1.1 Historical remarks

The Birkhoff conjecture was first stated in print by Poritsky [Reference Poritsky19], who proved it under an 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 [Reference Amiran1]. One of the most famous results on the Birkhoff conjecture is due to Bialy [Reference Bialy4], 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 [Reference Wojtkowski22]. Recently, Kaloshin and Sorrentino proved that any integrable deformation of an ellipse is an ellipse [Reference Kaloshin and Sorrentino13]. Very recently, Bialy and 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 [Reference Bialy and Mironov7]. For a detailed survey of the Birkhoff conjecture, see [Reference Bialy and Mironov5, Reference Bialy and Mironov7–Reference Glutsyuk9, Reference Kaloshin and Sorrentino13, Reference Kaloshin and Sorrentino14, Reference Tabachnikov21] and references therein.

Marvizi and Melrose have shown that the billiard ball map ${\mathcal T}$ in a planar domain bounded by a $C^\infty $ -smooth strictly convex closed curve $\gamma $ 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 $\gamma $ . Namely, there exists a $C^\infty $ -smooth function F on the closure of a domain as above such that the difference $F\circ {\mathcal T}-F$ is $C^\infty $ -smooth there, and it is flat at the points of the family of tangent lines to $\gamma $ ; see [Reference Marvizi and Melrose16, Theorem 3.2]; see also the statement of their result in Theorem 2.1.

Kaloshin and Koudjinan proved that for a non-integrable billiard bounded by a strictly convex closed curve, the Taylor coefficients of the normalized Mather $\beta $ -function are invariant under $C^\infty $ -conjugacies [Reference Kaloshin and Koudjinan12]. 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.

1.2 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^\infty $ -smooth one-dimensional submanifold $\gamma $ that is a curve parameterized by interval; it goes to infinity in both directions. Let U be a domain adjacent to $\gamma $ from the convex side. Consider a foliation ${\mathcal F}$ of the domain U by strictly convex smooth curves, with $\gamma $ being a leaf. We consider that it is a foliation by (connected components of) level curves of a continuous function h on $U\cup \gamma $ such that $h|_\gamma =0$ , $h|_U>0$ and h strictly increases as a function of the transversal parameter. We also consider that for every $x\in \gamma $ and every leaf ${\mathcal L}$ of the foliation ${\mathcal F}$ , there are at most two tangent lines to ${\mathcal L}$ through x. One can achieve this by shrinking the foliated domain U, since for every $x\in \gamma $ , the line $T_x\gamma $ is the only line through x tangent to $\gamma $ . Indeed, if there were another line through x tangent to $\gamma $ at a point $y\neq x$ , then the total increment of azimuth of the orienting tangent vector to $\gamma $ along the arc $xy$ would be greater than $\pi $ . However, the latter azimuth is monotonous, and its total increment along the curve $\gamma $ is no greater than $\pi $ , since $\gamma $ is convex and goes to infinity in both directions. The contradiction thus obtained proves uniqueness of tangent line through x.

Definition 1.6. We say that ${\mathcal F}$ is a foliation by caustics of the billiard played on $\gamma $ if its leaves are caustics, see Figure 3, in the following sense. Let $x\in \gamma $ , and let ${\mathcal L}$ be a leaf of the foliation ${\mathcal F}$ . If there exist two tangent lines to ${\mathcal L}$ through x, then they are symmetric with respect to the tangent line $T_x\gamma $ .

Figure 3 Foliation by caustics in an open strictly convex planar billiard. Here the ambient plane ${\mathbb R}^2$ is presented together with its boundary: the infinity line.

Remark 1.7. The above definition also makes sense in the case when $\gamma $ is just a strictly convex arc that needs not go to infinity. A priori, in this case for some $x\in \gamma $ , there may be more than two tangent lines through x to a leaf of the foliation, even for leaves arbitrarily close to $\gamma $ . This holds, e.g., if there is a line through x tangent to $\gamma $ at a point distinct from x. This may take place only in the case when the azimuth increment along $\gamma $ of the orienting tangent vector to $\gamma $ is bigger than $\pi $ . In this case, we modify the above definition as follows. Let $\mathcal H$ denote the space of triples $(x,y,z)$ , where $x\in \gamma $ and y, z lie in the same leaf ${\mathcal L}$ of the foliation ${\mathcal F}$ , $y\neq z$ , such that the lines $xy$ and $xz$ are tangent to ${\mathcal L}$ at the points y and z respectively. Set

$$ \begin{align*} \overline{\mathcal H}:={\mathcal H}\cup\Delta, \Delta:=\{(x,x,x) \mid x\in \gamma\}. \end{align*} $$

Let ${\mathcal H}_0$ denote the path-connected component of the space $\overline {\mathcal H}$ that contains $\Delta $ . We require that for every $(x,y,z)\in {\mathcal H}_0\setminus \Delta $ , the lines $xy$ and $xz$ be symmetric with respect to the line $T_x\gamma $ .

Below we extend Theorem 1.9 to the case of immersed (or closed) curve $\gamma $ .

Definition 1.11. Let $\gamma \subset {\mathbb R}^2$ be a strictly convex $C^{\infty }$ -smooth curve that is the image of an interval $(0,1)$ with coordinate x under an immersion $\psi :(0,1)\to \gamma $ . Let $V\subset (0,1)\times {\mathbb R}_+\subset {\mathbb R}^2$ be a domain adjacent to the interval ${J:=(0,1)\times \{0\}}$ . Fix a $C^{\infty }$ -smooth immersion $\Psi : V\cup J\to {\mathbb R}^2$ extending $\psi $ as a map $J\to \gamma $ , sending V to the convex side from $\gamma $ . Let $U\subset V$ be a domain adjacent to J and equipped with a foliation ${\mathcal F}$ by smooth curves parameterized by intervals, with J being a leaf. We consider that ${\mathcal F}$ is a foliation by level curves of a continuous function $h:U\to {\mathbb R}$ , $h|_J=0$ , $h|_U>0$ , such that h strictly increases as a function of the transversal parameter. We say that ${\mathcal F}$ is a foliation by lifted caustics of the billiard played on $\gamma $ if $\Psi $ sends each its leaf ${\mathcal F}_t=\{ h=t\}$ to a caustic of the billiard, see Figure 4. In more detail, let $\mathcal H$ denote the space of triples $(x,y,z)$ , where $x\in J$ and y, z lie in the same leaf ${\mathcal L}$ of the foliation ${\mathcal F}$ , $y\neq z$ , such that the lines $\Psi (x)\Psi (y)$ and $\Psi (x)\Psi (z)$ are tangent to the curve $\Psi ({\mathcal L})$ at the points $\Psi (y)$ and $\Psi (z)$ respectively. Set

$$ \begin{align*} \overline{\mathcal H}:={\mathcal H}\cup\Delta, \Delta:=\{(x,x,x) \mid x\in J\}. \end{align*} $$

Let ${\mathcal H}_0$ denote the path-connected component of the space $\overline {\mathcal H}$ that contains $\Delta $ . We require that for every $(x,y,z)\in {\mathcal H}_0\setminus \Delta $ , the lines $\Psi (x)\Psi (y)$ and $\Psi (x)\Psi (z)$ be symmetric with respect to the line tangent to $\gamma $ at $\Psi (x)$ .

Figure 4 An immersed foliation by immersed caustics and a foliation by lifted caustics.

A generalization of Theorems 1.9 and 1.12 for the so-called $C^{\infty }$ -lifted strongly billiard-like maps will be stated in §1.4.

1.3 Background material: symplectic properties of billiard ball map

Let $\gamma $ be a $C^{\infty }$ -smooth strictly convex oriented curve in ${\mathbb R}^2$ parameterized injectively either by an interval or by a circle. Let s be its natural length parameter respecting its orientation. We identify a point in $\gamma $ with the corresponding value of the natural parameter s. Set

$$ \begin{align*} \Gamma= \{(q,u) \mid q\in\gamma, u\in T_q{\mathbb R}^2, \lVert u \rVert = 1 \} \subset T{\mathbb R}^2_\gamma. \end{align*} $$

This is a two-dimensional surface parameterized diffeomorphically by $(s,\phi )\in \gamma \times S^1$ ; here $\phi =\phi (u)$ is the angle of a given unit tangent vector $u\in T_s{\mathbb R}^2$ with the orienting unit tangent vector $\dot \gamma (s)$ to $\gamma $ . The curve

$$ \begin{align*} \widetilde{\gamma}:=\{\phi=0\}=\{ (s,\dot\gamma(s)) \mid s\in\gamma\} \end{align*} $$

is the graph of the above vector field $\dot \gamma $ . For every $(q,u)\in \Gamma $ , set

$$ \begin{align*} L(q,u) := \text{the oriented line through } q \text{ directed by the vector } u. \end{align*} $$

We treat the two following cases separately.

Case (1). The curve $\gamma $ is either:

• • parameterized by an interval and goes to infinity in both directions (for a convex curve, this means either the existence of asymptotic rays in both directions, or converging (in both directions) to one and the same point in the infinity line in the ambient projective plane ${\mathbb {RP}}^2$ ); or

• • parameterized by a circle.

That is, $\gamma $ bounds a strictly convex unbounded (respectively, bounded) planar domain. Let $\Gamma ^0\subset \Gamma $ denote the neighborhood of the curve $\widetilde {\gamma }$ that consists of those $(q,u)\in \Gamma $ that satisfy the following conditions:

1. (a) the line $L(q,u)$ either intersects $\gamma $ at two points q and $q'$ , or is the orienting tangent line to $\gamma $ at q: $u=\dot \gamma (s)$ ; in the latter case, we set $q':=q$ ;

2. (b) the angle between the oriented line $L(q,u)$ and any of the orienting tangent vectors to $\gamma $ at q or $q'$ is acute. (In the case under consideration, this condition implies that the line $L(q,u)$ has acute angle with the orienting tangent vector $\dot \gamma $ at each point of the arc $qq'$ (for appropriately chosen arc $qq'$ in the case, when $\gamma $ is a closed curve).)

Let $u'$ denote the directing unit vector of the line $L(q,u)$ at $q'$ . Consider the two following involutions acting on $\Gamma ^0$ and $\Gamma $ respectively:

$$ \begin{align*} \beta:\Gamma^0 \to\Gamma^0,\quad \beta(q,u)=(q',u');\quad \beta^2=\mathrm{Id}; \qquad\\I:\Gamma \to\Gamma \text{ is the reflection from } T_q\gamma : I(q,u)=(q,u^*), \end{align*} $$

where $u^*$ is the vector symmetric to u with respect to the tangent line $T_q\gamma $ . Let $\Gamma ^0_+\subset \Gamma ^0$ denote the open subset of those pairs $(q,u)$ in which the vector u is directed to the convex side from the curve $\gamma $ .

Let $\Pi _\gamma $ denote the open subset of the space of oriented lines in ${\mathbb R}^2$ consisting of the lines $L(q,u)$ with $(q,u)\in \Gamma ^0_+$ . The mapping ${\Lambda } :(q,u)\mapsto L(q,u)$ is a diffeomorphism

$$ \begin{align*} {\Lambda} : \Gamma^0_+\to\Pi_\gamma. \end{align*} $$

It extends to the set $\Gamma ^0_+\cup \widetilde {\gamma }$ as a homeomorphism sending each point $(s,\dot \gamma (s))\in \widetilde {\gamma }$ to the tangent line $T_s\gamma $ directed by $\dot \gamma (s)$ .

Remark 1.18. Let ${\mathcal T}$ denote the billiard ball map given by reflection from the curve $\gamma $ acting on oriented lines. It is well known that the billiard ball map ${\mathcal T}$ restricted to $\Pi _\gamma $ is conjugated by ${\Lambda } $ to the product of two involutions

$$ \begin{align*} \widetilde{\delta}_+:=I\circ\beta={\Lambda} ^{-1}\circ{\mathcal T}\circ {\Lambda} :\Gamma^0_+\to\Gamma. \end{align*} $$

If the curve $\gamma $ is $C^{\infty }$ -smooth, then both involutions I and $\beta $ are $C^{\infty }$ -smooth on $\Gamma $ and $\Gamma ^0$ respectively. Their product is well defined and smooth on $\Gamma ^0$ and fixes the points of the curve $\widetilde {\gamma }$ . Both involutions preserve the canonical symplectic form $\sin \phi ds\wedge d\phi $ on $\Gamma \setminus \widetilde {\gamma }$ , which is known to be the ${\Lambda } $ -pullback of the standard symplectic form on the space of oriented lines. See [Reference Arnold2, Reference Arnold3, Reference Marvizi and Melrose16–Reference Melrose18, Reference Tabachnikov20]; see also [Reference Glutsyuk10, §7.1].

Let us recall another representation of the billiard ball map ${\mathcal T}$ in a chart where it preserves the standard symplectic form. To do this, consider the orthogonal projection ${\pi _\perp :(T{\mathbb R}^2)|_\gamma \to T\gamma }$ sending each vector $u\in T_q{\mathbb R}^2$ with $q\in \gamma $ to its orthogonal projection to the tangent line $T_q\gamma $ . It projects the unit tangent bundle $\Gamma $ to the unit ball bundle

$$ \begin{align*} T_{\leq1}\gamma:=\{(q,w) \mid q\in\gamma, w\in T_q\gamma, \lVert w\rVert \leq1\}. \end{align*} $$

A tangent vector $w = w ({\partial }/{\partial s})\in T_q\gamma $ will be identified with its coordinate $w=\pm \lVert w\rVert $ in the basic vector ${\partial }/{\partial s}$ . Thus, $\pi _\perp (s,\phi )=(s,\cos \phi )$ . Consider the following function and differential 2-form on $T\gamma $ :

(1.2) $$ \begin{align}y:=1-w;\quad \omega:=ds\wedge dy. \end{align} $$

The form $\omega $ coincides with the standard symplectic form on the tangent bundle $T\gamma $ of the curve $\gamma $ (considered as a Riemannian manifold equipped with the metric $\lvert ds\rvert ^2$ coming from the standard Euclidean metric on ${\mathbb R}^2$ ).

The curve $\widetilde {\gamma }=\{(s,\dot \gamma (s)) \mid s\in \gamma \}= \{ w=1\}=\{ y=0\}\subset T\gamma $ is a component of the boundary $\partial T_{\leq 1}\gamma $ . The projection $\pi _\perp $ sends $\Gamma ^0_+$ diffeomorphically to a domain in $T_{\leq 1}\gamma $ adjacent to $\widetilde {\gamma }$ . It extends homeomorphically to $\Gamma ^0_+\cup \widetilde {\gamma }$ as the identity map $\mathrm {Id}:\widetilde {\gamma }\to \widetilde {\gamma }$ . Let $\mu _+: \pi _\perp (\Gamma ^0_+\cup \widetilde {\gamma })\to \Gamma ^0_+\cup \widetilde {\gamma }$ be the inverse to the restriction of the projection $\pi _\perp $ to $\Gamma ^0_+\cup \widetilde {\gamma }$ . Set

(1.3) $$ \begin{align} \delta_+:=\pi_\perp\circ\widetilde{\delta}_+\circ\mu_+=\pi_\perp\circ {\Lambda} ^{-1}\circ{\mathcal T}\circ {\Lambda} \circ\mu_+. \end{align} $$

Proposition 1.20. [Reference Glutsyuk10, Proposition 7.5]

Let $\kappa (s)$ denote the (geodesic) curvature of the curve $\gamma $ . The involutions I, $\beta $ and the mappings $\widetilde {\delta }_+$ , $\delta _+$ admit the following (asymptotic) formulas:

(1.4) $$ \begin{align} I(s,\phi)&=(s,-\phi), \beta(s,\phi)=(s+2\kappa^{-1}(s)\phi+O(\phi^2), -\phi+O(\phi^2)), \end{align} $$

(1.5) $$ \begin{align} \kern-0.1pt\!\!\!\!\widetilde{\delta}_+(s,\phi) &=(s+2\kappa^{-1}(s)\phi+O(\phi^2), \phi+O(\phi^2)),\qquad\qquad\qquad\qquad \end{align} $$

(1.6) $$ \begin{align}\kern-0.1pt \!\!\!\delta_+(s,y)&=(s+2\sqrt 2\kappa^{-1}(s)\sqrt y+O(y), y+O(y^{3/2})).\qquad\qquad \qquad\end{align} $$

The asymptotics are uniform on compact subsets of points $s\in \gamma $ , as $\phi \to 0$ (respectively, as $y\to 0$ ).

Case (2). Let $\gamma $ be parameterized by an interval, but now it does not necessarily go to infinity or bound a region in the plane. Moreover, we allow $\gamma $ to be an immersed curve that may self-intersect. In this case, some lines $L(q,u)$ may intersect $\gamma $ at more than two points. Now the definition of the subset $\Gamma ^0\subset \Gamma $ should be modified to be the subset of those $(q,u)\in \Gamma $ for which either $u=\dot \gamma (q)$ or there exists a $q'\in \gamma \cap L(q,u)$ satisfying the condition (b) from Case (1) and such that the arc $qq'\subset \gamma $ is disjoint from the line $L(q,u)$ , injectively immersed (that is, without self-intersections) and satisfies the statement that the orienting tangent vector $\dot \gamma $ at each point has acute angle with $L(q,u)$ . (Here q and $q'$ may not be the only points of intersection $\gamma \cap L(q,u)$ . If $u=\dot \gamma (q)$ , then we set $q'=q$ .)

1.4 Generalization to $C^{\infty }$ -lifted strongly billiard-like maps

In this subsection and in what follows, we study the next class of area-preserving mappings introduced in [Reference Glutsyuk10] generalizing the billiard maps in equation (1.6).

Definition 1.22. (See [Reference Glutsyuk10, Definition 7.6])

Let $(a,b)$ be a (may be (semi) infinite) interval in ${\mathbb R}$ with coordinate s. Let $V\subset {\mathbb R}\times {\mathbb R}_{+}$ be a domain adjacent to the interval ${J:=(a,b)\times \{0\}}$ . A mapping $F:V\cup J\to {\mathbb R}\times {\mathbb R}_{\geq 0}\subset {\mathbb R}^2_{s,y}$ is called billiard-like if it satisfies the following conditions:

1. (i) $F: V\cup J\to F(V\cup J)$ is a homeomorphism fixing the points in J;

2. (ii) $F|_V$ is a diffeomorphism preserving the standard area form $ds\wedge dy$ ;

3. (iii) F has the asymptotics of the type

   (1.7) $$ \begin{align} F(s,y)=(s+w(s)\sqrt y+O(y),y+O(y^{3/2})) \quad\text{as } y\to0;\quad w(s)>0, \end{align} $$

   uniformly on compact subsets in the s-interval $(a,b)$ ;

4. (iv) the variable change

   $$ \begin{align*} (s,y)\mapsto (s,z), \quad z=\sqrt y>0 \end{align*} $$

   conjugates F to a smooth map $\widetilde {F}(s,z)$ (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 $\widetilde {F}$ is a product of two involutions I and $\beta $ fixing the points of the line $z=0$ ,

$$ \begin{align*} \widetilde{F}=I\circ\beta,\quad I(s,z)=(s,-z), \end{align*} $$

(1.8) $$ \begin{align} \beta(s,z)=(s+w(s)z+O(z^2), -z+O(z^2)),\quad \beta^2=\mathrm{Id}, \end{align} $$

then F will be called a (strongly) billiard-like map.

If F is strongly billiard-like, and the corresponding involution $\beta $ (or equivalently, the conjugate map $\widetilde {F}$ ) is $C^{\infty }$ -smooth and also $C^{\infty }$ -smooth at the points of the boundary interval J, then F is called $C^{\infty }$ -lifted. The above definitions make sense for F being a germ of map at the interval J.

Proof. Let F be a $C^{\infty }$ -lifted strongly billiard-like map and $\widetilde {F}=I\circ \beta $ be its lifting. Let $V\subset {\mathbb R}\times {\mathbb R}_+$ be a domain adjacent to J. Let F be defined on $V\cup J$ , and let $G:V\cup J\to G(V\cup J)\subset {\mathbb R}\times {\mathbb R}_{\geq 0}$ be a $C^{\infty }$ -smooth symplectomorphism as above. Let us denote $G(s,y)=(\widehat {s}(s,y), \widehat {y}(s,y))$ . One has $\widehat {y}(s,0)\equiv 0$ , ${\partial \widehat {s}}(s,0)/{\partial s}> 0$ , ${\partial \widehat {y}}(s,0)/{\partial y}>0$ , by definition and orientation-preserving property (symplecticity). Thus, $\widehat {y}(s,y)=yg(s,y)$ , where $g(s,y)$ is a positive $C^{\infty }$ -smooth function on a neighborhood of the interval J in $({\mathbb R}\times {\mathbb R}_{>0})\cup J$ . The lifting $\widetilde {G}$ of the map G to the variables $(s,z)$ , $z=\sqrt y$ , acts as follows:

(1.9) $$ \begin{align} \widetilde{G}:(s,z)\mapsto(\widehat{s}(s,z^2), \widehat{z}(s,z));\quad \widehat{z}=\sqrt{\widehat{y}(s,z^2)}=z\sqrt{g(s,z^2)}. \end{align} $$

The latter square root is well defined and $C^{\infty }$ -smooth. This implies that the map $\widetilde {G}$ is a $C^{\infty }$ -smooth diffeomorphism of domains with arcs of boundaries corresponding to $V\cup J$ and $G(V\cup J)$ . Hence, the lifting $\widetilde {G}\circ \widetilde {F}\circ \widetilde {G}^{-1}$ of the conjugate $F_G:=G\circ F\circ G^{-1}$ is a $C^{\infty }$ -smooth diffeomorphism that is the product of $\widetilde {G}$ -conjugates of the involutions I and $\beta $ . One has $\widetilde {G}\circ I\circ \widetilde {G}^{-1}=I$ , by equation (1.9); $F_G$ is a symplectomorphism, since so are F and G;

(1.10) $$ \begin{align} G(s,y)=(\widehat{s},\widehat{y})=(\widehat{s}(s,0)+O(y), g(s,0)y+O(y^2)), \end{align} $$

by diffeomorphicity. Substituting equations (1.10) and (1.7) to the expression $F_G=G\circ F\circ G^{-1}$ and denoting $(s,0):=G^{-1}(\widehat {s},0)$ , we get

$$ \begin{align*} F_G(\widehat{s},\widehat{y})=\bigg(\widehat{s}+ \frac{\partial\widehat{s}}{\partial s}(s,0)w(s)g^{-(1/2)}(s,0) (\widehat{y})^{1/2}+O(\widehat{y}), \widehat{y}+O(\widehat{y}^{3/2})\bigg). \end{align*} $$

This implies that the conjugate $F_G$ has type of equation (1.7) and hence is strongly billiard-like. This proves the proposition.

Convention 1.25. Let $J=(a,b)\times \{0\}\subset {\mathbb R}^2_{x,y}$ . Let $U\subset \{ y>0\}$ be a domain adjacent to J. Let $F:U\cup J\to {\mathbb R}\times {\mathbb R}_{\geq 0}$ be a map fixing all the points of the interval J. Let $\widetilde {h}:U\cup J\to {\mathbb R}_{\geq 0}$ be a $C^{\infty }$ -smooth F-invariant function, that is, $\widetilde {h}(z)=\widetilde {h}\circ F(z)$ whenever $z,F(z)\in U\cup J$ , and let

(1.11) $$ \begin{align} \widetilde{h}|_J\equiv 0,\quad \frac{\partial\widetilde{h}}{\partial y}>0. \end{align} $$

Let $\widetilde {h}=\mathrm {const.}$ denote the foliation by connected components of level curves of the function $\widetilde {h}$ . This is a $C^\infty $ -smooth foliation on $U\cup J$ , with J being a leaf. It will be called a foliation by F-invariant curves.

Theorem 1.27. For every function $\widetilde {h}$ from Theorem 1.26, replacing it by its post-composition with a $C^\infty $ -smooth function of one variable (which does not change the foliation $\widetilde {h}=\mathrm {const.}$ ) one can achieve that there exists a $C^{\infty }$ -smooth function $\tau =\tau (s,y)$ and a domain $U\subset \{ y>0\}$ adjacent to J such that $(\tau ,\widetilde {h})$ are symplectic coordinates on $U\cup J$ and in these coordinates,

(1.12) $$ \begin{align}F(\tau,\widetilde{h})=(\tau+\sqrt {\widetilde {h}},\widetilde{h}).\end{align} $$

The proof of Theorem 1.26 uses the result of Marvizi and Melrose [Reference Marvizi and Melrose16, Theorem 3.2] stating a formal analogue of Theorem 1.26: the existence of an F-invariant formal power series $\sum _{k=1}^{+\infty }h_k(s)y^k$ , see Theorem 2.1. It implies that in appropriate coordinates $(\tau ,h)$ , the map F takes the form $F(\tau ,h)=(\tau +\sqrt h+{\operatorname {flat}}(h), h+{\operatorname {flat}}(h))$ . Here, ${\operatorname {flat}}(h)$ is an h-flat function, see the above definition. In the coordinates $(\tau ,\phi )$ , $\phi =\sqrt h$ , the lifted map $\widetilde {F}$ takes the form

(1.13) $$ \begin{align}\widetilde{F}(\tau,\phi)=(\tau+\phi+{\operatorname{flat}}(\phi), \phi+{\operatorname{flat}}(\phi)). \end{align} $$

We prove the existence of a $C^{\infty }$ -smooth $\widetilde {F}$ -invariant function $\widetilde {\phi }$ with $\widetilde {\phi }-\phi ={\operatorname {flat}}(\phi )$ (the next theorem), and then deduce the existence statements in Theorems 1.26, 1.9, and 1.12.

1.5 Unique determination of jets. Space of germs of foliations. Non-uniqueness

The statement of Theorem 1.31 follows from the result of Marvizi and Melrose [Reference Marvizi and Melrose16, Theorem 3.2] recalled below as Theorem 2.1. For completeness of presentation, we will give a proof of Theorem 1.31 in §2.8.

Let us now describe the space of jets at J of foliations satisfying the statements of Theorem 1.26. Recall that there exist coordinates $(\tau ,\widetilde {h})$ in which $J=\{\widetilde {h}=0\}$ and F acts as in equation (1.12): $F:(\tau ,\widetilde {h})\mapsto (\tau +\sqrt {\vphantom{\widetilde {h^A}}\widetilde {h}},\widetilde {h})$ . Fix these coordinates $(\tau ,\widetilde {h})$ . Without loss of generality, we consider that $0\in J$ .

Consider the foliation $\widetilde {h}=\mathrm {const}$ . Let ${\mathcal F}$ be another $C^\infty $ -smooth foliation by F-invariant curves defined on a domain V in the upper half-plane $\{\widetilde {h}>0\}$ adjacent to J that extends $C^\infty $ -smoothly to J with J being a leaf. It is a foliation by level curves of an F-invariant function $g(\tau ,\widetilde {h})=\widetilde {h}+{\operatorname {flat}}(\widetilde {h})$ , which follows from Theorem 1.31. We can and will normalize g so that

(1.15) $$ \begin{align} g(\tau,\widetilde{h})=\widetilde{h}+{\operatorname{flat}}(\widetilde{h}), \quad g(0,\widetilde{h})\equiv \widetilde{h}. \end{align} $$

Theorems 1.31, 1.34, and 1.35 and Proposition 1.33 will be proved in §2.8. In §2.9, we deduce non-uniqueness statements of Theorems 1.9, 1.12, 1.13, 1.26, and 1.30 from Theorem 1.35 and the following proposition, which will be also proved there.

1.6 Corollaries on conjugacy of open billiard maps near the boundary

The results stated below and proved in §2.10 concern (symplectic) conjugacy of billiard maps near the boundary.

Here we deal with a strictly convex oriented $C^\infty $ -smooth curve $\gamma \subset {\mathbb R}^2$ that is not closed: parameterized by an interval. We consider that it is positively oriented as the boundary of its convex side. Let us first consider that $\gamma $ goes to infinity in both directions and bounds a convex open billiard. By $\widetilde {\gamma }$ , we denote the family of its orienting unit tangent vectors; $\widetilde {\gamma }$ lies in the space $\Gamma $ , which is the unit tangent bundle of the ambient plane restricted to $\gamma $ . Let s be a natural length parameter of the curve $\gamma $ . Let $\Gamma ^0_+\subset \Gamma $ be the open subset adjacent to $\widetilde {\gamma }$ defined in §1.3. It lies in the space of pairs $(s,v)$ , where $s\in \gamma $ and $v\in T_s{\mathbb R}^2$ is a unit vector directed to the convex side from the curve $\gamma $ . Recall that $\phi =\phi (v)$ denotes the angle between the vector $v$ and the unit tangent vector $\dot \gamma (s)$ . Let $(a,b)=(a_\gamma ,b_\gamma )\subset {\mathbb R}$ denote the length parameter interval parameterizing $\gamma $ . In the coordinates $(s,\phi )$ , the curve $\widetilde {\gamma }$ is the interval $J=J_\gamma =(a,b)\times \{0\}$ , and $\Gamma ^0_+$ is a domain in ${\mathbb R}\times {\mathbb R}_+$ adjacent to J. Set $y=1-\cos \phi $ , see equation (1.2). Recall that the billiard map $\mathcal T_\gamma $ acting by reflection from $\gamma $ of the above unit vectors is a $C^\infty $ -smooth diffeomorphism defined on $\Gamma ^0_+\cup \widetilde {\gamma }$ . In the coordinates $(s,y)$ , it is a symplectic map: a $C^\infty $ -lifted strongly billiard-like map defined on $\mathcal V\cup J$ , where $\mathcal V\subset {\mathbb R}_s\times ({\mathbb R}_+)_y$ is a domain adjacent to J.

The above statements remain valid in the case when the curve $\gamma $ in question is a subarc (parameterized by interval but not necessarily infinite) of a strictly convex $C^\infty $ -smooth curve; $\gamma $ also may be an immersed curve.

The results stated below on conjugacy of billiard maps near the boundary are corollaries of Theorems 1.27 and 1.30 on normal forms of $C^\infty $ -lifted strongly billiard-like maps and their liftings.

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.

Definition 1.42. Let $\gamma $ be a $C^2$ -smooth oriented planar curve, and let s be its length parameter defining its orientation. Let $J_\gamma =(a_\gamma ,b_\gamma )\subset {\mathbb R}_s$ denote the length parameter interval parameterizing $\gamma $ . Let $\kappa =\kappa (s)$ denote the geodesic curvature of the curve $\gamma $ as a function of s. The Lazutkin length of the curve $\gamma $ is the integral

(1.19) $$ \begin{align}{\mathcal L}(\gamma):=\int_{a_\gamma}^{b_\gamma}\kappa^{2/3}(s)\,ds,\end{align} $$

see [Reference Lazutkin15, equation (1.3)]. (While the length parameter interval is defined up to translation, the integral ${\mathcal L}(\gamma )$ is uniquely defined.)

Theorems 1.39 and 1.40 will be deduced from Theorem 1.43 using the following propositions on $C^\infty $ -lifted strongly billiard-like maps and lemma on curves with asymptotic line at infinity.

Remark 1.48. For a $C^\infty $ -smooth strictly convex planar curve going to infinity, the existence of a finite asymptotic tangent line is not a necessary condition for convergence of the improper integral in equation (1.19) defining the Lazutkin length. Namely, consider the graph $\{ y=x^r\}\subset [1,+\infty )\times [1,+\infty )$ , $r>1$ . One has

(1.22) $$ \begin{align} \int\kappa^{2/3}(s)\,ds<+\infty \quad\text{if and only if } r>2. \end{align} $$

Indeed, $ds=\sqrt {1+r^2x^{2(r-1)}}\,dx$ , $\kappa (s(x))= {r(r-1)x^{r-2}}/{(1+r^2x^{2(r-1)})^{3/2}}$ , see equation (2.59),

$$ \begin{align*} \kappa^{2/3}(s(x))\,ds=(r(r-1))^{2/3} \frac{x^{(2/3)(r-2)}}{\sqrt{1+r^2x^{2(r-1)}}}\,dx\simeq(r(r-1))^{2/3}x^\nu \,dx,\!\quad \nu=-\frac{r+1}3, \end{align*} $$

as $x\to +\infty $ . Therefore, the integral in equation (1.22) converges if and only if $\nu <-1$ , that is, $r>2$ . In the case of parabola $\{ y=x^2\}$ , the integral in equation (1.22) diverges.

1.7 Plan of the proof of main results

In §2.1, we recall the above-mentioned result of Marvizi and Melrose [Reference Marvizi and Melrose16, Theorem 3.2] (with proof) yielding $C^{\infty }$ -smooth coordinates in which a $C^\infty $ -lifted strongly billiard-like map F takes the form $F(\tau ,h)=(\tau +\sqrt h+{\operatorname {flat}}(h), h+{\operatorname {flat}}(h))$ . It implies that the lifted map $\widetilde {F}$ , written in the coordinates $(\tau ,\phi )$ , $\phi =\sqrt h$ , takes form of equation (1.13).

Statement (1) of Theorem 1.30 will be proved in §§2.2–2.4. To do this, first in §2.2, we construct a fundamental domain for the map $\widetilde {F}$ (a curvilinear sector $\Delta $ with vertex at a point in J) and an $\widetilde {F}$ -invariant function $\widetilde {\phi }$ defined on a bigger sector that is $\phi $ -flatly close to $\phi $ on the latter bigger sector. Then in §2.3, we construct its $\widetilde {F}$ -invariant extension along the $\widetilde {F}$ -orbits and show that it is well defined on a domain adjacent to J. In §2.4, we prove that the extended function $\widetilde {\phi }$ is $C^{\infty }$ -smooth and $\phi $ -flatly close to $\phi $ . This will prove statement (1) of Theorem 1.30. Statement (2) on normal form will be proved in §2.5.

The existence statement in Theorem 1.26 will be deduced from statement (1) of Theorem 1.30 in §2.6, where we will also prove Theorem 1.27. Existence in Theorems 1.9, 1.12, and 1.13 will be proved in §2.7. The results from §1.5 on jets and space of germs of foliations will be proved in §2.8. Proposition 1.36 and non-uniqueness statements in the main theorems will be proved in §2.9.

The results of §1.6 on conjugacy of billiard maps near the boundary will be proved in §2.10.