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Orbital invariants of billiards and linearly integrable geodesic flows G. V. Belozerova, A. T. Fomenkoab a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
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     § 1. IntroductionIn recent years there was a great interest in the realization by billiard systems (up to Liouville equivalence) of various integrable Hamiltonian systems with two degrees of freedom known in symplectic geometry and topology, theoretical mechanics, and mathematical physics: for instance, see papers by Fomenko, Dragović, Radnović, Vedyushkina (Fokicheva), Kibkalo, Kharcheva, Kobtsev, Pusovoitov, Belozerov, Zav’yalov and Karginova [1]–[26]. Liouville equivalence of integrable systems means that there exists a homeomorphism (a diffeomorphism) of the symplectic phase manifolds (or isoenergy surfaces, that is, surfaces of constant energy) that identifies leaves of the Liouville foliations for the systems under comparison. Here one considers Liouville-integrable systems. Regular leaves are Liouville tori, while singular ones have a more involved structure. In the case of nondegenerate (Bott) systems, in Liouville equivalent systems the closures of almost all integral trajectories are ‘the same’ (just these closures are regular Liouville tori). Recall that the billiard system in a plane domain bounded by arcs of confocal quadrics is integrable. More precisely, the straight lines containing edges of any polygonal trajectory are tangent to a certain quadric which is confocal to the quadrics on the boundary (see Birkhoff’s classical paper [27] and the book [28] by Kozlov and Treshchev). The parameter of this quadric, which is called the caustic, remains constant on the whole trajectory of the system. Vedyushkina has introduced a new class of integrable billiards by considering two-dimensional piecewise smooth surfaces glued of elementary billiard domains (for details, see [13]). When a point mass occurs on a gluing edge, it passes from one elementary billiard to another by the standard reflection law. This class of billiard systems can be extended by considering more involved CW-complexes glued of elementary billiard domain, including so-called billiard books. After that the class of integrable billiard systems was significantly extended by various authors, who considered so-called billiards with potentials (see [29] and [30]), billiards with magnetic field (see [31]), force (evolutionary) billiards (see [1]–[4]), billiards with slipping (see [23] and [24]), billiard games (see [9]), and multidimensional billiard systems realised inside multidimensional confocal quadrics (see [32] and [33]). Fomenko [34] has stated a six-part conjecture on modelling integrable nondegenerate systems with two degrees of freedom by topological billiards. Several parts of this conjecture have already been proved (see [19], [34] and [35]). In this paper we consider the part of this conjecture concerning orbital invariants of integrable systems. Two dynamical systems are said to be orbitally invariant if there exists a homeomorphism between phase (or isoenergy) manifolds that takes trajectories to trajectories (but time on trajectories need not be preserved). In the case of integrable systems with two degrees of freedom an important thing is to know orbital invariants describing the behaviour of trajectories on one-parameter families of regular Liouville 2-tori ‘inhabiting’ 3-dimensional isoenergy surfaces. Examples of such invariants are rotation vectors, which are calculated from the rotation functions arising on such one-parameter families of Liouville tori (see [36]). Note that, previously, authors considered mostly invariants of Liouville equivalence of integrable systems. Thus, in this paper we make the next step by proceeding to an analysis of orbital invariants of billiard systems. These invariants are in two classes. The first class has already been mentioned: these are invariants of one-parameter families of Liouville tori. The second class consists of orbital invariants classifying trajectories on singular leaves of Liouville foliations (so-called ‘atoms’). Now we focus on invariants of the first type (occasionally called edge invariants, that is, invariants on edges of the marked graph equal to the Fomenko–Zieschang invariant). By Fomenko’s initial conjecture the rotation functions of integrable billiards are monotone on one-parameter families on tori. The first examples when the rotation functions were calculated for some billiard systems confirmed this conjecture (see [37]). However, then it turned out that the class of integrable billiards is quite wide and rich, so that it also contains systems with nonmonotone rotation functions, so that the picture turned out to be significantly more interesting and substantive. An analysis of this picture is the subject of our paper. For this analysis we had to cope with the problem of stating an analogue of the famous Liouville theorem. In fact, the classical theorem relates to smooth systems, so that we can introduce smooth action-angle variables in neighbourhoods of Liouville tori. Integrable billiards are piecewise smooth systems, and we cannot use the ‘smooth Liouville theorem’ directly. We investigated a wide class of billiard systems and proved an analogue of Liouville theorem for them. After that we were able to define rotation functions and rotation vectors, that is, construct orbital invariants of systems. In particular, we managed to calculate the rotation functions of important series of integrable billiard systems, for instance, ones that (up to Liouville equivalence) realize an important class of linearly integrable geodesic flows on orientable two-dimensional surfaces. Now we briefly list the main results in this paper. For an arbitrary billiard book not containing free edges on the focal line we establish a full analogue of Liouville’s theorem. Namely, in a small neighbourhood of an arbitrary regular leaf we define action-angle variables consistently and establish some properties of these which are inherited from ‘smooth’ symplectic geometry. The proof of this theorem takes § 2 and § 3. On the basis of this analogue of Liouville’s theorem we find a general formula for the rotation functions of billiard books. This formula is derived in § 4. Next, using this formula, in § 5 we investigate the monotonicity of the rotation functions of three planar billiards symmetric about the coordinate axes. We find a close connection of the action variables of a billiard system inside an ellipse with Graves’ theorem (on a thread and an ellipse). We also show that for a billiard inside an elliptic annulus one of its rotation functions is not monotone. Thus, we disprove Fomenko’s conjecture that all rotation functions of a billiard system are strictly monotone. In addition, we find an interesting connection of the rotation functions of billiards inside an elliptic annulus with the well-known Euler $B$ function. Just this function turns out to ‘prescribe’ the relation between the semiaxes of the boundary ellipses under which the rotation functions on two edges of the molecule are monotone. We also found an interesting example of two billiards with different marked molecules that nevertheless have the same rough molecules and the same edge invariants. In § 6 we calculate the rotation functions of closed topological billiards glued of discs and (round) annuli. This class of billiards is important since it models the Liouville foliations for all geodesic flows on compact orientable surfaces with an additional first integral which is linear in the momenta. The rotation functions of such billiard systems turn out to be expressible in terms of elementary functions. However, they are not always monotone either. Nevertheless, by changing the parameters of the gluing boundaries of such billiard domains we can always obtain monotone rotation functions on all edges of the molecule. AcknowledgementsThe author are grateful to E. A. Kudryavtseva and V. V. Vedyushkina who made a number of valuable comments and showed their interest in this work. G. V. Belozerov is a recipient of a scholarship awarded by the Theoretical Physics and Mathematics Foundation “BASIS”. § 2. Integrable Hamiltonian systems. Liouville’s theorem. Billiard booksConsider a symplectic manifold $(M^{2n},\omega)$ and a smooth function $H$ on it. This function produces a smooth vector field $v=\operatorname{sgrad} H$ on $M^{2n}$, whose components in a local coordinate chart can be calculated by the formula
$$
\begin{equation*}
(\operatorname{sgrad} H)^i=\omega^{ij}\, \frac{\partial H}{\partial x^j}.
\end{equation*}
\notag
$$
Definition 1. The vector field $v=\operatorname{sgrad} H$ is called a Hamiltonian system, and $H$ is called the Hamiltonian function (or simply Hamiltonian) of this vector field. Recall that the form $\omega$ specifies a Poisson bracket $\{\,\cdot\,{,}\,\cdot\,\}$ on the space $C^{\infty}(M^{2n})$. In any local chart it can be defined by the formula
$$
\begin{equation*}
\{f,g\}=\omega^{ij}\, \frac{\partial f}{\partial x^i}\,\frac{\partial g}{\partial x^j} \quad \forall\, f,g\in C^{\infty}(M^{2n}).
\end{equation*}
\notag
$$
Definition 2. A Hamiltonian system $v=\operatorname{sgrad} H$ is said to be completely Liouville integrable if there exists a set $f_1,f_2,\dots,f_n$ of smooth functions on $M^{2n}$ such that A Liouville-integrable Hamiltonian system defines a foliation of the manifold $M^{2n}$ by connected components of common level surfaces of the functions $f_1,\dots,f_n$. This is usually called the Liouville foliation. The structure of this foliation in a neighbourhood of a regular level surface is described by Liouville’s theorem. Theorem 1 (Liouville). Let $v=\operatorname{sgrad} H$ be a completely Liouville integrable Hamiltonian system on $M^{2n}$ and $T_\xi$ be a connected component of a regular level surface of the first integrals $f_1,\dots,f_n$ (from Definition 2). Then: (1) $T_\xi$ is a smooth Lagrangian submanifold of dimension $n$, which is diffeomorphic to $\mathbb R^n/\mathbb Z^m$; in particular, if $T_\xi$ is compact, then it is diffeomorphic to an $n$-torus; (2) the Liouville foliation is trivial in a small neighbourhood $U$ of the torus $T^n$, that is, it is diffeomorphic to the Cartesian product of $T^n$ and the $n$-dimensional disc $D^n$; (3) in the neighbourhood $U=T^n\times D^n$ there exist smooth coordinate variables $(s_1,\dots,s_n,$ $\varphi_1,\dots,\varphi_n)$ which are called action-angle variables and have the following properties:
For a modern proof of this theorem, see [36]. Note that in Liouville’s theorem we need the Hamiltonian system $v$ on $M^{2n}$ to be smooth. Moreover, the form $\omega$ and the manifold $M^{2n}$ itself must be smooth. There exist many systems in mechanics that satisfy these conditions. Nonetheless, there are examples when smoothness fails. In particular, this holds for billiards in domains bounded by arcs of confocal quadrics or on billiard books introduced by Vedyushkina. Recall the definition of confocal quadrics. Definition 3. A family of confocal quadrics in the plane is the set of quadrics defined by an equation where $a>b>0$ are fixed numbers and $\lambda\in \mathbb R$ is a real parameter. A billiard book $D$ is a $2$-complex glued of elementary confocal tables $D_1,\dots,D_n$ (which are compact plane domains bounded by confocal quadrics, with corners of opening $\pi/2$ on the boundary), along some common segments of their boundaries. On each gluing piece a rearrangement $\sigma\in S(n)$ is fixed which corresponds to the transition of a point mass from one sheet to another. In addition, the following restrictions are imposed on the structure of the book.
The gluing segments of the boundaries of sheets of a billiard book are divided into four classes. If there is no gluing on a segment $l$, then it is called a free edge. If the rearrangement assigned to $l$ is a transposition, then $l$ is called a gluing edge. If such a rearrangement is neither trivial nor a transposition, then the segment in question is called a spine of the book. For a more detailed definition, see [19]. The dynamics of a point mass on a billiard book is defined by the following set of rules.
According to Vedyushkina [10]–[16] a classical billiard (a system with no exterior forces) on a billiard book is Liouville integrable in the piecewise smooth sense. Its first integrals are the kinetic energy $H$ and the parameter $\Lambda$ of the confocal quadric that is tangent to all straight-line parts of the trajectory of the point mass (or their extensions). Moreover, the following analogue of Liouville’s theorem holds. Theorem 2 (Vedyushkina). Let $T_\xi$ be a connected component of a nonsingular level surface of the first integrals $H$ and $\Lambda$ of the billiard system on a book $D$. Then: Remark 1. For billiard books all regular level surfaces are compact. This is because the billiard book itself is compact. Billiard systems on noncompact billiard tables can possess noncompact level surfaces (for instance, see [38]). Theorem 2 is a direct generalization of assertions (1) and (2) in the classical Liouville theorem (Theorem 1) to billiard books. A reasonable question is as follows: can we generalize assertion (3) of Liouville’s theorem to such systems? According to Lazutkin [39] and Kudryavtseva [40], for a rather wide class of billiard books, in a neighbourhood of each level surface on which nonconvex parts of the boundaries of sheets of the book cannot touch one another, the system can be smoothed. Hence we can use the classical Liouville theorem in these cases. However, this approach does not work for regular level surfaces on which smoothing is impossible. Nevertheless we will see in what follows that we can still consistently define action-angle variables on such levels. To state a full analogue of Liouville theorem for billiard books we must understand the structure of the symplectic form $\omega$ in a neighbourhood of a regular level surface. Let $M^4$ denote the phase space of the billiard system, and let $\Gamma_1,\dots,\Gamma_n$ be the boundaries of sheets $D_1,\dots,D_n$ of a billiard book $D$, respectively. Recall that the projection $\pi\colon M^4\to D$ taking a point-vector pair $(x,v)$ to $x\in D$ is well defined on $M^4$. On the cotangent bundle of the interior of each sheet $D_i$ we have the exact form $\omega=dp_x\wedge dx+dp_y\wedge dy=d\alpha$, where $\alpha=p_x\, dx+p_y \,dy$, and $p_x$ and $ p_y$ are the momenta dual to the coordinates $x$ and $y$, respectively. Definition 4. The complex on a Liouville torus that consists of the pairs ${(x,v) \in M^4}$ such that $x\in \Gamma_i$ for some $i$ is called the critical graph. A critical graph consists of a finite set of circular generators of the Liouville torus of two types (‘parallels’, or ‘circles of latitude’, and ‘meridians’). It partitions the torus into connected components of dimension $2$, which we call faces. Each face of the critical graph corresponds to a unique sheet $D_i$. On the other hand different faces and different Liouville tori can correspond to the same sheets. Remark 2. At foci, for $\Lambda=b$, we obtain a whole circle of directions of the velocity vector, while at the other points only four or fewer directions are possible. Since this case must be distinguished from the others, we consider only billiard books containing no free edges on the focal line. This condition ensures that on the level surface $\Lambda=b$, provided that this surface is regular, the region of possible motion contains no foci. Consider an arbitrary chamber of the image of the moment map for the first integrals $(H,\Lambda)$. Let $\widetilde{M}^4$ denote the restriction of the phase space of our system to this chamber. By Theorem 2, $\widetilde{M}^4$ is homeomorphic to a Cartesian product of a 2-disc by a 2-torus or a disjoint union of several such products. We assume without loss of generality that $\widetilde{M}^4$ consists of one connected component. On each sheet of the billiard book we consider the elliptic coordinates $\lambda_1$ and $\lambda_2$ and pull them back to $\widetilde{M}^4$ by means of $\pi$. Note that on each face of the critical graph the form $\alpha$ can be calculated in the variables $(\lambda_1,\lambda_2,H,\Lambda)$ by the formula
$$
\begin{equation}
\alpha=p_1\,d\lambda_1+p_2\,d\lambda_2, \quad\text{where } p_i^2=\frac{H(\Lambda-\lambda_i)}{2(a-\lambda_i)(b-\lambda_i)}.
\end{equation}
\tag{2.1}
$$
Here $a>b>0$ are the parameters of the family of confocal quadrics (see Definition 3). It is clear from this formula that after reflecting from a free edge, crossing a gluing edge or a spine of the book, or touching the caustic the momentum $p_i$ changes sign. On the other hand the direction of increase of the corresponding elliptic coordinate also changes. Hence $\alpha$ can be well defined almost everywhere on $\widetilde{M}^4$. However, this definition has some complicated points. They are as follows. 1. The domain of definition of the form $\alpha$. To define $\alpha$ consistently in the variables $(\lambda_1,\lambda_2,H,\Lambda)$ (in a neighbourhood of a Liouville torus) we must assume that on each sheet of the billiard book the boundary of the domain of possible motion varies smoothly. So we divide all regular leaves into two classes. Definition 5. A Liouville torus is said to be perfectly regular if, in a smooth neighbourhood of this torus, on each sheet of the billiard book the boundary of the domain of possible motion depends smoothly on $\Lambda$. Otherwise we say that the torus is pseudoregular. Remark 3. The simplest example of pseudoregular tori is as follows. Consider a billiard inside an elliptic annulus. Let $\lambda_0$ denote the parameter of the inner boundary ellipse. Then Liouville tori corresponding to $\Lambda=\lambda_0$ are pseudoregular. In fact, for ${\Lambda<\lambda_0}$ the domain of possible motion is bounded by the outer boundary ellipse and the caustic, while for $\lambda_0<\Lambda<b$ this domain coincides with the whole billiard table. Thus, the parameter of the inner boundary of the domain of possible motion is $\min\{\Lambda,\lambda_0\}$. However, this is not a smooth function for $\Lambda=\lambda_0$. Hence for $\Lambda=\lambda_0$ Liouville tori are pseudoregular. Remark 4. We will see in what follows that for a fixed value of the energy the manifold $\widetilde{M}^4$ can only contain a finite number of pseudoregular Liouville tori. 2. The consistency of the definition of $\alpha$. On perfectly regular tori ambiguity can only arise at points where one of the elliptic coordinates takes the value $\Lambda$, $a$ or $b$. To avoid ambiguity we introduce the ‘adjusted’ elliptic coordinates defined by
$$
\begin{equation*}
u_i=\int_{a_i}^{b_i}\frac{dt}{\sqrt{(\Lambda-t)(a-t)(b-t)}}.
\end{equation*}
\notag
$$
Here $a_1=\lambda_1$, $a_2=b$, $b_1=\Lambda$ and $b_2=\lambda_2$ for elliptic chambers (that is, for $\Lambda<b$) and $a_1=\lambda_1$, $a_2=\Lambda$, $b_1=b$ and $b_2=\lambda_2$ for hyperbolic ones (that is, for $\Lambda\in(b,a)$). Remark 5. The variables $(u_1,u_2)$ are defined everywhere on $\widetilde{M}^4$ and are smooth in a small neighbourhood of a perfectly regular leaf. Now from $(u_1,u_2)$ we pass to well-defined variables $(\psi_1,\psi_2)\ \operatorname{mod}2\pi$. For an accurate definition we observe that we can partition the Liouville torus into rectangular domains where the elliptic coordinates are well defined. Taking one of these domains as a starting location we glue to it the remaining domains is succession, changing on them the directions of increase and origins of the ‘adjusted’ elliptic coordinates so that these coordinates increase continuously. Next we divide each coordinate by the total length of its range and multiply by $2\pi$. It is clear from this construction that the coordinates $(\psi_1,\psi_2,H,\Lambda)$ are smooth in a neighbourhood of a regular leaf. Similar arguments show that the form $\omega$ is also well defined in a neighbourhood of each perfectly regular leaf on $\widetilde{M}^4$. Recall that in terms of the elliptic coordinates $\lambda_1$ and $\lambda_2$ and dual momenta $p_1$ and $p_2$ the form $\omega$ has the canonical expression $\omega=dp_1\wedge d\lambda_1+ dp_2\wedge d\lambda_2$. Since after a reflection, a transition to another sheet or touching the caustic the direction of growth of an elliptic coordinate and the corresponding momentum change sign simultaneously, the terms in this form are invariant. Remark 6. The restriction of $\omega$ to each perfectly regular leaf is zero. In fact, $\omega=0$ on a common level of the first integrals $H$ and $\Lambda$ of the geodesic flow in the plane. Since each face of the critical graph on a Liouville torus is a part of a common level of $H$ and $\Lambda$ for a system without reflections, the form $\omega$ vanishes on each face. It remains to note that the critical graph has measure zero on the Liouville torus, so that $\omega=0$ on the whole torus. We see that $\alpha$ and $\omega$ are well defined in a neighbourhood of each perfectly regular leaf. Moreover, they are smooth almost everywhere in this neighbourhood. Now we can state a complete analogue of Liouville’s theorem for billiard books. Theorem 3. Let $T_\xi$ be a connected component of a nonsingular level surface corresponding to values $(h,\lambda)$ of the pair of first integrals $(H,\Lambda)$ of a billiard system on a book $D$ not containing free edges on the focal line. Then the following hold: Parts (1) and (2) of the theorem are due to Vedyushkina. We devote § 3 to the proof of part (3). Remark 7. The question of reducing the form $\omega$ to a canonical form is well posed in the case when the system can be smoothened in a neighbourhood of a leaf. For systems with reflections Hamiltonian smoothing was considered by Lazutkin [39] and Kudryavtseva [40]. § 3. Proof of the theorem on action-angle variables for billiard booksIn this section we are concerned with the proof of part (3) of Theorem 3. First we prove it for perfectly regular leaves. Note that the definition of a perfectly regular leaf is quite abstract, so we re-formulate it in terms of the first integrals $H$ and $\Lambda$. However, first we recall that an elliptic Liouville torus corresponds to $\Lambda=\lambda_0<b$, while a hyperbolic Liouville torus corresponds to $\Lambda=\lambda_0>b$. Lemma 1 (criterion of perfect regularity). Let $T_\xi$ be a Liouville torus corresponding to values $(h_0,\lambda_0)$ of the first integrals $(H,\Lambda)$. Then it is pseudoregular if and only if the quadric with parameter $\lambda_0$ is a boundary component of at least one sheet of the book. Proof. Note that domains of possible motion of a point mass are independent of the (positive) value of the energy, so from $\widetilde{M}^4$ we can go over to the isoenergy surface $\widetilde{Q}^3_h=\{{(x,v)\in\widetilde{M}^4}\mid \|v\|^2=2h\}$, where $h$ is an arbitrary positive number.
First we prove this lemma for elliptic tori. Assume that $\Gamma_1$ (the boundary of the sheet $D_1$ of the billiard book) contains an arc of an ellipse with parameter $\lambda_0$. By Vedyushkina’s theorem on the classification of planar billiard tables, $D_1$ lies either inside or outside this ellipse (so that the particle can also occur on the ellipse). Let $D_1$ lie in the closed domain bounded by the ellipse with parameter $\lambda_0$; then for $\Lambda<\lambda_0$ the set $\pi(T)$ (here the torus $T\in\widetilde{Q}^3_h$ corresponds to the value $\Lambda$ of the parameter) is disjoint from $D_1$, so that the range of the first elliptic coordinate on $D_1$ has length zero. For $\Lambda>\lambda_0$ the first elliptic coordinate on the projection $\pi(T)$ onto $D_1$ ranges over the closed interval $[\lambda_0,\lambda]$. Hence in a neighbourhood of $\Lambda=\lambda_0$ the function $g(\Lambda)$ expressing the length of the range of values of the first elliptic coordinate on the projection $\pi(T)$ onto $D_1$ has the expression $g(\Lambda)=\max\{0,\Lambda-\lambda_0\}$. However, this function is not smooth at the point $\Lambda=\lambda_0$. Hence, by Definition 5 the torus $T_{\xi}$ is pseudoregular. We can show in a similar way that if $D_1$ lies outside the ellipse with parameter $\lambda_0$, while the ellipse itself is a part of $\Gamma_1$, then the torus $T_{\xi}$ is also pseudoregular. On the other hand, if an ellipse with parameter $\lambda_0$ does not lie on the boundary of any sheet of the book, then on the sheets corresponding to close tori the inner boundary components (if they exist) either remain the same or coincide with the caustic, which is disjoint from the boundaries of sheets and varies smoothly. Thus, the endpoints of the range of the first elliptic coordinate vary smoothly, while the endpoints of the range of the second coordinates are fixed. Hence $T_\xi$ is a perfectly regular torus, and for elliptic tori the proof is complete. The case of hyperbolic Liouville tori is treated by means of similar arguments. It remains to consider the case when $\lambda_0=b$. By the assumptions of Theorem 3 the boundaries of sheets of the billiard book contain no free edges on the focal axis. Hence, if the level $\Lambda=b$ is regular, then in a small neighbourhood of the Liouville torus in question the boundary of the domain of possible motion remains the same. Hence the torus is perfectly regular. Lemma 1 is proved. We have established a convenient criterion of regularity, which we use slightly below. Recall that the form $\alpha$ is well defined in a neighborhood of a perfectly regular leaf. Thus, we can introduce action-angle variables in the classical way. Consider an arbitrary cycle $\gamma$ on the Liouville torus, and let $s(\gamma)$ denote the integral of $\alpha$ over this cycle. It turns out that this integral ‘inherits’ one property from ‘smooth’ symplectic geometry. Lemma 2. Let $\gamma_0$ and $\gamma_1$ be homological cycles on a perfectly regular Liouville torus. Then $s(\gamma_0)=s(\gamma_1)$. Proof. Since the cycles $\gamma_0$ and $\gamma_1$ are homological, $\gamma_1-\gamma_0$ is the boundary of a set $K$. The faces of the critical graph partition this set into parts $K_1,\dots,K_m$.
We calculate $s(\gamma_1)-s(\gamma_0)$. To do this we add to this difference the integrals of $\alpha$ over the remaining parts of the boundaries of the $K_i$ (not involved in $s(\gamma_1)-s(\gamma_0)$). The orientation of these parts must be compatible with the orientation of the $K_i$. Note however, that the sum of these integrals vanishes because the boundaries of adjacent sets $K_i$ have opposite orientation. Thus, we obtain the formula
$$
\begin{equation*}
s(\gamma_1)-s(\gamma_0)=\sum_{i}\int_{\partial K_i}\alpha.
\end{equation*}
\notag
$$
Instead of the Liouville torus in question we consider the right-hand integrals on a common level of the integrals $H$ and $\Lambda$ of the system without reflections. This is admissible because each face of the critical graph corresponds to a part of the phase space of the system without reflections. Now we can apply Stokes’ formula to each integral and take advantage of the fact that $\omega$ on each face of the critical graph:
$$
\begin{equation*}
s(\gamma_1)-s(\gamma_0)=\sum_{i}\int_{\partial K_i}\alpha=\sum_{i}\int_{K_i}\omega=0.
\end{equation*}
\notag
$$
Hence $s(\gamma_0)=s(\gamma_1)$.
Lemma 2 is proved. Fix two basis cycles $\gamma_1$ and $\gamma_2$ drawn by coordinate lines of $\psi_1$ and $\psi_2$. We define the action variables $s_1$ and $s_2$ by
$$
\begin{equation*}
s_1=\frac{1}{2\pi}\int_{\gamma_1}\alpha\quad\text{and} \quad s_2=\frac{1}{2\pi}\int_{\gamma_2}\alpha.
\end{equation*}
\notag
$$
Next we introduce the angular variables $(\varphi_1,\varphi_2)$. Recall that in the smooth case coordinate lines of the $\varphi_i$ are integral curves of the vector fields $\operatorname{sgrad} s_i$. Since the form $\omega$ is nondegenerate, continuous and smooth almost everywhere in a neighbourhood of a perfectly regular leaf, the vector fields $\operatorname{sgrad} H$ and $\operatorname{sgrad} \Lambda$ are continuous and tangent to Liouville tori. Moreover, they are linearly independent and smooth almost everywhere. On a Liouville torus they can fail to be smooth only on several curves of the form $\psi_i=\mathrm{const}$ (that is, at points on the critical graph). Still, each of these fields can be extended from any face of the critical graph to a smooth field on the whole torus. Note also that these fields commute away from the critical graph. We observe that by (2.1) the functions $s_1$ and $s_2$ can be represented as follows:
$$
\begin{equation}
s_i=\frac{1}{2\pi}\sum_{j=1}^{N_i}\int_{a_{i,j}}^{b_{i,j}} \sqrt{\frac{H(\Lambda-t)}{2(a-t)(b-t)}}\,dt.
\end{equation}
\tag{3.1}
$$
In this formula $a_{i,j}$ and $b_{i,j}$ can accept the following values:
By Definition 5, in a small neighbourhood of a perfectly regular leaf the coefficients $a_{i,j}$ and $b_{i,j}$ are infinitely smooth function of the Liouville torus. Hence $s_1$ and $s_2$ depend smoothly on $H$ and $\Lambda$. Thus, the fields $\operatorname{sgrad}s_i$ can be expressed linearly in terms of $\operatorname{sgrad}H$ and $\operatorname{sgrad}\Lambda$ with coefficients constant on Liouville tori. Moreover, these vector fields have the same properties as $\operatorname{sgrad} H$ and $\operatorname{sgrad} \Lambda$. For a consistent definition of the angle variables $(\varphi_1,\varphi_2)$ the phase flows of the vector fields $\operatorname{sgrad} s_i$ must be well defined and commuting. We show that this is indeed the case. To do this we look at an auxiliary problem. Let $v_1$ and $v_2$ be smooth vector fields which are linearly independent in a small neighbourhood of a point $P\in\mathbb{R}^2(x_1,x_2)$. It is well known that if they commute, then there exist a neighbourhood of $P$ and smooth coordinates $(y_1,y_2)$ such that $v_1=\partial/\partial y_1$ and $v_2=\partial/\partial y_2$. Now we have to see what changes if we relax the assumption that the fields are smooth. Namely, let $v_1$ and $v_2$ be vector fields with the following properties:
Remark 8. The vector fields $\operatorname{sgrad} s_1$ and $\operatorname{sgrad} s_2$ have the above properties 1–5 on perfectly regular Liouville tori, provided that the critical graph is a system of disjoint circles. Lemma 3. Let $v_1$ and $v_2$ be vector fields with properties 1–5. Then their phase flows are well defined and commute, so there exists a $C^1$-smooth change of variables $\mathbf{y}=f(\mathbf{x})$ such that $ v_i=\partial/\partial y_i$. Moreover, this change of variable is $C^{\infty}$-smooth away from the line $x_2=0$. Proof. Note that conditions 1–3 ensure that the Cauchy problems $\dot{\mathbf{x}}=v_i(\mathbf{x})$ with initial data in the disc $D$ are uniquely solvable (because the assumptions of Picard’s theorem are fulfilled). Moreover, their solutions depend continuously on the initial data and parameters.
Let $P=(0,0)$. We show that in a small neighbourhood of this point the fields $v_1$ and $v_2$ can be straightened. We split the proof into two steps. Step 1. We start by straightening out one of the fields. Suppose that $v_1(P)$ is not parallel to $\partial/\partial x_1$ (we can assume this by property $4$). We describe the straightening procedure for it. Let $\Phi(t,x)$ denote the value of the solution of the Cauchy problem
$$
\begin{equation*}
\begin{cases} \dot{\mathbf{x}}=v_1(\mathbf{x}), \\ \mathbf{x}(0)=(x,0) \end{cases}
\end{equation*}
\notag
$$
at the time $t$.
By the continuous dependence of the solution on the initial data in a small neighbourhood of zero, $\Phi(t,x)$ is a continuous map. We show that it is continuously differentiable. By definition the vector function $\Phi(t,x)$ satisfies the differential equation
$$
\begin{equation*}
\frac{\partial \Phi(t,x)}{\partial t}=v_1(\Phi(t,x)).
\end{equation*}
\notag
$$
Since the vector field $v_1$ and the map $\Phi(t,x)$ are continuous, the partial derivative ${\partial \Phi(t,x)}/{\partial t}$ is defined and continuous in a small neighbourhood of zero.
Now we calculate the derivative of $\Phi$ with respect to the second argument. Were the vector field $v_1$ smooth, we would have
$$
\begin{equation*}
\frac{\partial\Phi(t,x)}{\partial x}= \begin{pmatrix}1\\0\end{pmatrix} +\int_0^t\frac{\partial v_1(\Phi(s,x))}{\partial x_1}\,ds.
\end{equation*}
\notag
$$
Note that this formula also holds in our case. In fact, as the field $v_1$ satisfies condition $3$ and is not parallel to $\partial/\partial x_1$ in a small neighbourhood of $P$ on the line $x_2=0$, the formula can be applied to $t>0$ and $t<0$. As the limits of this partial derivative at $t=0$ from the left and right coincide, the equality remains valid for sufficiently small $t$ and $x$. Hence the partial derivative ${\partial\Phi(t,x)}/{\partial x}$ is well defined and continuous.
Now, since the first partial derivatives of $\Phi(t,x)$ are well defined and continuous, the map $\Phi(t,x)$ itself is continuously differentiable in a small neighbourhood of zero. Now we calculate the Jacobian matrix of $\Phi(t,x)$ at $P$:
$$
\begin{equation*}
J=\begin{pmatrix} v_1^1(P) & 1\\ v_1^2(P) & 0 \end{pmatrix}.
\end{equation*}
\notag
$$
Since $v_1(P)$ is not parallel to $\partial/\partial x_1$, the determinant $\det J$ is distinct from zero. Hence the assumption of the inverse mapping theorem are fulfilled, and in the variables $(t,x)$ the field $v_1$ coincides with $\partial/\partial t$. Note also that for $x\neq0$ this change of variable is infinitely smooth. Hence conditions 1–5 also hold in the new coordinates.
Step 2. Now we straighten the second field. We re-denote the variables for convenience: let $t$ be denoted by $y$. By Step $1$ we have $v_1=\partial/\partial y$. We use property $4$. For $y\neq0$ we have Hence the field $v_2$ is independent of $y$ almost everywhere. Therefore, $v_2$ depends only on $x$. It also follows that $v_2$ is smooth.Note that if $\mathbf{r}(t)=(x(t),y(t))$ is a solution of the differential equation ${\dot{\mathbf{r}}(t)=v_2(\mathbf{r}(t))}$, then for each $C\in \mathbb{R}$ the vector function $(x(t),y(t)+C)$ also solves this equation. Thus, if we regard $\Psi(t,z)$ as the value of the solution of the Cauchy problem
$$
\begin{equation*}
\begin{cases} \dot{\mathbf{r}}=v_2(\mathbf{r}), \\ \mathbf{r}(0)=(0,z) \end{cases}
\end{equation*}
\notag
$$
at time $t$, then $\Psi(t,z)=\Psi(t,0)+z$. Similarly to Step $1$ we can show that $\Psi(t,z)$ is a continuously differentiable map which is invertible in a neighbourhood of the origin. Moreover, $\partial/\partial z=\partial/\partial y$. Hence the vector fields straighten up in the variables $(t,z)$ so that $v_1=\partial/\partial z$ and $v_2=\partial/\partial t$.
Lemma 3 is proved. We have verified that the phase flows of the $\operatorname{sgrad} s_i$ are well defined and commute almost everywhere on perfectly regular Liouville tori. Only points at which two circles of the critical graph intersect can be exceptional. However, there is a finite number of such points, and they cannot affect the commutativity of the phase flows. Hence the phase flows commute everywhere on Liouville tori. Since phase flows are continuous maps and the vector fields $v_1$ and $v_2$ are linearly independent, the angle variables are continuously differentiable on each perfectly regular Liouville torus. To define them consistently assume that the reference point is $\psi_{1,2}=0$. Again, the variables $(s_1,s_2,\varphi_1,\varphi_2)$ are continuous by the theorem on the continuous dependence of the solution of the Cauchy problem on the initial data and parameters. We have almost proved Theorem 3 for perfectly linear leaves. It remains to show that these angle variables are $2\pi$-periodic. Lemma 4. The variables $(\varphi_1,\varphi_2)$ are $2\pi$-periodic. Coordinate lines of $\varphi_1$ and $\varphi_2$ are homologous to coordinate lines of $\psi_1$ and $\psi_2$, respectively. Proof. First we note that on each perfectly regular Liouville torus the form $\alpha$ is cohomologous to $\alpha_1=s_1\,d\psi_1+s_2\,d\psi_2$. Moreover, $\alpha - \alpha_1$ is the differential of a continuously differentiable function $f$ in a neighbourhood of a perfectly regular leaf, which is $C^{\infty}$-smooth almost everywhere. Going over to a cohomologous form preserves integrals of forms. So we can assume that in a full neighbourhood of a perfectly regular torus we have where the functions $\beta_i$ are continuous in a small neighbourhood of the perfectly regular leaf and infinitely smooth almost everywhere.
Hence the matrix of t$\omega=d\alpha$ has the following form in the variables $(\mkern-1mu s_1\mkern-1mu,\mkern-1mu s_2,\mkern-1mu \psi_1\mkern-1mu,\mkern-1mu\psi_2\mkern-1mu)$:
$$
\begin{equation*}
\omega=\begin{pmatrix} 0 & \dfrac{\partial\beta_2}{\partial s_1}-\dfrac{\partial\beta_1}{\partial s_2} & 1-\dfrac{\partial\beta_1}{\partial \psi_1} & -\dfrac{\partial\beta_1}{\partial \psi_2} \\ -\dfrac{\partial\beta_2}{\partial s_1}+\dfrac{\partial\beta_1}{\partial s_2}& 0 & -\dfrac{\partial\beta_2}{\partial \psi_1} & 1-\dfrac{\partial\beta_2}{\partial \psi_2} \\ -1+\dfrac{\partial\beta_1}{\partial \psi_1} & \dfrac{\partial\beta_2}{\partial \psi_1} & 0 & 0 \\ \dfrac{\partial\beta_1}{\partial \psi_2} & -1+\dfrac{\partial\beta_2}{\partial \psi_2} & 0 & 0 \end{pmatrix}.
\end{equation*}
\notag
$$
Note that, since $\omega$ is continuous, the partial derivatives of the functions $\beta_i$ with respect to the variables $\psi_j$ are too. Hence $\beta_i\in C^1$ on each perfectly regular Liouville torus.
The system of equations for integral curves of the vector field $\operatorname{sgrad}s_1$ assumes the form
$$
\begin{equation}
\begin{cases} \dfrac{\partial \beta_2}{\partial \psi_1}\dot{\psi}_1-\biggl(1-\dfrac{\partial \beta_2}{\partial \psi_2}\biggr)\dot{\psi}_2=0, \\ \biggl(1-\dfrac{\partial \beta_1}{\partial \psi_1}\biggr)\dot{\psi}_1-\dfrac{\partial \beta_1}{\partial \psi_2}\dot{\psi}_2=1. \end{cases}
\end{equation}
\tag{3.2}
$$
Eliminating $t$ we obtain the equation of these curves:
$$
\begin{equation*}
\frac{\partial \beta_2}{\partial \psi_1}\,d\psi_1=\biggl(1-\frac{\partial \beta_2}{\partial \psi_2}\biggr)\,d\psi_2.
\end{equation*}
\notag
$$
We consider it on the cylinder with coordinates $(\psi_1,\psi_2)$, where each $\psi_1$ is considered modulo $2\pi$. This equation has the first integral $F=\psi_2 - \beta_2(\psi_1,\psi_2)$ on the cylinder. Since the function $\beta_2$ is continuous on the torus, it is bounded. Hence level curves of $F$ are closed and bounded, and therefore they are compact. Also note that, since $\omega$ is nondegenerate, the gradient of $F$ is nonzero. Hence level curves of $F$ are $C^1$-submanifolds by the implicit function theorem. Since a compact one-dimensional manifold is homeomorphic to a circle, the field $\operatorname{sgrad}s_1$ has closed trajectories.
Hence, as the form $\omega$ is nondegenerate, integral curves of this field are periodic. Let $T>0$ be the least period of an integral curve. We integrate the equations in (3.2) from 0 to $T$. From the first equation we obtain $\psi_2(0)=\psi_2(T)$, and from the second $\psi_1(T)=T+\psi_1(0)$. Hence $T=2\pi n$ for some positive integer $n$. It remains to show that $n=1$. By the unique solvability of the solution of a Cauchy problem integral trajectories make no self-intersections for $t\in[0,T)$. We show that if $n>1$ then a self-intersection must occur. Suppose that this is not so for some integral curve $\chi=(\psi_1(t),\psi_2(t))$. Consider this curve on the coordinate plane $(\psi_1,\psi_2)$, where we assume that $\psi_i(0)\in[0,2\pi)$. Note that $\chi$ is bounded in $\psi_2$ because $\psi_2(0)=\psi_2(T)$. We denote the minimum value of $\psi_2(t)$ by $m$ and the maximum value by $M$. Note that $\chi$ partitions the coordinate plane into two connected components one of which contains the half-plane $\psi_2<m$ and the other contains the half-plane $\psi_2>M$. In the coordinate plane $(\psi_1,\psi_2)$ we consider the new curve $\chi'=(\psi_1(t)+2\pi,\psi_2(t))$. If it intersects $\chi$ at some point $(\psi_1^0,\psi_2^0)$, then the point $(\psi_1^0+2\pi,\psi_2^0)$ also lies on $\chi$. However, these two points coincide on the torus, so we have a self-intersection at this point, which is impossible by assumption. Thus, the curves $\chi$ and $\chi'$ are disjoint. On the other hand these two curves have the same minima and maxima with respect to the second coordinate. Hence the points of minima and maxima of $\chi'$ with respect to $\psi_2$ lie in different components of the partition of the coordinate plane by $\chi$. It remains to note that $\chi'$ is continuous. Hence $\chi$ and $\chi'$ must intersect, which is impossible as shown above. This is a contradiction. Hence $n=1$ and coordinate lines $\psi_2=\mathrm{const}$ are holomogous to curves $\varphi_2=\mathrm{const}$. In the case of the vector field $\operatorname{sgrad}s_2$ and the coordinate $\varphi_2$ the argument is similar. Lemma 4 is proved. We have proved Theorem 3 for perfectly regular Liouville tori. It remains to construct the action-angle variables on pseudoregular Liouville tori. Lemma 5. The functions $s_i$ can be extended by continuity to $C^1$-functions of $H$ and $\Lambda$ on the whole of $\widetilde{M}^4$. Proof. On perfectly regular tori the functions $s_i$ are defined by (3.1). Since the boundaries of domains of possible motion on sheets of the billiard book depend continuously on the point in a chamber of the bifurcation diagram, the functions $a_{i,j}$ and $b_{i,j}$ (in (3.1)) are continuous on $\widetilde{M}^4$. Therefore, the $s_i$ can be defined consistently as continuous functions on $\widetilde{M}^4$. We show that they are $C^1$-functions of $H$ and $\Lambda$.
If a leaf $(h_0,\lambda_0)$ is pseudoregular, then by Lemma 1 some $a_{i,j}$ and $b_{i,j}$ have representations of the form $\min\{\lambda_0,\Lambda\}$ or $\max\{\lambda_0,\Lambda\}$. We look at one integral in (3.1). Assume that $b_{1,j}=\widehat{\lambda}=\mathrm{const}$ and $a_{1,j}=\min\{\lambda_0,\Lambda\}$. Since the factor of $\sqrt{H}$ can be moved outside the integral sign, the partial derivative with respect to $H$ is well defined and continuous everywhere. Now we calculate the partial derivative of this integral with respect to $\Lambda$. Note that for $\Lambda<\lambda_0$ the integrand vanishes at the lower limit of integration. Therefore, by the rule of the differentiation of an integral with variable limits the left and right partial derivatives with respect to $\Lambda$ coincide. Hence the $s_i$ are continuously differentiable functions of $H$ and $\Lambda$. Lemma 5 is proved. Thus we have constructed action variables on the whole of $\widetilde{M}^4$. It remains to define angle variables on pseudoregular tori. Let $T_0$ be the pseudoregular torus corresponding to the values $(h_0,\lambda_0)$ of the first integrals $(H,\Lambda)$. Fix $\varepsilon>0$ such that for $\lambda\in(\lambda_0-\varepsilon,\lambda_0)\cup(\lambda_0,\lambda_0+\varepsilon)$ the leaf corresponding to $(h,\lambda)$ in $\widetilde{M}^4$ (where $h$ is arbitrary) is perfectly regular. Then we can consider continuous extensions of the fields $\operatorname{sgrad} s_i$ to $\lambda<\lambda_0$ and $\lambda>\lambda_0$. In the variables $(\lambda_1,\lambda_2,H,\Lambda)$ these extensions coincide. (The proof is as in Lemma 5.) To complete the proof of part (3) of Theorem 3 it remains to notice that the commutativity property of the flows of $\operatorname{sgrad} s_i$ on pseudoregular Liouville tori and the $2\pi$-periodicity of trajectories is stable with respect to taking the limit. The other properties follow from the theorem on the continuous dependence of the solution of the Cauchy problem on the initial data and parameters. Theorem 3 is proved. § 4. Rotation functions of billiard books and a method for their calculation. Edge orbital invariantLet $v=\operatorname{sgrad} H$ be an integrable Hamiltonian system on a symplectic manifold $M^4$ with an additional first integral $F$. If an isoenergy surface $Q^3_h=\{x\in M^4\mid H(x)=h\}$ is compact and regular and the restriction of $F$ to $Q^3_h$ is a Bott function, then the Liouville foliation on $Q^3_h$ for this system is uniquely coded by the Fomenko–Zieschang invariant. Recall that the Fomenko invariant (or rough molecule) is the type of the base of the Liouville foliation on $Q_h^3$, which is a Reeb graph with vertices endowed with symbols of atoms describing the Liouville foliation in neighbourhoods of singular leaves. This graph classifies the integrable Hamiltonian system uniquely up to rough Liouville equivalence, that is, up to a homeomorphism between the bases of Liouville foliations that can be lifted to a neighbourhood of each leaf as a leafwise homeomorphism. The Fomenko–Zieschang invariant (or marked molecule) is the rough molecule endowed with the marks $r$, $\varepsilon$ and $n$ corresponding to the pattern of gluing of Liouville tori on the boundaries of adjacent atoms. It its turn, this invariant classifies all integrable Hamiltonian systems up to Liouville equivalence, that is, up to a leafwise homeomorphism between Liouville foliations (see [41]). Our aim is to understand the structure of the rotation functions for an arbitrary billiard book and to describe a method for their calculation. Recall the definition of the rotation function on an edge of the molecule of a smooth integrable Hamiltonian system. On the whole edge we choose coordinates $(F,\varphi_1,\varphi_2)$, where the $\varphi_i$ are the angle variable from Liouville’s theorem (see Theorem 1); by part (3) of that theorem, in the variables $(F,\varphi_1,\varphi_2)$ the flow $\operatorname{sgrad} H$ straightens up. In other words, trajectories of the system are linear windings on Liouville tori and
$$
\begin{equation*}
\operatorname{sgrad} H=u(H,F)\,\frac{\partial}{\partial \varphi_1}+v(H,F)\,\frac{\partial}{\partial \varphi_2}.
\end{equation*}
\notag
$$
Definition 6. The rotation function on an edge of a rough molecule is the function $\rho(F)$ equal to the tangent function of the winding angles on Liouville tori in the variables $(\varphi_1,\varphi_2)$. In other words, the rotation function is defined by Note that by Theorem 3 this definition of the rotation function is quite consistent in the case of billiard books. Remark 9. The rotation function is always defined up to a (rational) linear fractional transformation. Its form depends on the basis $\gamma_1$, $\gamma_2$ on the Liouville torus involved in the definition of the action variables. As in the proof of Theorem 3, throughout what follows we assume that $\gamma_1$ goes along the first elliptic coordinate and $\gamma_2$ along the second. It remains to find a quick way to calculate rotation functions on billiard books. To do this we recall that the projection $\pi$ (from the phase space of the billiard book onto the table complex) is well defined on the whole phase space. Hence almost everywhere on $\widetilde{M}^4$ we have
$$
\begin{equation}
\pi_*(\operatorname{sgrad} H)=u(s_1,s_2)\cdot\pi_*(\operatorname{sgrad} s_1)+v(s_1,s_2)\cdot\pi_*(\operatorname{sgrad} s_2).
\end{equation}
\tag{4.1}
$$
Here $\pi_*$ denote the differential of $\pi$. It remains to understand how to calculate $\pi_*$ of the skew gradient of a smooth function. Note that the functions $H$, $\Lambda$, $s_1$ and $s_2$ are smooth first integrals of the geodesic flow on the plane. Almost everywhere in a neighbourhood of a regular leaf of this system we can define the elliptic coordinate $\lambda_1$ and $\lambda_2$ and the conjugate momenta $p_1$ and $p_2$. In combination, these are canonical variables. Hence for each smooth function $f$ on the phase space the skew gradient vector can be calculated as follows in the coordinates $(\lambda_1,\lambda_2,p_1,p_2)$:
$$
\begin{equation*}
\operatorname{sgrad} f=\biggl(\frac{\partial f}{\partial p_1}, \frac{\partial f}{\partial p_2}, -\frac{\partial f}{\partial \lambda_1}, -\frac{\partial f}{\partial \lambda_2}\biggr).
\end{equation*}
\notag
$$
Since the projection $\pi$ discards the momentum coordinates we obtain
$$
\begin{equation}
\pi_*(\operatorname{sgrad} f)=\biggl(\frac{\partial f}{\partial p_1}, \frac{\partial f}{\partial p_2}\biggr).
\end{equation}
\tag{4.2}
$$
As the momenta are well defined almost everywhere in $\widetilde{M}^4$, the last formula holds almost everywhere. We use it to find explicit expressions for rotation functions of billiard books. By the proof of Theorem 3 on edges of the molecule of a billiard book the rotation functions are calculated by formula (3.1). We denote $H\cdot \Lambda$ by $F$; then
$$
\begin{equation*}
\begin{aligned} \, \frac{\partial s_i}{\partial p_k} &=\frac{1}{2\pi}\sum_{j=1}^{N_i}\int_{a_{i,j}}^{b_{i,j}} \frac{1}{2\sqrt{2(F-Ht)(a-t)(b-t)}}\biggl(\frac{\partial F}{\partial p_k}-t\,\frac{\partial H}{\partial p_k}\biggr)dt \\ &= A_i\,\frac{\partial F}{\partial p_k}-B_i\,\frac{\partial H}{\partial p_k}, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
A_i=\sum_{j=1}^{N_i}\int_{a_{i,j}}^{b_{i,j}}\frac{dt}{4\pi\sqrt{2(F-Ht)(a-t)(b-t)}}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
B_i=\sum_{j=1}^{N_i}\int_{a_{i,j}}^{b_{i,j}}\frac{t\,dt}{4\pi\sqrt{2(F-Ht)(a-t)(b-t)}}.
\end{equation*}
\notag
$$
Expressing $\partial H/\partial p_k$ in terms of $\partial s_i/\partial p_k$ and taking (4.1) and (4.2) into account we obtain
$$
\begin{equation*}
\pi_*(\operatorname{sgrad} H)=-\frac{A_2}{B_1 A_2-A_1 B_2}\, \pi_*(\operatorname{sgrad} s_1)+\frac{A_1}{B_1 A_2-A_1 B_2}\, \pi_*(\operatorname{sgrad} s_2),
\end{equation*}
\notag
$$
which yields $\rho=-{A_1}/{A_2}$. Changing the sign of $s_1$ and the direction of increase of $\varphi_1$ we obtain $\rho={A_1}/{A_2}$. Thus we have established the following result. Theorem 4. Rotation functions $\rho$ on edges of the molecule of a billiard book can be calculated by the formula where the $N_i, a_{i,j}$ and $b_{i,j}$ depend on the combinatorial structure of the book. The quantities $N_i$ are constant on edges, while the functions $a_{i,j}$ and $ b_{i,j}$ assume the following values: For a billiard inside an ellipse a similar formula was obtained by Dragović and Radnović [8]. Note that those authors published several papers devoted to periodic trajectories of integrable billiard systems (see [42]–[44]). Recall the definition of the orbital equivalence of dynamical systems. Definition 7. Two dynamical systems are said to be orbitally equivalent if there exists a homeomorphism between their phase spaces that takes trajectories of the first system to trajectories of the second. In the case when the rotation function of an edge of a molecule is well defined and takes each value only at a finite number of points we can generally speaking calculate the orbital invariant $R$ on this edge. Recall its definition. Let $\rho(F)$ be a function on $(0,1)$. We define a vector $R$ whose components are real numbers and the symbols $+\infty$ and $-\infty$. As the first component we take the limit of $\rho$ at zero from the right. Now, travelling along the interval $(0,1)$ we write down sequentially the values of $\rho$ at maxima, minima and poles. In the process, to each pole we assign the two symbols $\pm\infty$ whose signs correspond to the signs of the limits from the left and right at this point. At the end we write down the limit of $\rho$ at 1 from the left. The resulting vector is called the rotation vector on the edge (Figure 1).  Figure 1.An example of the calculation of the rotation vector $R$ from the rotation function $\rho$. It is well known that two integrable Hamiltonian systems are orbitally equivalent on edges of a molecule if and only if there exist bases in the first homology of the corresponding Liouville tori in which the rotation vectors of these systems coincide (see [36]). Thus, the orbital invariant $R$ described the structure of trajectories of an integrable Hamiltonian system on edges of its molecule uniquely up to a change of the basis on a torus. In the next section we present examples of the calculation of rotation functions and rotation vectors of some planar billiard systems and billiard books. § 5. Rotation functions and edge orbital invariants if planar billiards symmetric about the coordinate axesIn this section we look at three planar billiard tables which are symmetric relative to the coordinate axes. Their boundaries are formed by at most two confocal quadrics. This simplifies the analysis of the monotonicity patterns of rotation functions. 5.1. A plane billiard inside an ellipseConsider the billiard system inside an ellipse with parameter $0$. We show its Fomenko–Zieschang invariant in Figure 2. The two right-hand edges of the molecule correspond to $\Lambda<b$ (so that the caustic is an ellipse). We call these edges elliptic. The left-hand edge corresponds to $\Lambda\in(b,a)$ (when the caustic is a hyperbola), so we call this edge hyperbolic. Consider an elliptic edge of the molecule. Let $\gamma_1$ be the cycle on the torus going along the $\lambda_1$-coordinate, and let $\gamma_2$ go along the $\lambda_2$-coordinate, as shown in Figure 3, (a). The orientation of the cycles agrees with the orientation specified by the vector fields on these cycles. As shown in the proof of Theorem 3, the action variables $s_1$ and $s_2$ are calculated by the formulae
$$
\begin{equation}
s_1=\frac{1}{\pi}\int_{0}^{\Lambda}\sqrt{\frac{H(\Lambda-t)}{2(a-t)(b-t)}}\,dt\quad\text{and} \quad s_2=\frac{2}{\pi}\int_{a}^{b}\sqrt{\frac{H(\Lambda-t)}{2(a-t)(b-t)}}\,dt.
\end{equation}
\tag{5.1}
$$
Both integrals have a clear geometric interpretation. Since the projection $\pi$ of the cycle $\gamma_2$ is tangent to the caustic at each point, the integral of $\alpha$ over this cycle is just the length of the elliptic caustic times the square of the velocity vector, that is, where $E(\Lambda)$ is the length of the ellipse with parameter $\Lambda$, that is, the caustic.The situation with the integral $s_1$ is more peculiar. Note that $\gamma_1$ is homotopic to both of the cycles shown in Figure 3, (b). Hence, by Lemma 2 the integrals over these cycles coincide. Consider one of these cycles and denote it by $\gamma'_1$. Let $A$ and $B$ denote the endpoints of the arc obtained as the projection of this cycle, and let $O$ be a point on the boundary ellipse. However, as in the case of $\gamma_2$, the projection $\pi(\gamma'_1)$ is tangent to the direction field on the Liouville torus. Hence
$$
\begin{equation*}
s_2=\frac{\sqrt{2H}}{\pi}(AO+OB-\operatorname{arc} AB),
\end{equation*}
\notag
$$
where $\operatorname{arc} AB$ is the length of the smaller arc of the ellipse. Then $\dfrac{\pi}{\sqrt{2H}}(s_1+s_2)$ is the sum of the lengths of the intervals $OA$ and $OB$ and the greater arc $AB$. Since $O$ is an arbitrary point, the intervals $OA$ and $OB$ are tangent to the elliptic caustic and the quantity $s_1 + s_2$ is independent of $O$, we have the following. If we slung a rope loop of length $\dfrac{1}{\sqrt{2H}}(s_1+s_2)$ around the ellipse with parameter $\Lambda$, put a lead pencil inside the loop and draw a curve while stretching the rope, then we obtain the ellipse with parameter $0$. Since in place of the boundary and confocal ellipses we can have two arbitrary confocal ellipses, we have the following result due to Graves1 (1850). Theorem 5 (Graves). Given an ellipse of length $l$, if a rope loop of larger length is slung around it, a pencil lead is put inside the loop and a curve is drawn by stretching the rope, then this curve is an ellipse confocal with the given one (Figure 4). Note that, apart from ellipses, this theorem also holds for hyperbolae as well as parabolas. However, then the statement is slightly different. Theorem 6. Let $l$ and $l'$ be co-directed branches of two confocal hyperbolae or two codirected confocal parabolas. Then two tangents $PA$ and $PB$ to $l'$ (where $A,B\in l'$) can be drawn from each point $P\in l$ and the quantity $PA+PB-\operatorname{arc} AB$, where $\operatorname{arc} AB$ is the length of an arc on $l'$, is independent of $P$ (Figure 5). Now we return to the original problem and find the rotation functions on the elliptic edges. Taking (5.1) and Theorem 4 into account, this function can be calculated by the formula
$$
\begin{equation*}
\rho_1(\Lambda)=\frac{\displaystyle\int_{0}^{\Lambda}\Phi(t,\Lambda)\,dt} {\displaystyle2\int_{a}^{b}\Phi(t,\Lambda)\,dt}, \quad\text{where } \Phi(t,u)=\frac{1}{\sqrt{(u-t)(a-t)(b-t)}}.
\end{equation*}
\notag
$$
On the hyperbolic edge the rotation function looks differently:
$$
\begin{equation*}
\rho_2(\Lambda)=\frac{\displaystyle\int_{0}^{b}\Phi(t,\Lambda)\,dt} {\displaystyle\int_{\Lambda}^{a}\Phi(t,\Lambda)\,dt}.
\end{equation*}
\notag
$$
Note that our result is perfectly consistent with the following statement due to Vedyushkina (see [37]). Proposition 1 (Vedyushkina). The rotation functions $\rho_1$ and $\rho_2$ of the billiard inside an ellipse are monotone on the intervals $(0,b)$ and $(b,a)$, respectively. If the direction towards the saddle atom is selected on the edges, then on the elliptic edges the rotation vector $R$ is $(0,+\infty)$, while on the hyperbolic edge it is $(l(a,b),+\infty)$, where $l(a,b)$ is a constant depending on the parameters $a$ and $b$. Remark 10. Apart from the edge orbital invariant for the billiard inside an ellipse, Vedyushkina also calculated the marked $t$-molecule for this system. Her approach was based on the following fact. The geodesic flow on a triaxial ellipsoid converges to the billiard system inside an ellipse as the semiminor axis of the ellipsoid tends to zero (Figure 6). In the process the orbital invariants of the geodesic flow on the ellipsoid tend to the orbital invariant of the billiard inside the ellipse. Note that the marked $t$-molecule of the geodesic flow on an ellipsoid was calculated by Fomenko and Bolsinov (for instance, see [36]). 5.2. The billiard inside an elliptic annulusConsider a planar billiard inside an annulus bounded by the ellipses with parameters $0$ and $\lambda_0>0$. We show this billiard table and the Fomenko–Zieschang invariant of the corresponding billiard in Figure 7. 1. Elliptic edges. We calculate the rotation functions on elliptic edges for this billiard. Note that for an elliptic torus, in the domain of possible motion the first elliptic coordinate ranges on the closed interval $[0,\min\{\lambda_0,\Lambda\}]$ while the second takes all values in $[b,a]$. Furthermore, on the Liouville torus the first coordinate traverses the interval $[0,\min\{\lambda_0,\Lambda\}]$ twice, while the second coordinate traverses $[b,a]$ four times. Hence we can calculate the rotation function on an elliptic edge by the formula
$$
\begin{equation}
\rho_1(\Lambda)=\frac{\displaystyle\int_{0}^{\min\{\lambda_0,\Lambda\}}\Phi(t,\Lambda)\,dt} {\displaystyle2\int_{a}^{b}\Phi(t,\Lambda)\,dt}.
\end{equation}
\tag{5.2}
$$
Note that this function is continuous, but not smooth at $\Lambda=\lambda_0$. Now we look at its monotonicity pattern. Proposition 2. The function $\rho_1$ has a unique local maximum on $(0,b)$. If the elliptic edges are oriented towards the saddle atom, then $R=(0,\widehat{l}(a,b,\lambda_0),0)$, where $\widehat{l}(a,b,\lambda_0)$ is some constant. Proof. Note that for $\Lambda<\lambda_0$ the function $\rho_1(\Lambda)$ coincides with the rotation function of a billiard inside an ellipse. Hence the function $\rho_1$ is increasing for $\Lambda<\lambda_0$ and $\lim_{\Lambda\to 0+}\rho_1(\Lambda)=0$ (here we denote convergence to zero from the right by $0+$).
Now let $\Lambda>\lambda_0$. Then the limits of integration in (5.2) are stable. Note that if ${b>\Lambda_1>\Lambda_2>\lambda_0}$, then for each $t\in[0,\lambda_0]$ we have $\Phi(t,\Lambda_1)<\Phi(t,\Lambda_2)$. Hence
$$
\begin{equation*}
\int_0^{\lambda_0}\Phi(t,\Lambda_1)\,dt<\int_0^{\lambda_0}\Phi(t,\Lambda_2)\,dt.
\end{equation*}
\notag
$$
On the other hand, if $t\in[b,a]$, then for $b>\Lambda_1>\Lambda_2>\lambda_0$ we have $\Phi(t,\Lambda_1)>\Phi(t,\Lambda_2)$, so that
$$
\begin{equation*}
\int_b^a\Phi(t,\Lambda_1)\,dt>\int_b^a\Phi(t,\Lambda_2)\,dt.
\end{equation*}
\notag
$$
Thus, for $\Lambda\in(\lambda_0,b)$ the numerator in (5.2) is decreasing and the denominator is increasing. Hence $\rho_1$ decreases for $\Lambda>\lambda_0$. It remains to observe that the limit as $\Lambda\to b-$ of the numerator in (5.2) is finite and that of the denominator is $+\infty$. Hence $\lim_{\Lambda\to b-}\rho_1(\Lambda)=0$.
The proof is complete. Remark 11. Proposition 2 shows that, generally speaking, rotation functions for billiard systems can be quite nontrivial and, in particular, necessarily monotone. Thus, Fomenko’s conjecture that in billiard systems rotation functions are always monotone fails. 2. Hyperbolic edges. Now we consider the rotation function on a hyperbolic edge. It is easy to see that
$$
\begin{equation}
\rho_2(\Lambda)=\frac{\displaystyle\int_{0}^{\lambda_0}\Phi(t,\Lambda)\,dt} {\displaystyle2\int_{\Lambda}^{a}\Phi(t,\Lambda)\,dt}.
\end{equation}
\tag{5.3}
$$
Since $\lambda_0<b$, the limit of the numerator as $\Lambda\to b+$ is finite, while the limit of the denominator is $+\infty$. Hence $\lim_{\Lambda\to b+}\rho_2(\Lambda)=0$. We make the substitution $s=(t-\Lambda)/(a-\Lambda)$ under the integral sign in the denominator in (5.3):
$$
\begin{equation}
\int_{\Lambda}^{a}\Phi(t,\Lambda)\,dt=\int_0^1\frac{ds}{\sqrt{s(1-s)((a-\Lambda)s+\Lambda-b)}}.
\end{equation}
\tag{5.4}
$$
It particular, it follows that the limit of the denominator in (5.3) as $\Lambda\to a-0$ is finite and equal to ${2\pi}/{\sqrt{a-b}}$. The limit of the numerator is also finite and distinct from zero, so there exists a limit $\lim_{\Lambda\to a-}\rho_2(\Lambda)>0$. Can $\rho_2$ be monotone? This is a difficult question. We consider two limiting cases. Proposition 3. There exists $p>0$ such that for all $\lambda_0$ such that $b-\lambda_0<p$ the function $\rho_2$ has at least one local maximum on $(b,a)$. Remark 12. If $\lambda_0$ is close to $b$, so that the inner boundary of the table is strongly retractable to the focal interval (Figure 8, (a)), then the rotation function $\rho_2$ is not monotone. Proof of Proposition 3. In fact, if we set $\lambda_0=b$ in (5.2), then we obtain a function equal, up to a constant, with the rotation function on a hyperbolic edge for the billiard inside an ellipse. Furthermore, for each compact subinterval of $(b,a)$ the function $\rho_2(\Lambda)$ tends uniformly to the rotation function on a hyperbolic edge for the billiard inside an ellipse with parameter $0$. However, the latter function is decreasing in $\Lambda$, while in our case, if $\rho_2$ is monotone, then it is increasing (Figure 9). Hence for $\lambda_0$ close to $b$ the function $\rho_2$ is not monotone.
The proof is complete. Now assume that the two boundary ellipses are sufficiently close (see Figure 8, (b)), that is, $\lambda_0$ is close to zero. This case is rather easy to analyse; moreover, we have the following result. Proof. We denote the integral in the numerator in (5.3) by $f(\Lambda)$ and the one in the denominator by $g(\Lambda)$. By the formula for the derivative of a ratio, the sign of the derivative of $\rho_2$ is the sign of the expression $f'\cdot g-g'\cdot f$, which can be written as follows using (5.4): We set $t=t_0$ in the integrand and then integrate from $0$ to $1$ with respect to $s$. Taking out the nonvanishing constants depending on $\Lambda$ and $t_0$ we obtain the function If we show that for all $\Lambda$ the function $h_{0}(\Lambda)$ is negative, then $h_{t_0}$ is negative for small $\lambda_0$ and $t_0\in[0,\lambda_0]$. Hence the function $\rho_2(\Lambda)$ is monotonically increasing. On the other hand, if $h_0$ changes sign, then $\rho_2$ is not monotone.
Note that $\lim_{\Lambda\to b+}h_0(\Lambda)=-\infty$. Now we calculate the limit of this function at the right-hand endpoint of the interval. Up to a positive constant it is equal to
$$
\begin{equation*}
\int_0^1\frac{sa-b}{\sqrt{s(1-s)}}\,ds=a \cdot B\biggl(\frac32,\frac12\biggr)-b\cdot B\biggl(\frac12,\frac12\biggr)=\pi\biggl(\frac{a}{2}-b\biggr).
\end{equation*}
\notag
$$
Here $B(x,y)$ denotes the well-known Euler beta function.
Hence if $a>2b$, then for $\Lambda$ close to $a$ the function $h_0(\Lambda)$ assumes positive values. Thus, for $a>2b$ the function $h_0$ changes sign, and therefore $\rho_2$ is not monotone for small $\lambda_0$. Thus we have proved part 1 of the proposition. Next we prove part 2. We show that $h_{t_0}\leqslant 0$ for each $t_0\in[0,\lambda_0]$. Since for $a+\lambda_0<2b$ the limits of $h_{t_0}$ at the endpoints are negative, it is sufficient to show that $h'_{t_0}$ is monotonically increasing in $\Lambda$ for all $t_0\in[0,\lambda_0]$. In fact, we can represent $h'_{t_0}(\Lambda)$ as a sum of two integrals:
$$
\begin{equation*}
\begin{aligned} \, h'_{t_0}(\Lambda) &=-\frac{3}{2}\int_0^{(b-t_0)/(a-t_0)}\frac{(1-s)((a-t_0)s+t_0-b)}{\sqrt{s(1-s)((a-\Lambda)s+\Lambda-b)^5}}\,ds \\ &\qquad- \frac{3}{2}\int_{(b-t_0)/(a-t_0)}^1\frac{(1-s)((a-t_0)s+t_0-b)}{\sqrt{s(1-s)((a-\Lambda)s+\Lambda-b)^5}}\,ds. \end{aligned}
\end{equation*}
\notag
$$
Dividing by $-3/2$, in the second integral we make the substitution interchanging the endpoints:
$$
\begin{equation*}
\begin{aligned} \, & \int_0^{(b-t_0)/(a-t_0)}\frac{(1-s)((a-t_0)s+t_0-b)} {\sqrt{s(1-s)((a-\Lambda)s+\Lambda-b)^5}}\,ds \\ &\qquad\qquad +\int_0^{1-(b-t_0)/(a-t_0)}\frac{s((a-b)-(a-t_0)s)} {\sqrt{s(1-s)((a-b)-(a-\Lambda)s)^5}}\,ds. \end{aligned}
\end{equation*}
\notag
$$
Now we compare the moduli of the integrands on $[0,(a-b)/(a-t_0)]$. Note that, as $a+\lambda_0<2b$, we also have $a+t_0<2b$, so that the right-hand endpoint of the interval is smaller than $1/2$. First we compare the moduli of the numerators. To do this we collect the coefficients of like powers of $s$. Since the numerator of the left-hand integrand is negative, its modulus has the opposite sign:
$$
\begin{equation*}
\begin{gathered} \, (1-s)(b-t_0-(a-t_0)s)\vee s((a-b)-(a-t_0)s), \\ s^2-s\vee -\frac{b-t_0}{2(a-t_0)}. \end{gathered}
\end{equation*}
\notag
$$
The quadratic equation $s^2-s+\dfrac{b-t_0}{2(a-t_0)}=0$ has discriminant $1- \frac{2(b-t_0)}{(a-t_0)}$. This number is negative because $a-t_0<2(b-t_0)$. Hence the left-hand expression is larger.
Now we compare $(a-\Lambda)s+\Lambda-b$ with $(a-b)-(a-\Lambda)s$. First we collect the coefficients of like powers of $s$:
$$
\begin{equation*}
\begin{gathered} \, (a-\Lambda)s+\Lambda-b\vee (a-b)-(a-\Lambda)s, \\ 2(a-\Lambda)s\vee a-\Lambda, \\ s\vee \frac12. \end{gathered}
\end{equation*}
\notag
$$
Hence the expression under the root sign in the first integral is less than the one in the second. Thus, the modulus of the first expression is greater than that of the second for all $\Lambda\in[b,a]$. Since the first expression is negative on $[0,(b-t_0)/(a-t_0)]$, the sum of the integrals is everywhere negative. Hence the function $h_{t_0}(\Lambda)$ is strictly monotone and negative for all $t_0\in[0,\lambda_0]$. Therefore, $\rho_2(\Lambda)$ is strictly monotone.
Proposition 4 is proved. 5.3. A billiard in a domain bounded by an ellipse and a hyperbola and not containing fociConsider the billiard on a billiard table not containing foci and bounded by the ellipse with parameter $0$ and the hyperbola with parameter $\widehat{\lambda}$ (Figure 10, (a)). Its Fomenko–Zieschang invariant is shown in Figure 10, (b). It is clear from the shape of the table that the rotation functions $\rho_1$ and $\rho_2$ on elliptic and hyperbolic edges have the expressions
$$
\begin{equation}
\rho_1(\Lambda)=\frac{\displaystyle\int_0^\Lambda \Phi(t,\Lambda)\,dt}{\displaystyle2\int_{\widehat{\lambda}}^a \Phi(t,\Lambda)\,dt}\quad\text{and} \quad \rho_2(\Lambda)=\frac{\displaystyle\int_0^b \Phi(t,\Lambda)\,dt}{\displaystyle\int_{\max\{\widehat{\lambda},\Lambda\}}^a \Phi(t,\Lambda)\,dt},
\end{equation}
\tag{5.5}
$$
respectively. Proposition 5. The functions $\rho_1$ and $\rho_2$ are monotone on the intervals $(0,b)$ and $(b,a)$, respectively. Moreover, the edge orbital invariant for the billiard under consideration is the same as for the billiard inside the ellipse with the same parameter. Proof. We show that for $\Lambda \in (0,b)$ (that is, on elliptic edges) the rotation function is monotone. We represent $\rho_1$ as follows:
$$
\begin{equation}
\rho_1(\Lambda)=\frac{\displaystyle\int_0^\Lambda \Phi(t,\Lambda)\,dt}{\displaystyle2\int_{b}^a \Phi(t,\Lambda)\,dt} \cdot\left( 1+\frac{\displaystyle\int_{b}^{\widehat{\lambda}} \Phi(t,\Lambda)\,dt}{\displaystyle\int_{\widehat{\lambda}}^a \Phi(t,\Lambda)\,dt}\right).
\end{equation}
\tag{5.6}
$$
The left-hand fraction is the rotation function on an elliptic edge for a planar billiard inside an ellipse. By a theorem of Vedyushkina (see [37]) this function is monotonically increasing. We show that the fraction in parentheses is also monotonically increasing. Let $f(\Lambda)$ denote its denominator and $g(\Lambda)$ denote the numerator. The sign of the fraction coincides with that of the expression Since $s\geqslant t$, this expression is nonnegative. Hence $\rho_1$ is strictly monotone. It remains to observe that Therefore, on elliptic edges the rotation vector $R$ is $(0,+\infty)$.
Now we turn to the hyperbolic edge. Note that for $\Lambda>\widehat{\lambda}$ the function $\rho_2$ coincides with the rotation function on the hyperbolic edge for the billiard inside the ellipse with parameter $0$. So for $\Lambda>\widehat{\lambda}$ the function $\rho_2$ is decreasing and $\lim_{\Lambda\to a-}\rho_2(\Lambda)=l(a,b)$, where $l(a,b)$ is the same constant, depending on $a$ and $b$, as in § 5.1. Now let $b<\Lambda_1<\Lambda_2\leqslant\widehat{\lambda}$; then for each $t\in(0,b)$ we have $\Phi(t,\Lambda_1)>\Phi(t,\Lambda)$, while for $t\in(b,a)$ we have the reverse inequality. Hence the same inequalities hold for the integrals. Thus, $\rho_2(\Lambda_1)>\rho_2(\Lambda_2)$. We conclude that the function $\rho_2$ is strictly monotone on $(b,a)$. In addition, $\lim_{\Lambda\to b+}\rho_2(\Lambda)=+\infty$. Thus, the rotation vector $R$ on the hyperbolic edge is $(+\infty,l(a,b))$. Proposition 5 is proved. Remark 13. Although the billiards in Proposition 5 (on the billiard tables from §§ 5.1 and 5.3) are not Liouville equivalent, they are nevertheless roughly Liouville equivalent, and the corresponding edge invariants for them coincide. § 6. Billiards modelling linear geodesic flows on compact orientable surfacesIn the examples considered above (see § 5) the rotation functions can not be expressed in terms of elementary functions. This makes the search for their minima and maxima more complicated. In this section we consider billiards for which the rotation functions can be expressed as compositions of elementary functions. For example, such are the topological billiards on closed tables glued of (round) discs and annuli. Remark 14. By a joint result of Vedyushkina and Fomenko [45] such topological billiards realize the Liouville foliations of all geodesic flows on compact orientable surfaces possessing an additional integral linear in the momenta. Thus, this class of billiards is important for investigation. Recall that the configuration space of a topological billiard is a billiard book without spines, that is, without edges along which three or more sheets are glued together. Moreover, in our case the book has no free edges and its sheets are round discs and annuli. Hence there are precisely two classes of such billiard books: spherical books (homeomorphic to the 2-sphere) and toroidal books (homeomorphic to the 2-torus). We show them in Figure 11 and call such billiard books accordions. Remark 15. Spherical books are glued of several annuli and two discs, while toroidal ones are glued of annuli alone. 6.1. The simplest topological billiardConsider the topological billiard on a spherical accordion formed by two equal round annuli and two equal discs ( Figure 12, (a)). We denote the largest gluing radius of the sheets by $\mathcal R$ and the least one by $r$. Our billiard system has the first integrals
$$
\begin{equation*}
H=\sqrt{v_1^2+v_2^2}\quad\text{and} \quad f=\frac{v_1y-v_2x}{\sqrt{v_1^2+v_2^2}}.
\end{equation*}
\notag
$$
Note that $|f|$ is the radius of the circle tangent to all segments of the trajectory of a point mass. The sign of $f$ determines the direction in which this trajectory is ‘winding’. If $f>0$, then the point mass goes clockwise, and if the reverse inequality holds, then it goes counterclockwise. In the case under consideration $f$ takes values in the interval $[-\mathcal R,\mathcal R]$. If a level of energy $H=h>0$ is fixed, then the values of $f$ on the isoenergy surface of the system are $-\mathcal R$, $-r$, $r$ and $\mathcal R$. The Fomenko invariant of the billiard on $Q^3_h$ in question is shown in Figure 12, (b) First we calculate the rotation function on the edge of the atom corresponding to the interval $(-r, r)$ (this edge connects the atoms $B$). To do this we find the explicit expressions for the action variables in terms of $H$ and $f$. Set where $\gamma_1$ is a cycle (Figure 13). Note that in Cartesian variable momenta coincide with the corresponding velocities , so we see the form $\alpha$ under the integral sign in this expression.We parametrize the projection $\pi$ of the cycle $\gamma_1$ on the accordion as follows:
$$
\begin{equation*}
(\mathcal R\sin t,\mathcal R\cos t), \qquad t\in[0,2\pi].
\end{equation*}
\notag
$$
Note that the angle between the tangent vector to this projection and the vector field on it is a constant (which we denote by $\varphi$). Furthermore, $\cos{\varphi}=f/\mathcal R$. Hence we have the chain of equalities
$$
\begin{equation*}
\begin{aligned} \, s_1 &=\frac{1}{2\pi}\int_{\gamma_1}v_1\,dx+v_2\,dy=\frac{1}{2\pi}\int_{0}^{2\pi}\langle v(t),\dot{\gamma}(t)\rangle \,dt \\ &=\frac{1}{2\pi}\int_{0}^{2\pi}\|v(t)\|\cdot\|\dot{\gamma}(t)\|\cdot\cos\varphi\, dt=\frac{1}{2\pi}\int_{0}^{2\pi}h\cdot \mathcal R\cdot\frac{f}{\mathcal R}\,dt= f h. \end{aligned}
\end{equation*}
\notag
$$
Thus, the integral $s_1$ is the product of the length of the caustic and the energy. Since $f h=v_1y-v_2x$, the action function $s_1$ can be written as The system of Hamiltonian equations with Hamiltonian $s_1$ on the cotangent bundle of the plane has the following form:
$$
\begin{equation}
\begin{cases} \dot{x}=y,\\ \dot{y}=-x,\\ \dot{v}_1=v_2,\\ \dot{v}_2=- v_1. \end{cases}
\end{equation}
\tag{6.1}
$$
Let $u_1$ denote the projection of the velocity vector of the system at the point
$$
\begin{equation*}
P=\biggl(0,\mathcal R,h\frac{f}{\mathcal R},-h\sqrt{1-\frac{f^2}{\mathcal R^2}}\biggr)
\end{equation*}
\notag
$$
onto a disc in the accordion. Clearly, $u_1=(\mathcal R,0)$. Now we select the cycle $\gamma_2$ as shown in Figure 14, (a), and consider the function Now we express $s_2$ in terms of elementary functions. We cut the accordion into elementary domains. Then $\gamma_2$ is cut into six segments (each disc contains one segment, while each annulus contains two). After integrating $v_1\,dx+v_2\,dy$ over each segment of $\gamma_2$ we add the resulting expressions. We start with the discs and calculate the integral for one of them. By symmetry, on the other disc the calculations are similar. The projection of $\gamma_2$ onto the disc is tangent to the vector field along it at all points (Figure 14, (b)). Since the vector field along the projection of $\gamma_2$ has constant length $h$, the integral of $v_1\,dx+v_2\,dy$ over this segment of $\gamma_2$ is the length of the projection of $\gamma_2$ onto the disc times $h$. In other words, using the notation on Figure 14, (b), this integral is The triangle $AOB$ is right. Since $AO=\mathcal R$ and $OB=f$, we have $AB=\sqrt{\mathcal R^2-f^2}$. In a similar way $CD=\sqrt{\mathcal R^2-f^2}$. It remains to calculate $\operatorname{arc}BC$. Note that $\angle BOC=\pi-2\angle AOB$. On the other hand $\cos\angle AOB={f}/{\mathcal R}$. Hence $\angle AOB=\operatorname{arccos}({f}/{\mathcal R})$, which yields
$$
\begin{equation*}
\operatorname{arc}BC=f\biggl(\pi-2\operatorname{arccos}\frac{f}{\mathcal R}\biggr) =2f\operatorname{arcsin}\frac{f}{\mathcal R}.
\end{equation*}
\notag
$$
Thus, the integral of $v_1\,dx+v_2\,dy$ over the restriction of $\gamma_2$ to a disc in the accordion is
$$
\begin{equation*}
2h\biggl(\sqrt{\mathcal R^2-f^2}+f\operatorname{arcsin}\frac{f}{\mathcal R}\biggr).
\end{equation*}
\notag
$$
Now we find the contribution to $s_2$ of the annuli in the accordion. Let the point $M$ on the line segment $KL$ have coordinates $(0,y)$. We find the momentum vector at $M$ shown in Figure 15. The triangle $MON$ is right. Since $ON = f$ and $OM = y$, we obtain
$$
\begin{equation*}
\sin\angle NMO=\frac{f}{y}\quad\text{and} \quad \cos\angle NMO=\sqrt{1-\frac{f^2}{y^2}}.
\end{equation*}
\notag
$$
Hence the momentum vector at $M$ has the coordinates
$$
\begin{equation*}
\Biggl(-h\frac{f}{y},-h\sqrt{1-\frac{f^2}{y^2}}\, \Biggr).
\end{equation*}
\notag
$$
Thus, the integral in question is Using a trigonometric substitution we can obtain the following inequality:
$$
\begin{equation*}
h\int_r^\mathcal R\sqrt{1-\frac{f^2}{y^2}}\,dy=h\biggl(\sqrt{\mathcal R^2-f^2} +f\operatorname{arcsin}\frac{f}{\mathcal R}-\sqrt{r^2-f^2}-f\operatorname{arcsin}\frac{f}{r}\biggr).
\end{equation*}
\notag
$$
Thus, we have deduced an explicit formula for $s_2$:
$$
\begin{equation*}
s_2=\frac{2h}{\pi}\biggl(2\sqrt{\mathcal R^2-f^2}+2f\operatorname{arcsin}\frac{f}{\mathcal R} -\sqrt{r^2-f^2}-f\operatorname{arcsin}\frac{f}{r}\biggr).
\end{equation*}
\notag
$$
The system of Hamiltonian equations with Hamiltonian $s_2$ on the cotangent bundle of the plane has the form
$$
\begin{equation}
\begin{cases} \dot{x}=F(f,h)v_1+G(f)y,\\ \dot{y}=F(f,h)v_2-G(f)x,\\ \dot{v_1}=G(f)v_2,\\ \dot{v_2}=-G(f)v_1.\\ \end{cases}
\end{equation}
\tag{6.2}
$$
Here
$$
\begin{equation*}
\begin{gathered} \, F(f,h)=\frac{2}{h\pi}\bigl(2\sqrt{\mathcal R^2-f^2}-\sqrt{r^2-f^2}\,\bigr)\ \ \text{and}\ \ G(f)=\frac{2}{\pi}\biggl(2\operatorname{arcsin}\frac{f}{\mathcal R}-\operatorname{arcsin}\frac{f}{r}\biggr). \end{gathered}
\end{equation*}
\notag
$$
Let $u_2$ denote the projection of the velocity vector of the system at the point
$$
\begin{equation*}
P=\Biggl(0,R,h\frac{f}{\mathcal R},-h\sqrt{1-\frac{f^2}{\mathcal R^2}}\,\Biggr)
\end{equation*}
\notag
$$
onto a disc in the accordion. It is clear from the above system that
$$
\begin{equation*}
u_2=\frac{1}{2\pi}\Biggl(hF(f,h)\frac{f}{\mathcal R}+G(f)\mathcal R,-hF(f,h)\sqrt{1-\frac{f^2}{\mathcal R^2}}\,\Biggr).
\end{equation*}
\notag
$$
We represent the vector $w=(h{f}/{\mathcal R},-h\sqrt{1-f^2/\mathcal R^2})$ as a linear combination of $u_1$ and $u_2$, $w=\alpha u_1+\beta u_2$, and find the ratio ${\alpha}/{\beta}$. This is the required rotation function. By Cramer’s rule ${\alpha}/{\beta}={\Delta_1}/{\Delta_2}$, where
$$
\begin{equation*}
\Delta_1=\frac{G(f)}{2\pi}\mathcal R h\sqrt{1-\frac{f^2}{\mathcal R^2}}\quad\text{and} \quad \Delta_2=-2\pi \mathcal R h\sqrt{1-\frac{f^2}{\mathcal R^2}}.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\rho(f)=\frac{\alpha}{\beta}=-\frac{G(f)}{2\pi} =-\frac{2}{\pi}\biggl(2\operatorname{arcsin}\frac{f}{\mathcal R}-\operatorname{arcsin}\frac{f}{r}\biggr)
\end{equation*}
\notag
$$
is the required rotation function. Now we calculate the derivative of $\rho(f)$ and find the values of $\mathcal R$ and $r$ for which it is monotone. Thus,
$$
\begin{equation*}
\rho'(f)=\frac{2}{\pi}\biggl(\frac{1}{\sqrt{r^2-f^2}}-\frac{2}{\sqrt{\mathcal R^2-f^2}}\biggr).
\end{equation*}
\notag
$$
Note that $\rho'(f)\to+\infty$ as $f\to r-$. Therefore, if $\rho(f)$ is monotone on $(-r,r)$, then $\rho'(f)\geqslant0$ on this interval. In turn, this is equivalent to the inequality $\mathcal R^2-f^2\geqslant4(r^2-f^2)$ for $f\in(-r,r)$, which yields $3f^2\geqslant 4r^2-\mathcal R^2$. Hence $\rho(f)$ is monotone if and only if $\mathcal R\geqslant 2r$. Note that if $\mathcal R<2r$, then the function $\rho$ has a unique local minimum and a unique local maximum on $(-r,r)$. Thus we have proved the following result. Proposition 6. The rotation function $\rho(f)$ is monotone on $(-r,r)$ if and only if ${\mathcal R\geqslant2r}$. For $\mathcal R<2r$ $\rho(f)$ has a unique local minimum and a unique local maximum on $(-r,r)$. Remark 16. Using similar calculations we can show that on the other edges of the molecule in Figure 12, (b), the rotation function are strictly monotone. Thus we see that on the accordion under consideration, for $f\in(-r,r)$ the rotation function is monotone only if $2r\leqslant \mathcal R$. On the other hand, on the intervals $(r,\mathcal R)$ and $(-\mathcal R,r)$ the rotation functions are always monotone. Nonetheless, for a suitable choice of $\mathcal R$ and $r$ the rotation functions can be made monotone on all edges simultaneously. It turns out that all accordions have this property. We prove this in the next subsection. 6.2. The general caseWe start with a significant simplification. If two sheets of annulus-disc or annulus-annulus type are glued together, and after the isometric embedding in the plane they intersect only along the gluing boundary, then they can be combined into a single sheet. Performing this procedure as many times as possible we simplify the combinatorial structure of the accordion. First consider the spherical series. A spherical accordion $D$ can be specified uniquely by indicating the sequence of gluing radii of its sheets. This sequence can be split into two: the sequence of maxima $\{\mathcal R_i\}_{i=1}^n$ and the sequence of minima $\{r_j\}_{j=1}^{n-1}$. Here the sequence of gluing radii looks as follows: $\mathcal R_1,r_1,\mathcal R_2,r_2,\dots$, $r_{n-1},\mathcal R_n$. From it we construct a piecewise linear function $h_D(x)$ which describes uniquely the ‘profile’ of $D$. To do this we fix a set of points $0=x_0<x_1<\dots<x_{2n}$ on the real line. Set
$$
\begin{equation*}
h_D(x_k)=\begin{cases} 0 &\text{for }k=0 \text{ or } k=2n, \\ \mathcal R_i &\text{for } k=2i-1, \\ r_j &\text{for } k=2j. \end{cases}
\end{equation*}
\notag
$$
We extend this function to the rest of the interval $[0,x_{2n}]$ by linearity (Figure 16). As in § 6.1, the billiard on the accordion $D$ has the first integrals $H$ and $f$. On a nonsingular isoenergy surface $f$ has the critical values $\pm \mathcal R_i$ and $\pm r_j$, where $i=1,\dots,n$ and $j=1,\dots,n-1$. Furthermore, when we know the relative position of the $\mathcal R_i$ and $r_j$, we can uniquely calculate the Fomenko–Zieschang invariant of the billiard on $D$ (see [45]). In other words, the function $h_D$ determines unambiguously the topology of the Liouville foliation. Nevertheless, we need not know the relative position of absolutely all gluing radii to calculate the Fomenko–Zieschang invariant. Using the function $h_D(x)$ we can construct an oriented tree $G_D$. To do this, on the coordinate $Oxy$-plane we draw the graph of the function $h_D(x)$. Recall that this function is defined on $[0,x_{2n}]$. Let $M$ denote the compact set bounded by the graph of $h_D$ and the interval $[0,x_{2n}]$ of the $Ox$-axis. The lines $y=\mathrm{const}$ foliate it into (connected) line segments. The base of this fibration is a tree. On edges of this tree we indicate the direction of growth of $y$. We call the resulting graph the codifier of the accordion $D$. This tree has one root vertex and precisely $n$ end vertices. Each finite vertex corresponds to some maximum $\mathcal R_i$ (Figure 17). Definition 9. We say that two accordions $D_1$ and $D_2$ are structurally equivalent if there exists a homeomorphism of their codifiers which preserves the orientations of edges. Note that all spherical accordions glued of two annuli and two discs are pairwise structurally equivalent, although the relative positions of the maxima involved can be different. Furthermore, the Liouville foliations for the corresponding billiards have the same topology. In fact, the following result holds. See the proof in [45]. Remark 17. Note that if all maxima $\mathcal R_i$ are increased in an arbitrary way, while preserving the minima $r_j$, then the resulting accordion is structurally equivalent to the original one. In the example of a spherical accordion with two maxima and one minimum we saw that, generally speaking, for the molecules of billiards on accordions rotation functions on edges are not necessarily monotone. Nonetheless, by choosing suitable gluing radii, while preserving the topology of the Liouville foliation we can obtain strictly monotone rotation functions. Our aim is to prove the following analogous result for an arbitrary accordion. Theorem 7. Let $\{\mathcal R_i\}_{i=1}^n$ and $\{r_j\}_{j=1}^{n-1}$ be the sequences of maxima and minima for a spherical accordion $D$. Then on the edge of the rough molecule that corresponds to the maxima $\mathcal R_{i},\dots,\mathcal R_{i+k}$ the rotation function $f$ has the following form: Moreover, there exists an accordion $D'$ structurally equivalent to $D$ such that all rotation functions on edges of the molecule of $D'$ are strictly monotone. Proof. The explicit expression for the rotation function can be found in quite the same way as in the above example.
Note that $\operatorname{arcsin} x$ is a monotone function, so that on edges connecting the atom $A$ with saddle atoms the rotation function is monotone. Now consider the saddle edge corresponding to the maxima $\mathcal R_i,\dots,\mathcal R_{i+k}$. On this edge the modulus of $f$ ranges on the interval $[a_1,a_2]$, where $a_2=\min\{\rho_{i},\dots,\rho_{i+k}\}$ and $a_1$ is either the largest of the quantities $\rho_{i-1}$ and $\rho_{i+k}$ (if they are defined) or zero. Let us calculate the derivative of $\rho(f)$:
$$
\begin{equation*}
\rho'(f)=-\frac{2}{\pi}\left(\sum_{j=0}^k \frac{1}{\sqrt{\mathcal R_{i+j}^2-f^2}}-\sum_{j=0}^{k-1}\frac{1}{\sqrt{r_{i+j}^2-f^2}}\right).
\end{equation*}
\notag
$$
Note that $\lim_{|f|\to a_2+}\rho'(f)=-\infty$. We let $M$ denote the minimum of the second sum on the range of $f$ over the edge of the molecule. Note that $M>0$. We increase all the $\mathcal R_{i_j}$ so that the first sum is less than $M/2$ on this interval. Then the derivative of the rotation function is positive on the whole interval. Looking sequentially at all edges of the molecule and increasing the corresponding $\mathcal R_i$ we attain the monotonicity of all rotation functions of the accordion. On the other hand, since an increase of the maximum values does not change the structure of the accordion, this does not change the Liouville foliation either.
Theorem 7 is proved. Remark 18. For toroidal accordions the statement of Theorem 7 only changes on the two edges containing the level $f=0$. On them the rotation function is It is certainly monotone because it can be represented as the sum of the $n$ monotone functions of the form 
Citation in format AMSBIB
\Bibitem{BelFom24} |
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