 3 on I 0 2 . Set J 2 = I 0 2 \ π -1 gr (Λ) ⊂ I 0 2 .
and thus, j α (q) is contained in only one set Λ j , j = 0, 1. Hence, the germ at q of the surface S should consist of quadrilateral orbits of a pseudo-billiard of type 1) (see Case 2)). Thus, at least one of the neighbor mirror germs (a 1 , A 1 (q)), (a 2 , A 2 (q)) should be a line, while they both have non-zero curvature. The contradiction thus obtained proves Theorem 5.6. 2

Proof of Theorem 5.8. Suppose the contrary: there exists a C 4 -smooth pseudo-billiard where only some two neighbor mirrors, say a 1 , a 2 have skew reflection law and the set of 4-periodic orbits has positive Lebesgue measure.

In more detail, consider the set S of its 3-edge orbits A 1 A 2 A 3 A 4 : the reflection law is required only at A 2 , A 3 but not necessarily at A 1 , A 4 . This is a two-dimensional non-trivial C 3 -smooth surface, which contains a positive measure set of 4-periodic orbits. Thus, S is a C 3 -smooth pseudo-integral surface (in terms of [10, definition 13, p.291]) of the analytic distribution D 4 α , α = (-1, -1, 1, 1). Note that j -1 α (Λ) = ∅. Indeed, for every quadrilateral A 1 A 2 A 3 A 4 ∈ R 8 \ Σ consider the concordant lengths l j = |A j A j+1 | with respect to the interior bisectors at A 1 , A 2 and the exterior bisectors at A 3 , A 4 . The lengths l j , j = 2, 3, 4 have the same signs, while l 1 has opposite sign, see Example 2.15. Therefore, the equality l 1 l 3 = l 2 l 4 defining Λ is impossible. Hence, S ∩ j -1 α (Λ) = ∅. This together with Lemma 5.16 and [10, theorem 28, p.295] implies that D 4 α has an analytic non-trivial integral surface. Therefore, there exists an analytic 4-reflective pseudo-billiard with exactly two skew reflection laws at some neighbor mirrors, -a contradiction to the last statement of Theorem 5.6. Theorem 5.8 is proved. 2

Application 1: Tabachnikov's Commuting Billiard Conjecture

The following theorem solves the piecewise C 4 -smooth case of S.Tabachnikov's conjecture on commuting convex planar billiards [25, p.58]. It deals with two billiards in nested convex compact domains Ω

1 Ω 2 R 2 , set a = ∂Ω 1 , b = ∂Ω 2 .
We consider that both a and b are piecewise C 4 -smooth. For every Ω j consider the corresponding billiard transformation acting on the space of oriented lines in the plane. It acts as identity on the lines disjoint from Ω j . Each oriented line l intersecting Ω j is sent to its image under the reflection from the boundary ∂Ω j at its last intersection point x with ∂Ω j in the sense of orientation: the orienting arrow of the line l at x is directed outside Ω j . The reflected line is oriented by a tangent vector at x directed inside Ω j . This is a continuous dynamical system if the boundary ∂Ω j is smooth and piecewise-continuous (measurable) otherwise.

elliptic. Thus, b is a finite chain of adjacent confocal elliptic arcs. Therefore b is an ellipse: the above sum of distances is continuous on b and constant on each arc, and hence, on all of b. Subcase 1b): some boundary point A of the analytic arc a is singular, the curve a is not analytic at A. b (L). We orient the line Λ so that Λ = σ a (L). We claim that all the lines L * pass through the same point of the curve a. Indeed, the mapping σ a

• σ b = σ b • σ a is singular at each L * , since Λ = σ a (L) = σ a • σ b (L * ), σ b is regular (that is, a local analytic diffeomorphism) at L * and σ a is singular at L = σ b (L * ), as is A. On the other hand, σ b is regular at σ a (L * ), since Λ = σ b • σ a (L * ) and σ -1
b is a local analytic diffeomorphism at Λ: the points of intersection Λ ∩ b are regular and transversal (by convexity). Therefore, σ a is singular at L * , as is σ b • σ a . Hence, either all the lines L * pass through the same singular point A of the curve a (the set of its singular points is totally disconnected), or they are tangent to a. In the latter case one has

σ b • σ a (L * ) = σ b (L * ) = L, while σ a • σ b (L * ) = σ a (L) = L, since L = T A a , (T A a ) ⊥ ,
-a contradiction to commutativity. Hence, the tangency case is impossible, and the lines L * pass through the same point A . Therefore, b is a conical arc confocal to a with foci A, A . Thus, the conic a contains its own focus A, -a contradiction.

Case 2): the curve a is a convex polygon. For every its vertex A each line through A intersecting a C 4 -smooth arc b of the curve b is reflected from b to a line through another vertex A of the polygon a, as in the above discussion. This implies that each C 4 -smooth arc b is a conic with foci A, A . Finally, each vertex of the polygon a is a focus of each C 4 -smooth arc of the curve b. Therefore, a has at most two vertices and cannot be a convex polygon. The contradiction thus obtained proves Theorem 5. [START_REF] Rashevskii | Geometrical theory of partial differential equations[END_REF] Then we repeat this procedure. Let us assume that after a finite number k of reflections the output oriented line coincides with the input line R and will not hit the body any more. Then we say that the body B is invisible for the ray R, see Fig. 13. We call R a ray of invisibility with k reflections. Proof The existence of a measure 4-reflective body as above implies the existence of a measure 4-invisible planar C 4 -smooth pseudo-billiard (Proposition 5.27). This is a measure 4-reflective planar C 4 -smooth pseudo-billiard with skew reflection law at some pair of neighbor vertices and usual reflection law at the other vertices. This contradicts Theorem 5.8. 2

6 General case of complex k-reflective billiards: state of art First let us recall the next conjecture and partial positive results from [START_REF] Glutsyuk | On odd-periodic orbits in complex planar billiards[END_REF].

Conjecture A [8, p.295]. There are no k-reflective complex analytic (algebraic) planar billiards for odd k. Theorem 6.1 [8, p.295]. There are no 3-reflective complex analytic planar billiards. Theorem 6.2 [8, p.295]. For every odd k there are no k-reflective complex algebraic planar billiards whose mirrors avoid isotropic points at infinity. Conjecture B. For every k ≥ 3 there are no k-reflective complex analytic planar billiards a, . . . , a with all the mirrors coinciding with the same irreducible analytic curve a ⊂ CP 2 .

Recently a positive result was proved by the author (paper in preparation) for every irreducible algebraic curve a that either is smooth, or satisfies a mild condition on either singularities, or tangential correspondence. lines are distinct and non-isotropic. (In the case, when j = k, we replace j + s by s, s = 1, 2.) The closure of set of the latter (k -2)-orbits is an analytic subset V j ⊂ i =j,j+1 âi that has only two-dimensional irreducible components. Let U ⊂ â1 ו • •× âk denote the k-reflective set, see Subsection 2.3. For every j = 1, . . . , k let P j denote the product projection

P j : â1 × • • • × âk → r =j,j+1
âr , U j = P j (U ) ⊂ V j . Theorem 6.7 Let a 1 , . . . , a k be a k-reflective complex analytic planar billiard. Let U ⊂ â1 × • • • × âk be its k-reflective set. For every j = 1, . . . , k the set U j = P j (U ) is analytic: a union of irreducible components of the set V j . The projection P j : U → U j is proper and bimeromorphic.

Proof Without loss of generality we prove the theorem for the pair of neighbor indices 1, k: j = k.

1) Properness and analyticity. Let us show that the mapping P k : U → V k is proper: then U k is analytic by Proper Mapping Theorem. 1a) Case, when a 1 , a k are algebraic: the above statements are obvious. 1b) Case, when some of them, say a k is not algebraic. The properness will be deduced from Corollary 3.2. To do this, we consider the space P 1,k = â2 × . . . âk-1 × P 2 equipped with the distribution

H 1,k = T â2 ⊕ • • • ⊕ T âk-1 ⊕ H ⊕ H,
where P = P(T CP 2 ), H is the standard contact plane field on P, see Subsection 2.7. A point of the space P 1,k is a triple A 2 . . . A k-1 , (A 1 , L A 1 ), (A k , L A k ), where L A j ⊂ T A j CP 2 is a one-dimensional subspace. Let R 1,k ⊂ P 1,k denote the analytic variety defined by the conditions that for every j = 1, . . . , k one has A j ∈ C 2 = CP 2 \ C ∞ , A j = A j±1 , the lines A j A j±1 are symmetric with respect to the line T A j a j if 2 ≤ j ≤ k-1 (the line L A j if j ∈ {1, k}), the three lines A j A j±1 , T A j a j (respectively, L A j ) are distinct and non-isotropic. In addition, it is required that A j be not cusps of the curves a j for j = 1, k. The set R 1,k is an analytically constructible smooth variety, and its closure R 1,k is an analytic set. The distribution H 1,k induces a two-dimensional analytic distribution D 1,k on R 1,k that extends to a singular analytic distribution on R 1,k . An open dense subset U 1 ⊂ U lifts to the union of non-trivial integral surfaces of the distribution D 1,k , as in Proposition 2.43. Let M ⊂ P 1,k denote the minimal analytic subset containing all the non-trivial integral surfaces.

Suppose the contrary: the mapping P k : U → V k is not proper. Then dimM ≥ 3: if dimM = 2, then P k is proper, as in the proof of Corollary 3.4. In what follows we fix some at least three-dimensional irreducible component of the set M and denote by M the latter component. The restriction D M to M of the distribution D 1,k is two-dimensional and integrable, by Proposition 2.38. As in Subsection 3.1, there exists an analytic subset Σ ⊂ M with dense complement M 0 = M \ Σ ⊂ M such that D M is analytic on M 0 and its integral surface through each x ∈ M 0 represents an open set of k-periodic orbits of a k-reflective billiard a 1 (x), a 2 , . . . , a k-1 , a k (x). Some integral surface, which we will denote by S, represents a family of k-periodic orbits of the initial billiard. The projection µ 1,k : M → â2 × • • • × âk-1 is proper, and the image N = µ 1,k (M 0 ) ⊂ V k is a purely two-dimensional analytically constructible subset, by Chevalley-Remmert Theorem.

Claim 1.

There exists an open subset of points x ∈ M 0 such that the billiard a 1 , a 1 (x), a k (x), a k is 4-reflective. Proof Fix x 1 ∈ S and x 2 ∈ M 0 \ {x 1 } with p = µ 1,k (x 1 ) = µ 1,k (x 2 ). There exist neighborhoods Y = Y (p) ⊂ N , X 1 = X 1 (p) ⊂ S, X 2 = X 2 (x 2 ) ⊂ M 0 \ X 1 such that µ 1,k projects X 1 and each integral surface of the distribution D M | X 2 diffeomorphically onto Y (let us fix them). Fix an arbitrary x ∈ X 2 , let S(x) denote the integral surface of the distribution D M | X 2 through x. Each y = A 2 . . . A k-1 ∈ Y lifts to two points in X 1 and S(x), which correspond to k-periodic orbits A 1 . . . A k and A 1 A 2 . . . A k-1 A k of the billiards a 1 , . . . , a k and a 1 (x), a 2 , . . . , a k-1 , a k (x) respectively. The quadrilaterals A 1 A 1 A k A k corresponding to generic y ∈ Y form a two-parametric family of 4-periodic orbits of the billiard a 1 , a 1 (x), a k (x), a k , and the latter is 4-reflective, as in the proof of Proposition 3.8.

2

The above claim yields at least one-dimensional family of 4-reflective billiards with two fixed neighbor mirrors a 1 , a k , since dimM ≥ 3. The curves a 1 , a k are not both algebraic, by assumption. This contradicts Corollary 3.2 (or Theorem 1.7) and thus, proves properness of the projection P k : U → V k .

2) Bimeromorphicity. Suppose the contrary: the proper analytic set projection P k : U → U k = P k (U ) is not bimeromorphic. This means that its inverse has at least two distinct holomorphic branches on an open subset in the analytic set U k . In other words, each A 2 . . . A k-1 from an open subset in V k extends to two distinct k-periodic orbits A 1 A 2 . . . A k-1 A k , A 1 A 2 . . . A k-1 A k . Then the two-dimensional family of quadrilaterals A 1 A 1 A k A k (depending on generic A 2 . . . A k-1 ) are 4-periodic orbits of the billiard a 1 , a 1 , a k , a k with coinciding neighbor mirrors, as in [7, proof of lemma 3.1]. Hence, the latter billiard is 4-reflective, -a contradiction to Corollary 3.5 (or Theorem 1.7) forbidding 4-reflective billiards with coinciding neighbor mirrors. Theorem 6.7 is proved. 2 Corollary 6.8 Let a 1 , . . . , a k be a k-reflective complex analytic planar billiard. The subsequent (k -2)-orbit correspondence P j+1 • P -1 j : U j → U j+1 , A 1 . . . A j-1 A j+2 . . . A k → A 1 . . . A j A j+3 . . . A k is bimeromorphic. Its graph is projected epimorphically onto both U j and U j+1 . If it contracts a curve, then the latter is compact and the mirror a j+2 is algebraic.

Proposition 2 . 1

 21 Let in a 4-reflective billiard a, b, c, d the mirrors a and b be algebraic curves. Then all the mirrors are algebraic.

  ) 2 under the differential of the bundle projection. Thus, it suffices to show that for every (B, C) ∈ T A,H the space T (B,C) T A,H is transversal to L B ⊕ L C . Here L B , L C are arbitrary lines such that AB and BC are symmetric with respect to the line L B and AC, BC are symmetric with respect to the line L C . For every (B, C) ∈ T A,H finitely punctured lines AB × C and B × AC are contained in T A,H , by definition. We identify AB and AC with the corresponding one-dimensional subspaces in T B CP 2 and T C CP 2 respectively. Thus, AB ⊕ AC ⊂ T (B,C) T A,H and AB ⊕ AC is transversal to L B ⊕ L C in T (B,C) (CP 2 ×CP 2 ), since AB = L B and AC = L C by definition. This proves the proposition. 2

Definition 2 .

 2 13 Let A, B, C ∈ C 2 , B = A, C, and let the lines AB, BC be non-isotropic. Let L be a symmetry line of the pair of lines AB, BC. The symmetry σ : AB → BC with respect to the line L is an isometry with respect to the complexified Euclidean metric. For every line l = AB, BC the complex distance function l → C, x → |Bx| = dist(B, x) has two affine branches that differ by sign. Let us choose those affine distance functions on the lines AB, BC for which |Bσ(x)| ≡ -|Bx|. We then say that the distances |BA| = |AB|, |BC| = |CB| calculated with respect to the above affine distance functions are L-concordant. Remark 2.14 The L-concordant distances are well-defined up to simultaneous change of sign. Their ratio

  Let P 2 : M 0 A,H → C be the squared perimeter function from the above proposition. Let p ∈ C, S p be an irreducible component of the level set {P 2 = p} in M 0 A,H . The projections ν G : S p → CP 2 to the position of the vertex G = B, C have discrete preimages, and thus, are submersions on Zariski open dense subsets. The restriction to S p of the line field D A,H is sent by each projection to an SO(2, C)-invariant (multivalued) algebraic line field on CP 2 depending on the choice of G and called triangular line field centered at A with parameters H, p.

Proposition 2 . 19 2Figure 2 :

 2192 Figure 2: Family of degenerate orbits with A = D: the mirrors b and c are either spirals, or conics

Figure 4 :

 4 Figure 4: Coincidence of subsequent vertices and mirrors: r = k.

Figure 5 :

 5 Figure 5: Opposite degeneracy to tangency vertex: tangency or cusp.

Definition 2 .

 2 32 Let W be a complex manifold, M ⊂ W be an irreducible analytic subset, and let D be a singular analytic distribution on W , M ⊂ Sing(D). There exists an open and dense subset of those points 4 x ∈ M reg regular for D, for which the intersection D| M (x) = D(x) ∩ T x M has minimal dimension. Then we say that the subspaces D| M (x) form a singular analytic distribution D| M on M . It is regular on an open dense subset M 0 reg ⊂ M reg . Its singular set M \ M 0

Definition 2 .

 2 42 (cf.[10, definition 16]) Consider some of the above Birkhoff distributions, let us denote it D, and let M denote the underlying manifold (e.g., R 0,k , M 0 ab ,...) carrying D. Consider the projections of the manifold M to the positions of vertices of the k-gon (quadrilateral). A subspace E ⊂ D(x), x ∈ M , is said to be non-trivial, if the restriction to E of the differential of each above projection has positive rank. (Then the rank equals one.) An integral surface of the same distribution is non-trivial, if its tangent planes are non-trivial. For every analytic billiard a, b, c, d there exist natural analytic mappings Ψ a : â × b × ĉ × d → P a , Ψ ab : â × b × ĉ × d → P ab : Ψ a (ABCD) = (A, (B, T B b), (C, T C c), (D, T D d)); Ψ ab (ABCD) = (A, B, (C, T C c), (D, T D d)). (2.5) Proposition 2.43 Let a, b, c, d be a 4-reflective billiard. The mappings Ψ a , Ψ ab send the subset U 1 ⊂ U (see Remark 2.41) to M 0 a , M 0 ab , and the images of its connected components are non-trivial integral surfaces of the restricted Birkhoff distributions D a and D ab respectively. Vice versa, each non-trivial integral surface of any of the latter distributions is the image of an open set of quadrilateral orbits of a 4-reflective billiard a, b, c, d with given mirror a (respectively, given mirrors a and b).

2 3 3 . 1 Lemma 3 . 1 2 Remark 3 . 3 Corollary 3 . 4

 2313123334 is at least one-dimensional subspace. The vertices A(x) = ν a (x) ∈ â, B(x) = ν b (x) ∈ b are not cusps, since x ∈ M 0 ab by assumption. The line functions l a , l b have zero derivatives along K(x), as do the vertices A, B. The derivative along K(x) of at least one of the vertices C or D, say D is not identically zero. Then dν D (K(x)) = l a (x) = L D (x), since D ∈ l a and l a has zero derivative. Thus, dν D (D ab (x)) ⊂ L D (x), -a contradiction to the definition of the distribution D ab . The proposition is proved. Non-integrability of the Birkhoff distribution D ab and corollaries Main lemma, corollaries and plan of the proofIn the present section we prove the following lemma on the non-integrability of the two-dimensional Birkhoff distribution D ab and corollaries. For every pair of analytic curves a, b ⊂ CP 2 distinct from isotropic lines that are not both lines the corresponding Birkhoff distribution D ab is non-integrable. Moreover, there is no three-dimensional irreducible analytic subset M ⊂ M ab (see Convention 2.31) such that M ∩ M 0 ab = ∅ and the restriction D ab | M is two-dimensional and integrable. Corollary 3.2 In the conditions of Lemma 3.1 the union of all the integral surfaces of the Birkhoff distribution D ab in M 0 ab is contained in a twodimensional analytic subset in M ab . Proof The minimal analytic subset M ⊂ M ab containing all the integral surfaces is tangent to D ab , and the distribution D ab is integrable there (Proposition 2.38). Hence, dimM = 2, by Lemma 3.1. This proves the corollary. In the case, when a and b are lines, the statements of Lemma 3.1 and the corollary are false. In this case there exists a one-parametric family of 4-reflective billiards a, b, c, d of type 2) from Theorem 1.7. The corresponding open sets of quadrilateral orbits form a one-parametric family of integral surfaces of the distribution D ab . They saturate an open and dense subset in a three-dimensional analytic subset in M ab . Let Ψ = Ψ ab : â × b × ĉ × d → P ab be the mapping from (2.5). Let a, b, c, d be a 4-reflective complex planar analytic billiard, and let U be the 4-reflective set. Then the image Ψ(U ) ⊂ P ab is a twodimensional analytic subset lying in M ab . The natural projection U → â × b is a proper epimorphic mapping. Proof In the case, when both a, b are algebraic, the curves c, d are also algebraic (Proposition 2.1), and the statements of the corollary follow immediately. Thus, without loss of generality we consider that some of the curves a, b is not algebraic. It suffices to prove the first statement of the corollary. Then its second statement, which is equivalent to the properness and the epimorphicity of the analytic set projection ν ab : Ψ(U ) → â × b, follows from the properness of the projection P ab → â × b and Remmert's Proper Mapping Theorem [11, p.34]. The image Ψ(U ) lies in M ab , which follows from definition. Recall that U 1 ⊂ U denote the open and dense subset of non-degenerate orbits whose vertices are not marked points. Let S ⊂ P ab denote the minimal analytic subset containing Ψ(U 1 ), which obviously contains Ψ(U ). Each its irreducible component is two-dimensional, as is U 1 , by Corollary 3.2 and since Ψ(U 1 ) is a union of integral surfaces of the distribution D ab , see Proposition 2.43. The projections ν C , ν D : S → CP 2 to the positions of the vertices C and D have rank one on an open dense subset, and ν C (S) ⊂ c, ν D (S) ⊂ d: this holds on Ψ(U 1 ), and hence, on each irreducible component of the set S. Let Ŝ denote the normalization of the analytic set S, and π S : Ŝ → S denote the natural projection. The above projections lift to holomorphic mappings ν

2 Corollary 3 . 5

 235 by its analyticity (Proposition 2.10) and since the image U 1 of the set Ψ(U 1 ) (lifted to Ŝ) lies in U . This together with the inclusion Ψ(U ) ⊂ S implies that Ψ(U ) = S and proves the corollary. Let a, b, c, d be a 4-reflective planar analytic billiard, and none of the mirrors a, b be a line. Let a and b intersect at a point A represented by some non-marked points in â and b. Then a = c and a = b.

Remark 3 . 6 Remark 3 . 7

 3637 The images M C = ν ab,C (M ), M D = ν ab,D (M ) are irreducible analytic subsets in â× b×CP2 , by Remark 2.45, Remmert's Proper Mapping Theorem and irreducibility of the variety M . In Subsections 3.2 and 3.3 respectively we treat the following cases: -some of the projections ν ab,C , ν ab,D is not bimeromorphic (see Footnote 3 in Subsection 2.6);-both latter projections are bimeromorphic.In what follows, we denote Σ 1 ⊂ M 0 the subset of points x such that -either x is a singular point of the variety M , -or it is a singular point of the distribution D M , -or the restriction to D M (x) of the differential dν ab (x) has rank less than two, -or x is a critical point of the projection ν ab,D : a point where the rank of differential is not maximal, -or its image under the latter is a singularity of the image, -or the differential of the projection ν D : M 0 ab → CP 2 : y → D(y) vanishes on the distribution plane D M (x), -or one of the three latter statements holds with D replaced by C. Let Σ 0 be the same, as in (3.1). SetΣ = Σ 0 ∪ Σ 1 ⊂ M. (3.2)This is an analytic subset in P ab that has positive codimension in M . Its complement in M is contained in M 0 and dense in M . For every point x ∈ M \ Σ the corresponding germs (g, G(x)) of mirrors g = a, b, c(x), d(x), G = A, B, C, D, are regular, and the points G(x) are not isotropic tangency points. This follows from the definition of the set Σ 0 ⊂ Σ (for the mirrors a and b) and from the two last conditions in the definition of the set Σ 1 ⊂ Σ (for the mirrors c(x) and d(x)). The projection ν ab : S → â × b of each integral surface S of the distribution D M in M \ Σ is a local diffeomorphism, by the Addendum to Proposition 2.10.

Proposition 3 . 8 Figure 6 : 2 Proposition 3 . 9 2 Claim 1 . 2 Corollary 3 . 10

 386239212310 Figure 6: The 4-reflective billiard c(x), d(x), d(y), c(y): open set of quadrilateral orbits C x D x D y C y .

2 Proposition 3 . 12

 2312 ν D : y → D(y) is a submersion. Indeed, the contrary would imply that d = ν D (M \ Σ) is a curve. Therefore, D(y) is locally determined by (A, B) = (A(y), B(y)) as a point of intersection l a (A, B) ∩ d, hence dimM D = 2. But dimM D = dimM ≥ 3, by assumption and bimeromorphicity, -a contradiction. Consider the analytic subset Σ ab ⊂ â × b of those pairs (A, B), for which either π a (A) = π b (B), or some of the lines T A a or T B b is isotropic. Note that dimM D = 3 = dimM , by the latter inequality and since D ∈ l a (A, B) for every (A, B, D) ∈ M D with (A, B) / ∈ Σ ab . Increasing the "exceptional set" Σ, we will assume that ν D is a submersion on all of M \ Σ. Its level sets in M \ Σ are thus holomorphic curves. Proposition 3.11 Every x ∈ M \ Σ has a neighborhood V = V (x) ⊂ M \ Σ such that the lines L D (y), y ∈ V depend only on D(y) and thus, form a holomorphic line field Λ D on a neighborhood W ⊂ CP 2 of the point D(x). Proof Let V be a neighborhood of the point x regularly fibered by local level curves of the submersion ν D . For every y ∈ V there exists a holomorphic curve γ(y) ⊂ V that corresponds exactly to the family of quadrilateral orbits of the billiard a, b, c(y), d(y) with fixed vertex D = D(y). The line L D = L D (y) = T D(y) d(y) obviously remains constant along the curve γ(y). On the other hand, the curve γ(y) should obviously coincide with the level curve {ν D = D(y)}. This proves the proposition. The restriction to each projective line of the field Λ D from Proposition 3.11 is tangent to a pencil of lines through the same point. Proof For every (A, B) ∈ (â× b)\Σ ab denote by F D A,B = A×B ×l a (A, B) C the fiber over (A, B) of the set M D . The fiber F C A,B of the set M C is defined analogously. The projections ν ab,G , G = C, D and the mappings ν -1 ab,D : M D → M , R DC = ν ab,C • ν -1 ab,D : M D → M C are bimeromorphic. Hence, the indeterminacies of the mapping ν -1 ab,D (which include those of R DC ) form an analytic subset, whose complement to the projection preimage of the set Σ ab has codimension at least two, i.e., dimension at most one (Footnote 3 in Subsection 2.6). The image of its projection to â × b is an analytic subset Ind of dimension at most one (Proper Mapping Theorem). Set Reg = (â × b) \ (Σ ab ∪ Ind); it is an open and dense subset in â × b. Then for every (A, B) ∈ Reg the following statements hold: -there exists an invertible holomorphic (hence Möbius) mapping T A,B : l a (A, B) → l b (A, B) for which R DC (A, B, D) ≡ (A, B, T A,B (D)); (3.3) -the restriction to l a (A, B) of the line field Λ D is holomorphic and hence, the projective lines Λ D (y), y ∈ l a (A, B), form a rational curve Γ A,B ⊂ CP 2 * . The former statement follows from bimeromorphicity by definition. The global holomorphic family of lines Λ D (y) depending on y ∈ l a (A, B) F D A,B ⊂ M D from the latter statement is given by the holomorphic inverse ν -1 ab,D : F D A,B → M : y → (A, B, (C(y), Λ C (y)), (D(y), Λ D (y))). It suffices to prove the statement of Proposition 3.

Proposition 3 . 14 2

 3142 In the above conditions the mirror c is also algebraic. Proof The projection ν b : U → b of the 4-reflective set U is proper and epimorphic, by Corollary 3.4 and since a is algebraic. This implies that for an open and dense set of points B ∈ b the preimage ν -1 b (B) ⊂ U is a compact analytic curve with non-constant holomorphic projection to ĉ. Hence, ĉ is compact and c is algebraic. The proposition is proved. Now let us prove Theorem 1.9. Fix a non-marked point B ∈ b as in the above proof and a one-dimensional irreducible component Γ B of the compact analytic curve ν -1 b (B) ⊂ U . Its image ν D (Γ B ) ⊂ d ⊂ CP 2 is either an algebraic curve, or a single point (Proper Mapping and Chow Theorems). The former case is impossible, since d is non-algebraic. Hence, for an open and dense set of points B ∈ b the projection of the curve Γ B to the position of the vertex D is constant and is determined by B. Thus, there exists a mapping b → d, B → D B , defined on an open set V ⊂ b such that for every fixed B ∈ V and variable A ∈ â the lines AB and AD B are symmetric with respect to the tangent line T A a.

Claim 1 .

 1 a = b. Proof Suppose the contrary: a = b. Then U contains a one-parametric analytic family T of quadrilaterals AACD with variable A, C, D, by the above epimorphicity statement. A generic quadrilateral in T is forbidden by Proposition 2.25. The contradiction thus obtained proves the claim. 2

Claim 2 .

 2 Every quadrilateral AACD ∈ Γ is a single-point quadrilateral: the mirrors c and d pass through the same point A; π c (C) = π d (D) = π a (A). Proof Suppose the contrary: say, π c (C) = π a (A). The projection U → â × b is open on a neighborhood of the point AACD, since it contracts no curve to (A, A) by assumption. Therefore, each converging sequence

B k j b

 j converge to two distinct limits H j , j = 1, 2, by assumption and since A is not a marked point of the curve b. The lines H 1 = H 2 pass through the same two points π a (A) = π c (C), by construction. The contradiction thus obtained proves the claim. 2 Thus, Γ is a finite set of points corresponding to the single-point quadrilateral AAAA. Fix one of them and denote it AAAA: the corresponding vertices C ∈ π -1 c (A) and D ∈ π -1 d (A) will be denoted by A. One has c = d, as in Claim 1. The projection U → ĉ × d is open on a neighborhood of the point AAAA, as in the above discussion. Let γ ⊂ ĉ × d be an irreducible germ at (A, A) of analytic curve consisting of pairs (C , D ) with variable C and D for which C ∈ T D d. The germ γ lifts to an irreducible germ γ of analytic curve through AAAA in U . The curve γ consists of quadrilaterals A B C D ∈ U for which B , D ≡ C (Proposition 2.25). Therefore, A ≡ C , by Corollary 2.26 and since A is not a marked point of the mirror b. Hence, a = c. Corollary 3.5 is proved.

Proposition 4 . 1 2 Corollary 4 . 2

 41242 There exists a complement â0 ⊂ â to a discrete subset in â such that for every A ∈ â0 the projection ν B : W A → CP 2 is epimorphic.Proof For every A ∈ â the image of the projection ν B : W A → CP 2 is either the whole projective plane, or an algebraic subset of dimension at most one (Remmert's Proper Mapping and Chow's Theorems). Either ν B (W A ) = CP 2 for all but a discrete set of points A (and then the statement of the proposition obviously holds), or it is at most one-dimensional algebraic set for an open and dense set Q of points A ∈ a, by analyticity. The latter case cannot happen, since otherwise for every non-markedA ∈ ν a (S) ∩ Q the set ν B (W A ∩ S)would be an open subset in the non-algebraic curve b that simultaneously lies in at most one-dimensional algebraic set ν B (W A ), -a contradiction. This proves the proposition. For every open dense subset N ⊂ M whose complement M \ N ⊂ M is an analytic subset there exists an open dense subset âN ⊂ â such that for every A ∈ âN the intersection ν B (W A ∩ N ) ∩ a contains the π a -image of an open dense subset in â: a complement to a discrete subset. Proof Let â0 be the same, as in Proposition 4.1. There exists an open dense subset âN ⊂ â0 such that for every A ∈ âN the subset W N A = W A ∩ N ⊂ W A is open and dense, since the complement M \ N is an analytic subset and all the W A are hypersurfaces. Then for the same A the subset W A \ W N

Remark 4 . 3

 43 The integral surface of the distribution D M through each point x ∈ M is non-trivial, by(4.1). The germ of its image under each one of the projections ν G , G = B, C, D, is a germ of analytic curve at its non-marked point. The regularity of germ follows by definition from the inequalities in (4.1). The non-isotropicity of tangent line to the germ follows from the definition of the distribution D M and non-degeneracy of the quadrilateral corresponding to x. Thus, the above integral surface corresponds to an open set of quadrilateral orbits of a 4-reflective billiard a, b(x), c(x), d(x) (Proposition 2.46), and the above germs are germs of mirrors at non-marked points.

Proposition 4 . 5

 45 For every analytic billiard a, b, c, d with a non-algebraic mirror b in every one-parametric family of quadrilateral orbits ABCD with fixed non-isotropic vertex A = I 1 , I 2 the vertex C is non-constant.

Convention 5 . 5

 55 Given two germs of smooth curves (a, A), (c, C) in R 2 ⊂ RP 2 , we say that a = c, if they lie in the same analytic curve in RP 2 . Theorem 5.6 A germ of C 4 -smooth real planar pseudo-billiard (a, A), (b, B), (c, C), (d, D) is 4-reflective, if and only if it has one of the following types: 1) a = c is a line, the curves b, d = a are symmetric with respect to it; 2) a, b, c, d are distinct lines through the same point O ∈ RP 2 , the line pairs (a, b),(d, c) are transformed one into the other by rotation around O (translation, if O is an infinite point), see Fig.8;3) a = c, b = d, and they are distinct confocal conics: either ellipses, or hyperbolas, or ellipse and hyperbola, or parabolas.

Figure 7 : 4 -

 74 Figure 7: 4-reflective pseudo-billiards symmetric with respect to a line mirror

Theorem 5 . 8

 58 There exist no measure 4-reflective C 4 -smooth planar pseudobilliard with exactly two skew reflection laws at a pair of neighbor mirrors.

Figure 8 : 4 -Figure 9 : 2 5. 2 8 Figure 10 :Figure 11 :

 8492281011 Figure 8: 4-reflective pseudo-billiards on two positively-isometric line pairs

  -directed parabolas: the other one with skew law b) oppositely directed parabolas: skew law at both of them.

Figure 12 :

 12 Figure 12: Open set of orbits on confocal parabolas: one type

Remark 5 . 14

 514 The above collection of lengths |A 1 A 2 |, . . . , |A k A 1 | (if exists) will be called concordant. It is unique up to simultaneous change of sign. The arguments below are analogous to Yu.G.Kudryashov's arguments from [10, section 2] on the Cartan prolongations of the Birkhoff distributions.

  Fix A and a subarc b of a conical arc of the curve b such that for every B ∈ b the interval (A, B) intersects a, the line AB is distinct from the lines T A a , T ⊥ A a , and the line Λ = Λ(B) symmetric to AB with respect to T A a intersects the curve b at its analyticity points. Let us orient the line L = AB from B to A. Consider the one-parametric analytic family of lines L * = L * (B) = σ -1

Figure 13 :Figure 14 :

 1314 Figure 13: A body invisible for one ray.

Definition 6 . 3 ADefinition 6 . 4

 6364 combination of complex analytic billiards α = (a 1 , . . . , a l ), β = (b 1 , . . . , b m ) is a billiard α • s β = (a 1 , . . . , a s , b 1 , . . . , b m , a s+1 , . . . , a l ), s ∈ {1, . . . , l}. For every collection of analytic curves δ = (d 1 , . . . , d t ) in CP 2 distinct from isotropic lines the billiard α • s,δ β = (a 1 , . . . , a s , d 1 , . . . , d t , b 1 , . . . , b m , d t , . . . , d 1 , a s+1 , . . . , a l ) will be called a combination with mirror adding of the billiards α and β. (The previous combination corresponds to δ = ∅.) Let α = (a 1 , . . . , a l ), β = (b 1 , . . . , b m ) be complex planar billiards such that for some 1 ≤ s < min{l, m}, one has a j = b m-j+1 for j = 1, . . . , s. Then the billiard α• [1,s] β = (a s+1 , . . . , a l , b 1 , . . . , b m-s ) is called a combination with mirror erasing of the billiards α, β.

  . . . A k ∈ U each triple A j-1 A j A j+1 satisfies the complex reflection law from Definition 2.8 with respect to the line T A j a j , and each projection U → âj × âj+1 is a submersion on an open dense subset in U .

	Proposition 2.10 [7, proposition 2.14]. The k-reflective set U is an an-
	alytic (algebraic) subset in â1 × • • • × âk . The billiard is k-reflective, if
	and only if the k-reflective set U is non-empty; then each its irreducible
	component is two-dimensional. If the billiard is k-reflective, then for every
	point A 1

  |AB| |BC| is uniquely defined. Example 2.15 Let in the above conditions A, B, C ∈ R 2 , and let |AB|, |BC| be the Euclidean distances. Then |AB|, |BC| are L-concordant, if L is the exterior bisector of the angle ∠ABC. Otherwise L is the interior bisector and |AB|, -|BC| are L-concordant, see Figure 1.Proposition 2.16The above squared perimeter P 2 is a first integral of the line field D A,H . The space M A,H , the line field D A,H and P 2 are invariant with respect to the complex rotation group SO(2, C) fixing A.

		L C
		C
		B
	L	B
		A
	Figure 1: Concordant lengths in the real case: here P = |AB|+|BC|-|AC|.
	Proof The invariance follows from construction. The statement saying
	that P 2 is a first integral is a complexification of the classical statement on
	the real perimeter and the real Birkhoff distribution, see, e.g., [2, section 2].
	Take an arbitrary point ((B, L B ), (C, L C )) ∈ M 0 A,H . We identify the lines
	L B and L C with the corresponding projective lines in CP 2 . Let us normalize
	the distances |AB|, |BC|, |CA| to make |AB| and |BC| L B -concordant and
	|BC|, |CA| L C -concordant. The perimeter P = |AB| + |BC| + |CA| thus
	constructed is well-defined up to sign, and its square P 2 is a well-defined
	holomorphic function on M 0 A,H , see Figure 1.

Its proof is analogous to that in the real case. The proposition is proved. 2 Proposition 2.17

  This subsection is devoted to Plakhov's Invisibility Conjecture: the analogue of Ivrii's conjecture in the invisibility theory[17, conjecture 8.2]. We recall it below and show that in planar case it follows from a conjecture saying that no finite collection of germs of smooth curves can form a measure k-reflective billiard for appropriate "invisibility" reflection law. Both Plakhov's and Ivrii's conjectures have the same complexification, see [8, subsection 5.2, proposition 8]. We prove the C 4 -smooth case of planar Plakhov's Invisibility Conjecture for four reflections as an immediate corollary of Theorem 5.8. Definition 5.23[17, chapter 8] Consider a perfectly reflecting (may be disconnected) closed bounded body B in a Euclidean space. For every oriented line (ray) R take its first intersection point A 1 with the boundary ∂B and reflect R from the tangent hyperplane T A 1 ∂B. The reflected ray goes from the point A 1 and defines a new oriented line, as in the previous subsection.

	.	2
	5.4 Application 2: planar Plakhov's Invisibility Conjecture
	with four reflections

Everywhere in the paper by cusp we mean the singularity of an arbitrary irreducible singular germ of analytic curve, not necessarily the one given by equation x

= y

+ . . . in appropriate coordinates.

Real triangular spirals were introduced in[10, p.320], where a real version of Proposition 2.19 was proved. Our proof of Proposition 2.19 is analogous to arguments from loc. cit.

Recall that a mapping V → W of complex manifolds (or analytic sets in complex manifolds) is meromorphic, if it is well-defined and holomorphic on an open and dense subset in V , and the closure of its graph is an analytic subset in V ×W , see Convention 2.31. It is well-known that if W is compact and V is irreducible, then the set of indeterminacies of every meromorphic mapping V → W is contained in the union of the singular set of V (which has codimension at least two in V , if V is normal) and an analytic subset in V of codimension at least two. A mapping is bimeromorphic, if it is meromorphic together with its inverse.

Everywhere below for an analytic set M by Mreg (Msing) we denote the set of its smooth (respectively, singular) points

Everywhere below by analytic variety we mean an analytic subset in a complex manifold

The edge lengths denoted by Lj in[START_REF] Glutsyuk | No planar billiard possesses an open set of quadrilateral trajectories[END_REF] are denoted here by lj.

Acknowledgements

I am grateful to Yu.S.Ilyashenko, Yu.G.Kudryashov, A.Yu.Plakhov and S.L.Tabachnikov for attracting my attention to Ivrii's Conjecture, invisibility and Commuting Billiard Conjecture. I wish to thank Yu.G.Kudryashov, to whom this paper owes much: some his arguments from our joint paper [10] were used here in a crucial way. I am grateful to them and to E.M.Chirka, E.Ghys, A.L.Gorodentsev, Z.Hajto, B.Sevennec, V.V.Shevchishin, J.-C.Sikorav, M.S.Verbitsky, O.Ya.Viro, V.Zharnitsky for helpful discussions.

‡ Supported by part by RFBR grants 10-01-00739-a, 13-01-00969-a, 16-01-00748, 16-01-00766 and NTsNIL a 10-01-93115 (RFBR-CNRS grant), by ANR grant ANR-13-JS01-0010.

Proof The projection of the variety R 0,k to the space of k-gons has 2 k preimages: at each vertex A j we can choose a symmetry line L A j between the lines A j A j±1 in two ways; the two symmetry lines are orthogonal.

Claim. For every j = 1, . . . , k every configuration ((A 1 , L A 1 ), . . . , (A k , L A k )) ∈ R 0,k is connected by path in R 0,k to the same configuration with simultaneously changed lines L A j , L A j+1 . Proof Let us identify the complex infinity line C ∞ with the space of complex lines through the point A j . Its complement C * = C ∞ \ {I 1 , I 2 } to the isotropic points has fundamental group Z. Consider a closed path φ : [0, 1] → C * starting at the line A j A j+1 and disjoint from the line A j A j-1 that represents its generator. This yields a closed path ψ(t) = A 1 . . . A j A t j+1 A j+2 . . . A k in the space of non-degenerate complex k-gons (in the sense of Definition 2.40): A t j+1 = φ(t) ∩ A j+1 A j+2 , A 0 j+1 = A j+1 . The line L A j = L A j (0) deforms into a family L A j (t) of symmetry lines between the lines A j A j-1 and φ(t) that is analytic along the path ψ. One has L A j (1) = L A j (0). Indeed, let z j-1 , z j (t), z j+1 (t) denote the points of intersection of the infinity line with the lines A j A j-1 , L A j (t), A j A t j+1 = φ(t) written in the standard affine coordinate z on C ∞ : z(I 1 ) = 0, z(I 2 ) = ∞.

One has z j+1 (t) = z 2 j (t) z j-1 , by symmetry and [7, proposition 2.4, p.249]. The point z j (t) makes half-turn around zero, by the latter formula and since z j+1 (t) makes one turn by assumption. Hence, L A j (1) = L A j (0). We similarly consider the deformation L A j+1 (t) = L A t j+1 of the line L A j+1 and get

The claim implies that any two configurations ((A 1 , L A 1 ), . . . , (A k , L A k )) and ((A 1 , L A 1 ), . . . , (A k , L A k )) in R 0,k with even number of pairs of different lines L A j = L A j are connected by path in R 0,k . Therefore, the variety R 0,k consists of at most two irreducible components R ± 0,k , each of them contains the images j α (R 2k \ Σ) for all α ∈ Ψ ± k . Each real (1, . . . , 1)-framed configuration has a concordant collection of real lengths (Example 2.15). The existence of concordant complex length collection is invariant under continuous deformations in R 0,k , which follows from definition. Therefore, each configuration from the component R + 0,k has a concordant length collection. On the other hand, each real (1, . . . , 1, -1)-framed configuration has no concordant length collection, which follows from Example 2.15. Hence, this is true for every configuration from the component R - 0,k , and R + 0,k = R - 0,k . Proposition 5.13 is proved.

2

Lemma 5. [START_REF] Kuranishi | Cartan's prolongation theorem of exterior differential systems[END_REF] The projection π gr : J 2 → R + 0,4 \ Λ is a regular fibration by holomorphic curves. The restriction to J 2 of the distribution F 3 is regular and three-dimensional. For every y ∈ J 2 the corresponding subspace F 3 (y) contains a unique integral 2-plane E(y) of the distribution F 3 . The planes E(y) form a two-dimensional analytic distribution on J 2 .

Lemma 5.17 The restriction to Λ \ (Λ 0 ∩ Λ 1 ) of the distribution D 4 + has at most one non-trivial integral 2-plane at each point.

The two lemmas are proved below. In their proofs we use the notations introduced in [10, subsection 2.5.1, p.297] 6 . The analytic extensions to R + 0,4 of the complexified 1-forms θ j , ν j from loc. cit. are regular and doublevalued: well-defined up to sign. The forms ν 1 , . . . , ν 4 yield a coordinate system on the subspaces of the distribution D 4 + , as in loc. cit. The tangent functions t j from loc. cit. are holomorphic single-valued on R + 0,4 . Let us show this in more detail. Recall that for a real convex quadrilateral A 1 . . . A 4 equipped with its exterior bisectors L A j one has t j = tan ∠(L ⊥ A j , A j A j+1 ). Let z be the above standard affine coordinate on C ∞ : z(I 1 ) = 0, z(I 2 ) = ∞. Let z j , w j denote respectively the z-coordinates of the intersection points

It follows from definition that t j = i(z j -w j ) z j + w j .

The latter formula defines the analytic extension of the functions t j to R + 0,4 . One has z j = ±w j on R + 0,4 . Indeed, otherwise either A j A j+1 = L ⊥ A j , or A j A j+1 = L A j ; in both cases A j A j-1 = A j A j+1 , which is impossible by non-degeneracy. Thus, t j are non-vanishing holomorphic functions on R + 0,4 . Proof of Lemma 5.16. The real Pfaffian system D 4,2 α corresponding to the usual real Birkhoff distribution D 4 α , α = (1, . . . , 1), was studied by Yu.G.Kudryashov in [10, subsection 2.5]. The proof of the statement of the lemma for the integral planes of the system D 4,2 α is presented in [10, subsection 2.5.4, pp. 299-300]. The calculations presented there extend analytically to all of R + 0,4 \ Λ and apply without changes. 2

Proof of Lemma 5.17. Kudryashov's calculations in [10, p.298] extended analytically to complex domain show that for every x ∈ R + 0,4 each non-trivial integral plane E 2 ⊂ D 4 + (x) has a basis of the following type:

the two raws represent vectors in D 4 + (x) written in the coordinates given by the forms ν j from loc. cit. Let x ∈ Λ \ (Λ 0 ∩ Λ 1 ). Then ηη = l 2 l 4 -l 1 l 3 = 0. Thus, either η = 0, or η = 0.

Case 1): η = 0. Then the inclusion E 2 ⊂ T x Λ implies that Finally, the quadrilateral

Case 2): η = 0. We similarly get that x ∈ Λ 1 , -a contradiction. Lemma 5.17 is proved. 2

Remark 5.18 As it was shown in loc.cit., an integral plane E 2 ⊂ T x Λ may exist only for x from an algebraic subset in Λ. For example, in the case, when η = 0, it exists only if t 1 (x) = t 3 (x).

Remark 5. [START_REF] Plakhov | Fractal bodies invisible in 2 and 3 directions[END_REF] The distributions D 4 + | Λ j , j = 0, 1, are three-dimensional. One can easily show that the restriction to Λ j of the Pfaffian system D 4,2 + is involutive. Each its complex integral surface is an open set of quadrilateral orbits of a 4-reflective complex billiard of type 1): if, say, j = 0, then a 1 = a 3 is a line, the curves a 2 and a 4 are symmetric with respect to it. Similar statement holds for smooth integral surfaces.

Then the set of analyticity points of the surface S is open and dense in S.

Proof The image j α (S) ⊂ R + 0,4 is a totally real integral surface of the complex distribution

⊂ S is open and dense. Hence, it suffices to prove the corollary in each one of the two following separate cases:

Case 1): j α (S) ∩ Λ = ∅. The complex span of each tangent plane to j α (S) is a complex integral plane of the distribution D 4 + . Therefore, j α (S) lifts to a totally real C 2 -smooth integral surface S ⊂ J 2 of the distribution E from Lemma 5.16. Let M ⊂ J 2 denote the minimal complex analytic subset containing S. The restriction to M of the distribution E is two-dimensional and integrable (cf. Proposition 2.38). Without loss of generality we can and will assume that S is contained in the regular part of the analytic set M . Then S is contained in a complex analytic integral surface of the distribution E on M and coincides with its intersection with the real part of the variety I 2 . Therefore, S is real analytic, and hence, so is S.

Case 2):

denote the minimal complex analytic subset that contains j α (S). Recall that for every x ∈ Λ \ (Λ 0 ∪ Λ 1 ) there exists at most one non-trivial integral plane at x of the restriction D 4 + | Λ (Lemma 5.17). The union of the latter integral planes is an analytic subset in Gr 2 (T R + 0,4 )| Λ\(Λ 0 ∪Λ 1 ) with proper projection to Λ \ (Λ 0 ∪ Λ 1 ). Its image under the latter projection contains M , since it contains j α (S) and is analytic (Proper Mapping Theorem). Therefore, for every y ∈ M there exists a unique integral plane in D 4 + | Λ (y). The restriction to M of the singular distribution formed by the latter planes is two-dimensional and integrable (cf. Proposition 2.38). Afterwards without loss of generality we assume that j α (S) lies in the regular part of the set M and deduce that S is analytic, as in the above case. α (Λ 0 ∪Λ 1 ) is non-empty. Let us show that S is analytic at p. This will imply that the above pseudo-billiard is of type either 2), or 3) (Theorem 5.6, analytic case). Suppose the contrary. Then the open subset of analyticity points in S contains a connected component W ⊂ j -1 α (Λ 0 ∪ Λ 1 ) with non-empty boundary, by Corollary 5.20. Fix a boundary point x ∈ ∂W and a path ψ ⊂ W going to x such that for every j one has A j (y) = A j (x) for y ∈ ψ arbitrarily close to x. The surface S is not analytic at x. The subset W is an open set of quadrilateral orbits of an analytic pseudo-billiard. The latter is not of type 1), since W ⊂ j -1 α (Λ 0 ∪Λ 1 ). Hence, it is of type either 2) or 3): its mirrors are either all lines, or all confocal conics. Thus, the mirrors a j are analytic at A j (y), y ∈ ψ and some of them, say a 2 is not analytic at A 2 (x) = lim y→x;y∈ψ A 2 (y). Let us show that this is impossible. To do this, let us fix a y ∈ ψ close to x with A 1 (y) = A 1 (x). Consider the smooth deformation of a quadrilateral orbit A 1 (y)A 2 (y)A 3 (y)A 4 (y) with fixed A 1 . That is, the family of quadrilateral orbits

A s 1 = A 1 (y), s 0 = ∠A 2 (y)A 1 (y)A 4 (y), A s 0 j = A j (y).

Let (s -, s + ) be the maximal interval of analyticity of the family of quadrilaterals Q(s) as a function of s. For some of s ± , say s + , some vertices A s + j should be singular points of the corresponding mirrors and coincide with A j (x), by definition. At least two points A s + j should be singular, see [10, lemma 41 and its proof, pp. 305-306]. If two neighbor vertices are singular, e.g., A

is the point of intersection close to A 1 (x) of the curve a 1 with the line A 1 (x)A 2 (x) symmetric to A 2 (x)A 3 (x) with respect to the line T A 2 (x) a 2 . This contradicts the assumption A 1 (y) = A 1 (x). The contradiction thus obtained shows that

Let a 0 j ⊂ a j denote the arcs saturated by the vertices A j (y), y ∈ ψ. Recall that the arcs a 0 j are analytic, and they are either all lines, or all confocal conics, see the above discussion. The above statement implies that for A = A(y) ∈ a 0 1 the lines AA 2 (x), AA 4 (x) are symmetric with respect to the line T A a 0 1 . Hence, this is true for every A ∈ a 0 1 , by analyticity. Therefore, a 0 1 is either the symmetry line between the points A 2 (x), A 4 (x), or a conic with foci at them [7, proposition 2.32]. In the case, when a 0 j are confocal conics, the conic a 0 2 would contain its own focus A 2 (x), -a contradiction. In the case, when they are lines, they should intersect at one point (the pseudobilliard is of type 2)). This implies that the lines a 0 2 , a 0 4 are symmetric with respect to the line a 0 1 , as are A 2 (x), A 4 (x), and hence, the pseudo-billiard is of type 1), -a contradiction. Hence, S is analytic.

Case 2): j α (S) ⊂ Λ j for some j = 0, 1. Then we obviously get a pseudobilliard of type 1) (see Remark 5.19).

Case 3):

α (Λ j ), j = 0, 1, are non-empty. Let us show that this case is impossible. Suppose the contrary: the latter assumptions hold. Then p corresponds to a rhombus with interior bisectors. Some of germs of mirrors a j , j = 2, 4 at the points A j (p) is not a line, since otherwise, a 2 = a 4 and we obviously get that j α (S) ∈ Λ 1 , -a contradiction. The similar statement holds for the mirrors a 1 and a 3 . Finally, some two germs of neighbor mirrors, say a 1 and a 2 are not lines. The billiard under question being 4-reflective, the surface S contains a point q corresponding to a 4-reflective orbit A 1 (q)A 2 (q)A 3 (q)A 4 (q) with A 1 (q) ∈ a 1 , A 2 (q) ∈ a 2 being points of non-zero curvature with nonorthogonal tangent lines. Hence, the latter quadrilateral is not a rhombus It is known that confocal elliptic billiards commute [24, p.49, corollary 4.6]. The next theorem shows that the converse is also true. The proposition follows from definition. Proof of Theorem 5.21. The interior of the set of quadrilateral orbits from the proposition contains an open and dense subset of those quadrilaterals ABCD for which the germs of curves a, b at the vertices are C 4 -smooth and form a 4-reflective pseudo-billiard. The latter has thus some of types 1)-3) from Theorem 5.6. It cannot be of type 1) with germs (b, B), (b, D) lying on the same line and (a, A), (a, C) being symmetric with respect to it (see Addendum 1). This is impossible by convexity and the obvious inclusion ABCD ⊂ Ω 2 . Therefore, either (a, A), (a, C) lie in the same line and the pseudo-billiard is of type 1), or its has type 2) or 3). This implies that the curve a contains an open dense union of analytic pieces, each of them being either a line segment, or an arc of conic. In what follows the reflections from the curves a, b acting on the space of oriented lines will be denoted by σ a and σ b respectively.

Case 1): some analytic arc a of the curve a is a conic. Then the above germs of pseudo-billiards with A ∈ a have type 3). Thus, the curve b contains an open dense union of analytic arcs lying in conics confocal to a . Subcase 1a): the curve a = a is analytic, and hence, is an ellipse. Note that the tangent lines to b are disjoint from a and hence, from the focal segment, by convexity. This implies that each conical piece of the curve b is not a hyperbola, and thus, is an ellipse. Recall that b is a finite union of j are symmetric with respect to the line a, that has at least one symmetric 2l-periodic orbit. Its 2l-reflectivity is obvious. It is obtained by subsequent combinations with mirror erasing of 4-reflective billiards of type 1): on each step we combine billiards α j and β j = (b * j , a, b j , a) erasing one mirror a, which is the last mirror a in β j identified with the first one in α j .

-Every billiard formed by an even number of complex confocal conics, some of them coincide and each conic is taken even number of times; no two neighbor mirrors coincide. It is obtained by subsequent combinations (usual ones and those with mirror erasing) of 4-reflective billiards of type 3).

-Every billiard formed by an even number of non-isotropic complex lines such that the product of the corresponding symmetries is the identity. It is obtained by subsequent combinations (usual ones and those with mirror erasing) of 4-reflective billiards of type 2). This easily follows from the complexification of [26, theorem 1.B, p.3].

The next small technical results, which generalize Corollary 3.4, might be useful in studying the general case. They are immediate consequence of the results of Section 3. To state them, consider a complex k-reflective billiard a 1 , . . . , a k . For every j = 1, . . . , k let us introduce the corresponding space of (k -2)-orbits: collections A 1 . . . A j-1 A j+2 . . . A k ∈ i =j,j+1 âi , A i ∈ âi , such that for every i = j -1, j, j + 1, j + 2 one has A i = A i±1 , the lines A i A i±1 are symmetric with respect to the line T A i a i and the three latter