On infinitely many foliations by caustics in strictly convex open billiards

Abstract Reflection in a strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reflected by the billiard to lines tangent to C. The famous Birkhoff conjecture states that the only strictly convex billiards with a foliation by closed caustics near the boundary are ellipses. By Lazutkin’s theorem, there always exists a Cantor family of closed caustics approaching the boundary. In the present paper, we deal with an open billiard, whose boundary is a strictly convex embedded (non-closed) curve 
$\gamma $
 . We prove that there exists a domain U adjacent to 
$\gamma $
 from the convex side and a 
$C^\infty $
 -smooth foliation of 
$U\cup \gamma $
 whose leaves are 
$\gamma $
 and (non-closed) caustics of the billiard. This generalizes a previous result by Melrose on the existence of a germ of foliation as above. We show that there exists a continuum of above foliations by caustics whose germs at each point in 
$\gamma $
 are pairwise different. We prove a more general version of this statement for 
$\gamma $
 being an (immersed) arc. It also applies to a billiard bounded by a closed strictly convex curve 
$\gamma $
 and yields infinitely many ‘immersed’ foliations by immersed caustics. For the proof of the above results, we state and prove their analogue for a special class of area-preserving maps generalizing billiard reflections: the so-called 
$C^{\infty }$
 -lifted strongly billiard-like maps. We also prove a series of results on conjugacy of billiard maps near the boundary for open curves of the above type.


Introduction and main results
The billiard reflection from a strictly convex smooth planar curve γ ⊂ R 2 (parametrized by either a circle, or an interval) is a map T acting on the subset in the space of oriented lines that consists of those lines that are either tangent to γ, or intersect γ transversally at two points. (In general, the latter subset is not T -invariant. In the case, when γ is a closed curve, the latter subset is T -invariant and called the phase cylinder.) Namely, if a line is tangent to γ, then it is a fixed point of the reflection map. If a line L intersects γ transversally at two points, take its last intersection point B with γ (in the sense of orientation of the line L) and reflect L from T B γ according to the usual reflection law: the angle of incidence is equal to the angle of reflection. By definition, the image T (L) is the reflected line oriented at B inside the convex domain adjacent to γ. The reflection map T is called the billiard ball map. See Fig. 1.
The space of oriented lines in Euclidean plane R 2 x,y is homeomorphic to cylinder, and it carries the standard symplectic form ω = dφ ∧ dp, It is well-known that -the symplectic form ω is invariant under affine orientation-preserving isometries; -the billiard reflections from all planar curves preserve the symplectic form ω. Definition 1.1 A curve C is a caustic for the billiard on the curve γ, if each line tangent to C is reflected from γ to a line tangent to C. Or equivalently, if the curve of (appropriately oriented) tangent lines to C is an invariant curve for the billiard ball map. See Fig. 1.
The famous Birkhoff Conjecture deals with a planar billiard bounded by a strictly convex closed curve γ. Recall that such a billiard is called Birkhoff integrable, if there exists a topological annulus adjacent to γ from the convex side foliated by closed caustics, and γ is a leaf of this foliation. See Figure  2. It is well-known that the billiard in an ellipse is integrable, since it has a family of closed caustics: confocal ellipses. The Birkhoff Conjecture states the converse: the only integrable planar billiards are ellipses.   Lazutkin (1973) states that each strictly convex bounded planar billiard with boundary smooth enough has a Cantor family of closed caustics. But Lazutkin's caustic family does not extend to a foliation in general.
The main result of the paper presented in Subsection 1.1 shows that the other condition of the Birkhoff Conjecture stating that the caustics in question are closed is also important: the Birkhoff Conjecture is false without closeness condition. Namely we show that any open strictly convex C ∞smooth planar curve γ has an adjacent domain U (from the convex side) admitting a foliation by caustics of γ that extends to a C ∞ -smooth foliation of the domain with boundary U ∪ γ with γ being a leaf. Moreover, we show that U can be chosen so that there exist infinitely many (continuum of) such foliations, and any two distinct foliations have pairwise distinct germs at every point in γ. We prove analogous statement for a non-injectively immersed curve γ and "immersed foliations" by immersed caustics. We state and prove an analogue of this statement in the special case, when γ is a closed curve. Remark 1.3 Consider the map T of billiard reflection from a strictly convex planar oriented C ∞ -smooth curve γ that is a one-dimensional submanifold in R 2 parametrized by interval. Let γ denote the family of its orienting tangent lines. Then the points of the curve γ are fixed by T . The map T is a well-defined area-preserving map on an open subset adjacent to γ in the space of oriented lines. The latter subset consists of those lines that intersect γ transversally and are directed to the concave side from γ at some intersection point. Each caustic close to γ corresponds to a T -invariant curve (the family of its tangent lines chosen with appropriate orientation) and vice versa. Thus, a foliation by caustics induces a foliation by T -invariant curves. In Subsection 2.7 we prove the converse: each C ∞ -smooth foliation by T -invariant curves on a domain adjacent to γ from appropriate side (with γ being a leaf) induces a C ∞ -smooth foliation by caustics (with γ being a leaf).
We show that the billiard map has infinite-dimensional family of C ∞smooth foliations by invariant curves (including γ) in appropriate domain adjacent to γ with pairwise distinct germs at each point of the curve γ. This together with Remark 1.3 implies existence of infinite-dimensional family of foliations by caustics.
In Subsection 1.3 we state the generalization of the above result on foliations by invariant curves to a special class of area-preserving maps: the so-called C ∞ -lifted strongly billiard-like maps, for which we prove existence of infinite-dimensional family of C ∞ -smooth foliations by invariant curves with pairwise distinct germs at each point of the boundary segment. In Subsection 1.4 we describe one-to-one correspondence between germs of the latter foliations and germs at S 1 × {0} of C ∞ -smooth h-flat functions ψ(t, h) on the cylinder S 1 × R ≥0 such that ψ(0, h) ≡ 0. This yields a one-to-one correspondence between foliations by caustics and the above germs of flat functions on cylinder. Theorem 1.28 stated in Subsection 1.4 asserts that all the foliations by caustics (invariant curves) corresponding to a given billiard (map) have coinciding jets of any order at each point of the boundary curve.
The results of the paper mentioned below are motivated by the following open question attributed to Victor Guillemin: Let two billiard maps corresponding to two strictly convex closed Jordan curves be conjugated by a homeomorphism. What can be said about the curves? Are they similar (i.e., of the same shape)?
Theorem 1.24 presented in Subsection 1.3 states that each C ∞ -lifted strongly billiard-like map is C ∞ -smoothly symplectically conjugated near the boundary (and up to the boundary) to the normal form (t, z) → (t + √ z, z) restricted to U ∪ J, where J ⊂ R × {0} is an interval of the horizontal axis and U ⊂ R × R + is a domain adjacent to J. In particular, this holds for the billiard map corresponding to each C ∞ -smooth strictly convex (immersed) curve. As an application, we obtain a series of results on (symplectic) conjugacy of billiard maps near the boundary for billiards with reflections from C ∞ -smooth strictly convex curves parametrized by intervals. These conjugacy results are stated in Subsection 1.5 and proved in Subsection 2.10. One of them (Theorem 1.36) states that for any two strictly convex open billiards, each of them being bounded by an infinite curve with asymptotic tangent line at infinity in each direction, the corresponding billiard maps are C ∞ -smoothly conjugated near the boundary.
The results of the paper are proved in Section 2. The plan of proofs is presented in Subsection 1.6. The corresponding background material on symplectic properties of billiard ball map is recalled in Subsection 1.2. A brief historical survey is presented in Subsection 1.7.

Main result: an open convex arc has infinitely many foliations by caustics
Consider an open planar billiard: a convex planar domain bounded by a strictly convex C ∞ -smooth one-dimensional submanifold γ that is a curve parametrized by interval; it goes to infinity in both directions. Let U be a domain adjacent to γ from the convex side. Consider a foliation F of the domain U by strictly convex smooth curves, with γ being a leaf. We consider that it is a foliation by (connected components of) level curves of a continuous function h on U ∪ γ such that h| γ = 0, h| U > 0 and h strictly increases as a function of the transversal parameter. We also consider that for every x ∈ γ and every leaf L of the foliation F there are at most two tangent lines to L through x. One can achieve this by shrinking the foliated domain U , since for every x ∈ γ the line T x γ is the only line through x tangent to γ. Indeed, if there were another line through x tangent to γ at a point y = x, then the total increment of azimuth of the orienting tangent vector to γ along the arc xy would be greater than π. But the latter azimuth is monotonous, and its total increment along the curve γ is no greater than π, since γ is convex and goes to infinity in both directions. The contradiction thus obtained proves uniqueness of tangent line through x.
Remark 1.4 In the above conditions for every compact subarc γ ′ ⊂ γ and every leaf L of the foliation F close enough to γ for every x ∈ γ ′ there exist exactly two tangent lines to L through x. This follows from convexity.
Definition 1. 5 We say that F is a foliation by caustics of the billiard played on γ, if its leaves are caustics, see Fig. 3, in the following sense. Let x ∈ γ, and let L be a leaf of the foliation F. If there exist two tangent lines to L through x, then they are symmetric with respect to the tangent line T x γ.

Remark 1.6
The above definition also makes sense in the case, when γ is just a strictly convex arc that needs not go to infinity. A priori, in this case for some x ∈ γ there may be more than two tangent lines through x to a leaf of the foliation, even for leaves arbitrarily close to γ. This holds, e.g., if there is a line through x tangent to γ at a point distinct from x. This may take place only in the case, when the azimuth increment along γ of the orienting tangent vector to γ is bigger than π. In this case we modify the above definition as follows. Let H denote the space of triples (x, y, z), where x ∈ γ and y, z lie in the same leaf L of the foliation F, y = z, such that the lines xy and xz are tangent to L at the points y and z respectively. Set Let H 0 denote the path-connected component of the space H that contains ∆. We require that for every (x, y, z) ∈ H 0 \ ∆ the lines xy and xz be symmetric with respect to the line T x γ.
Definition 1.7 Let γ ⊂ R 2 be a smooth curve parametrized by an interval. Let U ⊂ R 2 be a domain adjacent to γ. A collection of C ∞ -smooth foliations on U ∪ γ with γ being a leaf is said to be an infinite-dimensional family of foliations with distinct boundary germs, if their germs at each point in γ are pairwise distinct, and if their collection contains a C ∞ -smooth N -parametric family of foliations for every N ∈ N.
Consider an open billiard bounded by a strictly convex C ∞ -smooth curve γ ⊂ R 2 : a one-dimensional submanifold parametrized by interval. There exists a simply connected domain U adjacent to γ from the convex side that admits a foliation by caustics of the billiard that extends to a C ∞ -smooth foliation on U ∪ γ, with γ being a leaf. Moreover, U can be chosen to admit an infinite-dimensional family of foliations as above with distinct boundary germs. See Fig. 3. Figure 3: An open strictly convex planar billiard and its caustics. Here the ambient plane R 2 is presented together with its boundary: the infinity line.
2) The above statements remain valid in the case, when γ is just an arc: a strictly convex curve parametrized by an interval such that each its point has a neighborhood V whose intersection with γ is a submanifold in V . Remark 1.9 It follows from R.Melrose's result [17, p.184, proposition (7.14)] that each point of the curve γ has an arc neighborhood α ⊂ γ for which there exists a domain U adjacent to α from the convex side such that U ∪ α is C ∞ -smoothly foliated by caustics of the billiard played on γ. The new result given by Theorem 1.8 is the statement that the latter holds for the whole curve γ and there exist infinitely many foliations by caustics with distinct boundary germs.
Below we extend Theorem 1.8 to the case of immersed (or closed) curve γ. Definition 1.10 Let γ ⊂ R 2 be a strictly convex C ∞ -smooth curve that is the image of an interval (0, 1) with coordinate x under an immersion ψ : (0, 1) → γ. Let V ⊂ (0, 1) × R + ⊂ R 2 be a domain adjacent to the interval J := (0, 1) × {0}. Fix a C ∞ -smooth immersion Ψ : V ∪ J → R 2 extending ψ as a map J → γ, sending V to the convex side from γ. Let U ⊂ V be a domain adjacent to J and equipped with a foliation F by smooth curves parametrized by intervals, with J being a leaf. We consider that F is a foliation by level curves of a continuous function h : U → R, h| J = 0, h| U > 0, such that h strictly increases as a function of the transversal parameter. We say that F is a foliation by lifted caustics of the billiard played on γ, if Ψ sends each its leaf F t = {h = t} to a caustic of the billiard, see Fig. 4. In more detail, let H denote the space of triples (x, y, z), where x ∈ J and y, z lie in the same leaf L of the foliation F, y = z, such that the lines Ψ(x)Ψ(y) and Ψ(x)Ψ(z) are tangent to the curve Ψ(L) at the points Ψ(y) and Ψ(z) respectively. Set Let H 0 denote the path-connected component of the space H that contains ∆. We require that for every (x, y, z) ∈ H 0 \ ∆ the lines Ψ(x)Ψ(y) and Ψ(x)Ψ(z) be symmetric with respect to the line tangent to γ at Ψ(x). Theorem 1.11 Let γ, ψ, Ψ, J, V be as above. There exists a domain U ⊂ V adjacent to J on which there exists a foliation by lifted caustics that extends to a C ∞ -smooth foliation on U ∪J, with J being a leaf. The above U can be chosen so that it admits an infinite-dimensional family of foliations as above with distinct boundary germs. See Fig. 4. Theorem 1.12 Let γ be a strictly convex closed curve bijectively parametrized by circle. Fix a topological annulus A adjacent to γ from the convex side. Let π : A = R × [0, ε) → A be its universal covering, set J := R × {0}; π : J → γ is the universal covering over γ. There exists a domain U ⊂ A \ J adjacent to J that admits a foliation by lifted caustics of the billiard in γ that extends to a C ∞ -smooth foliation on U ∪ J, with J being a leaf. Moreover, one can choose U so that there exist an infinite-dimensional family of foliations as above with distinct boundary germs. Remark 1.13 In general, in Theorem 1.12 the projected leaves are caustics that need not be closed, may intersect each other and may have selfintersections. Each individual caustic may have a finite length. However the latter finite length tends to infinity, as the caustic in question tends to γ.
A generalization of Theorems 1.8, 1.11 for the so-called C ∞ -lifted strongly billiard-like maps will be stated in Subsection 1.3.

Background material: symplectic properties of billiard ball map
Let γ be a C ∞ -smooth strictly convex oriented curve in R 2 parametrized injectively either by an interval, or by circle. Let s be its natural length parameter respecting its orientation. We identify a point in γ with the corresponding value of the natural parameter s. Let Γ := T =1 R 2 | γ ⊂ T R 2 γ denote the restriction to γ of the unit tangent bundle of the ambient plane R 2 : It is a two-dimensional surface parametrized diffeomorphically by (s, φ) ∈ γ × S 1 ; here φ = φ(u) is the angle of a given unit tangent vector u ∈ T s R 2 with the orienting unit tangent vectorγ(s) to γ. The curve is the graph of the above vector fieldγ. For every (q, u) ∈ Γ set L(q, u) := the oriented line through q directed by the vector u.
We treat the two following cases separately.
Case 1): the curve γ either is parametrized by an interval and goes to infinity in both directions, or is parametrized by circle. That is, it bounds a strictly convex infinite (respectively, bounded) planar domain. Let Γ 0 ⊂ Γ denote the neighborhood of the curve γ that consists of those (q, u) ∈ Γ that satisfy the following conditions: a) the line L(q, u) either intersects γ at two points q and q ′ , or is the orienting tangent line to γ at q: u =γ(s); in the latter case we set q ′ := q; b) the angle between the oriented line L(q, u) and any of the orienting tangent vectors to γ at q or q ′ is acute 1 Let u ′ denote the directing unit vector of the line L(q, u) at q ′ . Consider the two following involutions acting on Γ 0 and Γ respectively: where u * is the vector symmetric to u with respect to the tangent line T q γ. Let Γ 0 + ⊂ Γ 0 denote the open subset of those pairs (q, u) in which the vector u is directed to the convex side from the curve γ.
Remark 1.14 The domain Γ 0 is β-invariant. It is a topological disk (cylinder), if γ is parametrized by an interval (circle). The domain Γ 0 + is a topological disk (cylinder) adjacent to γ.
Let Π γ denote the open subset of the space of oriented lines in R 2 consisting of the lines L(q, u) with (q, u) ∈ Γ 0 + . The mapping Λ : (q, u) → L(q, u) is a diffeomorphism Λ : Γ 0 + → Π γ It extends to the set Γ 0 + ∪ γ as a homeomorphism sending each point (s,γ(s)) ∈ γ to the tangent line T s γ directed byγ(s).

Remark 1.15
Let T denote the billiard ball map given by reflection from the curve γ acting on oriented lines. It is well-known that the billiard ball map T restricted to Π γ is conjugated by Λ to the product of two involutions If the curve γ is C ∞ -smooth, then both involutions I and β are C ∞ -smooth on Γ and Γ 0 respectively. Their product is well-defined and smooth on a neighborhood of the curve γ and fixes the points of the curve γ. Both involutions preserve the canonical symplectic form sin φds ∧ dφ on Γ \ γ, which is known to be the Λ-pullback of the standard symplectic form on the space of oriented lines. See [2,3,16,17,18,20]; see also [10, subsection 7.1].
Case 2). Let γ be parametrized by an interval, but now it does not necessarily go to infinity or bound a region in the plane. Moreover, we allow γ to be an immersed curve that may self-intersect. In this case some lines L(q, u) may intersect γ in more than two points. Now the definition of the subset Γ 0 ⊂ Γ should be modified to be the subset of those (q, u) ∈ Γ for which there exists a q ′ ∈ γ ∩ L(q, u) satisfying the condition b) from Case 1) and such that the arc qq ′ ⊂ γ is disjoint from the line L(q, u), injectively immersed (i.e., without self-intersections) and satisfies the statement of Footnote 1: the orienting tangent vectorγ at each its point has acute angle with L(q, u). (Here q and q ′ may be not the only points of intersection γ ∩ L(q, u).) Remark 1.18 For any given (q, u) ∈ Γ 0 the point q ′ satisfying the conditions from the above paragraph exists, whenever u is close enough toγ(q) (dependently on q). Whenever it exists, it is unique. All the statements and discussion in the previous Case 1) remain valid in our Case 2). Now the mapping Λ is a local diffeomorphism but not necessarily a global diffeomorphism: an oriented line intersecting γ at more than two points (if any) may correspond to at least two different tuples (q, u) ∈ Γ 0 + .

Generalization to C ∞ -lifted strongly billiard-like maps
In this subsection and in what follows we study the next class of areapreserving mappings introduced in [ is a homeomorphism fixing the points in J; (ii) F | V is a diffeomorphism preserving the standard area form ds ∧ dy; (iii) F has the asymptotics of the type uniformly on compact subsets in the s-interval (a, b); (iv) the variable change (s, y) → (s, z), z = √ y > 0 conjugates F to a smooth map F (s, z) (called its lifting) that is also smooth at points of the boundary interval J; thus, w(s) is continuous on (a, b).
If, in addition to conditions (i)-(iv), the latter mapping F is a product of two involutions I and β fixing the points of the line z = 0, then F will be called a (strongly) billiard-like map.
If F is strongly billiard-like, and the corresponding involution β (or equivalently, the conjugate map F ) is C ∞ -smooth, and also C ∞ -smooth at the points of the boundary interval J, then F is called C ∞ -lifted. The above definitions make sense for F being a germ of map at the interval J.

Proposition 1.21
The class of (germs at J of ) C ∞ -lifted strongly billiardlike maps is invariant under conjugacy by (germs at J of ) C ∞ -smooth sym- Proof Let F be a C ∞ -lifted strongly billiard-like map, F = I • β be its lifting. Let V ⊂ R × R + be a domain adjacent to J. Let F be defined on V ∪ J, and let G : , be a C ∞ -smooth symplectomorphism as above. Let us denote G(s, y) = ( s(s, y), y(s, y)). One has y(s, 0) ≡ 0, ∂ s ∂s (s, 0) > 0, ∂ y ∂y (s, 0) > 0, by definition and orientationpreserving property (symplecticity). Thus, y(s, y) = yg(s, y), where g(s, y) is a positive C ∞ -smooth function on a neighborhood of the interval J in (R × R >0 ) ∪ J. The lifting G of the map G to the variables (s, z), z = √ y, acts as follows: G : (s, z) → ( s(s, z 2 ), z(s, z)); z = y(s, z 2 ) = z g(s, z 2 ). (1.9) The latter square root is well-defined and C ∞ -smooth. This implies that the map G is a C ∞ -smooth diffeomorphism of domains with arcs of boundaries corresponding to V ∪ J and G(V ∪ J). Hence, the lifting This implies that the conjugate F G has type (1.7) and hence, is strongly billiard-like. This proves the proposition. Let h = const denote the foliation by connected components of level curves of the function h. This is a C ∞ -smooth foliation on U ∪ J, with J being a leaf. It will be called a foliation by F -invariant curves.
Theorem 1.23 For every C ∞ -lifted strongly billiard-like map F there exists a domain U adjacent to J such that U ∪ J admits a C ∞ -smooth F -invariant function h satisfying (1.11); thus, h = const is a foliation by F -invariant curves. Moreover, U can be chosen so that there is an infinite-dimensional family of foliations as above with distinct boundary germs.
Theorem 1.24 For every function h from Theorem 1.23, replacing it by its post-composition with a C ∞ -smooth function of one variable (which does not change the foliation h = const) one can achieve that there exists a C ∞smooth function τ = τ (s, y) and a domain U ⊂ {y > 0} adjacent to J such that (τ, h) are symplectic coordinates on U ∪ J and in these coordinates tends to zero with all its partial derivatives, as y → 0, and the latter convergence is uniform on compact subsets in the s-interval (a, b) for the function f and for each its individual derivative.
(Here we do not assume any area-preserving property.) 1) There exists a domain W ⊂ V adjacent to J and an F -invariant (1.14) 3) There exist continuum of functions φ satisfying Statement 1) such that the corresponding foliations φ = const are C ∞ -smooth on the same subset W ∪ J and form an infinite-dimensional family of foliations with distinct boundary germs. Consider the foliation h = const. Let F be another C ∞ -smooth foliation by F -invariant curves defined on a domain V in the upper half-plane { h > 0} adjacent to J that extends C ∞ -smoothly to J with J being a leaf. It is a foliation by level curves of an F -invariant function g(τ, h) = h + flat( h), which follows from Theorem 1.28. We can and will normalize g so that (1.15) Remark 1.29 The above normalization can be achieved by replacing the function g by its post-composition with a function φ + flat(φ) of one variable φ. Each foliation from Theorem 1.23 admits a unique F -invariant first integral g as in (1.15) and vice versa: for every F -invariant function g as in (1.15) the foliation g = const satisfies the statement of Theorem 1.23.
Here we treat ψ(t, h) as a function of two variables that is 1-periodic in t.

Corollaries on conjugacy of open billiard maps near the boundary
The results stated below and proved in Subsection 2.10 concern (symplectic) conjugacy of billiard maps near the boundary.
Here we deal with a strictly convex oriented C ∞ -smooth curve γ ⊂ R 2 that is not closed: parametrized by an interval. We consider that it is positively oriented as the boundary of its convex side. Let us first consider that γ goes to infinity in both directions and bounds a convex open billiard. By γ we denote the family of its orienting unit tangent vectors; γ lies in the space Γ, which is the unit tangent bundle of the ambient plane restricted to γ. Let s be a natural length parameter of the curve γ. Let Γ 0 + ⊂ Γ be the open subset adjacent to γ defined in Subsection 1.2. It lies in the space of pairs (s, v) where s ∈ γ and v ∈ T s R 2 is a unit vector directed to the convex side from the curve γ. Recall that φ = φ(v) denote the angle between the vector v and the unit tangent vectorγ(s). Let (a, b) = (a γ , b γ ) ⊂ R denote the length parameter interval parametrizing γ. In the coordinates Recall that the billiard map T γ acting by reflection from γ of the above unit vectors is a The above statements remain valid in the case, when the curve γ in question is a subarc (parametrized by interval but not necessarily infinite) of a strictly convex C ∞ -smooth curve; γ also may be an immersed curve. Definition 1.34 Let γ 1 , γ 2 ⊂ R 2 be strictly convex C ∞ -smooth planar curves parametrized by intervals (they are allowed to be immersed), positively oriented as boundaries of their convex sides. Let J γ i ⊂ R × {0}, i = 1, 2, be the corresponding intervals defined above. We say that the billiard maps T γ i are C ∞ -smoothly conjugated near the boundary in the In the case, when the billiard maps are conjugated in the (s, y)-coordinates, and the conjugating diffeomorphism H is a symplectomorphism, we say that they are C ∞ -smoothly symplectically conjugated near the boundary.

Remark 1.35
Smooth conjugacy of billiard maps near the boundary in the coordinates (s, y) implies their smooth conjucacy in the coordinates (s, φ). This follows from the fact that for every two intervals J 1 , J 2 ∈ R s × {0} and every two domains U 1 , U 2 ⊂ R s × (R + ) y adjacent to J 1 and J 2 respectively every diffeomorphism H : U 1 ∪ J 1 → U 2 ∪ J 2 lifts to a diffeomorphism of the corresponding domains in the (s, φ)-coordinates (taken together with adjacent intervals J i ). The latter statement follows from [10, lemma 3.1] applied to the second component of the diffeomorphism H.
The results stated below on conjugacy of billiard maps near the boundary are corollaries of Theorems 1.24 and 1.27 on normal forms of C ∞ -lifted strongly billiard-like maps and their liftings. Theorem 1.36 Let γ 1 , γ 2 be strictly convex C ∞ -smooth one-dimensional submanifolds in R 2 parametrized by intervals (thus, going to infinity in both directions) and positively oriented as boundaries of their convex sides. Let in addition, the curves γ i have finite asymptotic tangent lines at infinity: as x ∈ γ i tends to infinity (in each direction), the tangent line T x γ i converges to a finite line. Then the corresponding billiard maps are C ∞ -smoothly conjugated near the boundary in (s, y)-(and hence, in (s, φ)-) coordinates.
Theorem 1.37 The statement of Theorem 1.36 on conjugacy of billiard maps corresponding to C ∞ -smooth strictly convex curves γ i remains valid in the case, when each γ j is either a submanifold going to infinity in both directions, or a (may be immersed) subarc of an (immersed) C ∞ -smooth curve, and the two following statements hold: 1) as the length parameter s of the curve γ j goes to an endpoint of the length parameter interval, the corresponding point of the curve γ j tends either to a finite limit (endpoint of γ j ) where γ j is C 2 -smooth, or to infinity; 2) in the latter case, when the limit is infinite, the tangent line T s γ j has a finite limit: a finite asymptotic tangent line. Remark 1.38 V. Kaloshin and C.E. Koudjinan [12] proved continuous conjugacy near the boundary of two billiard maps corresponding to two arbitrary ellipses. For any two ellipses with two appropriate points deleted in each of them they have also proved smooth conjugacy of the corresponding billiard maps on open domains adjacent to the corresponding boundary intervals J in the (s, φ)-coordinates.
Below we state a more general result and provide a sufficient condition of symplectic conjugacy of billiard maps in the coordinates (s, y). To this end, let us recall the following definition. The same criterium also holds for C ∞ -smooth conjugacy near the boundary in (s, φ)-coordinates. Theorems 1.36 and 1.37 will be deduced from Theorem 1.40 using the following propositions on C ∞ -lifted strongly billiard-like maps and lemma on curves with asymptotic line at infinity.
1) The diffeomorphism H is orientation-preserving, H(J) ⊂ R×{0}, and the restriction H 1 (s, 0) to J of its first component is an increasing function.
2) If H is symplectic, then, up to additive constant, (1.20) 3) If H is not necessarily symplectic, then Proposition 1.43 Let F , U , J be the same, as in Proposition 1.42. Let F be the lifting of the map F to the coordinates (s, ψ), Therefore, the integral in (1.22) converges, if and only if ν < −1, i.e., r > 2.
In the case of parabola {y = x 2 } the integral (1.22) diverges.

Plan of the proof of main results
In Subsection 2.1 we recall the above-mentioned Marvizi -Melrose result [16, theorem 3.2] (with proof) yielding C ∞ -smooth coordinates in which . It implies that the lifted map F , written in the coordinates (τ, φ), φ = √ h, takes form (1.13). Theorem 1.27, Statement 1) will be proved in Subsections 2.2-2.4. To do this, first in Subsection 2.2 we construct a fundamental domain for the map F (a curvilinear sector ∆ with vertex at a point in J) and an F -invariant function φ defined on a bigger sector that is φ-flatly close to φ on the latter bigger sector. Then in Subsection 2.3 we construct its F -invariant extension along the F -orbits and show that it is well-defined on a domain adjacent to J. In Subsection 2.4 we prove that thus extended function φ is C ∞ -smooth and φ-flatly close to φ. This will prove Statement 1) of Theorem 1.27. Its Statement 2) on normal form will be proved in Subsection 2.5.
The existence statement in Theorem 1.23 will be deduced from Statement 1) of Theorem 1.27 in Subsection 2.6, where we will also prove Theorem 1.24. Existence in Theorems 1.8, 1.11 and 1.12 will be proved in Subsection 2.7. The results from Subsection 1.4 on jets and space of germs of foliations will be proved in Subsection 2.8. Proposition 1.33 and non-uniqueness statements in main theorems will be proved in Subsection 2.9.
The results of Subsection 1.5 on conjugacy of billiard maps near the boundary will be proved in Subsection 2.10.

Historical remarks
The Birkhoff Conjecture was first stated in print by H. Poritsky [19], who proved it under additional condition that for any two nested closed caustics the smaller one is a caustic of the billiard played in the bigger one; the same result was later obtained in [1]. One of the most famous results on the Birkhoff Conjecture is due to M. Bialy [4], who proved that if the phase cylinder of the billiard is foliated by non-contractible invariant closed curves, then the billiard boundary is a circle; see also another proof in [23]. Recently V. Kaloshin and A. Sorrentino proved that any integrable deformation of an ellipse is an ellipse [13]. Very recently M. Bialy and A. E. Mironov proved the Birkhoff Conjecture for centrally-symmetric billiards having a family of closed caustics that extends up to a caustic tangent to four-periodic orbits [7]. For a detailed survey of the Birkhoff Conjecture see [13,14,8,5,7,9,22] and references therein.
Existence of a Cantor family of closed caustics in every strictly convex bounded planar billiard with sufficiently smooth boundary was proved by V. F. Lazutkin [15] using KAM type arguments.
R. Melrose proved that for every C ∞ -smooth germ γ of strictly convex planar curve there exists a germ of C ∞ -smooth foliation by caustics of the billiard played on γ, with γ being a leaf [17, p.184, proposition (7.14)]. S. Marvizi and R. Melrose have shown that the billiard ball map T in a planar domain bounded by a C ∞ -smooth strictly convex closed curve γ always has an asymptotic first integral on a domain with boundary in the space of oriented lines: a domain adjacent to the family of tangent lines to γ. Namely, there exists a C ∞ -smooth function F on the closure of a domain as above such that the difference F • T − F is C ∞ -smooth there, and it is flat at the points of the family of tangent lines to γ; see [16, theorem (3.2)]; see also statement of their result in Theorem 2.1 below.
(Strongly) billiard-like maps were introduced and studied in [10], where results on their dynamics were applied to curves with Poritsky property.
V. Kaloshin and E.K.Koudjinan proved that for a non-integrable billiard bounded by a strictly convex closed curve, the Taylor coefficients of the normalized Mather β-function are invariant under C ∞ -conjugacies [12]. They also obtained a series of results on conjugacy of elliptic billiard maps, showing in particular that global topological conjugacy implies similarity of underlying ellipses.

Construction of foliation by invariant curves.
Proofs of main results

Marvizi-Melrose construction of an "up-to-flat" first integral
Here we recall the following Marvizi-Melrose theorem with proof. Though it was stated in [16] for billiard ball maps, its statement and proof remain valid for C ∞ -lifted strongly billiard-like maps. 2) The analogue of the above statement holds if J is replaced by the coordinate circle S 1 = S 1 × {0}, S 1 := R s /Z, lying in the cylinder C := S 1 × [0, ε) equipped with the standard area form and F is a strongly billiardlike map C → S 1 × R ≥0 . In this case the coefficients h k (s) of the above normalized series are 1-periodic and C ∞ -smooth.
3) Let h be the function normalized as in Statement 1). Let τ denote the time function for the Hamiltonian vector field with the Hamiltonian function h. In the coordinates (τ, h) (which are symplectic) the map F takes the form Proof The lifting F (s, z), z = √ y, of the map F (s, y) is C ∞ -smooth and has the form where q(s) is a C ∞ -smooth function on (a, b). This follows from (1.7) and C ∞ -liftedness. The map F (s, z) admits an asymptotic Taylor series in z.
The map F has the form F (s, y) = (s + w(s) √ y + O(y), y + q(s)y by (2.2), and it admits an asymptotic Puiseux series in y involving powers 0, 1 2 , 1, 3 2 , 2, . . . . The coefficients of both series are C ∞ -smooth functions in s. Therefore, the mapping F acts by the formula h → h • F not only on functions, but also on formal Puiseux series. It transform each power series h = +∞ k=1 h k (s)y k with coefficients being C ∞ -smooth functions on (a, b) to a Puiseux series of the above type. Our goal is to find an F -invariant power series (or equivalently, an F -invariant even power series +∞ k=1 h k (s)z 2k ) and then to choose its C ∞ -smooth representative. To do this, we use the following formula for the function q(s) in (2.3), see [15, formula (1.2)], [10, formula (7.18)], which follows from area-preserving property: Step 1: constructing an even series +∞ k=1 g k (s)z 2k whose F -image is also an even series. We construct its coefficients g k by induction as follows.
Induction base: k = 1. Let us find a function g 1 (s) such that the Fimage of the function g 1 (s)z 2 contains no z 3 -term. This is equivalent to the statement saying that the function g 1 (s + w(s)z)(z + q(s) 2 z 2 ) 2 contains no z 3 -term, which is in its turn equivalent to the differential equation which has a unique solution g 1 (s) = w 2 3 (s) up to constant factor. (Note that w 2 3 (s)y is a well-known function: the second Lazutkin coordinate [15,16].) Induction step in the case, when J = (a, b) × {0} is an interval. Let we have already found an even Taylor polynomial G n−1 (s, z) := n−1 k=1 g k (s)z 2k , n ≥ 2, such that the asymptotic Taylor series in z of the function G n−1 • F contains no odd powers of z of degrees no greater than 2n − 1. Let us construct g n (s), set G n (s, z) := n k=1 g k (s)z 2k , so that Note that G n • F − G n obviously cannot contain odd powers of degrees less than 2n. Let b(s)z 2n+1 denote the degree 2n + 1 term in the Taylor series of the function G n−1 • F . Condition (2.5) is equivalent to the differential equation which always has a solution g n (s) well-defined on the interval (a, b).
Step 2. Constructing an F -invariant series. The mapping F is the product I • β of two involutions: I(s, z) = (s, −z) and β. Let g := +∞ k=1 g k (s)z 2k be the series constructed on Step 1. One has since the series g is even. The series (2.7) is even (Step 1). Hence, the series t := g + g • β is even and β-invariant by construction. Therefore, it is F -invariant. Its first coefficient is equal to 2g 1 (s) = 2w 2 3 (s) > 0, by construction. We denote the F -invariant series thus constructed by t := +∞ k=1 t k (s)z 2k .
Step 3: symplectic coordinates and normalization. Let t(s, y) be a function representing the series +∞ k=1 t k (s)y k , which is obtained from the latter series (given by Step 2) by the variable change y = z 2 . It is defined on a domain W adjacent to J and C ∞ -smooth on W ∪ J; t| J ≡ 0, ∂t ∂y | J > 0. Let H t denote the corresponding Hamiltonian vector field. Fix an arbitrary C ∞ -smooth function θ such that dθ(H t ) ≡ 1, θ| s=0 = 0: a time function for the vector field H t . Then (θ, t) are symplectic coordinates for the form ω = dx ∧ dy: ω = dθ ∧ dt. Shrinking W (keeping it adjacent to J) we can and will consider that they are global coordinates on W ∪ J. The difference t • F − t is t-flat, by construction, and hence, so is dF (H t ) − H t . Therefore, in the coordinates (θ, t) the symplectic map F takes the form F : (θ, t) → (θ + ξ(t), t) + flat(t). (2.8) In the new coordinates (θ, t) the map F is C ∞ -lifted strongly billiard-like, as in the old coordinates (s, y), by Proposition 1.21.

Step 1. Construction of an invariant function on a neighborhood of fundamental domain
Here we give the first step of the proof of Theorem 1.27. We consider a fundamental sector ∆ for the map F that is bounded by the segment K = [0, η 2 ] of the φ-axis, by its F -image and by the straightline segment connecting their ends. We construct an F -invariant function φ that is φflatly close to φ on a sectorial neighborhood S χ,η of ∆ \ {(0, 0)}. See Without loss of generality we consider that the τ -interval contains the origin: a < 0 < b. Fix a number χ, 0 < χ < 1 2 . Consider the sectors The domain S χ,η will be the above-mentioned neighborhood of fundamental sector, where we construct an F -invariant function.
(ii) The domains S χ,2η and F 2 (S χ,2η ) are disjoint; the latter lies on the right from the former.
(iii) The segment K := {0} × [0, η 2 ] ⊂ R 2 τ,φ and its image F (K) intersect just by the origin; F (K) lies on the right from K. The domain ∆ ⊂ S χ,2η bounded by K, F (K) and the straightline segment connecting the endpoints of the arcs K and F (K) distinct from (0, 0) is a fundamental domain for the map F . See Fig. 5.

Step 2. Extension by dynamics
Here we show that an F -invariant function φ constructed above on a neighborhood of the fundamental domain ∆ extends along F -orbits to an Finvariant function on a domain W adjacent to J = (a, b) × {0} ⊂ R 2 τ,φ . The fact that it is C ∞ -smooth on W ∪ J and coincides with φ up to φ-flat terms will be proved in the next subsection. It suffices to prove that the function φ extends as above to a rectangle (a ′ , b ′ ) × [0, η ′ ) adjacent to arbitrary relatively compact subinterval J ′ = (a ′ , b ′ ) × {0} ⋐ J. A union of the above rectangles corresponding to an exhaustion of J by a sequence of subintervals J ′ yields a domain W adjacent to all of J, where the extended function is defined. Therefore, we make the following convention.

Convention 2.5
Everywhere below we identify the interval J = (a, b)×{0} with (a, b) and sometimes we denote J = (a, b) ⊂ R. We consider that J is a finite interval: a, b are finite. We will consider that there exists a δ > 0 such that F ±1 are diffeomorphisms of the rectangle J × [0, δ) ⊂ R 2 τ,φ onto its images, and the φ-flat terms in asymptotic formula (1.13) are uniformly φ-flat: the difference F (τ, φ) − (τ + φ, φ) converges to zero uniformly in τ ∈ J, and every its partial derivative (of any order) also converges to zero uniformly, as φ → 0. Indeed, the flat terms in question are uniform on compact subsets in J. Hence, one can achieve their uniformity replacing J by its relatively compact subinterval. Under this assumption the above difference and its differential are both uniformly o(φ m ) in τ ∈ J for each individual m ∈ N.
The next proposition describes asymptotics of two-sided F -orbits.
Proof Consider two lines and segments through x:

Claim 2.
For every x = (τ 0 , φ 0 ) ∈ J × [0, 2η) with φ 0 small enough e) the image F (λ ± ) is disjoint from λ ± and lies on its right; f ) the image F −1 (λ ± ) is disjoint from λ ± and lies on its left. g) the right sector S + (x) bounded by the right subintervals in λ ± with vertex x is F -invariant; h) the left sector S − (x) bounded by the left subintervals in λ ± with vertex x is F −1 -invariant. Proof If η is small enough, then F ±1 are well-defined on J × [0, 3η). If φ 0 is small enough, then each λ ± is projected to all of J, and the φ-coordinates of all its points are uniformly asymptotically equivalent to φ 0 (finiteness of J). The map F moves a point z = (τ, φ) ∈ λ ± to y := (τ + φ, φ) up to a φ-flat term, which is o(φ m 0 ) for every m ∈ N. On the other hand, the distance of the latter point y to the line L ± is equal to φ ≃ φ 0 times the | sin | of the azimuth of the line L ± . The latter | sin | is asymptotic to φ 4 0 , and hence, is greater than 1 2 φ 4 0 , whenever φ 0 < 1 is small enough. Thus, dist(y, L ± ) ≥ 1 3 φ 5 0 . Therefore, adding a term o(φ m 0 ), m ≥ 5, to y will not allow to cross L ± , and we will get a point lying on the same, right side from the line L ± , as y. The cases of lines L ∓ and inverse iterates are treated analogously. Statements e) and f) are proved. They immediately imply statements g) and h). ✷ Let η ∈ (0, 1 8 ) be small enough so that F is defined on the rectangle Π := J × [0, 3η) and for every x ∈ Π with φ 0 = φ(x) ∈ [0, 2η] the sector S + (x) contains the points x j = F j (x) until they go out of Π (Claim 2 g)).
2) Let ∆ denote the fundamental domain (curvilinear triangle) for the map F from Proposition 2.2, Statement (iii). Let ∆ denote the complement of the closure ∆ to the union of its vertex (0, 0) and the opposite side. If η > 0 is small enough, then the domain W saturated by the above two-sided orbits of points in ∆ lies in J × [0, 2η 3 ) and contains the strip J × (0, η 4 ).
3) The orbit of each point in W contains either a unique point lying in the fundamental domain ∆, or two subsequent points lying in its lateral boundary curves (glued by F ). 4) Each F -invariant function φ on ∆ extends to a unique F -invariant function on W as a function constant along the latter orbits.
The corollary follows immediately from Proposition 2.6. Step 2 is done.

Step 3. Regularity and flatness. End of proof of Theorem 1.27, Statement 1)
Here we will prove the following lemma, which will imply Statement 1) of Theorem 1.27.
Lemma 2.8 Let in Corollary 2.7 the function φ on ∆ be the restriction to ∆ of a C ∞ -smooth F -invariant function defined on a neighborhood of ∆. Let the function φ(τ, φ) − φ be flat on ∆: it tends to zero with all its partial derivatives, as (τ, φ) ∈ ∆ tends to zero. Consider its extension to the above domain W from Corollary 2.7, Statement 4), and let us denote the extended function by the same symbol φ. The difference φ(τ, φ) − φ is C ∞ -smooth on W ∪ J, and it is uniformly φ-flat (see Convention 2.5).
Proof For every point x = (τ, φ) ∈ W there exists a N = N (x) ∈ Z such that F N (x) ∈ ∆. The latter image F N (x) lies in the definition domain of the initial function φ (which is defined on a neihborhood of ∆), and φ(x) = φ N (x) := φ( F N (x)), by definition. This immediately implies C ∞smoothness of the extended function φ on W . Let us prove its φ-flatness. This will automatically imply C ∞ -smoothness at points of the boundary interval J. To do this, we use the asymptotics We study the derivatives of the functions φ N −φ, N = N (x), at the point x = (τ, φ), as functions in x with fixed N chosen as above for this concrete x. To prove uniform flatness, we have to show that all its partial derivatives tend to zero uniformly in τ ∈ J, as φ → 0. We prove this statement for the first derivatives (step 1) and then for the higher derivatives (step 2).
Without loss of generality everywhere below we consider that N ≥ 1, i.e., x lies on the left from the sector ∆: for negative N the proof is analogous.
Step 1: the first derivatives. The initial function φ defined on a neighborhood of the set ∆ is already known to be φ-flat on ∆. The differential of the composition φ N = φ • F N at the point x, N = N (x), is equal to (2.17) Proposition 2.9 For every sequence of points Proposition 2.9 implies uniform convergence to zero of the first derivatives. In its proof (given below) we use the following asymptotics of differential d F ( F j (x)) and technical proposition on matrix products. We denote M (τ, φ) := the Jacobian matrix of the differential d F (τ, φ). Proof Formula (2.18) follows from (2.15) and Proposition 2.6, part c). ✷

Proposition 2.11 Consider arbitrary sequences of numbers
Here the latter asymptotics is uniform in j = 1, . . . , N k for each individual m, as k → ∞. Then the products of the matrices M j;k have the asymptotics Proof Conjugation by the diagonal matrix H k := diag(1, φ −1 0k ) transforms the matrices M j;k and their product respectively to the following matrices: Claim 3. One has Proof Without loss of generality we can and will consider that N k φ 0k → C ∈ R ≥0 , passing to a subsequence, since N k = O( 1 φ 0k ), by assumption. Let U T ⊂ GL 2 (R) denote the one-parametric subgroup of unipotent upper triangular matrices. Consider the tangent vector Let us extend it to a left-invariant vector field on GL 2 (R), which is tangent to the U T -orbits under right multiplication action. Take a small transverse section S ⊂ GL 2 (R) passing through the identity and consider the subset W ⊂ GL 2 (R) foliated by arcs of phase curves of the field V starting in S and parametrized by time segment [0, 2C]. The subset W is a bordered domain (flowbox) diffeomorphic to the product S × [0, 2C] via the diffeomorphism sending a point y ∈ W to the pair (s(y), t(y)) such that the orbit issued from the point s(y) ∈ S arrives to y in time t(y). Fix an arbitrary m ≥ 3. In the new chart (s, t) the multiplication by a matrix M j;k = B k + o(φ m 0k ) from the right moves a point (s, t) to the point (s, t + φ 0k ) up to a small correction of order o(φ m 0k ). Therefore, the multiplication by N k ≃ C φ 0k similar matrices M j;k with the o(φ m 0k ) in their asymptotics being uniform in j moves a point (s, t) to a point (s, t+N k φ 0k ) up to a correction of order The string of the first partial derivatives of the function by φ-flatness of the initial function φ on ∆ and by the uniform asymptotics φ j = φ 0 (1 + o(1)), j = 1, . . . , N (Proposition 2.6, Statement c)). Take arbitrary sequence of points x(k) := (τ 0k , φ 0k ), τ 0k ∈ J, φ 0k → 0, as k → ∞. Set The sequence of collections of Jacobian matrices M j+1;k := M (τ jk , φ jk ), j = 0, . . . , N k − 1, satisfy the conditions of Proposition 2.11, by (2.16) and (2.18). Therefore, their product M k , which is the Jacobian matrix of the differential d F N k (x(k)), has asymptotics (2.20): Thus, the matrix-string of the differential d φ N k (τ 0k , φ 0k ) is the product (2.16). For m = 2 we get that the differential d( φ N k (τ, φ) − φ) taken at the point x(k) tends to zero, as k → ∞. This proves Proposition 2.9. ✷ Step 2: the higher derivatives. For a smooth function f defined on a neighborhood of a point x by j ℓ x (f ) we will denote its ℓ-jet at x. Below we prove the following proposition.

Proposition 2.12
In the conditions of Proposition 2.9 for every ℓ ∈ N the ℓ-jet at x(k) of the difference φ • F N k − φ tends to zero, as k → ∞. Proposition 2.12 will imply C ∞ -smoothness and φ-flatness of the extended function φ at the points of the boundary interval J × {0}.
For every ℓ ∈ N and x ∈ R 2 let J ℓ x denote the space of ℓ-jets of functions at the point x. The map F induces a transformation of functions, g → g • F . This induces linear operators in the jet spaces, We identify the space of ℓ-jets at each point in R 2 with the ℓ-jet space at the origin, which in its turn is identified with the space P ≤ℓ of polynomials in two variables of degrees no greater than ℓ. Thus, we consider the operator D ℓ F (x) as acting on the above space P ≤ℓ . One has Linear changes of variables (τ, φ) act on the space P ≤ℓ and induce an injective linear anti-representation ρ : GL 2 (R) → GL(P ≤ℓ ). Let A denote the unipotent Jordan cell, see (2.19).  Here the latter asymptotics is uniform in j = 1, . . . , N k for each individual m, as k → ∞. Then the product of the matrices M j;k has the asymptotics Proof Conjugating the matrices M j;k by ρ(H k ), H k := diag(1, φ −1 0k ), transforms them to matrices It suffices to show that the product of the matrices M j;k has asymptotics ρ(B N k k ) + o(φ m 0k ) for every m ∈ N, as in Claim 3. This is done by applying the arguments from the proof of Claim 3 to the left-invariant vector field on GL(P ≤ℓ ) whose time t flow map acts by right multiplication by ρ(A t ). ✷ Formula (2.25) is deduced from Proposition 2.14 and formulas (2.24), (2.27), as formula (2.23). ✷ Proof of Proposition 2.12. The polynomial representing the ℓ-jet of the initial function φ at a point z ∈ ∆ tends to the linear polynomial P (τ, φ) = φ, as z → 0, so that its distance to P (τ, φ) is o(φ m ) for every m ∈ N, by flatness of φ on ∆. This together with Proposition 2.6, Statement c) implies that the distance of its ℓ-jet at the point F N k (x(k)) to the polynomial φ is asymptotic to o(φ m 0k ). The image of the latter ℓ-jet under the operator D ℓ F N k (x(k)) is also o(φ m 0k )-close to φ for every m ∈ N. This follows from the previous statement, formula (2.25), the fact that ρ(A) fixes φ and the asymptotics ). Finally we get that the difference of the ℓ-jet of the function φ at x(k) and the ℓ-jet j ℓ x(k) ( φ • F N k ) of the extended function tends to zero, as k → ∞. Proposition 2.12 is proved. Let φ be a function from Statement 1) of Theorem 1.27. The vector function (τ, φ) has non-degenerate Jacobian matrix at J. Hence, shrinking W , we can and will consider that (τ, φ) are C ∞ -smooth coordinates on W ∪ J. In these coordinates F : (τ, φ) → (τ + g(τ, φ), φ), g(τ, φ) = φ + flat( φ).

Proof of existence in Theorem 1.23. Proof of Theorem 1.24
Let F be a C ∞ -lifted strongly billiard-like map. Let (τ, h) be the coordinates from Theorem 2.1. Set φ = √ h. Let F denote the map F written in the coordinates (τ, φ), which is C ∞ -smooth and takes the form (τ, φ) → (τ + φ + flat(φ), φ + flat(φ)) (Theorem 2.1). There exists a F -invariant function φ = φ+flat(φ) (Theorem 1.27). The function h := φ 2 is F -invariant, C ∞ -smooth, and h = h + flat(h); hence ∂ h ∂h > 0 on J and on some domain adjacent to J. The existence in Theorem 1.23 is proved. Non-uniqueness of the function h will be proved in Subsection 2.9.
Let us now prove Theorem 1.24. Let us fix a function h constructed above. Let θ denote the time function of the Hamiltonian vector field with the Hamiltonian function h, normalized to vanish on the vertical axis {τ = 0}. (We consider that (0, 0) ∈ J, shifting the coordinate τ .) The coordinates (θ, h) are symplectic. In these coordinates F (θ, h) = (θ + ξ( h), h) for some function ξ( h) = hψ( h) in one variable, since F preserves the symplectic area; ψ is C ∞ -smooth and ψ(0) > 0, as in Claim 1 in Subsection 2.1. Afterwards modifying the functions h and θ, as at the end of Subsection 2.1, we get new coordinates (τ, h) (with new τ ) in which F takes the form (1.12). Theorem 1.24 is proved. First let us consider the case, when γ is a strictly convex curve injectively parametrized by interval and bounding a domain in R 2 (conditions of Theorem 1.8).
Let W denote the domain in the space of oriented lines that consists of lines intersecting γ twice and satisfying condition b) from the beginning of Subsection 1.2. Let γ denote the curve given by the family of orienting tangent lines of γ. The domain W is adjacent to γ. The billiard ball map is well-defined on W ∪ γ. Each line L close to a tangent line ℓ of γ carries a canonical orientation: the pullback of the orientation of the line ℓ under a projection L → ℓ close to identity. The billiard ball map acting on thus oriented lines close to tangent lines of γ and intersecting γ twice will be treated as a map acting on non-oriented lines: we will just forget the orientation.
Let us fix a natural length parameter s on the curve γ and identify each point in γ with the corresponding length parameter value. Let us introduce the following tuples of coordinates on the domain W . For every line L ∈ W let s 1 = s 1 (L) and s 2 = s 2 (L) denote the length parameter values of its intersection points with γ. Let φ j denote the oriented angles between L and the tangent lines to γ at the points s j . To each L we put into correspondence the pair (s 1 , φ 1 ) where s j are numerated so that s 1 < s 2 . Set see (1.2). Any of the pairs (s 1 , φ 1 ) or (s 1 , y 1 ) defines L uniquely. Recall that (s 1 , y 1 ) are symplectic coordinates on W , see the discussion after Remark 1.15. Let V ⊂ R × R + denote the domain W represented in the coordinates (s 1 , y 1 ). It is adjacent to an interval J = (a, b) × {0} representing γ. Let V ⊂ R×R + denote the same domain represented in the coordinates (s 1 , φ 1 ).

Proposition 2.15
In the coordinates (s 1 , y 1 ) the billiard ball map is a C ∞lifted strongly billiard-like map F defined on V ∪J. In the coordinates ( Proof The statements of the proposition follow from Proposition 1.17 and Example 1.20. ✷ Proposition 2.16 Shrinking V (without changing its boundary interval J), one can achieve that there exists a C ∞ -smooth F -invariant function G(s 1 , y 1 ) on V ∪ J such that G| J ≡ 0, ∂G ∂y 1 > 0.
Proof The proposition follows from Theorem 1.23 (existence).
✷ From now on by W we denote the domain of those lines that are represented by points of the (shrinked) domain V from Proposition 2.16.
The level curves of the function G are F -invariant and form a C ∞ -smooth foliation. Lifting everything to the domain W in the space of lines we get a foliation by invariant curves under the billiard ball map. Each its leaf is a smooth family of lines. Its enveloping curve is a caustic of the billiard in γ.
To prove that γ and the caustics in question form a C ∞ -smooth foliation of a domain adjacent to γ, we use the following lemma.

Lemma 2.17
The above function G is C ∞ -smooth as a function on the domain with boundary W ∪ γ in the space of lines. It has non-degenerate differential on W ∪ γ. Thus, its level curves form a C ∞ -smooth foliation of W ∪ γ with γ being a leaf.

Remark 2.18
The function s 1 (L) is smooth on W but not on W ∪ γ: it is not C 1 -smooth at points of the curve γ. Therefore, a priori a function smooth in (s 1 , y 1 ) is not necessarily smooth on W ∪ γ.
Proof of Lemma 2.17. The function G is C ∞ -smooth on W and has non-degenerate differential there, by Proposition 2.16. Let us prove that this also holds at points of the boundary curve γ. The function G lifts to an F -invariant function G(s 1 , φ 1 ) = G(s 1 , 1 − cos φ 1 ).
The map (s 1 , φ 1 ) → (s 1 , s 2 ) is a diffeomorphism defined on V ∪ J. The analogous statement holds for the diffeomorphism (s 2 , φ 2 ) → (s 1 , s 2 ). One has (which are C ∞ -diffeomorphic coordinates on V ∪ J) the function G is invariant under sign change at β. Hence, G is a C ∞ -smooth function in (α, β 2 ), the function G is C ∞ -smooth on the domain in R α × (R + ) ψ adjacent to J and corresponding to W , and it is also smooth at points of the boundary J. . It is invariant under pertumation of the coordinates s 1 , s 2 , and its restriction to each connected component of the complement to the diagonal is a diffeomorphism, by convexity. Equivalently, it is C ∞ -smooth in the coordinates (α, β) and invariant under sign change at β. Hence, it is smooth in (α, ψ). Its differential is non-degenerate at those points, where s 1 = s 2 , or equivalently, ψ = 0. It remains to check that it has non-degenerate differential at the points of the line {ψ = 0}. To do this, consider yet another tuple of coordinates (α * , ψ * ) on W ∪ γ defined as follows: for every L ∈ W ∪ γ -the point α * = α * (L) ∈ (a, b) is the unique point in the curve γ where the tangent line to γ is parallel to L (it exists by Rolle Theorem); -the number ψ * = ψ * (L) is the distance between the line L and the above tangent line.

Proposition 2.20
The coordinates (α * , ψ * ) are C ∞ -smooth coordinates on W ∪ γ. Proposition 2.20 follows from definition and strict convexity of γ. Consider now (α * , ψ * ) as functions of (α, ψ). One obviously has Here κ is the curvature of the curve γ. Indeed, as s 1 , s 2 → α 0 , the line L through s 1 and s 2 is parallel to a line tangent to γ at a point α * that is o(s 1 − s 2 )-close to α = s 1 +s 2 2 . The distance between the two lines is asymptotic to 1 2 κ(α)(α * − s 1 ) 2 , by [10, formula (2.1)]. This together with the equality α − s 1 = β ≃ α * − s 1 implies (2.38), which in its turn implies that ∂ψ * ∂ψ (α, 0) > 0. Together with (2.37), this implies non-degeneracy of the Jacobian matrix of the vector function (α * (α, ψ), ψ * (α, ψ)) at the line {ψ = 0}. This proves Proposition 2.19. ✷ The function G(α, ψ) = G(s 1 , φ 1 ) is smooth in (α, ψ), as was shown above. Hence, it is smooth on W ∪ γ (Proposition 2.19). It remains to show that it has non-zero differential at each point x ∈ γ; then shrinking W we get that the differential is non-zero at each point in W ∪ γ. Indeed, it is smooth in the coordinates (s 1 , y 1 ) (in which y 1 | γ ≡ 0), and one has G(s 1 , φ 1 ) ≃ a(s 1 )y 1 (1 + o(1)), as y 1 → 0, a(s) > 0, (2.39) by Proposition 2.16. On the other hand, as s 1 , s 2 → s, one has y 1 , φ 1 → 0 and Hence, y 1 ≃ 1 2 κ 2 (s)ψ. This together with (2.39) implies that in the coordinates (α, ψ) one has ∂ G ∂ψ (α, 0) > 0. Together with the above discussion, this proves Lemma 2.17. ✷ Proof of existence in Theorem 1.8. The function G defined on the set W ∪ γ in the space of lines is invariant under the billiard ball map. Therefore, its level curves are invariant families of lines. They form a C ∞smooth foliation of W ∪ γ, with γ being a leaf. Let us denote the latter foliation by F. The enveloping curves of the curve γ and of its leaves are respectively the curve γ and caustics of the billiard on γ. Let us show that they lie on its convex side and there exists a domain U ⊂ R 2 adjacent to γ from the convex side such that the latter caustics form a C ∞ -smooth foliation of U ∪ γ, with γ being a leaf. Fix a projective duality sending lines to points, e.g., polar duality with respect to the unit circle centered at a point O in the convex domain bounded by γ. Let us shrink W so that its points represent lines that do not pass through O. Then the duality represents the subset W ∪ γ in the space of lines as a domain in the affine chart R 2 ⊂ RP 2 with a boundary curve. The latter domain and curve will be also denoted by W and γ respectively. The curve γ is dual to γ.

Proposition 2.21
The curve γ is strictly convex, and the domain W lies on its concave side.
Proof The curve γ is strictly convex, being dual to the strictly convex curve γ. Each point x ∈ W is dual to a line intersecting γ twice. Therefore, there are two tangent lines to γ through x. Hence, x lies on the concave side from γ. ✷ For every x ∈ W let F x ⊂ R 2 ⊂ RP 2 denote the leaf through x of the foliation F (represented in the above dual chart), and let L x denote its projective tangent line at x. The enveloping curve of the family of lines represented by the curve F x (treated now as a subset in the space of lines) is its dual curve F * x . It consists of points L * y dual to the lines L y for all y ∈ F x . Recall that the boundary curve γ is a strictly convex leaf.

Proposition 2.22
Shrinking the domain W adjacent to γ one can achieve that the map x → L * x be a C ∞ -smooth diffeomorphism of the domain W ∪ γ onto a domain U ⊂ R 2 ⊂ RP 2 taken together with its boundary arc γ. The domain U lies on the convex side from the curve γ.
Proof The curve γ is strictly convex. No its tangent line passes through O, being dual to a point of the curve γ (which is a finite point). Therefore, every compact arc in γ has a neighborhood in R 2 whose intersection with each leaf of the foliation F is a strictly convex curve. Thus, shrinking W we can and will consider that each leaf L is strictly convex and no its tangent line passes through O. Hence, each line L tangent to L is disjoint from the leaves lying on the convex side from L. Thus, L is disjoint from γ and O / ∈ L. Let U denote the set of points dual to lines tangent to leaves in W . In the dual picture the latter statements mean that U ⊂ R 2 and for every A = L * ∈ U there are no tangent lines to γ passing through A. The set U is path-connected, disjoint from γ, and it accumulates to all of γ. Therefore, it approaches γ from the convex side, by the previous statement. Hence, it lies entirely on its convex side. Let us now prove that shrinking W one can achieve that the map x → L * x be a diffeomorphism W ∪ γ → U ∪ γ. Fix a compact arc exhaustion For every k fix a flowbox Π k ⊂ W of the foliation F adjacent to γ k and lying in W whose leaves are strictly convex. We construct the flowboxes Π k with decreasing heights, which means that for every k each leaf of the flowbox Π k+1 crosses Π k . Now replace W by the union ∪ k Π k , which will be now denoted by W . The leaves of the foliation on W ∪ γ are strictly convex and connected, by construction. We claim that the map x → L x , and hence, x → L * x is a C ∞ -smooth diffeomorphism. Indeed, it is a local diffeomorphism by strict convexity of leaves. It remains to show that L x = L y for every distinct x, y ∈ W . Indeed, fix an x ∈ W , let L denote the leaf of the foliation F through x. Set L = L x . Fix a k such that x ∈ Π k . Every leaf in the flowbox Π k that does not lie in its leaf through x either intersects L transversally, or is disjoint from L, by convexity. Then the latter statement also holds for every other flowbox Π ℓ , by construction and convexity. This implies that L can be tangent to no other leaf in W . It cannot be tangent to the same leaf L at another point y = x, by convexity and the above statement. This proves diffeomorphicity of the map x → L * x . ✷ The above C ∞ -smooth diffeomorphism x → L * x sends W onto a domain U ⊂ R 2 adjacent to γ. It sends leaves of the foliation F to the corresponding caustics of the billiard on γ. Hence, the caustics together with the curve γ form a C ∞ -smooth foliation of U ∪ γ. Constructing the above flowboxes Π k narrow enough in the transversal direction (step by step), we can achieve that for every x ∈ γ and every leaf L of the foliation F there are at most two tangent lines through x to the caustic L * . Indeed, each leaf L of the foliation F is a leaf of some flowbox Π k . Its dual caustic L * will satisfy the above tangent line statement, if the total angle increment of its tangent vector is no greater than π. The latter angle increment statement holds for the curve γ. Hence, it remains valid for the caustics dual to the leaves of the flowbox Π k , if Π k is chosen narrow enough. The existence statement of Theorem 1.8 is proved. ✷ The proof of the existence in Theorem 1.11 repeats the above proof of the existence in Theorem 1.8 with obvious changes. The existence statement of Theorem 1.12 follows from that of Theorem 1.11.  ψ(s, h) is C ∞ -smooth and h-flat on a cylinder S 1 × [0, δ), S 1 = R s /Z, for a small δ > 0. This follows from smoothness and h-flatness of the function g(τ, h) − h. Conversely, consider an h-flat function ψ(s, h) that is 1-periodic in s and such that ψ(0, h) = 0. Then the function

2.8
is C ∞ -smooth, F -invariant and its difference with h is h-flat, by construction. Statement 1) of Proposition 1.30 is proved. Its Statement 2) can be reduced to Statement 1) and also can be proved analogously. ✷ Theorem 1.31 follows immediately from Proposition 1.30, and in its turn, it immediately implies Theorem 1.32.

Proof of Proposition 1.33 and non-uniqueness in main theorems
Proof of Proposition 1.33. Let us prove the statement of Proposition 1.33 for a map F of type (1.13). We prove it for line fields: for other objects the proof is analogous. Without loss of generality we can and will consider that the map F takes the form (τ, φ) → (τ + φ, φ): see Statement 2) of Theorem 1.27. Let G 1 and G 2 be two F -invariant line fields on W with distinct germs at J. This means that there exists a sequence of points x(k) = (τ (k), φ(k)) with φ(k) → 0 and τ (k) lying in a compact subset in J such that the lines G 1 (x(k)), G 2 (x(k)) ⊂ T x(k) R 2 are distinct. Taking a subsequence, we can and will consider that x(k) → x = (τ 0 , 0), as k → ∞. The twosided orbit of a point x(k) with big k consists of points with φ-coordinate φ(k) whose τ -coordinates form an arithmetic progression with step φ(k) converging to zero. At each point of the orbit the lines of the fields G 1 and G 2 are distinct, since this holds at x(k) and by F -invariance. Therefore, passing to limit, as k → ∞, we get that for every point z ∈ J there exist points z ′ arbitrarily close to z with G 1 (z ′ ) = G 2 (z ′ ). Hence, the germs at z of the line fields G 1 and G 2 are distinct. The first statement of Proposition 1.33, for a map F of type (1.13), is proved. Its second statement, for a C ∞ -lifted strongly billiard-like map F : V ∪ J → F (V ∪ J) ⊂ R 2 follows from its first statement and the fact that F is conjugated to the map (τ, φ) → (τ + φ, φ) by a homeomorphism that is smooth on the complement to the boundary interval J. The latter conjugating homeomorphism is the composition of a diffeomorphism conjugating F to the map (τ, h) → (τ + √ h, h) (Theorem 1.24) and the map (τ, h) → (τ, φ), φ = √ h. Proposition 1.33 is proved. ✷ Proof of non-uniqueness in Theorems 1.27, 1.23, 1.8, 1.11, 1.12 Existence of continuum of distinct germs of foliations satisfying the statements of any of the above-mentioned theorems follows immediately from Proposition 1.33 and Theorem 1.32, which states that there are as many distinct boundary germs, as many flat functions on the cylinder S 1 × R ≥0 with distinct germs at S 1 × {0}. It remains to show that there exists a domain adjacent to the boundary interval (or the curve γ) which admits an infinite-dimensional family of corresponding foliations with distinct germs.
Case of Theorem 1.27. Fix coordinates (τ, φ) in which F (τ, φ) = (τ + φ, φ). Recall that the coordinates (τ, φ) are defined on W ∪ J, where W ⊂ R × R + is a domain adjacent to the interval J = (a, b) × {0}. Fix a C ∞smooth h-flat finction ψ on the cylinder S 1 ×R ≥0 , S 1 = R/Z, with non-trivial germ at S 1 × {0}: For every ε > 0 the function is F -invariant, C ∞ -smooth and well-defined on W ∪ J. Let us show that shrinking the domain W one can achieve that the foliation g ε = const is regular, that is g ε has no critical points on W , whenever ε is small enough. Claim 1. Replacing W by a smaller domain adjacent to J, one can achieve that each partial derivative of the function χ (of any order) be bounded on W . For any given m and every δ > 0 shrinking W (dependently on m and δ) one can achieve that all its order m partial derivatives have moduli less than δ. Proof The modulus of each partial derivative of order at most m admits an upper bound by a quantity Here the latter o(φ k ) is uniform in τ , as φ → 0. Estimate (2.46) follows from 1-periodicity and flatness of the function ψ(s, h) and chain rule for calculating derivatives. Let us now replace the domain W by a smaller domain adjacent to J on which the right-hand side in (2.46) is bounded for each m and is less than δ for a given m. First let us replace W by the connected component adjacent to J of its intersection with the strip {a < τ < b}. In the case, when (a, b) is a finite interval, the right-hand side in (2.46) is uniformly bounded on W and tends to zero uniformly in τ ∈ (a, b), as φ → 0: the asymptotics ψ ℓr (τ, φ) = o(φ 3m+3 ) kills polynomial growth of the function φ −2(m+1) . Therefore, shrinking W one can achieve that for given m and δ, the right-hand side in (2.46) be less than δ on W .
In the case, when some (or both) of the boundary points a or b is infinity, take an exhaustion of the interval (a, b) by segments [a k , b k ]. By the above argument, we can take a rectangle Π k = (a k , b k ) × (0, d k ) ⊂ W on which for all m the right-hand sides in (2.46) be bounded, and for some given m the same right-hand side be less than a given δ. Replacing W by ∪ k Π k , we achieve that the two latter inequalities hold on W ∪ J. ✷ Let W satisfy the statements of the above claim so that each first partial derivative of the function χ has modulus less than 1 2 . Then for every ε ∈ [0, 1] the foliation g ε = const is regular on W ∪ J, since for those ε one has ∂gε ∂φ = 1+ε ∂χ ∂φ > 1 2 on W ∪J. All its leaves are F -invariant. For distinct values of the parameter ε the germs of the corresponding foliations are distinct at each point of the interval J, by Theorem 1.32 and Proposition 1.33. This yields a one-dimensional family of foliations from Theorem 1.27 with pairwise distinct germs at each point in J. Now let us apply the above argument with the expression εχ in (2.45) being replaced by an arbitrary linear combination Recall that |ψ| < 1 8 . This inequality together with the above assumption that the first partial derivatives of the function χ(τ, φ) = ψ( τ φ , φ) have moduli less than 1 2 on W imply that for every ε as in (2.47) the module of each first partial derivative of the function χ ε is less than 1 2 on W ∪J. This implies that the foliation by level curves of the function g ε (τ, φ) = φ+ χ ε is a C ∞ -smooth foliation on W ∪ J. We get a N -dimensional family of foliations on W ∪ J depending on (ε 1 , . . . , ε N ) ∈ [0, 1] N with pairwise distinct germs at J, and hence, at each point of the curve J (Proposition 1.33). The non-uniqueness statement of Theorem 1.27 is proved.
Case of Theorem 1.23. Its non-uniqueness statement follows from that of Theorem 1.27 and also from the above arguments.
Case of Theorem 1.8. Let us consider the billiard ball map acting on lines as a C ∞ -lifted strongly billiard-like map F . Let us introduce new (symplectic) coordinates (τ, h) in which F (τ, h) = (τ + √ h, h), see (1.12). The map F is defined on W ∪ J, where J = (a, b) × {0} parametrizes the family of lines tangent to γ and W ⊂ R × R + is a domain adjacent to J. Representing lines as points in RP 2 via a projective duality RP 2 * → RP 2 transforms J to a strictly convex curve γ * ⊂ RP 2 dual to γ, and W to a domain adjacent to γ * from the concave side. See Subsection 2.7. We can and will consider that γ * and W lie in an affine chart R 2 , as in the proof of the existence in Theorem 1.8 in Subsection 2.7. In what follows we identify J with γ * . Consider the foliation h = const by F -invariant curves. Let us construct a family of foliations using a C ∞ -smooth h-flat function ψ(s, h) on S 1 × R ≥0 with non-trivial germ at S 1 × {0}, as in (2.44). Namely, set The functions g ε are F -invariant. The germs of any two foliations g ε 1 = const, g ε 2 = const, ε 1 = ε 2 , are distinct at each point in J, by Theorem 1.32 and Proposition 1.33. It remains to prove their regularity and regularity of the dual foliations by caustics on one and the same domain. To do this, we use the following claim. Claim 2. Shrinking the domain W adjacent to J = γ * one can achieve that for every ε ∈ [0, 1] the foliation g ε = const is regular on W ∪ J, its leaves are strictly convex curves, as is γ * , and the map Λ ε : x → L x,ε sending a point x ∈ W to the projective line L x,ε tangent to the level curve {g ε = g ε (x)} at x is a diffeomorphism on W . Proof Consider the function h and the above function χ as functions on W ∪ γ * as on a domain in R 2 ⊂ RP 2 . The curve γ * = {h = 0} is strictly convex. Hence, shrinking W we can and will consider that each level curve {h = const}∩W is strictly convex. Consider the rectangles Π k ⊂ W from the proof of the above Claim 1 (in the coordinates (τ, h)) with decreasing heights. They are represented as curvilinear quadrilaterals in R 2 ⊂ RP 2 . Choosing them with heights small enough, we can achieve that ||∇χ|| < 1 2 ||∇h|| on Π k . Let us now replace W by ∪ k Π k . Then ∇g ε = 0 on W , and hence, the foliation g ε = const is regular for all ε ∈ [0, 1]. Choosing Π k with heights small enough (step by step) one can also achieve that each level curve {g ε = const} ∩ Π k be strictly convex for every ε ∈ [0, 1], by strict convexity of the boundary curve γ * and h-flatness of the function ψ. In more detail, let (x, y) be coordinates on the ambient affine chart R 2 . Strict convexity of level curves {g ε = const} is equivalent to non-vanishing of the Hessian 2 H(g ε ): H(g ε ) = 0, H(g) := ∂ 2 g ∂x 2 ∂g ∂y 2 + ∂ 2 g ∂y 2 ∂g ∂x 2 − 2 ∂ 2 g ∂x∂y ∂g ∂x ∂g ∂y .
The Hessian H(g ε ) is the sum of the Hessian H(h) (which is non-zero on W ∪ γ * , since the curves {h = const} are strictly convex) and a finite sum of products; each product contains ε, at least one derivative of the function χ and at most two derivatives of the function h; each derivative is of order at most two. Choosing the rectangles Π k with heights small enough, we can achieve that the module of the latter sum of products be no greater than 1 2 ε|H(h)| for ε ∈ [0, 1]. This follows from convexity of the curve γ * and h-flatness of the function ψ: shrinking W , one can achieve that all the first and second derivatives of the function χ have moduli bounded by arbitrarily small δ (Claim 1). Then H(g ε ) = 0 on W , hence, the curves {g ε = const} ∩ W are strictly convex. Now for every k we choose smaller rectangles Π k ⊂ Π k with decreasing heights and with the lateral (i.e., vertical) sides lying in the lateral sides of the bigger rectangles Π k that satisfy the following additional statement. For every ε ∈ [0, 1] let Π k,ε denote the minimal flowbox for the foliation g ε = const with lateral (i.e., transversal) sides lying in the lateral sides of Π k that contains Π k . This is the union of arcs of leaves that go from one lateral side of Π k to the other one and cross Π k . For every k we can and will subsequently choose Π k with heights small enough (i.e., narrow enough in the transversal direction) so that for every ε ∈ [0, 1] the flowbox Π k,ε lies in Π k , and the heights of the flowboxes Π k,ε be decreasing in k: more precisely, for every k each local leaf in Π k+1,ε crosses Π k,ε , as in the proof of Proposition 2.22. Then the map Λ ε : x → L x,ε is a diffeomorphism on W ε := ∪ k Π k,ε for every ε ∈ [0, 1], as at the end of the proof of Proposition 2.22. Hence, it is a diffeomorphism on W := ∪ k Π k . (2.48) The claim is proved. ✷ Claim 3. Consider the foliation by caustics of the billiard on γ that is dual to the foliation g ε = const. There exists a domain U ⊂ R 2 adjacent to γ from the convex side where the above foliation by caustics is C ∞ -smooth (and also smooth at the points of the curve γ) for every ε ∈ [0, 1]. Moreover, shrinking U one can achieve that for every x ∈ γ and every ε ∈ [0, 1] there are at most two tangent lines through x to any given leaf of the corresponding foliation by caustics on U . Proof Let W be the domain (2.48) constructed above. For every ε ∈ [0, 1] the map Λ * ε : x → L * x,ε sending x to the point dual to the corresponding line L x,ε is a diffeomorphism, since so is Λ ε . It sends the domain W foliated by level curves of the function g ε onto a domain U ε adjacent to γ and foliated by their dual curves: caustics of the billiard on γ. They form a C ∞ -smooth foliation on U ε ∪ γ. For the proof of the first statement of Claim 3 it remains to show that there exists a domain U adjacent to γ that lies in the intersection ∩ ε U ε (and hence, for each ε it is smoothly foliated by the corresponding caustics). To do this, we construct the above W and a smaller domain W ′ ⊂ W adjacent to γ * so that the following statement holds: (*) for every p ∈ W ′ and every ε ∈ [0, 1] there exists a q = q(p, ε) ∈ W such that the projective line L p,0 tangent to the curve {h = h(p)} at p is tangent to the leaf of the foliation g ε = const at q. Statement (*) implies that the image U = Λ 0 (W ′ ) is contained in all the domains U ε and regularly foliated by caustics dual to level curves of the function g ε for every ε ∈ [0, 1].
Take the rectangles Π k and Π k from the proof of Claim 2. Let us call their sections h = const horizontal and transversal sections τ = const vertical. For every k fix two vertical sections ℓ 1,k and ℓ 2,k crossing the interior Int(Π k ) that lie in the 1 2 k -neighborhoods of the corresponding lateral sides of the rectangle Π k . We can and will choose a rectangle Π ′ k ⊂ Π k with lateral sides lying on ℓ 1,k and ℓ 2,k and height small enough so that for every ε ∈ [0, 1] and every p ∈ Π ′ k there exists a q = q(p, ε) ∈ Π k satisfying statement (*). This is possible by flatness of the function ψ and strict convexity of the curve γ * . Then statement (*) holds for the domain W ′ = ∪ k Π ′ k ⊂ W . This together with the above discussion proves the first statement of Claim 3. One can achieve that its second statement (on tangent lines) hold as well by choosing the above rectangles Π ′ k with heigth small enough, as in the proof of the existence in Theorem 1.8 at the end of Subsection 2.7. by (2.52) and since H 2 (s, 0) ≡ 0, which yields ∂H 2 ∂s (s, 0) = 0. This proves orientation-preserving property of the diffeomorphism H and increasing of the function H 1 (s, 0).
Fix small ε, δ > 0 such that Fix a small η > 0 and a y 0 ∈ (0, η 4 ), set q 0 = (s 0 , y 0 ). Let q −N − , . . . , q −1 , q 0 , q 1 , . . . , q N + , q j = (s j , y j ), denote the F -orbit of the point q 0 in the rectangle [s 0 −ε, s * 0 +ε]×[0, η]. Here N ± = N ± (y 0 ). It is known that the s-coordinates of its points form an asymptotic arithmetic progression s j = s(q j ), and their y-coordinates are asymptotically equivalent: whenever y 0 is small enough (dependently on ε). See [10, lemma 7.13]. The image of the above orbit under the map H should be an orbit of the map Λ : (t, z) → (t + √ z, z). The abscissas of its points, x j := H 1 (q j ), form an arithmetic progression: x j+1 − x j = √ z 0 , z 0 = z(H(q 0 )). We claim that this yields a contradiction to the inequality ℓ = ℓ * and (2.54). Indeed, one has by (2.54) and the Lagrange Increment Theorem. On the other hand, take a family of indices k = k(y 0 ) such that s k = s k (y 0 ) → s * 0 , as y 0 → 0: it exists, since the asymptotic progression s j has steps uniformly decreasing to 0, it starts on the left from s 0 and ends on the right from s * 0 > s 0 , see (2.55). Repeating the above argument for x k and x k+1 yields The claim together with (2.59) imply convergence of the improper integral defining the Lazutkin length. This proves Lemma 1.44. ✷ Proof of Theorem 1.36. The Lazutkin lengths of both curves γ 1 and γ 2 are finite, since they have asymptotic tangent lines at infinity in both directions and by Lemma 1.44. This together with Theorem 1.40 implies C ∞ -smooth conjugacy of the corresponding billiard maps near the boundary and up to the boundary. Theorem 1.36 is proved. ✷ The proof of Theorem 1.37 is analogous to the above proof of Theorem 1.36.