On semi-global invariants for focus focus singularities

Transcription

1 Topology 4 (003) On semi-global invariants for focus focus singularities Vũ Ngȯc San Institut Fourier, Unite mixte de recherche CNRS-UJF 558, BP 74, 3840-Saint Martin d Heres Cedex, France Received March 00; received in revised form 4 June 00; accepted June 00 Abstract This article gives a classication, up to symplectic equivalence, of singular Lagrangian foliations given by a completely integrable system of a four-dimensional symplectic manifold, in a full neighbourhood of a singular leaf of focus focus type.? 00 Elsevier Science Ltd. All rights reserved. MSC: 37J35; 37J5; 70H06; 37G0; 53D; 53D; 70H5 Keywords: Symplectic geometry; Lagrangian bration; Completely integrable systems; Morse singularity; Normal forms. Introduction In the study of completely integrable Hamiltonian systems, and more generally for any dynamical system, nding normal forms is often the easiest way of understanding the behaviour of the trajectories. Normal forms generally deal with a local issue. However, the locality here depends on one s viewpoint: one can be local near a point, an orbit, or any invariant submanifold. If F =(H ;:::;H n ) is a completely integrable system on a n-symplectic manifold M (meaning that {H j ;H i } = 0), several normal forms hold: Near a point m where dh j (m); j=;:::;n are linearly independent, one can construct Darboux Caratheodory coordinates: a neighbourhood of m is symplectomorphic to a neighbourhood of the origin in R n with its canonical coordinates (x; ), in such a way that H j H j (m)= j. Tel.: ; fax: address: san.vu-ngoc@ujf-grenoble.fr (S. Vũ Ngȯc) /03/$ - see front matter? 00 Elsevier Science Ltd. All rights reserved. PII: S (0)0006-X

2 366 S. Vũ Ngȯc / Topology 4 (003) If c is a regular value of F, one has near any compact connected component c of F (c) the Liouville Arnold theorem which states that the system is symplectomorphic to a neighbourhood of the zero section of T (T n ) in such a way that there is a change of coordinates in R n such that F =( ;:::; n ). Here, T n is the torus R n =Z n and the cotangent bundle T (T n ) is equipped with canonical coordinates (x; ). The rst one is typically a local normal form, while I would refer to the Liouville Arnold theorem as a semi-global result, for it classies a neighbourhood of a whole invariant Lagrangian leaf c. These two statements above are now fairly standard. They can be extended in dierent directions: (a) trying to globalise: what can be said at the level of the whole - bration of regular bres c? This of course involves more topological invariants, as described in Duistermaat s paper [4] and (b) including critical points, which is the main incentive for this article. A Morse Bott like theoretical study of critical point of completely integrable Hamiltonian systems exists, which yields a local symplectic classication of non-degenerate singularities (see [5]). These results have been used by Nguyên Tiên Zung [8] (extending previous results by Fomenko) to obtain a topological semi-global classication of the singular foliation. This work does not give the corresponding smooth symplectic classication, where new semi-global invariants show up, as demonstrated in the one degree of freedom (-D, i.e. n = ) case by Dufour et al. [3]. The point of our present article is to extend the results of [3] to the -D case of focus focus singularities. Note that our arguments could easily be applied in the -D case, thus supplying for the lack of proofs in [3]. Between the pure topological classication of the singular foliation and the exact symplectic classication, some other interesting notions of equivalence have been introduced (see e.g. []), which are all weaker than what we shall present here. The semi-global viewpoint seems to be able to shed some new light in semi-classical mechanics, where a quantum state is associated to a Lagrangian submanifold. Quantum states associated to singular manifolds have a particularly rich structure, strongly linked to the local (for this, see []) and semi-global symplectic invariants of the foliation. We expect to return on this in a future paper.. Statement of the result In this article, (M;!) is a four-dimensional symplectic manifold, equipped with the symplectic Poisson s bracket { ; }. Any smooth function H on M gives rise to a Hamiltonian vector eld denoted by X H. The word smooth always means of C category and a function f is said at at a point m if f and all its derivatives vanish at m. Denition.. A map F =(H ;H ) dened on some open subset U of M with values in R is called a momentum map if df is surjective almost everywhere in U and {H ;H } =0.

3 S. Vũ Ngȯc / Topology 4 (003) Denition.. A singular Liouville foliation F is a disjoint union of connected subsets of M called leaves for which there exists a momentum map F dened in some neighbourhood of F such that the leaves of F are the connected components of the level sets F (c), for c in some open subset of R. The total space of the foliation is also denoted by F. The above denition implies that F is an open subset of M. Denition.3. Let m F. The maximum of the set {rank(df(m));f dening F} is called the rank of m. m is called regular if its rank is maximal (= ). Otherwise it is called singular. If m is a regular point, then there is an open neighbourhood of m in which all points are regular, and if F F are associated momentum maps near m, one has F = F, for some local dieomorphism of R (these facts come from the local submersion theorem). Note that the condition {H ;H } = 0 implies that the leaves are local Lagrangian manifolds near any regular point. However, the foliation near a regular leaf (= a leaf without any singular point) is not the most general Lagrangian foliation (which would be dened as a foliation admitting locally associated momentum maps), since the latter does not necessarily admit a global momentum map (see []). In what follows, the word Liouville is often omitted. If m F, we denote by F m the leaf containing m. Denition.4. A singular Liouville foliation F is called of simple focus focus type whenever the following conditions are satised:. F has a unique singular point m;. the singularity at m is of focus focus type; 3. the leaf F m is compact. The leaf F m is called the focus focus leaf. Recall that the second condition means that there exists a momentum map F =(H ;H ) for the foliation at m such that the Hessians of H and H span a subalgebra of quadratic forms that admits, in some symplectic coordinates (x; y; ; ), the following basis: q = x + y; q = x y: This implies that focus focus points are isolated, which ensures that the above denition is non-void. Note that focus focus singularities are one of the four types of singularities of Morse Bott type in dimension 4, in the sense of Eliasson [6]. Denition.5. Two singular foliations F and F in the symplectic manifolds (M;!) and ( M;!) are equivalent if there exists a symplectomorphism : F F that sends leaves to leaves. ()

4 368 S. Vũ Ngȯc / Topology 4 (003) Denition.6. Let F and F be singular foliations in M, and m F F such that F m = F m. The germs of F and F at F m are equal if and only if there exists a saturated neighbourhood of F m in F such that F = F. The classication of germs of Liouville foliations near a compact regular leaf is given by the Liouville Arnold theorem that asserts that they are all equivalent to the horizontal bration by tori of T T n. The presence of singularities imposes more rigidity, and we have the following theorem (which is natural in view of [3]): Theorem.. The set of equivalence classes of germs of singular Liouville foliations of focus focus type at the focus focus leaf is in natural bijection with R<X; Y = 0 ; where R<X; Y = is the algebra of real formal power series in two variables; and R<X; Y = 0 is the subspace of such series with vanishing constant term. This formal statement does not contain the most interesting part of the result, which is the geometric description of the power series involved (it is essentially the Taylor series of a regularisation of some action integral). The rest of the paper is devoted to this description which is the sense of the theorem, and to the proof of the sense, for which we provide a normal form corresponding to any given power series in R<X; Y = 0. The articles ends up with a sketchy argument as to how the result can be extended to handle the case of several focus focus points in the singular leaf. 3. The regularised action Let F be a singular foliation of simple focus focus type. Then in some neighbourhood U of the focus focus point m, the following linearisation result holds [5]: there exist symplectic coordinates in U in which the map (q ;q ) (dened in ()) is a momentum map for the foliation. Notice therefore that, contrary to what the picture of Fig. may suggest, F m is dieomorphic near m to the union of two two-dimensional planes transversally intersecting at m. Let A bea point in F m U \{m}, and be a small two-dimensional surface transversal to the foliation at A, and be the open neighbourhood of F m consisting of leaves intersecting. In what follows, we restrict the foliation to. Let F be a momentum map for the whole foliation F satisfying the hypothesis of Denition.4. In a neighbourhood of ; F and q =(q ;q ) are regular local momentum maps, hence q = F, for some local dieomorphism of R. Now let F = F. It is a global momentum map for F that extends q. We denote F =(H ;H ) and c = F (c). Near m, the Hamiltonian ow of q is -periodic, and assuming U to be invariant with respect to this ow the associated S -action is free in U \{m}. Since this action commutes with the ow of H, the H -orbits must be periodic of primitive period for any point in a (non-trivial) trajectory of X H. On the leaf F m = 0, these trajectories are homoclinic orbits for the point m, which implies that the ow of H generates an S -action on a whole neighbourhood of F m (see [0] for details).

5 S. Vũ Ngȯc / Topology 4 (003) Fig.. Construction of the periods j (c). For any point A c ;caregular value of F, let (c) 0 be the time of rst return for the X H -ow to the X H -orbit through A, and (c) R=Z the time it takes to close up this trajectory under the ow of X H (see Fig. ). These times are independent of the initial point A on c. For any regular value c of F, the set of points (a; b) R such that ax H + bx H has a -periodic ow on c is a sublattice of R called the period lattice [4]. The vector elds X H + X H and X H both dene -periodic ows, hence ( ; ) and (0; ) form a Z-basis of the period lattice (see Remark 3.3). As we shall see, the classication we are looking for relies on the behaviour of this basis as c tends to 0. One immediate fact is that the cycle associated to X H shrinks to a point (vanishing cycle). On the other hand, the coecients of the rst vector eld display a logarithmic divergence, as stated in the following proposition. Proposition 3.. Let ln c be the some determination of the complex logarithm; where c =(c ;c ) is identied with c +ic. Then the following quantities (c)= (c)+r(ln c); (c)= (c) I(ln c) extend to smooth and single-valued functions in a neighbourhood of 0. The dierential -form := dc + dc is closed. Proof. As before, let U be the neighbourhood of m found using Eliasson s result, with canonical coordinates (x; y; ; ). In U, we use the complex coordinates z =(z ;z ) with z = x +iy and z = +i, so that q (z)+iq (z)=z z. The ow of q is (z (t);z (t))=(e t z (0)); e t z (0); () while the ow of q is the S -action given by (z (t);z (t))=e it (z (0);z (0)): (3)

6 370 S. Vũ Ngȯc / Topology 4 (003) Fix some small 0. Then, the local submanifolds u = {z = ; z small} and s = { z small;z = ; } are transversal to the foliation c = {(z ;z ); z z = c}; therefore, the intersections A(c):= u c and B(c):= s c are smooth families of points. The S -orbits of u=s form two small hypersurfaces transversal to the ow of q ; therefore one can uniquely dene A;B (c) as the time of rst hit on s for the X H -ow starting at A(c) (and hence owing outside of U), and A;B (c) as the time it takes to - nally reach B(c) under the X H -ow. A;B (c) and A;B (c) are smooth functions of c in a neighbourhood of 0. Interchanging the roles of A and B and thus of u and s, the times B;A j (c) for j =; are dened in the same way. However, since the corresponding ows now take place inside U, where a singular point occur, B;A j (c) is not dened for c = 0. On the other hand, Eqs. () and (3) yield the following explicit formula: B;A (c)+i B;A (c)=ln z (A) z (B) =lnz (A)z (B) ln c =ln ln c: (4) Writing now (c)+i (c)=( A;B (c)+ B;A (c))+i( A;B (c)+ B;A (c)) using (4), and the fact that ln c =ln c iarg c, we obtain that (c)+i (c)= A;B (c)+i A;B (c)+ln ; which proves the rst statement of the proposition. Let us show now that for regular values of c the -form (c)dc + (c)dc is closed. For this we x a regular value c 0 and introduce the following action integral, for c in a small ball of regular values around c 0 : A(c):= ; (5) c where is any -form on some neighbourhood of c in M such that d =! (which always exists since c is Lagrangian), and c c is a smooth family of loops on the torus c with the same homology class as the trajectory of the joint ow of (H ;H ) during the time ( (c); (c)). A simple argument (see for instance [0, Lemma 3:6]) shows j = c j, where j is the closed -form on c dened by XHi = i;j. In other words, the integral of j along a trajectory of the ow of H j measures the increase of the time t j along this trajectory. This means that da(c)= (c)dc + (c)dc (6) and thus proves the closedness of the right-hand side. Another way of proving this fact would be to apply the Liouville Arnold theorem, which ensures that any -form adc + bdc, where a, b depend smoothly on c near a regular value, such that (a; b) is in the period lattice, is closed (see Remark 3.3). Adding the fact that ln (c)dc is closed as a holomorphic -form, we obtain the closedness of at any regular value of c, and hence at c = 0 as well.

7 S. Vũ Ngȯc / Topology 4 (003) Remark 3.. From this proposition, one easily recovers the result of [9] stating that the monodromy of the Lagrangian bration around a focus focus bre is generated by the matrix ). Notice that the function c (c) is dened modulo the addition of a xed constant in Z. We shall from now on assume that (0) [0; [. This amounts to choosing the determination of the complex logarithm in accordance with the determination of. Denition 3.. Let S be the unique smooth function dened in some neighbourhood of 0 R such that ds = and S(0) = 0. The Taylor expansion of S at c = 0 is called the symplectic invariant of Theorem.. It is denoted by (S). Remark 3.. Using Eq. (6), one can interpret S as a regularised action integral: S(c)=A(c) A(0) + R(c ln c c): Remark 3.3. Formula (6) denes the -form = dc + dc independently of the choice of the coordinate system (c ;c ). Another (standard) way of viewing this is the following. First let B be the set of regular leaves of F, and be the projection (which is a Lagrangian bration) F B. The choice of a particular semi-global momentum map F:=(H ;H ) for the system (near a Lagrangian leaf c := (c), for some c B) is equivalent to the choice of a local chart for B near c: F =. Then for each c B, Tc B acts naturally on c by the time- ows of the vector elds symplectically dual to the pull backs by of the -forms in Tc B. This action extends to a Hamiltonian action in a neighbourhood of c if and only if we restrict to closed -forms on B. (In the local coordinates (c ;c )ofb given by the choice of a momentum map F =(H ;H ), the constant -forms dc, dc act by the ows of X H, X H, respectively.) The stabiliser of this action form a particularly interesting lattice in Tc B, which is another representation of the period lattice [4]. It is the main point of the Liouville Arnold theorem to show that, as c varies, the points of this lattice are associated to closed -forms, called period -forms. (Indeed, in action-angle coordinates, the period -forms have constant coecients.) In our case, the period lattice is computed using a local chart given by Eliasson s theorem. First, we see that this lattice has a privileged direction given by the S -action of q. Then, we construct a minimal basis of this lattice by choosing the generator of this S -action (i.e. dc ) together with the smallest transversal vector that has positive coecients on dc and dc. This is what we have done in this section. ( 0 4. Uniqueness In order to show that the above invariant (S) is indeed symplectic and uniquely dened by the foliation, we need to prove that it does not depend on any choice made to dene them. A priori, (S) =(S) (F;) depends on the foliation F and on the choice of the chart that puts a neighbourhood of the focus focus point m into normal form. It follows from the denition

8 37 S. Vũ Ngȯc / Topology 4 (003) that if is a symplectomorphism sending F to F, then (S) ( F; )=(S) (F; ). So (S) is well dened as a symplectic invariant of F if and only if, for any choice of two chart and putting a neighbourhood of m into normal form, (S) (F;)=(S) (F; ). This is guaranteed by the following lemma: Lemma 4.. If is a local symplectomorphism of (R 4 ; 0) preserving the standard focus focus foliation {q:=(q ;q )=const} near the origin; then there exists a unique germ of dieomorphism G : R R such that q = G q and G is of the form G =(G ;G ); where G (c ;c )= c and G (c ;c ) c is at at the origin; with j = ±. Remark 4.. This uniqueness statement about Eliasson s normal form does not appear in [5]. Proof of Lemma 4.. The existence of some unique G satisfying (7) is standard (because the leaves of the focus focus foliation are locally connected around the origin). What interests us here are the last properties. As before, we use the complex coordinates (z ;z ) C = R 4, and c =z z C = R. Let 0 be such that is dened in the box B = { z 6 ; z 6 }. Since the ow of q is -periodic, (7) implies that the Hamiltonian vector G X q G X q is also -periodic (with as a primitive period). However, on 0 the only linear combinations of X q and X q that are periodic are the integer multiples of X q. G (0) = 0 G (0) = ±. The ow of q on 0 is radial: any line segment ]0;A[ for some A 0 is a trajectory. Then, by (7) its image by must be a trajectory of G q. Since is smooth at the origin, the image of ]0;A[ for A B close enough to 0 lies in some proper sector of the plane 0 containing (A) ( is either {z =0} or {z =0}). However, the only linear combinations of X q and X q which yield trajectories that are conned in a proper sector of are the multiples of X q. G (0) = 0. It follows now from the previous paragraph G (0) 0 (since G is a local dieomorphism). preserves the critical set of q; since left composition of by the symplectomorphism (z ;z ) ( z ;z ) leaves (7) unchanged (except for the sign of G ), we may assume that each axis ({z =0} and {z =0}, respectively) is preserved by. But then {z =0} is the local unstable manifold for both q and G (q ;q ), which says G (0) 0. Using () and (3), it is immediate to check that the joint ow of (q ;q ) taken at the joint time ( ln c= ; arg c) sends the point ( c; ) to the point (; c), and hence extends to a smooth and single-valued map from a neighbourhood of (0;) to a neighbourhood of (; 0). sends a neighbourhood of (0;)=(0;a) to a neighbourhood of (; 0) = (b; 0) and, because of (7), it is equal in the complement of the singular leaf 0 to the joint ow of G q at the joint time ( ln c= ; arg c), which is equal to the joint ow of q at the joint time G ln c= G arg G ln c= G arg c): (7)

9 S. Vũ Ngȯc / Topology 4 (003) Since is smooth at the origin, we obtain by restricting the rst component of this map to the Poincare surface {(c; a) with c near 0 in C} that the map: c G )ln c G arg c +i((@ G ) arg G ln c )) (8) is single-valued and smooth at the origin. (We have factored out the terms exp(@ j G ln ), j =;, which are obviously smooth.) The single-valuedness of (8) implies G 0 G Z. G = ±. Now the smoothness of (8) says that the following two functions c G )ln c and G ln c are smooth at the origin, which easily implies that G ) G are at at the origin, yielding the result. Suppose we dene two semi-global invariants (S) (F;) and (S) (F; ) by choosing two dierent charts and which put a neighbourhood of the focus focus point into normal form. As before, one denes the momentum maps F and F, which are the extensions to F of q and q, and computes the corresponding period -forms and. Then, we can invoke the lemma to =, and the conclusions apply to G = FF. Suppose that j =, j =;, i.e. G is innitely tangent to the identity. Then, the same type of arguments as above (a logarithm cannot compete against a at term) shows that, since the vector elds X Hj and XH j are innitely tangent to each other, and must dier by a at term. Actually, since by Remark 3.3 G is also a period -form associated with the momentum map F, one has = G. This implies that (c)=(c)+r(ln cdc) and =(G ) dier by a at form, hence (S) (F;)=(S) (F; ). If =, it suces to compose with the symplectomorphism (x; ) ( x; ), which sends (q ;q )to(q ; q ) and leaves invariant (both and dc change sign). An analogous remark holds with the symplectomorphism (z ;z ) ( z ;z ), which sends (q ;q )to( q ;q ) and leaves invariant, while changing the sign of. 5. Injectivity Let F and F are two singular foliations of simple focus focus type on the symplectic manifolds (M;!) and ( M;!). Assume that they have the same invariant (S) (F)=(S) ( F) R<X; Y = 0. We shall prove here that F and F are semi-globally equivalent, i.e. there exists a foliation preserving symplectomorphism between some neighbourhoods of the focus focus leaves. For each of the foliations F and F, we choose a chart of Eliasson s type around the focus focus point, and thus dene the period -forms and on (R \{0}; 0). The hypothesis implies that there is a smooth closed -form = dc + dc on (R ; 0) whose coecients are at functions of c at the origin such that = + : Lemma 5.. One can chose symplectic charts of Eliasson s type at the focus focus points in such a way that =0; i.e. = :

10 374 S. Vũ Ngȯc / Topology 4 (003) Proof. () We rst prove that there exists a local dieomorphism G of (R ; 0) isotopic to the identity such that (G ) =. We wish to realise G as G where G t is a ow satisfying Gt ( + t)=: This amounts to nding the associated vector eld Y t which must satisfy d( Yt ( + t)) = : We can write = dp for some smooth function P which is at at 0. Assume we look for a eld Y t of the form Y t = f t (c)@=@c. We obtain the following equation: f t (c)= P(c) P(c) = : (c)+t ln c (c)+t Since P is at at 0, the right-hand-side is indeed a (at) smooth function depending smoothly on t, and the result is proved. () Notice also that G is innitely tangent to the identity, and moreover leaves the second variable c unchanged. Now we show that for any dieomorphism G of (R ; 0) sharing these properties (which are those of Lemma 4.) there exists a symplectomorphism near the focus focus point m such that G(q ;q ) =(q ;q ): Here, again we seek as the time- map of the ow of some vector eld X t. Of course, we shall look now for a Hamiltonian vector eld X t = X ft to ensure the symplecticity of t. Then, the requirement t q t = q 0 ; where q t =(q t; ;q t; ) def = tg(q ;q )+( t)(q ;q ), leads to the following system: {f t ;q t; } = g ; {f t ;q t; } =0 with (g ; 0)=(q ;q ) G(q ;q ). By hypothesis g is a at function at the origin, and the fact that {q t; ;q t; } 0 implies that {g ;q t; } = 0. Moreover, the quadratic part of q t is q 0,sowe know (see [5]) that such a system admits a solution f t. It remains to put all our remarks together: Point () shows that left composition by of the Eliasson chart we have chosen at m is again an admissible chart of Eliasson s type, yielding the new momentum map G(q ;q ). Using the G obtained at Point () and in view of the naturality property (Remark 3.3), the new period -form (denoted by again) satises =. We are now in position to construct the required equivalence. Applying the lemma we get a local symplectomorphism that allows us to identify some neighbourhoods U and Ũ of the focus focus points m and m, and two momentum maps F and F (both equal to (q ;q ) inside their respective neighbourhoods of the focus focus points) which dene the same closed -form on (R ; 0). We denote c = F (c) and c = F (c). Let U be an open ball strictly contained in U, let u U be a transversal section as dened in the proof of Proposition 3., and construct in the same way u for the foliation F (so

11 S. Vũ Ngȯc / Topology 4 (003) that u and u are identied by the above symplectomorphism). Reduce F (and F) tothe neighbourhoods of the focus focus leaves composed of the leaves intersecting u (or u ). We construct our equivalence by extending the identity outside U. Let x c \ U, and dene t(x) ]0; (c)[ to be the smallest time it takes for the point u c to reach the X H -orbit of x. (Recall that H generates an S action.) Now dene s(x) R=Z as the remaining time to nally reach x under the X H -ow. To this x we associate the point x F obtained from the point u c by letting the joint ow of F act during the times (t(x);s(x)). This map let us call it is well dened because of the equality =. It is a bijection since the inverse is equally well dened just by interchanging the roles of F and F. Between U and Ũ, is a symplectomorphism since through Eliasson s charts, it is just the identity. Concerning now the symplecticity of in the complement of the singular points, one can prove it for c 0 (which is sucient by continuity) by invoking the Liouville Arnold theorem, which shows that is symplectically conjugate to a translation in the bres. Then, the symplectic property near the singular points implies that this translation must be symplectic everywhere. A similar argument using the less sophisticated Darboux Caratheodory theorem would also do. However, the simplest is may be the following. It is clear from the construction that is equivariant with respect to the joint ows of our Hamiltonian dynamics: (t ;t ); t;t = t;t ; (9) where t;t and t;t are the joint ows of F and F at the joint time (t ;t ). Using (9) together with the fact that t;t is symplectic, we see that t ;t (!)=!; in other words,! is invariant under the joint ow t;t. Since! has the same property, so has!!. Since!! = 0 near m, it must vanish as well on the whole F. 6. Surjectivity We prove here that any formal power series (S) R<X; Y = 0 is the symplectic invariant in the sense of Denition 3. of some Liouville foliation of simple focus focus type. More precisely, we construct a foliation F together with a local chart that puts a neighbourhood of the focus focus point into normal form such that (using the notation of Section 4) (S) =(S) (F;). Another proof of this result has been proposed by Castano Bernard []. Using the same notations as before, we let (q ;q )=z z be the standard focus focus bration R 4 C C R dened in (). The joint ow will be denoted by t;t. Invoking Borel s construction, let S C (R ) be a function vanishing at the origin and whose Taylor series is (S). We shall denote by S, S the partial X S Y S, respectively. Let us dene two Poincare surfaces in C by means of the following embeddings of the ball D = B(0;) C, for some ]0; [: (c)=(c; ); (c)=(e S(c)+iS(c) ;ce S(c)+iS(c) ): Notice that for each c, the points j (c), j =; belong to the (non-compact) Lagrangian submanifold c :={z z = c}. j (D ), j =; are smooth two-dimensional manifolds constructed in

12 376 S. Vũ Ngȯc / Topology 4 (003) such a way that for any c 0, (c) is the image of (c) by the joint ow of (q ;q )atthe time (S (c) ln c ;S (c) + arg(c)). Let be this dieomorphism, dened on all j (D ) by the embeddings: (D ) and (D ) are transversal to the Lagrangian foliation, and can be extended uniquely to a dieomorphism between small neighbourhoods of (D ) and (D ) requiring that it commute with the joint ow: ( t;t (m)) = t;t ((m)): (0) Lemma 6.. is a symplectomorphism. Proof. One can write in terms of and and check the result by explicit calculation. However, the reason why it works is the following: Since we already know that is smooth, it is enough to prove the lemma outside of the singular Lagrangian 0.Soxc 0 0; we can construct a Darboux Caratheodory chart (x; ) R 4 in a connected open subset of c0 containing both (c 0 ) and (c 0 ). In these coordinates, the momentum map is ( ; ) and the ow is linear: t;t is the translation by (t ;t )inthex variables. Through this chart, is by construction a bre translation : where (x; )=(x + f();); f()=(s ();S ()) + (ln ; arg()): () Now, it is easy to check that () denes a symplectomorphism if and only if the -form f ()d + f ()d is closed. In our case the closedness is automatic since S dx + S dy = ds. Let j, j =; bethes -orbit of j (D ). Construct a four-dimensional cylinder C by letting the q -ow take to, namely C:= C c ; c D \{0} where C c c is the two-dimensional cylinder spanned by t;t ( (c)), for (t ;t ) [0;S (c)+ ln c ] [0; ]. Finally, let M be the symplectic manifold obtained by gluing the two ends j ()

13 S. Vũ Ngȯc / Topology 4 (003) Fig.. Construction of the symplectic manifold M. of the cylinder C using the symplectomorphism (Fig. ). Since preserves the momentum map (q ;q ), the latter yields a valid momentum map F on M. The corresponding Lagrangian foliation F (c) is given by C c with its two ends identied by. In particular, all leaves are compact and the foliation is of simple focus focus type. The S action is unchanged, while the transversal period ( (c); (c)) on F (c) is b y construction the time it takes for the joint ow to reach (c) from (c), i.e. ( (c); (c))=(s (c) ln c ;S (c) + arg(c)): Then by Denition 3. the symplectic invariant of the foliation is given by the Taylor expansion of the primitive of the -form S dc + S dc vanishing at 0, i.e. (S). 7. Further remarks Multiple focus focus. Assume now that the singular bre 0 carries k focus focus points m 0 ;:::;m k. Then 0 is a k-times pinched torus, and Theorem. can be generalised. In this case, the regularisation of the action integral S must take into account all the singular points. In order to do this, one has to consider k local invariants, which are also formal power series in R<X; Y =, and which measure the obstruction to construct a semi-global momentum map that is in Eliasson normal form simultaneously at two dierent singular points. Here follows a sketch of the argument. Let F be a semi-global momentum map. At each point m j one has a local normal form F j = G j (q ;q ). Due to Lemma 4., one can extend q to a periodic Hamiltonian on a whole neighbourhood of 0, and one can always assume that j is orientation preserving this means we x once and for all the sign of the j.ifnowf if of the form (H ;q ), then G j takes the form G j (q ;q )=(F j (q ;q );q ). By the implicit function theorem, F j is locally invertible with respect to the variable q. Let (F j ) be this inverse, and dene G i;j =(F i ) F j. Again by Lemma 4., the Taylor expansions of G i;j are invariants of the foliation.

14 378 S. Vũ Ngȯc / Topology 4 (003) Fig. 3. The multi-pinched torus. Assume the points m i are ordered according to the ow of H, with indices i Z=kZ. Similarly to the case k =, one can dene a regularised period -form by the following formula: with k := i=0 (G 0 G i) ( i;i+ (c)dc + i;i+ (c)dc ) (3) i;i+ (c)= i;i+ (c)+r(ln c); i;i+ (c)= i;i+ (c) I(ln c); (4) where ( i;i+ (c); i;i+ (c)) are the smallest positive times needed to reach A i+ (c) from A i (c) under the ow of (G i ) F which is the momentum map (q ;q ) in the normal form coordinates near point A i (Fig. 3). Here, we have chosen a point A i (c) in a Poincare section of each local stable manifold near m i. Of course, i;i+ j (c) depends heavily on the choice of A i and A i+, but the sums appearing in (3) does not, and the resulting -form is closed. Notice that the denition of depends on the choice of a start point m 0. Thus, we are here classifying a singular foliation with a distinguished focus focus point m 0. Let (S) be the Taylor series of the primitive of vanishing at the origin. Then, (S) and the k ordered invariants (G i;i+ ) are independent and entirely classify a neighbourhood of the critical bre 0 with distinguished point m 0. The arguments of the proof are similar to the ones of the case k =. An abstract construction of a foliation admitting a given set of invariants is proposed in Fig. 4. There the local pictures are described by canonical coordinates, respectively, given by (q ;q ), (G ; (q ;q );q ), (G ; (G ;3 (q ;q ); q );q ), etc., and the gluing dieomorphisms i;i+ are constructed as in Section 6 using the following functions, respectively: S 0; = S ; = = S k ;k = 0 and S 0;k is a resummation of (S).

15 S. Vũ Ngȯc / Topology 4 (003) Fig. 4. Multiple gluing. Remark 7.. We can regard the reduced space 0 =S as a cyclic graph G whose vertices are the focus focus points m i, and which is oriented by the ow of H. For each edge [i; i + ] one can dene a -form i;i+ :=(G 0 G i) ( i;i+ dc + i;i+ dc ) (D) (for some xed small disc D around the origin in R ). This denes a -cocycle on G with values in the vector space (D). If one varies the points A j, this cocycle is easily seen to change by a coboundary; hence the set of { i;i+ } naturally denes a well-dened cohomology class on G. With the same argument as in the case k = (i.e. essentially Arnold Liouville s theorem) this class is closed, in the sense that the cochain { i;i+ }, modulo some coboundary, can be chosen to consist only of closed -forms. Hence, we end up with a class [] H (G;H (D)). Since G is homeomorphic to a circle, H (G;H (D)) H (D) and [] is represented by the de Rham cohomology class of the closed -form = i;i+ dened in (3). Now, the functor that produces Taylor series of -forms can be applied to the coecients of this cochain, yielding a cocycle with values in formal closed -forms and whose class is represented by the dierential of our invariant (S). Exact version. If one intends to extend the results to a semi-classical setting, general symplectomorphism do not suce: one needs to control the action integrals (in the standard semi-classical pseudo-dierential theory, a potential for the symplectic form: d =! is part of the data). In view of Remark 3., this is naturally done by including the constant term in the Taylor series of S as being the integral S 0 := ; 0 where 0 is the generator of H ( 0 ). Acknowledgements This article answers a question that J.J. Duistermaat asked me when I was in Utrecht in 998. I wrote then a short and incomplete draft, and that was it. Two years after I nally read the note by Toulet et al. [3], and a fruitful discussion with Richard Cushman made me realise that I had the result at hand. I wish to thank him for this. I would also like to thank

16 380 S. Vũ Ngȯc / Topology 4 (003) Ricardo Castano-Bernard for interesting discussions, and for showing me an alternative proof of the surjectivity part using general arguments developed for mirror symmetry via special Lagrangian brations. After I wrote this article, P. Molino informed me of an unpublished work of his (in collaboration with one of his students [7]) concerning the same problem. They dened a similar invariant, and the classication result was conjectured. References [] A.V. Bolsinov, A.T. Fomenko, Orbital equivalence of integrable hamiltonian systems with two degrees of freedom, a classication theorem I. Russian Acad. Sci. Sb. Math. 8() (995) (translated from the Russian). [] R. Castano-Bernard, 000. personal communication. [3] J.-P. Dufour, P. Molino, A. Toulet, Classication des systemes integrables en dimension et invariants des modeles de Fomenko, C. R. Acad. Sci. Paris Ser. I Math. 38 (994) [4] J.J. Duistermaat, On global action-angle variables, Comm. Pure Appl. Math. 33 (980) [5] L.H. Eliasson, Hamiltonian systems with Poisson commuting integrals, Ph.D. Thesis, University of Stockholm, 984. [6] L.H. Eliasson, Normal forms for hamiltonian systems with Poisson commuting integrals elliptic case, Comment. Math. Helv. 65 (990) [7] D. Grossi, Systemes integrables non degeneres invariants symplectiques d une singularite focus focus simple, Memoire de DEA, universite de Montpellier, 995. [8] Z. Nguyên Tiên, A topological classication of integrable hamiltonian systems, in: R. Brouzet (Ed.), Seminaire Gaston Darboux de geometrie et topologie dierentielle, Universite Montpellier II, , pp [9] Z. Nguyên Tiên, A note on focus focus singularities, Dierential Geom. Appl. 7 () (997) [0] S. Vũ Ngȯc, Bohr Sommerfeld conditions for integrable systems with critical manifolds of focus focus type, Comm. Pure Appl. Math. 53 () (000) [] S. Vũ Ngȯc, Formes normales semi-classiques des systemes completement integrables au voisinage d un point critique de l application moment, Asymptotic Anal. 4 (3,4) (000) [] A. Weinstein, Symplectic manifolds and their lagrangian submanifolds, Adv. Math. 6 (97)