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This article is cited in 3 scientific papers (total in 3 papers) Realization of geodesic flows with a linear first integral by billiards with slipping V. V. Vedyushkina, V. N. Zav'yalov Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
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     § 1. IntroductionGeodesic flows on closed two-dimensional closed surfaces that have an additional first integral which is polynomial in the components of the momentum are the subject of many papers (the history of this question was quite comprehensively reviewed in [1], vol. 2, Chs. 2 and 3). We only mention an important result, due to Kozlov [2], [3], that a geodesic flow on a compact Riemannian 2-manifold has no additional integrals (in the analytic category) if this manifold has genus two or higher. Moreover, the condition that the manifold is analytic is superfluous when we discuss polynomial integrability, that is, when we look for integrals that are polynomials in the components of the momentum (see [1], vol. 2, Comment 2 to Theorem 2.1). For integrals of degrees 1 and 2 (that is, for linearly or quadratically integrable geodesic flows) a number of classification problems for such flows has been solved. Recall that such flows can be classified from the standpoints of the canonical form of the metric, the topology of the Liouville foliation (Liouville equivalence), or orbital or geodesic equivalence (see [4]). For integrals of degree higher than two the question is open (there are examples of geodesic flows with first integrals of degrees 3 and 4 on 2-manifolds.). The important results of Kolokol’tsov [5] were subsequently developed and extended by Babenko, Nekhoroshev, and Matveev (see [1] and also [5] and [6]). In particular, Matveev obtained a topological classification of linearly integrable geodesic flows on nonorientable surfaces by considering an involution in the phase space of the geodesic flow defined over the corresponding orientable surface. In this paper we model geodesic flows with linear first integrals on the projective plane and the Klein bottle using billiards from the class of billiards with slipping recently introduced by Fomenko (see [7] and also [8]), by defining slipping along boundary arcs of the topological billiards introduced by Vedyushkina [9]). The classes of topological billiards and their generalizations, billiard books, introduced by Vedyushkina [10] make it possible to model many integrable systems used in applications and model singularities of such systems. The table complexes on which a billiard ball moves consist of elementary billiard domains (2-dimensional cells; see [11]) glued along edges, with some permutations describing the passage from one sheet to another across glued edges (one-dimensional cells). Further generalizations of billiards on table complexes arise, for example, when additional structures are introduced on their planar sheets: a potential (see [12]), a magnetic field (see [13]), a Minkowski metric (see [14]), slipping along boundaries, or some combinations of these (see, for instance, [15]). Billiard foliations on three-dimensional confocal tables (see [16] and [17]) are also an object of active investigations. Billiard books were instrumental in making a considerable advance toward the proof of Fomenko’s conjecture on billiards (see [7]), which states the problem of realizing smooth and real analytic integrable Hamiltonian systems by means of suitable integrable billiards, from the point of view of the topology of the corresponding Liouville foliations. The Fomenko-Zieschang invariant, a graph molecule with numerical marks, classifies integrable Hamiltonian systems with nondegenerate singularities on three-dimensional energy levels in the sense of Liouville equivalence. In parts A and B of this conjecture the problem of realizing atoms-bifurcations and rough molecules was stated; it was solved by Vedyushkina and Kharcheva. Their algorithm for constructing a billiard book that models a prescribed 3-atom or a prescribed rough molecule was described in [18] and [19], respectively; also see [20]. Part C of Fomenko’s conjecture is the most general one. It states the problem of realizing an arbitrary Fomenko-Zieschang invariant by billiards. A complete answer here is not yet clear, but the following significant steps, in addition to realizing atoms and rough molecules, have been made. In the investigation of the ‘local’ version of this conjecture in [21], arbitrary numerical marks were realized (see [22] and [23] for more details; also see [24], where a combination of the local conjecture and part B of the general conjecture was established). Part D of the conjecture concerns realizing 3-manifolds as energy levels of some integrable billiards. Vedyushkina described algorithmically a billiard book whose manifold $Q^3$ is homeomorphic to a an arbitrary prescribed connected sum of lens spaces and products $S^1 \times S^2$; see [25]. This shows, in particular, that the manifold $Q^3$ realized by a billiard is not necessarily a Seifert manifold: the connected sum of three copies of $\mathbb{R}P^3=L(1,2)$ is not a Seifert manifold (see [26]). In [27] topological billiards and billiard books were used to realize arbitrary geodesic flows on orientable 2-surfaces (namely, a sphere and a torus) that have a first integral which is linear or quadratic in the momentum. Our paper is devoted to the extension of this result to the case of nonorientable surfaces and linear first integrals. Vedyushkina and Fomenko also realized foliations associated with many well-known integrable systems in mechanics and mathematical physics in suitable energy ranges [28]–[30]. Further generalizations of integrable billiards allow one to turn to realizing the Liouville foliations on four-dimensional subsets of the phase space by means of billiards. For instance, the class of evolutionary force billiards introduced by Fomenko (when the geometry of the table varies with the energy of the billiard ball) enables one to model integrable systems in several nonsingular energy ranges simultaneously: see [31]–[33]. On the other hand, adding a Hooke potential to a topological billiard or a billiard book has made it possible to advance significantly in the problem of modelling nondegenerate local and semilocal singularities of rank 0, that is, neighbourhoods of an equilibrium of a system and the leaf of the foliation containing it, by means of billiards [34]–[36]. § 2. Topological billiards with slipping and the main resultStandard billiard system describe the motion of a point in a domain with natural bouncing at the boundary, on which a finite number of corners with angle ${\pi}/{2}$ are allowed. Fix coordinates $(x, y)$ in the plane $\mathbb{R}^2$. Consider the family of concentric circles with centre at the origin A billiard bounded by concentric circles is integrable because the angle $\varphi$ between the segment of the trajectory and the tangent to the boundary circle remains the same. In what follows we consider domains bounded by circles.Let $F$ be the isometry of the boundary taking a point $x$ to the diametrically opposite point $y$. In terms of the circle this means the rotation of the radius vector of $x$ through $\pi$. We call a billiard in the disc with this isometry of the circle a billiard with slipping (Figure 1). As shown in [8], such a billiard system is also integrable with the same additional integral: the rotation through $\pi$ preserves the property of being tangent to the concentric circle corresponding to the fixed value of the integral. It [8] the authors also considered the case of a disc (modelling the gluing of two solid tori into a lens space $L_{4,1}$ on an iso-energy level) corresponding to a projective plane, and the case of two annuli glued together along the inner circles, corresponding to a Klein bottle. It turns out that these billiards with slipping model geodesic flows on the projective plane and Klein bottle that have additional integrals which are linear or quadratic in the momentum. We recall the construction of the gluing of two billiard tables along a common part of their boundary which was introduced by Vedyushkina in [9]. Let two billiard domains have a common part of the boundary and let them lie on the same side of this part of boundary in the plane. We glue the billiard domains together along this common part. After hitting the glued part of the boundary, a point mass moving on one sheet continues its motion in the other billiard domain. In the case when we glue together planar billiard domains and the result of gluing is an orientable manifold, this is the definition of gluing for topological billiards introduced by Vedyushkina. Consider domains bounded by one or two concentric circles in the plane $\mathbb{R}^2$. We denote a domain homeomorphic to a disc (and a billiard inside it) by $D$. We always set the radius of the outer circle to be one. Consider a domain bounded by two concentric circles (which we denote by $C$), and let the smaller circle have the equation $x^2+y^2=\lambda$, where $0<\lambda\leqslant1$. Definition 1. When there is slipping on the outer circle of the domain $C$, then the billiard is said to be ‘involutive’ and denoted by $A_\lambda$, and when there is slipping on the smaller circle (the nonconvex one relative to the billiard table of type $C$), then the billiard is denoted by $A_\lambda^*$. We call a domain $C$ without slipping an intermediate domain and denote it by $B_\lambda$. We show the four domains $C$, $D$, $A_\lambda$ and $A_\lambda^*$ mentioned above in Figure 2, where slipping is indicated by the letters $a$ and $b$. Statement 1 (Matveev; see [1], vol. 2, Ch. 3). Let the Klein bottle be endowed with an $(L,g)$-metric $g(y)(dx^2+dy^2)$, where $g$ is a nonconstant smooth function. Then the Liouville foliation of the geodesic flow on an iso-energy surface $Q$ is described by the molecule shown in Figure 3, b. Let the projective plane be endowed with a metric $ds^2= d\theta^2+f(\theta)\,d\varphi^2$. Then the molecule $\widetilde{W}$ corresponding to the geodesic flow is shown in Figure 3, a. These graph molecules only atoms $A$, $B_n$, $B^*_n$ and $B^{**}_n$, and their numerical marks are as follows. $\bullet$ Inner edges of $\widetilde{W}(f)$ and $\widetilde{W}(g)$. The marks $r$ on edges connecting saddle atoms are equal to infinity. The marks $r$ between saddle atoms and atoms $A$ are equal to zero, except when an atom $A$ corresponds to the fixed point of an involution taking an iso-energy surface of a geodesic flow on a $2$-torus to an iso-energy surface of a geodesic flow on a Klein bottle. In this case the mark $r$ is $1/2$. $\bullet$ On the unique central edge connecting the two copies of the molecule $\widetilde{W}(f)$ or $\widetilde{W}(g)$ the mark $r$ is equal to infinity, while the mark $\varepsilon$ is $-1$. The unique family existing there coincides with the whole of $\widetilde{W}$ without the endpoint atoms $A$. The mark $n$ on this family is $-2$. $\bullet$ The total number of stars and marks $1/2$ is two for a geodesic flow on the projective plane and four for a flow on the Klein bottle. The main result of this paper is the following theorem. Theorem 1. A geodesic flow on a two-dimensional nonorientable manifold (a Klein bottle or a projective plane) that has an additional first integral which is linear in the momentum is Liouville equivalent to a billiard with slipping formed by planar billiards bounded by concentric circles. The linear integral on this billiard reduces to the canonical integral on the billiard, which is equal to the angle between the trajectory and the boundary of any of the billiard domains. 2.1. An algorithmic construction of the billiard table for $ {\mathbb{R}}P^2$Let $R$ denote the billiard obtained by gluing sequentially a domain with slipping ($A_{\lambda_0}$ or $A^*_{\lambda_0}$), a domain $D$ and a finite number of intermediate domains $B_{\lambda_i}$, $i=1,\dots,n$. We examine trajectories in such billiards. With the value $\varphi=0$ of the integral we associate the motion of the point mass along arcs of circles of radius $1$, that is, along convex glued arcs of the billiards forming $R$. Assume that in this motion the point mass encircles the origin clockwise. Values $0<\varphi<\pi/2$ of the integral correspond to a clockwise motion and $\pi/2<\varphi\leqslant\pi$ correspond to an anticlockwise motion of the point mass; furthermore, $\varphi=\pi$ corresponds to a motion along circles of radius $1$. Finally, trajectories on the level $\varphi=\pi/2$ consist of line segments passing through the origin. It is clear that, topologically, the levels of the integral for $\varphi<\pi/2$ and $\varphi>\pi/2$ are the same: after reverting the direction, trajectories of the first type turn to trajectories of the second. From the billiard $R$ we construct a piecewise linear function $\widehat{f}$. If $R$ contains an involutive domain $A_{\lambda_0}$, then we consider the points with coordinates $(i+1/2,\lambda_i)$, where the $\lambda_i$ are the radii of the smaller circles of domains with slipping and intermediate domains $B_{\lambda_i}$. We add the point $(n+3/2,0)$ to this set and call all these points minimum points. Now consider the points with coordinates $(i,1)$ lying between minimum points. Connect all these point sequentially by a polygonal line so that minimum points become minima of the arising graph of a function. Add the reflection of this polygonal line with respect to the $Oy$-axis. Let $\widehat{f}$ denote the function whose graph is this symmetric line. In the case of a domain $A^*_{\lambda_0}$ we put minimum points at $(i,\lambda_i)$ and maximum points at $(i/2,1)$ and add the minimum point $(n+1,0)$. We also call the union of the polygonal line with these vertices and its reflection the graph of $\widehat{f}$. In fact, the graph of this function is the profile that a straight line through the origin cuts in the billiard domains forming $R$. Moreover, from any such graph of a function we recover $R$ uniquely. Consider an integrable geodesic flow on the projective plane. It is uniquely defined by a smooth even function $f$ vanishing an the endpoints on its interval of definition and satisfying certain smoothing conditions at these endpoints. Consider the graph of this function and replace it by the polygonal line going through the maximum and minimum points. Next, preserving the mutual position of the maxima and minima we modify the graph so that
As a result, we obtain a function $\widehat{f}$ (see an example in Figure 4). Given this function, we can construct a billiard with slipping $R$. This is the billiard that is Liouville equivalent to the prescribed geodesic flow. To see this it remains to show that the Fomenko-Zieschang invariant calculated for $R$ coincides, after the replacement of $f$ by $\widehat{f}$, with the invariant presented in Figure 3. Remark 2. The construction of $\widehat{f}$ can be made more geometrically transparent. Consider the graph of $\widehat{f}$ and glue the points symmetric relative to the $Oy$-axis; the point on this axis is fixed under gluing. Then we rotate the graph of this ‘half-function’ about the $Ox$-axis. On the circle obtained by the rotation of the fixed point of the gluing we glue together the diametrically opposite points. The resulting surface of revolution (‘accordion’) is isometric to the billiard $R$ from which we have constructed $\widehat{f}$. Proposition 1. After the identification of the arrows on the boundaries of involutive domains, the billiard table $R$ produced by this construction becomes homeomorphic to the projective plane. Proof. Intermediate domains are annuli, homeomorphic each to a cylinder. Involutive domains $A_\lambda$ and $A_\lambda^*$, after the identification of the boundary arrows denoting slipping (see Figure 2), become homeomorphic to Möbius bands. Thus, the resulting billiard table $R$ is homeomorphic to a cylinder (formed by the intermediate domains $B_\lambda$) with a Möbius band attached to one boundary circle and a disc $D$ attached to the other. It is obvious that $R$ is homeomorphic to the projective plane. The proof is complete.
§ 3. Proof of the main theorem for $ {\mathbb{R}}P^2$3.1. Step 1. Calculating the rough moleculeConsider the graph of a continuous nonnegative function $f$ on $[a,b]$ that takes equal values at the endpoints. In addition, we assume that the function can vanish only at the endpoints. If its value at the endpoints is zero, then we fibre the subgraph of the function by the line segments obtained as the intersections of the lines $y=\mathrm{const}$ with this subgraph. We contract each segment to a point. Then the domain between the graph and the line $y=0$ turns to a tree. To vertices of this graph we assign atoms as follows. To points that are the images of maxima of $f$ we assign atoms $A$. To all other vertices we assign atoms $B_k$, where $k$ is the number of minima occurring on the corresponding horizontal line segment (for the case of an arbitrary positive function $f$, see Figure 5). We denote this graph by $W(f)$. If the value at the endpoints is distinct from zero, then we glue the subgraph into a cylinder along the line segments $[(a,0),(a,f(a))]$ and $[(b,0),(b,f(b))]$ parallel to the $Oy$-axis. The intersections of lines $y=\mathrm{const}$ with the subgraph of the function fibre this domain into circles and line segments. If the line $y=\mathrm{const}$ lies below both the minimum of $f$ and the line through the endpoints of the graph of $f$, then we obtain a circle in the intersection. In all other cases we obtain line segments in intersection. We contract each segment and each circle into a point. To maxima of the function we assign atoms $A$. To bifurcations of segments to segments we assign atoms $B_k$, where $k$ is the number of minima occurring on the corresponding horizontal line segment. To bifurcations of circles into segments we assign atoms $C_k$, where $k$ is the number of segments into which the circle bifurcates. We denote this graph by $W_2(f)$. Now we look at the graph of $\widehat{f}$. As $\widehat{f}$ can have value zero only at the endpoints of its interval of definition, from $\widehat{f}$ we construct the graph $W(\widehat{f})$. In our case it is symmetric because $f$ is symmetric. Next we go over to a graph $\widetilde{W}(\widehat{f})$ by taking the quotient by this symmetry. Assume that the symmetry axis goes through an atom $B_{m}$. a. Assume that the axis of symmetry goes through a maximum (Figure 6, a). As there is no minimum on the symmetry axis, the number of minima is even and we have an atom $B_{2k}$. After taking the quotient, the number of minima reduces twofold and we obtain an atom $B_{k}$. b. Assume that this axis goes through a minimum (Figure 6, b). Since there is an even number of minima away from the symmetry axis, we have an atom $B_{2k+1}$. The minimum point on the symmetry axis is fixed, and the number of minima away from the axis reduces twofold after taking the quotient, so we obtain an atom $B_k^*$. This coincides with the construction of the graph $\widetilde{W}( f )$ for the geodesic flow on the projective plane described by the function $f$. Fix a level $\varphi_0<\pi/2$ of the integral. It corresponds to the horizontal line $y=c<1$, where $c$ is the radius of the circle that is tangent to the trajectories on this level of the integral. Consider the level surface corresponding to this value of the integral and its projection onto the billiard table. This projection can be obtained by removing the interior of the circle of radius $c$ from each billiard domain forming $R$. As a result, $R$ falls into several connected components. All these components (except maybe one of them) are homeomorphic to cylinders obtained by gluing together domains of type $C$ obtained by removing interior parts from domains of types $D$, $B_\lambda$ and $A_\lambda^*$ (provided that $\lambda<c$). The exceptional domain (if exists) is homeomorphic to a Möbius band. It is obtained by removing the interior of the circle of radius $c$ from an involutive domain $A_\lambda$ or $A_\lambda^*$ (provided that we must remove the interior of a circle of smaller radius that the radii of the boundary circles of the domain $A_\lambda^*$). Note that the number of connected components is equal to the number of intervals of the line $y=c$ lying under the graph of $\widehat{f}$ for $x>0 $ (because the part of the graph of $\widehat{f}$ in this half-plane is the billiard profile). Assume that a line $y=c$ does not go through minima of $\widehat{f}$. Then the inverse image of each cylinder or Möbius band is a 2-torus (by analogy with topological billiards and the simplest billiards with slipping: see [8] and [9]). Assume that a line $y=c$ goes through several minima of the function but not through the possible minimum on the $Oy$-axis. Then, similarly to topological billiards, the bifurcation on this level corresponds to gluing several tori into one along the critical circles corresponding to nonconvex gluings along circles of radius $c$. It is easy to see that this bifurcation is described by an atom $B_k$, where $k$ is the number of positive minima occurring on the line $y=c$. Assume that a line $y=c$ goes through several minima of the function, including the minimum on the $Oy $-axis (in this case $R$ contains an involutive domain $A_\lambda^*$, where $\lambda=c$). Then we can show that the bifurcation is described by an atom $B_k^*$, where $k$ is the number of positive minima at which the line $y=c$ is tangent to the graph of the function. If we cancel slipping on the inner surface of the involutive domain $A_c^*$, then the corresponding bifurcation is described by an atom $B_k$. On the singular leaf of $B_k$ we distinguish the singular circle, the preimage of the inner boundary circle of $A_c^*$. (This is just one circle, because on the singular leaf trajectories are tangent to a circle of radius $c$. For values of integral smaller than $c$, such a preimage is empty, and for larger values it consists of two circles.) We cut the 3-atom along the preimage of this circle, and then we re-introduce slipping on this circle. As a result, this distinguished circle on the singular leaf of $B_k$ is glued to itself. Extending this gluing to the whole atom (that is, to non-singular leaves) we obtain an atom $B_k^*$, similarly to how gluing a torus along the distinguished cycle (so that the cycle goes twice along itself) results in the formation of a singular leaf of an atom $A^*$. As a result, we obtain a rough molecule of the form $\widetilde W(\widehat{f})$, as required. 3.2. Step 2. Calculating the marks on the marked moleculeProposition 2. Edges between saddle atoms carry the marks $r=\infty$, and ones between saddle atoms and atoms $A$ carry $r=0$, except when an atom $A$ corresponds to a fixed point of the involution (see [1]; in terms of the above construction this means that the maximum corresponding to this atom $A$ lies on the $Oy$-axis). In that case the mark is $r=1/2$. Proof. When selecting cycles on boundary tori of 3-atoms we use their projections onto the billiard table $R$. We draw curves on the billiard table and supply them with suitable velocity vectors.
Recall the rule for the choice of cycles. On boundary tori of atoms $A$ we choose as $\lambda$ the unique cycle that can be contracted to a point inside the solid torus $A$. As $\mu$ we take an arbitrary cycle complementing it to a basis. As we approach the singular leaf, $\mu$ tends to the critical circle, so that we can consistently define an orientation on $\mu$ (coinciding with the orientation of the critical circle). On boundary tori of saddle atoms we take cycles $\lambda$ homologous to the fibres of the Seifert fibration. For atoms without stars this means that, when the tori attain the singular leaf, these cycles must be homologous to (and codirected with) the critical circle. For atoms without stars we take for $\mu$ the boundary circles of the two-dimensional atom obtained as a cross-section of the 3-atom which is transversal to the critical circle. It we proceed in this way for an atom with star, then the following situation can (and occasionally does) occur. The arising cycles $\widehat{\mu}$ can intersect the cycle $\lambda$ in two points. In the case of an atom $B_{k}^*$, which is under consideration, on one torus $\widehat{\mu}$ intersects $\lambda$ in one point, but on the other torus they intersect in two points. Then on the torus with one-point intersection, as a basis cycle $\mu_s$ we take $\widehat{\mu}$, and on the other torus we take $(\widehat{\mu}+\lambda)/2$ (see details in Bolsinov and Fomenko [1]). Assume that $R$ contains only an involutive domain $A_\lambda$. Then the projections of cycles onto planar billiard domains forming $R$ and the velocity vectors are as shown in Figure 7. The arrows attached to points on curves show the velocity vectors on their preimages under the projection of the torus onto a billiard domain. We explain the choice of the cycle $\mu$ on the top torus of an atom $B_k$ (we assume that the molecule is oriented from top to bottom). After traversing the preimage of $A_\lambda$, the cycle goes over to the preimage of an intermediate billiard domain $B_\lambda$ precisely when the curve in its projection attains the glued edge between two billiard domains. Also note the following. Assume that the radius of the tangent circle is less than the radii of the inner circles of $A_\lambda $ and all domains $B_\lambda$. Then there is a single torus on this level. We can show that $\mu$ is homologous to the cycle that, for $\varphi=\pi/2$, turns to the motion along a diameter of the circle. Assume that $R$ contains only an involutive domain $A_\lambda^*$. Then the projections of cycles onto billiard domains forming $R$ and the velocity vectors on cycles are as shown in Figure 8. The arrows attached to points on curves show the velocity vectors on their preimages under the projection of the torus onto the billiard domain. On atoms $B_k$ positioned in the molecule below the atom with star cycles are selected just as above. We explain the choice of cycles on the boundary tori of the atom with star. As the cycle $\widehat{\mu}$ we can take the preimage of a segment of a straight line through the origin. At the lowest level of the first integral the cycle is the preimage of two segments of a line through the origin; each of these preimages intersects $\lambda$ in one point (see Figure 8). This cycle intersects $\lambda$ in two points, so we take half the sum of $\widehat\mu$ and $\lambda$ as $\mu_s$. We show the resulting cycle in Figure 8, (g), (h) and (i). (Note that its projection onto an intermediate billiard domains $B_\lambda$ looks as before: it lies on the fixed radius of the family of circles.) We must also take the same pairs $\lambda$, $\mu_s$ as cycles on the boundary tori of atoms $B_k$ lying over the level of the atom with star. Now we calculate the marks. If the projection of the torus lies away from the domains $A_\lambda$, then the gluing matrices on edges between saddle atoms and atoms $A$ have the form $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. The marks are $r = 0$ and $\varepsilon = 1$, and there is no contribution to the mark $n$. If the projection lies in an involutive domain $A_\lambda$, then the gluing matrices on edges between saddle atoms and atoms $A$ have the form $\begin{pmatrix} -1 & 2 \\ 0 & 1 \end{pmatrix}$. The marks are $r=1/2$ and $\varepsilon=1$, and the contribution of the gluing matrix to the mark $n$ on the family is $[-1/2]=-1$. On the other hand, if the projection lies in $A_\lambda^*$, then the gluing matrices on edges between saddle atoms and atoms $A$ have the form $\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}$. The marks are $r=0$ and $\varepsilon=1$, and the contribution of the gluing matrix to the family is $[-1/2]=-1$. The gluing matrix on an edge between two saddle atoms is $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ because all cycles $\lambda$ are homologous and co-oriented. As a result, the marks are $r=\infty$ and $\varepsilon=1$. These inner edges make no contribution to the mark $n$. Proposition 3. On the single central edge of $\widetilde W(f)$ the marks are $r=-\infty$ and $\varepsilon= -1$. This molecule contains a unique family, with mark $-2$ on it. Proof. We have discussed a clockwise and an anticlockwise motion, which turn one to the other when we go over the value $\varphi=\pi/2$. The cycles $\lambda$ are oppositely oriented, while the cycles $\mu$ transform one into the other. Thus we obtain the gluing matrix $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ and the marks $r=\infty$ and $\varepsilon=-1$ on the edge. One family exists, and only the incoming edges of the domains $A_{\lambda}$ make contributions to the mark on it. Hence $n=-2$ and the proof is complete. Remark 3. Now we explain our notation for domains. The involutive domain $A_{\lambda}$ is denoted in this way because the 3-atom $A$ whose projection lies there has a nontrivial mark on the outgoing edge. The domain $A^*_{\lambda}$ obviously models a bifurcation atom with star on the singular level. The proof of Theorem 1 for the projective plane is complete. § 4. Proof of the main theorem for $\mathrm{KL}^2$Let $K$ denote the billiard obtained by gluing in succession two (not necessarily similar) domains with slipping ($A_{\lambda_0}$ or $A^*_{\lambda_0}$) and a finite number of intermediate domains $B_{\lambda_i}$, $i \in \{1,\dots,n\}$. From $K$ we construct a piecewise linear function $\widehat{g}$ describing the profile of the billiard. If $K$ contains at least one involutive domain $A_{\lambda_0}$, then we position minima at points with coordinates $(i+1/2,\lambda_i)$ so that the last point corresponds to the other domain with slipping. The points with coordinates $(i,1)$ correspond to maxima. If the second domain with slipping is a domain of type $A_{\lambda_0}$, then a maximum point goes last, while if it is a domain of type $A^*_{\lambda}$, then this will be a minimum point. If both domains with slipping in $K$ are of type $A^*_{\lambda}$, then we position the minima at the points $(i,\lambda_i)$, and the maxima at the points $(i/2,1)$. Note that the last point is a minimum point. We connect these points sequentially by a polygonal line so that the minimum points are the minima of the arising graph of a function, and we add the reflection of this graph with respect to the $Oy$-axis. We denote the function with the resulting symmetric polygonal graph by $\widehat{g}$. An integrable geodesic flow on the Klein bottle is uniquely defined by a smooth even periodic function $g$. We look at its graph and replace it by a polygonal line by connecting the minimum and maximum points. Then, preserving the relative arrangement of the minima and maxima we modify the graph so that
Consider the part of the graph constructed over an interval of the period which intersects the $Oy$-axis. It does not necessarily coincide with the graph of $\widehat{g}$, but after shifting $g$ by half the period the resulting function coincides with $\widehat{g}$ (see [1], Theorem 2.17). As a result, we obtain the function $\widehat{g}$, from which one can construct a billiard with slipping $K$. This is the required billiard that is Liouville equivalent to the original geodesic flow. To see this it remains to show that, after the replacement of $g$ by $\widehat{g}$, the Fomenko-Zieschang invariant calculated for $K$ coincides with the invariant in Figure 3. Proposition 4. The billiard table $K$ constructed as described above, after the identification of the arrows on the boundaries of involutive domains becomes homeomorphic to the Klein bottle. Proof. The billiard table $K$ is obtained from $R$ by removing the domain $D$ homeomorphic to a disc and then gluing in a domain homeomorphic to a Möbius band along the boundary. This is how a Klein bottle is constructed. The proof is complete. 4.1. Step 1. Calculating the rough moleculeConsider the graph of $\widehat{g}$. As $\widehat{g}$ is positive on its interval of definition, we can construct the corresponding graph $W_2(\widehat{g})$. It is symmetric because $g$ is even. Next we consider the graph $\widetilde{W_2}(\widehat{g})$ by taking the quotient of the molecule obtained by this symmetry. The difference in taking the quotients of $W(\widehat{f})$ and $W_2(\widehat{g})$ is that the subgraph of $\widehat{f}$ is homeomorphic to the disc and we have one symmetry axis. In the case of $W_2(\widehat{g})$ the subgraph of $\widehat{g}$ is glued into a cylinder and we have two symmetry axes, $Oy$ and the straight line parallel to $Oy$ and going through the point of the graph over the endpoint of the interval. Hence we take the quotient with respect to the plane passing through these two symmetry axes of the cylinder. Now we describe the quotient of an atom $B_m$ assigned to a vertex of the graph $\widetilde{W_2}(\widehat{g})$. a. Assume that we take the quotient of an atom $B_k$ located on a level containing no minima on the symmetry axes. Then the number of minima is even and axes of symmetry are taken to themselves. Thus the quotient of $B_{2k}$ is an atom $B_k$. b. Assume that we take the quotient of an atom $B_k$ located on a level containing one minimum on one symmetry axis. As the number of minima outside this axis is even we see that the atom with this symmetry axis is $B_{2k+1}$. The point in question on the axis is fixed and the number of minima outside the axis decreases twofold, so the quotient of the atom is $B_k^*$. It remains to describe the quotients of atoms $C_m$. c. Assume that some $\lambda_{i}$, $0 < i < n+1$, is the smallest number of the $\lambda_{i}$, $0 \leqslant i \leqslant n+1$; then no minima can occur on the symmetry axis on this level and the total number of minima is even. Hence the number of minima decreases twofold and the axis is taken to itself. Thus, the quotient of our atom $C_{2k}$ is an atom $B_k$. After this only cases a. and b. are possible, because line segments are taken to line segments. Consider the case when no $\lambda_{i}$ for $0 < i < n+1$ is the smallest of the numbers $\lambda_{i}$, $0 \leqslant i \leqslant n+1$. Assume that the $Oy$ symmetry axis passes through a minimum. Then the second symmetry axis also goes through a minimum. d. Assume that $\lambda_{n+1}$ and $\lambda_0$ are different, so there is an odd number of minimum points. Then on the level $\min(\lambda_{n+1},\lambda_0)$ we have an atom $C_{2k+1}$, which is transformed into $B_{k}^*$ (Figure 6, d) since a fixed point is unique. On the level corresponding to the larger quantity of $\lambda_{n+1}$ and $\lambda_0$ we have case b, because line segments are taken to line segments. Also, Figure 6, (d) illustrates the case when the $Oy$ symmetry axis passes through a maximum point and the second symmetry axis passes through a maximum. Then the number of maxima on the level $\lambda_{n+1}$ is odd, so that on the level $\lambda_{n+1}$ we have an atom $C_{2k+1}$, whose quotient is $B_{k}^*$. In the other cases line segments are taken to line segments as described in case a. e. Assume that $\lambda_{n+1}$ and $\lambda_0$ coincide, so that the number of minimum point is even. Then in $W_2(\widehat{g})$ we have an atom $C_{2k}$ on this level. Since there are two minimum points on the symmetric axes, they are invariant under the involution, so the quotient of $C_{2k}$ is $B_k^{**}$ (Figure 6, e). The above construction is the same as the one producing the graph $\widetilde{W_2}(g)$ for the geodesic flow corresponding to the function $g$ on the projective plane. Now fix a level $\varphi_0<\pi/2$ of the first integral. It corresponds to the horizontal line $y=c<1$ such that $c$ is the radius of the circle tangent to the trajectories at this energy level. Consider the level surface for this value of the integral and its projection onto the billiard table. This projection can be obtained by removing the interior of the circles of radius $c$ from each billiard domain forming $K$. As a result, $K$ falls into several connected components. All of these (with the possible exception of two) are homeomorphic to cylinders obtained by gluing together domains $C$ obtained by removing the interior of a circle from domains $A_\lambda$, $B_\lambda$ and $A_\lambda^*$ (in the case when $\lambda<c$). The other domains (if exist) are homeomorphic to Möbius bands. They result from removing the interior of the circle of radius $c$ from involutive domains $A_\lambda$ or $A_\lambda^*$ (in the case when this circle has a smaller radius than the boundary circles of the domain $A_\lambda^*$). Note that the number of connected components is the number of segments on the line $y=c$ lying under the graph of $\widehat{g}$ for $x>0$ (because this half of the graph of $\widehat{g}$ is the profile of the billiard). Assume that the line $y=c$ does not pass through minima of $\widehat{g}$. Then the preimage of each cylinder and each Möbius band is a 2-torus. Assume that $y=c$ passes through several minima of this function but not through the ones occurring on the symmetry axes. Then, similarly to topological billiards, the bifurcation on this level of energy describes the gluing of several tori into one along the critical circles corresponding to nonconvex gluings along circles of radius $c$. It is easy to see that such a bifurcation is described by an atom $B_k$, where $k$ is the number of minima for $x>0$ at which the line $y=c$ is tangent to the graph. Assume that the line $y=c$ goes through several minima of the function, including one of the minima occurring on the symmetry axes $Oy$ and $(n+1,y)$ (in this case $K$ must contain an involutive domain $A_\lambda^*$, where $\lambda=c$). Then we can show that the bifurcation is described by an atom $B_k^*$, where $k$ is the number of minima for $x>0$ at which the line $y=c$ is tangent to the graph of the function. If we cancel slipping on the inner circle of the involutive domain $A_c^*$, then the corresponding bifurcation is described by an atom $B_k$. On the singular leaf of $B_k$ we distinguish the singular circle, the preimage of the inner boundary circle of $A_c^*$. (This is just one circle because on the singular leaf trajectories are tangent to circles of radius $c$. For values of integral smaller than $c$ such a preimage is empty, while for values larger than $c$ it consists of two circles.) We cut the 3-atom along the preimage of this circle and then re-introduce slipping on the circle. As a result, the distinguished circle on the singular level of $B_k$ is glued to itself. Extending this gluing to the whole atom (that is, to nonsingular levels) we obtain an atom $B_k^*$ in a similar way to how gluing along a distinguished cycle (so that this cycle goes twice along itself) results in forming the singular leaf of an atom $A^*$. Assume that the line $y=c$ passes through several minima of the function, including both minima occurring on the $Oy$-axis and the line $(n+3/2,y)$ (in this case $K$ must contains two domains $A_c^*$). Then the bifurcation is described by an atom $B_k^{**}$. The resulting rough molecule has the form $\widetilde W_2(\widehat{g})$, as required. 4.2. Step 2. Calculating marks on the moleculeProposition 5. The edges between saddle atoms carry the marks $r=\infty$. The ones between saddle atoms and atoms $A$ carry the marks $r=0$, except when an atom $A$ corresponds to a fixed point of the involution (see [1]; in the context of the above construction this means that the maximum corresponding to this atom $A$ lies on the $Oy$-axis). In this case the mark is $r=1/2$. Proof. The projections of cycles onto billiard domains forming $K$ were fully described in Proposition 2 because the billiard $K$ is formed by billiard domains from the series also used for the billiard table $R$. Hence the marks away from the central edge are the same. If the projection of a torus lies away from the domains $A_\lambda$ and $A^*_\lambda$, then the gluing matrices on the edges between saddle atoms and atoms $A$ have the form $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. The marks there are $r=0$ and $\varepsilon=1$, with no contribution to the mark $n$. If the projection lies in an involutive domain $A_\lambda$, then the gluing matrices on the edges between saddle atoms and atoms $A$ have the form $\begin{pmatrix} -1 & 2 \\ 0 & 1 \end{pmatrix}$, and the marks are $r=1/2$ and $\varepsilon=1$. This matrix makes a contribution of $[-1/2]=-1$ to the mark $n$ on the family. If the projection lies in a domains $A_\lambda^*$, then the gluing matrices on the edges between saddle atoms and atoms $A$ are $\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}$, and the marks are $r=0$ and $\varepsilon=1$. This matrix makes a contribution of $[-1/1]=-1$ to the mark on the family. On an edge between two saddle atoms the gluing matrix is $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$: all cycles $\lambda$ are homologous and co-oriented. This results in the marks $r=\infty$ and $\varepsilon=1$. These inner edges of the family make no contribution to the mark $n$. The proof is complete. Proposition 6. On the single central edge of the molecule $\widetilde{W_2}(g)$ the marks are $r=-\infty$ and $\varepsilon=-1$. This molecule contains a unique family, with mark $-2$ on it. Proof. We must describe the transition of cycles across the level $\Lambda\!=\!\pi/2$. The cycles $\lambda$ transform one into the other with opposite orientation. Consider the cycle $\delta$ shown in Figure 9. As the angle of reflection from the boundary tends to zero, clockwise- and anticlockwise-oriented cycles $\delta$ transform one into the other. Here we have $\delta=\lambda - \mu$. If $\varphi<\pi/2$, then we denote the cycles $\lambda$ and $\mu$ constructed above by $\lambda_+$ and $\mu_+$, respectively, while if $\varphi>\pi/2$, then we denote them by $\lambda_-$ and $\mu_-$, respectively. This shows that $\lambda_++\mu_+=\lambda_-+\mu_-$. Hence $\mu_+= 2\lambda_-+\mu_-$, so that the gluing matrix is $\begin{pmatrix} -1 & 0 \\ 2 & 1 \end{pmatrix}$. The contribution to the mark $n$ is $2$. Adding all the contributions of edges we obtain $n=-4+2=-2$. The marks on the central edge are $r=\infty$ and $\varepsilon=-1$. The proof is complete.
Theorem 1 for the Klein bottle is proved. Remark 4. Here is yet another significant difference between the graph molecules of integrable geodesic flows on the projective plane and Klein bottle. After reversing the orientation of the iso-energy manifold $Q^3$, the mark $n$ transforms into $n'=-n -l -\sum s$, where $l$ is the number of edges with noninteger mark $r$ that are outgoing edges of the family in question, and $s$ is the total number of stars on atoms in this family. Thus, for the projective plane the mark $n=-2$ transforms into $n'=0$, while for the Klein bottle it transforms into $n'=-2$.
Citation in format AMSBIB
\Bibitem{VedZav22} |
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