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CLASSIFICATION OF FOUR-DIMENSIONAL INTEGRABLE HAMILTONIAN SYSTEMS AND POISSON ACTIONS OF

IN EXTENDED NEIGHBORHOODS OF SIMPLE SINGULAR

POINTS. II This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1994 Russ. Acad. Sci. Sb. Math. 78 479 (http://iopscience.iop.org/1468-4802/78/2/A13) View the table of contents for this issue, or go to the journal homepage for more

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Pocc. Aicaa. HayK MaTeM. C6OPHHK TOM 184 (1993), I * 4

Russian Acad. Sci. Sb. Math. Vol. 78 (1994), No. 2

CLASSIFICATION OF FOUR-DIMENSIONAL INTEGRABLE HAMILTONIAN SYSTEMS AND POISSON ACTIONS OF R2 IN EXTENDED NEIGHBORHOODS OF SIMPLE SINGULAR POINTS. II UDC 514.7

L. M. LERMAN AND YA. L. UMANSKIl ABSTRACT. Integrable Hamiltonian systems and Poisson actions of the group R2 with simple singular points on a smooth (C°° or real-analytic) four-dimensional symplectic manifold (Μ, Ω) are studied, where Ω is a symplectic 2-form. Bibliography: 6 titles.

This is a direct continuation of the paper [1], in which the necessary definitions and results were given. PART IV. A SINGULAR POINT OF SADDLE TYPE 1. THE DEGENERATION SET AND THE BIFURCATION DIAGRAM

Let (XH , K) be an IHVF, ρ a singular point of it of saddle type, U a local neighborhood of ρ at which the assertion of Proposition 1.1 holds, (χι, Χ2, yi, yi) the corresponding coordinates in U. Throughout this section, without loss of generality, we shall assume that λ\, λ2 > 0, Δ > 0. In fact, for Αι < 0, under the symplectic change of variables (x|, x2, y[, y2) = (—j>i, x2, x\, y2), the monomial Xiyi goes to the monomial — x[y[, i.e., λ\ changes sign. If Δ < 0, then the symplectic change of variables (xj, x'2, y\, y'2) = (x 2 ,x\,yi,y\) changes the sign of Δ while preserving the signs of λ\ and λ2 . We first recall the structure of the orbits of the vector field XH on the degeneration set in U. This set consists of two disks Xi = y\ = 0 and x2 = y2 = 0 which intersect 1 transversally at the point ρ . We consider the disk Σ, ^. : x2 = y2 = 0. The restriction of XH to it is a Hamiltonian vector field with one degree of freedom, having a saddle singular point at ρ . Its local stable and unstable manifolds (separatrices) are the segments Xi = 0 and >Ί = 0. We note that all the trajectories of XH on Σ ^ (except the point p) are one-dimensional (hyperbolic) orbits of the action; every unstable separatrix from Σ,^ under continuation in time along the trajectories of XH, because Η is Morse, returns to Σ ^ and merges with a stable separatrix [4], i.e., the disk is associated to two loops γ\, γ2- Suppose that a trajectory of XH exits from Σ/^ close to a loop, say γι, and does not lie on it. Using small disjoint transversals to the separatrices and the continuous dependence of the trajectories on the initial point, it is easy to prove that such a trajectory must return to Σ/^, since it is simultaneously a one-dimensional orbit of the action. The value of the integral 1991 Mathematics Subject Classification. Primary 58F05, 58F14; Secondary 58F16, 58F21. This is a direct continuation of the paper [1], in which the necessary definitions and results were given. ©1994 American Mathematical Society 1064-5616/94 $1.00+ $.25 per page 479

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L. M. LERMAN AND YA. L. UMANSKIl

Η along a trajectory is preserved, so that there are only two possibilities: a) the trajectory becomes closed immediately after going around a loop (a short singular periodic trajectory); b) the trajectory becomes closed after going around γ\ first and then y2 (a long singular periodic trajectory). Thus Z/oc can be extended along periodic trajectories in a neighborhood of a global piece of the loops y\ and y2, as a result of which we obtain a two-dimensional symplectic submanifold Σ ι , whose boundary consists of two short singular periodic trajectories and one long periodic trajectory, on which Η — ±h*. In an analogous way one constructs a symplectic submanifold Σ2 and the loops γι, y 4 . The local bifurcation diagram in U consists of two smooth curves l\ = μ(Σι) and h — μ(^2), which are the graphs of the functions k — ki(h) = (ι>,\λ,·)Α + O(h2), i = 1,2. This follows from the description of Σι (x2=y20) and Σ 2 (χι =yi = 0) in U and Proposition 1.1. The curves h and l2 intersect transversally at the point ( 0 , 0 ) , which follows from the condition Δ Φ 0. 2. TYPES OF LOOPS AND SINGULAR PERIODIC TRAJECTORIES

The aim of this section is to study the possible types of singular periodic trajectories (SPT) of the vector field XH from the point of view of the orientability of their stable and unstable manifolds. It turns out that these properties are completely determined by the corresponding properties of the loops. It follows from Proposition 1.1 that in the neighborhood U there are four XHinvariant Lagrangian disks that pass through ρ, two of which W^. {y\ =y2 = 0) and W£. (χι = x2 — 0) are the local stable and unstable manifolds of the point ρ, and the restriction of XH to the other two, Wf (χι = y2 — 0) and W2C (x2 = yx = 0), has a saddle at ρ . The union of these four disks forms the complete set of solutions of the system Η = 0, Κ = 0 in U. From this we see that Wfa. intersects W{ in the segment x2 = yi — Χι = 0, and W2C in the segment y\ — X\ = x2 = 0. Analogously, W^. intersects Wf in the segment X\ — y\ = y2 = 0, and W{ in the segment x2 = y\ = y2 - 0. We now consider a loop γ, for example, lying on Σ ι , and assume that its intersection with W^ is the half-line X\ = x2 = y2 = 0, y\ > 0, while its intersection with H^ c is the half-line y\ — y2 = x2 = 0, x\ > 0. According to [2], the following transversality condition holds: the extension of W^. along the trajectories of XH close to γ coincides with W2C, and under continuation of trajectories in time reversed along trajectories close to γ, W^. coincides with W{ .(') We denote the line W&. Π W2C by d. It cuts W&. and W2C into two half-disks. We choose the pair of these four half-disks that intersects γ . This pair of half-disks forms a topological disk DQ , glued along d, containing the initial and final segments of the loop. To DQ we glue a strip B^ , formed by the segments of the trajectories of XH that are close to the loop and exit from the half-disk D^ η W^. and land on the half-disk D^ η W2C (Figure 1). The resulting two-dimensional topological manifold Co' = D^UBQ is homeomorphic to either a cylinder or a Mobius band. Definition 4.1. A loop γ is said to be orientable if Cj is homeomorphic to a cylinder, and nonorientable if it is homeomorphic to a Mobius band. Remark 4.1. An analogous construction for the loop γ can be realized using W^. and Wf . It is easy to show that the type of orientability of the loop defined by this method coincides with the type introduced in Definition 3.1. (')This condition means that on the level Η = 0 containing the point ρ and W^, W^., W{, W^ , the intersection of the global stable manifold Ws and the global unstable manifold Wu of the point ρ is transversal along the loop y .

FOUR-DIMENSIONAL INTEGRABLE HAMILTONIAN SYSTEMS. II

481

FIGURE 1

Remark 4.2. For the second loop on Σι the manifold CQ = U BQ is constructed, where D\ is formed by the second pair of half-disks from ^' and w WS, 2 , and Bl is a strip containing the segment of the second loop between W^. and W{. Our next problem will be to describe the topology of the separatrix set W of the garland of the point ρ. We number the loops in cyclic order 71, 72, 73, 74 on W^., and 7i follows 74 in this order. For definiteness we shall now assume that 71 is a loop on Σ[ exiting in the direction x{ — x2 = y2 = 0, y\ > 0; then 73 is the second loop on Σ ι , 72 is the loop on Σ 2 exiting in the direction x{ = x2 = yx = 0, y2 > 0, and 74 is the second loop on Σ2 . On any of W^., Wf, W£ the segments of the loops lying on the same symplectic manifold are not adjacent, so that on each of these the cyclic order of loops is preserved. Lemma 4.1. W is a two-dimensional CW-complex having one zero-dimensional cell ρ, four one-dimensional cells 71, ... , 74, and four two-dimensional cells L\, ... , L4, which are two-dimensional orbits of the action. Proof. The set W contains the point ρ and is a connected component of the set of points that solve the system Η = 0, Κ = 0. In particular, W contains the loops 7i, ... , 74, and WnU coincides with the four Lagrangian disks W^c, W{, W£, W^. For example, we consider the disk W^.. The segments of loops 71, ... , 74 divide it into four open quadrants. We shall call these quadrants unstable. Analogously, there C are four stable quadrants on W^. and eight saddle quadrants on W{ U PF2 . For definiteness, we choose a quadrant between the loops 71 and 72 and consider two trajectories of the field XH passing through the quadrant; one, τ\, is close to 71, and the other, τ2 , is close to 72. From the transversality of Ws and Wu along γχ on the level Η = 0, mentioned above, it follows that the trajectory τ\ goes close to 7i and again hits U, passing through a saddle quadrant on W£ and leaves it close to the loop 72 (or 74), and, moving along it, hits some stable quadrant. Analogously with τ2 ; however, τ2, while moving along 72, passes through a saddle quadrant on the disk W[, since 72 enters at ρ along the eigendirection corresponding to the eigenvalue λ2. Since no one-dimensional trajectory passes through a quadrant, it follows that τ ι and τ2 go into one stable quadrant. Therefore, the trajectories of XH passing through one quadrant form one two-dimensional orbit Li whose boundary consists of a set of several loops and the point ρ. Thus, one stable and one unstable quadrant enter each such orbit, as do two saddle quadrants belonging

482

L. M. LERMAN AND YA. L. UMANSKH

to different W{, W%. Therefore, in W there are a total of four orbits L\, ... , L4 . The CW-property is easily obtained from the local structure of the orbits around a one-dimensional hyperbolic orbit and a singular point (see the argument above and §1 of Part I). This proves the lemma. We give a canonical description of the set W. For this we divide W along the loops 7,·, i.e., we consider the connected components of the set W\{ji , 72, 73, 74} . Each of the resulting four components L ; coincides with a two-dimensional orbit of the action. Now we close each L ; in the Caratheodory topology [3] by adjoining to Li all the boundary points accessible from L,. This closure L* gives a set homeomorphic to a rectangle, whose angles are the four quadrants belonging to the orbit Li. The boundary of the set L, will consist of four segments, each of which corresponds uniquely to one of the loops Yi, ••• , 7A , and all the vertices of the rectangle correspond to the point ρ. The canonical representation of W is obtained from four such rectangles by gluing their boundaries along the segments corresponding to the loops of the same name with orientations induced by the vector field XH , and by identifying the vertices to a single point. This representation of W allows us to give an intuitive interpretation of the notions of an orientable and a nonorientable loop. For this we take the loop y\ , say, and the two rectangles that contain the two quadrants of W^. adjoining γ ι . We glue these rectangles along γ ι. It follows from the transversality condition that, moving near γ ι from W^., we hit W{, on which the two quadrants of W{ adjoin γι . From this it is clear that the set D\ U BQ is obtained by gluing the dashed part in Figure 1 along the segment d. On W£, 72 and 74 will also be adjacent to γι. The loop γι will be nonorientable if the loops on W£ are located as shown in Figure 4 (see §5), and orientable if 72 and 74 change places on Wf. Lemma 4.2. If two loops that are adjacent in numbering are nonorientable, then all the loops on W are nonorientable. Proof. Let γι and 72 be nonorientable loops, and assume that 73 is orientable. We choose a rectangle Κ from the canonical representation of W, in which 71 and 72 lie on the boundary of the unstable quadrant. We let a trajectory of XH close to 71 issue into Κ. This trajectory exits from the unstable quadrant, then hits the saddle quadrant whose boundary contains the loops γι and 74 , since 71 is nonorientable. Then this trajectory hits the stable quadrant whose boundary contains 74. We now let another trajectory, close to 72, issue into Κ. This trajectory goes through the saddle quadrant bounded by 72 and 73, since 72 is nonorientable, and hits the stable quadrant whose boundary contains the loops 72 and 73, since 73 is orientable. This contradicts the fact that there is only one stable quadrant in Κ. This proves the lemma. An immediate consequence of Lemma 4.2 is Corollary 4.1. There are only four possibilities for loops in W: 1) all the loops are orientable, 2) all the loops are nonorientable; 3) three loops are orientable; one is nonorientable; 4) the loops on one symplectic manifold are orientable, and those on the other are nonorientable. Proposition 4.1. Let (XH, K), (XHi, K') be two IHVFs with Morse Hamiltonians, each having a singular point of saddle type. Then their separatrix sets W and W are homeomorphic if and only if the same case of Corollary 4.1 is realized for both of them. (The types of sets W are shown in Figure 2.)

FOUR-DIMENSIONAL INTEGRABLE HAMILTONIAN SYSTEMS. II

ι

483

>

ft y

η

Λ

Ά Ϊ3

2. The edges of the same name are glued together, and the vertices are identified to a point. FIGURE

Proof. The proposition follows immediately from the canonical representation of W. We now establish a connection between the type of orientation of loops on Σ, and the type of orientation of periodic trajectories of XH on Σ,. Indeed, using the definition of orientable and nonorientable loops and the smooth dependence on ε of the stable and unstable manifolds of a SPT on compact subsets, it is not hard to prove the following lemma. and 73 be loops on Σ ι . Then: Lemma 4.3. Let 1. If γι and are orientable, then both the short and the long SPTs on are orientable. 2. If γ\ and 73 are nonorientable, then the short SPTs on Σι are nonorientabie and the long ones are orientable. 3. If 7, is orientable and y, is nonorientable, i φ j , then the long SPTs are nonorientable, the short ones freely homotopic to 7, are orientable, and the short ones freely homotopic to jj are nonorientable. An analogous assertion holds for the loops 72 and 74 on Σ2. 3. CONSTRUCTION OF THE EXTENDED NEIGHBORHOOD

We now construct the extended neighborhood V, which is the union of all the orbits that intersect U . We shall describe its structure. In V we define a smooth Riemann metric that coincides with the Euclidean metric in U. We note that each of the sets Σ\\ΙΙ and Σ2\£/ consists of two strips Q\, Qj , i = 1, 2, fibered into segments y//(e) of trajectories of XH ; 7ij(e) lies on the level Η — ε, i.e., Qj is diffeomorphic to the product IH χ {7ij(0)}, where IH = [-hm, A»], 7,7(0) is a segment of the loop on Σ,\(7 . At the same time, the segments of trajectories on Qj are segments of hyperbolic one-dimensional orbits of the action. Therefore, each Qj is the transversal intersection of the two three-dimensional submanifolds formed by the two-dimensional local submanifolds for 7,7(ε) (see§l). These three-dimensional submanifolds are diffeomorphic to Qj χ I, Qj χ {0} = Qj. In each Qj χ I we choose two sections ξ = ±δ , ξ e / . Then on each level Η = h on the corresponding twodimensional local submanifold for the segment of a curve from Qj we have the segments ξ = ±δ , which single out the rectangle containing 7, 7 (ε). We denote this

484

L. M. LERMAN AND YA. L. UMANSKII

pair of segments on one manifold by n\ (ε), ηι(ε), and on the other by s\ (ε), ε e In. We extend the segments «,·(ε), 5,(ε) along the trajectories of the vector field Υ. On each level Η — ε we obtain four rectangles Ν{(ε), Si(e), respectively, transversal to the local submanifolds of the orbits γ^(ε). The rectangles iV)-(e), S, (ε) are transversal to orbits of the action, and thus on the level Η = e we can single out a neighborhood Rij(e) of the union of the submanifolds of the segment of 7,_/(ε) on this level, defined by the inequality \k - Α:,(ε)| < d, d > 0, where &,·(ε) is the value of the function Κ on 7, ; (ε). It is obvious that d can be chosen to be independent of e. We write Ru = \Je Ru(e). The part of W that lies outside of Vc = U U ((J i?,7) consists of four disks, each of which belongs to its two-dimensional orbit (see §2). It is not hard to show that there exists a δ small enough that for \H\ < δ, \Κ\ < δ the set V\ Vc is fibered into two-dimensional disks whose boundaries belong to Vc. From this it follows, in particular, that the limit set of any two-dimensional orbit belonging to V lies on ΣιϋΣ2 . Therefore, the neighborhood V is compact, μ(¥) = μ(ϋ), and the bifurcation diagram for V coincides with the local bifurcation diagram for U. In the domain μ(¥) we choose a rectangle σ defined by the conditions \K\ < k*, \H\ < h,, where A» is chosen so small that \ki{h)\ < k\ for \h\ < A». Now the extended neighborhood of the point ρ is understood to be the set μ~ι(σ) Π V. We use the notation V for this as well. As above, Ve = V Π Με. We write d V+ = V η {Κ = Κ} = U d K+ , 9 V- = V η {Κ = - Κ } = U e d K~~ • From the compactness and invariance of dV+, dV~ , and also the property dV± Π (Σι U Σ2) = 0 (since \ki(h)\ < K) it follows that the dV± consist of a finite number of Liouville tori. 4. AN INVARIANT, AND STATEMENT OF THE EQUIVALENCE THEOREM

The fact that W and W are homeomorphic for two IHVFs is not sufficient for the isoenergetic equivalence of the vector fields in their extended neighborhoods V and V . This is related to the fact that for given types of loops for ε Φ 0 a different number of saddle SPTs can lie on VE. From the structure of the restriction of the field XH to Σ, it follows that Vt Π Σ, may contain either one long SPT or two short SPTs. Therefore, the following two cases are possible: A) for one sign of ε there are four SPTs on Ve (and all of them are short), and there are two SPTs on V-e (both long); B) for all ε / 0 there are three SPTs on VE (one long and two short). Thus, each of the cases l)-4) (Corollary 4.1) describing the types of loops is subdivided into two: A and Β. Corresponding to these we obtain eight different cases, which are numbered «A and «B, « = 1 , 2 , 3 , 4 . To decrease the number of possibilities to consider, we indicate a standard representative for each of the cases «A, «Β, η = 1, 2, 3, 4, to which any IHVF reduces up to isoenergetic equivalence. We shall call upper symplectic manifold the one of Σ ι , Σ 2 on which the function Κ takes the larger value for a fixed ε > 0. We note that the IHVFs {XH, Κ), {ΧΗ, -Κ), and (X-H , K) are obviously isoenergetically equivalent. Therefore, after replacing (ΧΗ,Κ) by (XH, -K) if necessary (in the Β cases), or (XH, K) by (X-H, K) if necessary (in the A cases), we may assume that for ε > 0 there are two short SPTs on the upper symplectic manifold in Ve. After this, by changing (XH , K) to (XH , —K) if necessary (in the A cases), or (XH, K) to (X-H , K) (in the Β cases), we find that the nonorientable loops in cases 3) and 4) lie on the upper symplectic manifold. The system obtained in this way is again denoted by (XH , K) and is called the canonical representative of the case. Below we consider only these representatives. As was noted above, in some local symplectic coordinates, Η and Κ have the form as given in Proposition 1.1, where λ\ > 0, λι > 0, and Δ > 0. Therefore,

FOUR-DIMENSIONAL INTEGRABLE HAMILTONIAN SYSTEMS. II

485

it is easy to see that Σ2 is the upper symplectic manifold. We shall also assume that in fact the loop γ2 on Σ 2 in the cases 3A and 3B is nonorientable. This can always be achieved by a local symplectic change of coordinates (χι, x2, y\, yi) —• (χι, -x2, >>2, -yi) > while preserving all the other conditions. Remark 4.3. From the construction of the standard representatives it follows that on Σ 2 for ε > 0 there are always two short SPTs, so that the loop 72 enters ρ in the direction χλ = yx - y2 — 0, x2 > 0, and 74 enters ρ from the direction •*i = y\ — yi = 0, xi < 0. Correspondingly, the loop y\ enters ρ from the direction *2 — y\ = yi = 0, xi > 0, in the A cases, and from the direction x 2 = y2 = y\ = 0, Xi < 0 in the Β cases. Let (XH , K), (Xji', K') be two IHVFs with Morse Hamiltonians, having singular points ρ and p' of saddle type, respectively, and let V, V be the extended neighborhoods of these points. Theorem 7. The field (XH , K) on V is isoenergetically equivalent to the field (XH> , K') on V if and only if the same case «A or «B occurs for both fields. The necessity of the conditions of the theorem is obvious, and the proof of sufficiency will be given in §7 below. 5. THE STRUCTURE OF AN AUXILIARY SYSTEM AND ITS INVARIANT FOLIATIONS

In an extended neighborhood U we consider an auxiliary gradient system Υ. For this we define a Euclidean Riemannian metric in U via the coordinates (χι, Χ2, y\, y2)-ln U the field Υ has the form ,4l)

j>i = xxh2{xl

xx=yxh2{xl+yl)R, X2 =-yihiixj

2

+ y i)R,

+ y\)R,

h = -X2hi(xf + yj)R,

where ft,- = ^h|^ξi, i - 1, 2, R = (hxk2 - h2kx)l[h} • (x\ + y\) + h\ • (xf + y\)], and we may assume that R > 0 in U, so that the factor R denotes a change of time. One can check directly that the system has three invariant foliations: the Hfoliation Η = const, the 6,-foliation 6, = (xf -yf)/2 = const, i = 1,2, and the disks Σ ^ , Σ ^ consist of singular points. In order to describe the geometry of the &i-foliations, i = 1, 2, we make a quadratic transformation π : (4.2)

ti = xm,

bi = (xf-yf)/2.

The field (4.1) in these variables has the form 1

'

ξ[ = 4Α2(ί? + 6f)'/ 2 (i 2 2 + bl)l'2R,

61=0,

£2 = -4h^\

b'2 = 0,

+

b\Yl^l

+ blYl>R,

where 2{ξ} + bf)l/2 is substituted in place of xf +yf in R. The field (4.3) will be considered in a domain G, where |ξ,| < ρ, |6,·| < δ, / = 1,2, and ρ is chosen small enough that hi > λ,/2, h\k2 — h2k\ > Δ/2. Obviously, from ρ, δ one can choose h*, Κ so that these inequalities for ξι, b\ hold in U. The first two equations in (4.3) define a Hamiltonian system on the (ξ\, ξ2) plane, "spoiled" by a change of time; the system depends on the parameters b\, b2 . Since A is a local integral, we obtain a foliation into curves h^\, ξ2) = ε . The parameters b\, b2 influence the speed of motion along these curves. For b\ Φ 0, b2 ψ 0 the system has no singular points in a neighborhood of ( 0 , 0 ) . For b\ = 0, b2 Φ 0 there is a line of singular points ξ\ = 0, and for b\ Φ 0, b2 = 0 there is a line of singular points ξ2 = 0. For b\ = 0, b2 = 0 there is a "cross" of singular points,

486

L. M. LERMAN AND YA. L. UMANSKll

defined by the equation £1 · £2 = 0, and on any curve Λ(£ι, ξ2) = ε, ε Φ 0, there are two singular points. On the (£ι, ξ2) plane the projection of the set n{U) is a curvilinear Quadrilateral d e f i n e d b y t h e i n e q u a l i t i e s \h(£\ , ξ ι ) \ < Κ , ψ { ξ χ , ξ 2 ) \ < Κ . Now we consider a set Gc in n(U), invariant relative to (4.3) and diffeomorphic to the cube, defined by the relations h{£\ , ξ2) = ε , |fc(& , ξι)\ < ^* > where |ε| < A», l^fl < ί = 1,2. Expressing & = ρ(^ι, ε) from the equation /ζ(£ι , ξ2) = ε we see that the coordinates in GE are (& , b\, b2). The integral curves of the field (4.3) in Ge are the lines b\ = const, b2 = const. The set of singular points of the field (4.3) in G£ consists of two lines: C2(e) = {£»i = 0 , ί ι = 0 } and C,(e) = {b2 = 0, ξ1 = i,(e)}, where ξι(ε) = λγιε + Ο(ε2) is a solution of the equation φ{ξ\, ε) = 0. It is easy to see that C,-(e) = π(Σ, η Ue). We see from (4.3) that ξχ > 0, and ξχ = 0 only on Cj(e), C 2 (e). Thus, for ε φ 0 there is a trajectory going from one line of singular points to the other, which is defined by the relations b\ — 0, b2 = 0, 0 < £1 < ξι (ε) for ε > 0, and by the inequality ξι (ε) < ξι < 0. The stable manifold for C2(e) is the set 61 = 0, ξι < 0 for ε < 0, or bx = 0, ξι < 0 without the line bi=b2 = 0, 0. The unstable manifold for C2(8) is the set bi = 0, ξι < 0 for ε > 0, and Z>i = 0, ξι > 0 without the line bi — b2 = 0, ξ > ξι (ε) for ε > 0. Analogously, the stable and unstable manifolds for Cj (ε) lie on the plane b2 = 0. We denote by Π + (ε) the face of Gc defined by the equality 1ι(ξι, φ(ξι, ε)) = Κ , and by Π~(ε) the face 1 0 on Π + (ε) and £1 > 0, ξ2 < 0 on Π~(ε) by virtue of the conditions λι > 0, λ2 > 0, Δ > 0. We now return to U. From the formulas for π it follows that the mapping π is a ramified covering U —» n(U) with ramification set the union of the planes bx = ξι = 0 and b2 = ξ2 — 0. Over all the points of n(U) except the points of these planes, there are four preimages, over the points of the planes there are two preimages, and over the point ( 0 , 0 , 0 , 0 ) there is one preimage, the point ρ . The sets Π ^ ε ) do not intersect the curves C,-(e) (since ξι·ξ2φ0 on them), and therefore π~1(Π+(ε)) is the four curvilinear quadrilaterals lying in Ve on the level Κ = k», and π~ 1 (Π~(ε)) is four such quadrilaterals in Ve on the level Κ = -Κ . We renumber them as follows: ITf(e) denotes the quadrilateral on which Χι > 0, yi < 0, x2 > 0, y2 > 0; Π£(β) the one on which Χι > 0, yi < 0, x2 < 0, y2 < 0; Π£(β) the one on which xx < 0, j ^ > 0, x 2 > 0, y2 > 0; Π+(ε) the one on which JCI < 0, yi < 0, x 2 < 0, y 2 < 0. Analogously, ITf(e) denotes the quadrilateral on which JCI > 0, yi > 0, x 2 > 0, y 2 < 0; Π^(ε) the one on which Χι > 0, >Ί > 0, x 2 < 0, y 2 > 0; IT^ (ε) the one on which χι < 0, yi < 0, x 2 > 0, y2 < 0; Π^~(ε) the one on which Xi < 0, yx > 0, x 2 < 0, y 2 > 0. In order to describe the structure of π in a neighborhood of the ramification set Σ,'Ο(. υ Zj^ we restrict π to Ue; then n{Ue) = Gc. In Ge, ε Φ 0, we choose a small rectangular neighborhood Da of a e C 2 (8), not containing points of Ci(e), on the plane b2 = const. The set π" 1 (α) consists of two points Αι, A2 lying on two curves belonging to Σ ^ . , defined by the equation Η(Ο,ξ2) = ε (these curves form π ~ ' ( ^ ( ε ) ) ) . Then, as follows from (4.2), n~1(Da) is two disjoint two-dimensional smooth disks, one D(Ai) in a neighborhood of the point Αι, the other D{A2) in a neighborhood of A2, and the restriction of π to such a disk is a ramified covering over Da of the type ζ ^ z2 over z = 0, ζ e C. Here D(Ai) is invariant for the field Υ, and the restriction

FOUR-DIMENSIONAL INTEGRABLE HAMILTONIAN SYSTEMS. II

487

J

FIGURE 3

of Υ to D(Aj) has a saddle singular point at At; its two stable separatrices are projected by a mapping π into one half-trajectory of the system (4.3), entering at the point a, and the two unstable separatrices are projected to the half-trajectory exiting from a. In an analogous way π is constructed in a neighborhood of the inverse image n~l(C\(e)) of the set C\(e) on the leaf b\ = const for ε Φ 0. For ε = 0 the line segments C\(0) and C2(0) intersect at the point π(ρ) : b\ = b2 = ξ\ = 0. For the points of these curves different from n{p), the structure of π is the same as described above. We now consider the leaf £2 = 0 in Go, containing Ci(0). The half of Ci(0) corresponding to b < 0 is the image of x\ = 0 on Σ ^ , and the other half of C\ (0), where b\ > 0, is the image of y\ — 0 on Σ,1^. Thus, n~l(C\(0)) is a "cross" on Σ['0{., denned by the equation x\y\ = 0. Under the mapping π the points (±X\, 0) are mapped into the same point on Q ( 0 ) ; analogously for the points (0, ± y i ) . On Go one half-trajectory b\ = 0, bi = 0, ξ\ < 0 enters at the point π(ρ). Its inverse image in UQ consists of four half-trajectories, entering at the point ρ . Each such half-trajectory passes through the corresponding point (2+(0) of the square Π+(0). A smooth curve τ+ passes through the point Q+(0) on n t ( 0 ) , whose image on Π+(0) is the trace of the stable manifold of Ci(0). Therefore, the set of all halftrajectories passing through the points of the curves r t (/ = 1 , 2 , 3 , 4 ) forms the stable manifold of the "cross" lying on Σ,'0(.. Using (4.3) it is not hard to check that the ω-limit set of the half-trajectories passing through the curves τ\ and τ\ is the two segments of the "cross" yx = 0, x\ > 0, and X\ = 0, y\ < 0, and the ω-limit set of those passing through τ\ and x\ is the segments of the cross yx = 0, X\ < 0, and Χι = 0, y\ > 0 (Figure 3). Analogously, on the squares Π~(0) there are points Q ; r (0) and curves zj . Through them pass the half-trajectories that form the unstable manifold of the "cross" lying on Σ ^ , and the α-limit set of the trajectories passing through x\ and τ^" is formed by the segments of the "cross" yx = 0, x\ > 0, and JCI = 0, y\ > 0, while the α-limit set at those passing through τ^ and T 0 two trajectories of the vector field Υ exit from the points z\, z 2 ) and the trajectories exiting from one point Zi go to different points z[ and z 2 . For ε < 0 the trajectories go from z\ to z\ and z 2 .

We now draw an auxiliary segment Λ on the plane Z>2 = 0: ξ\=ξ\, intersecting the trajectory f between the points^ ζ and z'. An orientation is induced on it by the coordinate b\. The segment Λ divides the leaf bi = 0 into two rectangles Ri, Ri, where R\ contains the point ζ, and i? 2 contains the segment C\{s). The sets π~X{R\), n~l(R2) have already been described: the geometry of n~l(Ri) is the same as for the leaf bi = b\ Φ 0, and n~x{Rj) consists of two components, each of which is the union of the local stable and unstable manifolds of the segment ί/,(ε), i.e., two curvilinear quadrilaterals that intersect transversally along */,-(e). For ε > 0 the pair of segments on π - 1 (Λ) is the pair of opposite sides of the stable manifold for one i/,-(e), and the second pair of segments on π~'(Λ) is the pair of opposite sides of the stable manifold for dj(e), i φ j . The orientation of Λ defines an orientation on all the segments π~ι(Α). The boundary of the curvilinear quadrilateral that is the local stable manifold for 6?,(ε) can be oriented, and, as follows from (4.2), the two orientations defined by the two oriented segments are opposite. The same four segments of π^'(Λ) occur pairwise in the boundary of each component of the set n~l(R\). They are transversal segments for the unstable separatrices of a saddle singular point on its component of π~' (R\). The boundary of this component is a simple closed curve, and its orientations, defined by each of the two oriented segments of π " 1 ^ ) , coincide. For ε > 0 the description of the inverse images of the rectangles R{, i? 2 is the same with the words stable and unstable interchanged. From Lemma 4.4 it follows that there is a gluing along these segments. The resulting set is shown in Figure 4; it consists of a smooth two-dimensional annulus with a piecewise smooth boundary and of two smooth rectangles (the unstable manifolds of the curves π~'((ΓΊ(ε))) that intersect the annulus transversally.

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As ε —> + 0 , the inverse image of the leaf b-ι = 0 is deformed so that the heteroclinic contour is contracted to the point ρ , the two segments n~1(C\(e)) are glued at a point, becoming a "cross" on Σ,1^,. Thus, we obtain the union of the stable and unstable manifolds of the "cross", as described above. Under passage to e < 0, a heteroclinic contour is born from ρ , only now the smooth annulus contains the unstable manifolds for the curves π~1(θ2(ε)). Below, for convenience, the resulting foliation (with singularities) in Ue will also be called the 6i-foliation (respectively, the bi-foliation). Remark 4.4. On ΙΤ^ε) there are two transversal foliations into the segments b2 — b\ and b\—b\. Since the restriction of the mapping π to Π*(ε) is a diffeomorphism, then on these squares there are two smooth transversal foliations (these are the traces of the b\ -foliation and the ^-foliation on the surface Κ — ±k,, Η = ε). On each square Π*(ε) there are two distinguished leaves b\ = 0 and b~L = 0 (denoted by rf and Cf above, and their intersection point was denoted by Qf). For ε > 0 (respectively, ε < 0) the leaf b\ = 0 (respectively, bi — 0) on Π+(ε) is the trace of the stable manifold of a CST lying on Σι (respectively, on Σ2), and the trajectory that tends, as t -> 00, to a point on Σ 2 (respectively, on Σ ι ) , passes through the point Qf(s) from which two heteroclinic trajectories exit. The leaf bi = 0 (respectively, b{ = 0 ) , except for the point Q+(e), is the trace of the stable manifolds of the curves of singular points of Σι (respectively, Σ2). The distinguished leaves on Π~(ε) are described analogously. Below, in §7, we shall need assertions concerning the orientations at the points Q+(e) of 3-frames formed by two vectors tangent to the curves b\ = 0, &2 = 0 on nt(fi) and the vector of the field Υ at the point Qf(e). More precisely, h t e\ be the unit tangent vector to the curve b-ι = 0 on n t ( e ) in the direction of increasing b\, e'2 the unit tangent vector to the curve b\ = 0 on Π+(ε) in the direction of increasing b2 , and e\ the normalized vector of the field Υ at the point β+(β). +

L e m m a 4.5. In UE all the frames at the points β, ( ε ) , / = 1 , 2 , 3 , 4 , define the same orientation. Proof. To determine the orientation of these frames we compute the orientation of 4-frames at the points Q/"(e) in U, adding another (first) vector VH to the 3frames that is computed relative to the Euclidean metric in U. The computation in the coordinates (JCI , xj, y>\, y{) shows that the signs of the determinants formed

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from the coordinates of the four vectors are the same at all points, i.e., all these frames define the same orientation in U. For any e the set UE is orientable and its orientation can be defined by the vectors VH (everywhere, except the point p). Therefore, the 3-frames tangent to U£ also define the same orientation. The lemma is proved. Thus, we have described the structure of the vector field Υ in U. We now consider some properties of Υ in V\U. In Ve the points of (ΣιυΣ 2 )\ί7 form four disjoint segments of CSTs of the field Υ. It follows from Proposition 1.2 that there are stable and unstable manifolds of these CSTs. The following assertion holds for these manifolds. Lemma 4.6. In Ve\U the stable and unstable manifolds of the curves of singular points do not intersect (i.e., all the heteroclinic trajectories lie in U). Proof. The lemma follows easily from the monotonicity of the function Κ on a trajectory of the field Υ and from the theorem on the smooth dependence of the stable and unstable manifolds of CSTs in V\U. 6. THE TOPOLOGY ΟΈ dVE

In this section we shall describe the boundary d Ve of the manifold with boundary Ve, and, in particular, we determine the number of boundary tori. This boundary is the joint level Η = ε and Κ = ±k,, and since Κ = ±k» does not intersect Σι υ Σ2 , the boundary consists of some number of Liouville tori, and for all e, |ε| < A*, the number of boundary tori is the same. Hence, it suffices to determine the number of boundary tori on dV0. Obviously, dV0 = dV+ υ dV0~ , where Κ = k, on dV0+, where Κ = kt on dV+ and Κ = -k, on dV~ . We shall formulate the main properties of the trajectories of Υ on VQ , following from the fact that the only limit sets of trajectories of the vector field Υ in V are singular points filling up Σ ( υ Σ 2 . Proposition 4.2. 1) On VQ the set of singular points of the field Υ coincides with Εχ U E2 (twofigure-eightsformed by the loops γι, ... , y 4 ). 2) All the trajectories of WS(EX) U WS(E2) not belonging to Ex UE2 intersect d Vo+ , and the trajectories from WU(E\) U W"(E2) intersect dVQ~ . s S 3) The mapping along trajectories of the field Υ from dV+\(W (Ey) U W (E2)) onto W\(EX U E2) is a diffeomorphism. Analogously, the mapping along trajectories from W\(EX\JE2) onto dV-\(W"(E{)OW«(E2)) is also a diffeomorphism. s s u u 4) The sets dV+\(W (Ex)\jW (E2)) and dV^\(W (Ex)OW (E2)) consist of four disjoint open disks. Proof. Assertions l)-3) follow directly from the structure of the vector field Υ described in §3, and assertion 4) follows from 3) and the topology of the set W. We shall describe the structure of the traces of the stable and unstable manifolds of the figure-eights E\, E2 on d V^, d Vo~ , respectively. The stable manifold of the figure-eight £, is obtained by gluing two strips along the global part of the loop on Σ, (see Figure 3) to its local part in U. The gluing is defined by the following construction. For definiteness we consider the loop yi . On the level Η = 0 we choose two-dimensional local disks 0s and D" in the neighborhood U, which are transversal, respectively, to the terminal and initial segments of the loop. The traces of the local stable and saddle manifolds of the singular point ρ of the vector field XH are cut out on Ds, while the traces of the local unstable and the other saddle

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manifolds of ρ (see §2) are cut out on Du . These traces intersect transversally on their respective disks. Going along trajectories of XH over the global part of the loop, taking account of its orientability type, defines a diffeomorphism of gluing D" with D" described in §2. Obviously, the traces of the stable and unstable manifolds of the CST γι of the field Υ intersect Ds and Du transversally. From the structure of Υ in a neighborhood of a saddle CST (Proposition 1.2) and the monotonicity of Κ along the trajectories of Υ it follows that the traces of the manifolds of XH and the manifolds of Υ on Ds, Du form two "crosses" with a common vertex, rotated relative to each other by an acute angle. Therefore, the extension of the stable and unstable manifolds of a CST along the global part of γι is uniquely determined by the diffeomorphism of gluing Ds and D", since the manifolds of different fields intersect nowhere except for points on γι . Actual knowledge of the action of the diffeomorphism at all points is not required; it is only necessary to know the correspondence of the sectors into which the disks Ds and D" are partitioned by the traces of the manifolds of the vector field XH . This correspondence is completely determined by the orientability type of the loop. The boundaries of the manifolds are smooth curves lying on dV^ (for Ws(Et)) and dV^~ (for W" (is,·)), where the trace of IVs(Ει) transversally intersects the trace of WS(E2) at four points Q\ , ... , β | . Analogously, the traces of WU(E{) and W"(E2) intersect transversally at four points (2 Γ > • · · > QA • Omitting the elementary computations for each loop, depending on the case under consideration, we obtain the following proposition. Proposition 4.3. The traces of the stable and unstable manifolds of the figure-eights on dVJ and dV^ have the form shown in Figure 5 (next page). Analogous arguments prove the following assertion. Proposition 4.4. If a closed CST of the vector field Υ corresponds to an orientable (respectively nonorientable) SPT of the vector field XH, then the local stable and unstable manifolds of the CST are cylinders (respectively, Mobius bands). There is a similar assertion in [4]. Proposition 4.5. The number of tori on dV^ (respectively, dV^) is equal to the number of components of the set (WS(E\)O Ws(E2))ndVj (respectively, (Wu(Ei)U U W (E2)) Π dV^), and a unique component lies on each torus. +

Proof. Let η be the number of tori on d Fo , and m the number of connected S s components of the set (W (E\) U W (E2))f\dVj. Assume that n> m. Then there 5 s exists a torus Τ c dVj that does not intersect W (Ei)uW (E2), which contradicts Proposition 4.2, 4). Suppose that η < m. Then there exists a torus Γ c θΚ0+ containing at least two components. In this case the set T\(WS(E\) U WS(E2)) contains a component that is not simply connected, which contradicts Proposition 4.2, 4). This proves the proposition. Proposition 4.6. For fixed i the curves forming the trace of Ws(Ei) on dV£ (or of Wu(Ei) on dV^) and lying on the same torus are not homotopic to zero and are isotopic to each other. Two curves lying on the same torus, one of which belongs to the trace of WS(E{) and the other to that of WS(E2), are not isotopic; analogously for Wu(Ei) and WU(E2). Proof. This proposition can be proved by exhaustion of all the cases, using the topology of the torus, the intersection number of curves on the torus, part 4) of Proposition 4.2, and Proposition 4.3.

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0 there are four heteroclinic trajectories on Ws(Zi) Π Vt; all the remaining trajectories intersect dVe+ . Therefore, Ws(Ll)ndV+ is the union of four nonclosed smooth curves with endpoints Qf{e), i = 1, ... , 4. The trajectories of Ws(^2) Π Ve pass through these points. The trace of the leaf bi = 0 passing through Qf(e) is a segment, all of whose points, except Qf(e), belong to Ws(Li). Therefore, the closure of Η^(Σι) ndV+ is the union of a number of closed curves. Thus, A\ is the union over ε of smooth closed curves. The curves are formed from the segments of the local leaves bj = 0 near the points β+(ε) and global pieces that are the traces of WS{E\\U) on dVe+. It follows from Proposition 1.2 that the traces of the leaves of WS{EX\U) on dV+ depend smoothly on ε, and the local pieces depend smoothly on ε because of the smooth dependence of the 62-foliation in 9 F + on e. Hence, A\ is a smooth twodimensional submanifold with boundary, smoothly fibered into smooth closed curves. The assertion of the proposition follows from this. 7. PROOF OF THE EQUIVALENCE THEOREMS

In this section we prove the sufficiency of the conditions of Theorem 7. Let {XH , K), (Xjji, K') be two IHVFs with Morse Hamiltonians having saddle singular points ρ and p', respectively, and V, V their extended neighborhoods, σ = μ(Κ), σ' = μ'(Κ'), where μ, μ' are the moment maps. We assume that the standard representatives of the fields (XH, K) and (XH> , K') have been chosen. We construct a homeomorphism f\V—*V realizing an isoenergetic equivalence of vector fields. Let / = [-A*, A*], h» > 0, be the range of Η in V, and /' = [-Λ,, h'r], h't > 0, the corresponding interval for H'. Analogously, j = [_fc., k.], J' = [-K , K] are the ranges of Κ and K'. We construct an auxiliary homeomorphism q: σ —» σ'. For this we first define an arbitrary homeomorphism q\: / —• / ' , #i(0) = 0, q\{h*) = A». Now we define a function ^(A, k) with the following properties: 1) qi{h, ±/e») = ±k't; 2)