singularities of integrable hamiltonian systems - CiteSeerX

SINGULARITIES OF INTEGRABLE HAMILTONIAN SYSTEMS ALEXEY V. BOLSINOV [email protected] Department of Mechanics and Mathematics, Moscow State University Moscow 119992 Russia

ANDREY A. OSHEMKOV [email protected] Department of Mechanics and Mathematics, Moscow State University Moscow 119992 Russia

The main subject of the paper is the classification problem for nondegenerate singularities of integrable Hamiltonian systems. We survey known results, discuss some new constructions and open questions.

Contents Introduction . . . . . . . . . . . . . . . . . . 1. Basic definitions . . . . . . . . . . . . . . . 1.1. Integrable Hamiltonian system . . . . . . . 1.2. Liouville foliation . . . . . . . . . . . . . . 2. General statement of the problem . . . . . 2.1. Types of equivalence . . . . . . . . . . . . 2.2. Notion of non-degeneracy . . . . . . . . . . 2.3. Stability of singularities . . . . . . . . . . . 3. Basic singularities . . . . . . . . . . . . . . 3.1. Elliptic and hyperbolic singularities. Atoms 3.2. Focus-focus singularities . . . . . . . . . . 4. Singularities as almost direct products . 4.1. Basic definitions in the semi-local case . . . 4.2. Direct and almost direct products . . . . . 4.3. Decomposition theorem . . . . . . . . . . .

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MSC2000 : 37C15, 70H06, 37J30 Keywords: Hamiltonian system, singularity, integrability, Liouville foliation, topological invariant, symplectic invariant, torus action, Lagrangian fibration The authors were partially supported by the Russian Foundation for Basic Research (Grants 05-01-00978 and 04-01-00682).

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5. Examples . . . . . . . . . . . . . . . . . . 5.1. Hyperbolic singularities of corank 1 . . . 5.2. Hyperbolic singularities of rank 0 . . . . 5.3. Splittable singularities . . . . . . . . . 6. Symplectic invariants . . . . . . . . . . 6.1. Actions and affine structure . . . . . . . 6.2. Hyperbolic case (one degree of freedom) 6.3. Focus-focus case . . . . . . . . . . . . . 7. Global aspects . . . . . . . . . . . . . . 7.1. One degree of freedom systems . . . . . 7.2. Fomenko–Zieschang invariant . . . . . . 7.3. Loop molecules . . . . . . . . . . . . . 7.4. Non-singular Liouville foliations . . . . 7.5. Torus action . . . . . . . . . . . . . . . 7.6. Almost toric manifolds . . . . . . . . . 7.7. Zung’s classification theorems . . . . . . 7.8. Topology of the set of singularities . . . References . . . . . . . . . . . . . . . . .

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I NTRODUCTION The main object considered in the paper is the integrable Hamiltonian system. Starting from classical papers written more than 200 years ago, integrable Hamiltonian systems were studied in many different directions: looking for new integrable cases and their analytical solutions, investigating the algebraic nature of the integrability, topological analysis of integrable systems, and so on. An integrable Hamiltonian system can be described as a symplectic manifold M of dimension 2n (phase space) with n commuting Hamiltonian vector fields independent almost everywhere on M . In other words, an integrable Hamiltonian system can be viewed as a Hamiltonian action of the commutative Lie group Rn on a 2n-dimensional symplectic manifold. Orbits of an (arbitrary) action of Rn are direct products of torus T k and Rl ; in particular, the orbits of maximal dimension are T k ×Rn−k . Thus, in the case when the phase space M is compact, almost all orbits are n-dimensional tori T n (Liouville tori), and classical Liouville theorem says that the Hamiltonian action takes a very simple standard form in some symplectic coordinates (existing in a neighborhood of each Liouville torus). So, the topological or analytical structure of an integrable Hamiltonian system is quite clear in some neighborhood of a Liouville torus. But there are singular orbits. The topology of their neighborhoods (that are invariant under the action) is more complicated. In some sense, all topological (and even some analytical) properties of the system are determined by the structure of its singularities. The aim of this paper is to discuss singularities of integrable Hamiltonian systems in various aspects, in particular, questions connected with their local and global classification under some restrictions.

singularities of integrable hamiltonian systems

1. B ASIC

3

DEFINITIONS

Let us recall some necessary definitions (see, for example, [6]).

1.1. I NTEGRABLE H AMILTONIAN

SYSTEM

A symplectic manifold (M, ω) is a smooth 2n-dimensional manifold M with a non-degenerate closed 2-form ω on it, which is called a symplectic form (or symplectic structure). Each smooth function H on a symplectic manifold (M, ω) defines a vector field, which is sometimes called the skew-gradient of the function H (the notation is sgrad H ). It is simply the vector field dual to the differential of the function H with respect to the symplectic form ω. The one-parametric group of diffeomorphisms ΦtH : M → M defined by the vector field sgrad H is called a Hamiltonian flow. Namely, ΦtH (x) is the shift of the point x by time t along the trajectory of the field sgrad H passing through x. We will assume that the field sgrad H is complete; in this case these shifts are well-defined for all t ∈ R. The dynamical system defined by such a vector field is called a Hamiltonian system with Hamiltonian H on the phase space M . (If dim M = 2n, one says that the Hamiltonian system has n degrees of freedom.) In local ∂H −1 is the inverse coordinates it can be written as x˙ i = (ω −1 )ij ∂x j , where ω tensor for the symplectic form ω. A Hamiltonian system also can be written in terms of Poisson bracket as x˙ i = {xi , H}, where the Poisson bracket is the bilinear skew-symmet∂f ∂g ric operation on C ∞ (M ) defined by the formula {f, g} = (ω −1 )ij ∂x i ∂xj . This operation endows the space C ∞ (M ) with the structure of a Lie algebra. A function f on the phase space M is called a first integral (or, simply, integral) of the Hamiltonian system with Hamiltonian H if f is constant along trajectories of the system (i. e., along trajectories of the vector field v = sgrad H ). This means that v(f ) = 0 or, equivalently, {f, H} = 0. If the Poisson bracket of two functions is equal to zero, such functions are called commuting with respect to this bracket. So, the integrals of a Hamiltonian system are exactly the functions commuting with the Hamiltonian. Definition 1. A Hamiltonian system on a 2n-dimensional symplectic manifold M (i. e., a Hamiltonian system with n degrees of freedom) is called integrable in the sense of Liouville (or Liouville integrable, or, simply, integrable) if there are n pairwise commuting functionally independent integrals f1 , . . . , fn for this system such that the vector fields sgrad fi are complete on M . Note that the condition of completeness for vector fields on a manifold M is automatically satisfied if M is compact.

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The following classical theorem makes clear the sense of this definition. Theorem 1 (Liouville theorem). Let v = sgrad H be a Liouville integrable Hamiltonian system on a symplectic manifold (M, ω) with integrals f1 , . . . , fn . Then each compact connected component of a regular level surface of the integrals f1 , . . . , fn is diffeomorphic to the n-dimensional torus T n . Moreover, some neighborhood U of such torus is diffeomorphic to the direct product of T n and the n-dimensional disk D n , and there exist coordinates s1 , . . . , sn on Dn and angle coordinates ϕ1 , . . . , ϕn on T n such that in the coordinates s1 , . . . , sn , ϕ1 , . . . , ϕn on U the symplectic structure ω is n ! dsi ∧ dϕi , and the system takes the form s˙ i = 0, ϕ˙ i = ci (s1 , . . . , sn ). i=1

Tori T n from the above theorem are called Liouville tori. Coordinates s1 , . . . , sn , ϕ1 , . . . , ϕn are called action-angle variables.

Remark 1. Although the choice of integrals is not unique, usually (unless otherwise stated) by an integrable Hamiltonian system we will understand all the ingredients (M, ω, H, f1 , . . . , fn ) together. Moreover, as a rule, we will omit the Hamiltonian H and speak of an integrable Hamiltonian system (M, ω, f1 , . . . , fn ) assuming that the Hamiltonian H is some function of f1 , . . . , fn . Note that the choice of a Hamiltonian is important if we interested in the dynamic of the system on invariant manifolds. Such approach (the orbital classification of integrable Hamiltonian systems) is developed in detail in the book [6]. In the present paper we restrict ourselves to “more rough” investigation of invariant manifolds ignoring the dynamic on them. Given an integrable Hamiltonian system (M, ω, f1 , . . . , fn ), one can associate with it the following action of the commutative group Rn on the phase space M generated by Hamiltonian flows of the first integrals. Definition 2. Any element λ = (λ1 , . . . , λn ) ∈ Rn defines the diffeomorphism Φ1fλ : M → M that is the shift by unit along skew-gradient of the function fλ = λ1 f1 + · · · + λn fn . Since the vector fields sgrad f1 , . . . , sgrad fn commute and are complete on M , the mapping λ %→ Φ1fλ is a homomorphism ρ : Rn → Diff M and, hence, defines an action ρ of the group Rn on the manifold M . We call this action the Hamiltonian action of the group Rn defined by the functions f1 , . . . , fn (or associated with the system). An integral f of a Hamiltonian system is called periodic if all trajectories of the field sgrad f are periodic with the same period. A periodic integral f defines the action of the group S 1 by diffeomorphisms Φtf , which is called the Hamiltonian circle action generated by f . Similarly, commuting functionally independent periodic integrals f1 , . . . , fk define the action of the compact commutative group T k on the phase space, which is called the Hamiltonian torus action. (Note that an integrable Hamiltonian system can have no periodic integrals.) As is seen from the definition, the Hamiltonian action of Rn associated with an integrable Hamiltonian system (M, ω, f1 , . . . , fn ) is uniquely defined.


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The converse passage from a Hamiltonian action of Rn on a symplectic manifold (M, ω) to an integrable Hamiltonian system (M, ω, f1 , . . . , fn ) is equivalent to the choice of a basis in Rn (of course, if the choice of a Hamiltomian is not important; see Remark 1). In this sense, the notions of an integrable Hamiltonian system and a Hamiltonian action of Rn are equivalent.

1.2. L IOUVILLE

FOLIATION

An integrable Hamiltonian system (M, ω, f1 , . . . , fn ) defines a mapping F : M → Rn taking each point x ∈ M to the point (f1 (x), . . . , fn (x)) ∈ Rn . This mapping is called the momentum mapping. The set of critical points K for the momentum mapping F is defined as {x ∈ M | rk dF(x) < n}, and the set of critical values Σ = F(K) (the image of K ) is called the bifurcation diagram. For a point x ∈ K , the number rk dF(x) is called the rank of the singular point x. (Also, the number n − rk dF(x) is called the corank of x.) If the phase space is compact, then the Liouville theorem states that the preimage under the momentum mapping of a point y '∈ Σ (i. e., of a non-critical value y of the momentum mapping) is the union of a number of Liouville tori. For critical values y of the momentum mapping, their preimages are more complicated. The main object of investigation is described in the following definition. Definition 3. The decomposition of the phase space of an integrable Hamiltonian system into connected components of F −1 (y) (i. e., the connected components of common levels of first integrals f1 , . . . , fn ) is called the Liouville foliation corresponding to this system. Leaves of a Liouville foliation that contain no critical points of the momentum mapping are called regular. (By the Liouville theorem, all compact regular leaves are Liouville tori.) All other leaves are called singular. Each regular leaf of the Liouville foliation corresponding to an integrable Hamiltonian system with n degrees of freedom is exactly an n-dimensional orbit of the associated Hamiltonian action. Singular leaves can be the union of a number of orbits (of different dimensions). Note that the Liouville foliation, which is uniquely defined by a system (or by the associated Hamiltonian action), does not define a system in unique way. Roughly speaking, the Liouville foliation corresponding to an integrable Hamiltonian system contains less information than the system itself, but all what we wish to retain under a classification of systems.

2. G ENERAL

STATEMENT OF THE PROBLEM

Now we can say that under topological (or, qualitative) investigation of an integrable Hamiltonian system we mean the study of the topology of the corresponding Liouville foliation. In this terminology, one can say that the Liouville theorem completely describes the Liouville foliation in a neighborhood of a Liouville torus, but says almost nothing about its structure near singular leaves.

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The main problem in this theory is to classify Liouville foliations in a certain sense. Thus, in order to classify integrable Hamiltonian systems, first of all, it is necessary to define an equivalence relation for them. The second step is to distinguish a certain class of “non-degenerate” singularities for which the classification can be obtained in reasonable terms (in particular, this means that one can explicitly describe the complete list of such singularities of given “complexity”). Of course, many variants of these problems could be considered. Let us describe some possible approaches.

2.1. T YPES

OF EQUIVALENCE

Definition 4. Two integrable Hamiltonian systems on U1 and U2 are called Liouville equivalent if there exists a homeomorphism Ψ : U1 → U2 taking each leaf of the Liouville foliation on U1 to a leaf of the Liouville foliation on U2 . The Liouville classification, i. e., the classification up to the Liouville equivalence, is the main problem from the topological point of view. One can vary the definition considering another class of the mappings Ψ. For example, one can replace “homeomorphism” by “diffeomorphism”. Taking into account that the phase spaces are symplectic, it is natural to require that Ψ is a symplectomorphism. This leads to the problem of the symplectic classification of Liouville foliations. Definition 5. Two integrable Hamiltonian systems on (U1 , ω1 ) and (U2 , ω2 ) are called symplectically equivalent if there exists a symplectomorphism Ψ : U1 → U2 taking each leaf of the Liouville foliation on U1 to a leaf of the Liouville foliation on U2 . Another aspect of the classification is the choice of the sets U1 and U2 in the above definitions. They could be, for example, – a neighborhood of a singular point, – a neighborhood of a singular leaf of the Liouville foliation, – the whole phase space. Thus we have, respectively, – local, – semi-local, – global problems of classification. Also, one can consider global classification not for the whole phase space, but for an isoenergy surface of the system, i. e., for a level surface of the Hamiltonian (see Section 7.2). In fact, in the local and semi-local cases, one speaks about classification of “singularities” of integrable Hamiltonian systems. More precisely, under local and semi-local equivalence we mean the following. Definition 6. Let x1 and x2 be critical points of the momentum mappings for integrable Hamiltonian systems on (M1 , ω1 ) and (M2 , ω2 ) respectively, and let L1 ( x1 and L2 ( x2 be singular leaves of the correspond-


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ing Liouville foliations. Then we will say that these singularities are locally (resp. semi-locally) equivalent if there are neighborhoods U1 and U2 of the points x1 and x2 (resp. of the leaves L1 and L2 ) such that the systems on U1 and U2 are equivalent, where the equivalence is understood in one of the above senses, and the mapping Ψ from Definitions 4 and 5 takes the point x1 to the point x2 (resp. the leaf L1 to the leaf L2 ). If a mapping Ψ defines an equivalence of integrable Hamiltonian systems (i. e. is fiberwise as in Definitions 4 and 5) we will also call it simply homeomorphism (diffeomorphism) or symplectomorphism of the corresponding Liouville foliations. In particular, in the local and semi-local cases, we will say that Liouville foliations (or, simply, singularities) are locally symplectomorphic, semi-locally homeomorphic, etc.

2.2. N OTION

OF NON - DEGENERACY

There is a general definition of non-degeneracy for singularities of arbitrary rank. First, let us explain this notion for singular points of rank 0. Consider an integrable Hamiltonian system (M ,ω,f1 , . . . ,fn ). Let x ∈ M be a singular point of rank 0, i. e., dfi (x) = 0 for all i = 1, . . . , n. Then for each function fi one can define a linear operator Afi : Tx M → Tx M by the formula Afi = ω −1 d2fi (x). This operator can be considered as an element of the Lie algebra sp(Tx M ) (Lie algebra of the group of linear symplectic transformations of Tx M ). Indeed, the vector field sgrad fi defines a one-parameter group of symplectomorphisms. Since x ∈ M is a fixed point, the differentials of these symplectomorphisms at the point x form a one-parameter subgroup in the group of linear symplectic transformations Sp(Tx M ). The operator Afi is exactly the tangent vector for this subgroup, i. e., an element of the Lie algebra sp(Tx M ). In other words, Afi is the linearization of the vector field sgrad fi at the singular point x. Under the linearization at a singular point, the commutator of vector fields is transformed to the commutator of corresponding linear parts, i. e., the linearization induces a homomorphism from the Lie algebra of vector fields with singularity at the point x to the Lie algebra gl(Tx M ). Images of Hamiltonian vector fields under this homomorphism are elements of the subalgebra sp(Tx M ). Moreover, [Af , Ag ] = A{f,g} since the mapping f %→ − sgrad f is also a homomorphism of Lie algebras. Since the integrals f1 , . . . , fn commute, i. e., {fi , fj } = 0, one has [Afi , Afj ] = 0. Thus, each singular point x of rank 0 defines a commutative subalgebra hx in sp(Tx M ) generated by the operators Af1 , . . . , Afn . Definition 7. A singular point x ∈ M 2n of rank 0 is called non-degenerate if hx ⊂ sp(Tx M ) is a Cartan subalgebra. Note that, unlike the complex case, there exist non-conjugate Cartan subalgebras in the real symplectic Lie algebra sp(2n, R). The classes of conjugate Cartan subalgebras in sp(2n, R) are described by the Williamson theorem.

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It is convenient to formulate this theorem in terms of homogeneous quadratic polynomials taking into account that there is a natural isomorphism between the Lie algebra of homogeneous quadratic polynomials on a symplectic linear space (V, ω) and the symplectic Lie algebra sp(V ). Indeed, as above, each operator A ∈ sp(V ) defines the quadratic form fA (v) = 12 ω(v, Av) such that this correspondence A %→ fA is a homomorphism of Lie algebras, i.e., {fA , fB } = f[A,B] , where { , } is the Poisson bracket corresponding to the form ω. Theorem 2 (Williamson theorem [56, 57]). Let h ⊂ sp(2n, R) be a Cartan subalgebra. Then there exist linear coordinates q1 , . . . , qn , p1 , . . . , pn n ! in R2n such that ω = dpi ∧ dqi and the following n quadratic polynomials i=1 form a basis in h: ell

1) Pi

= p2i + qi2

hyp

2) Pi

= pi q i

(elliptic type)

i=1, 2, . . . , m1

(hyperbolic type)

i=m1 +1, m1 +2, . . . , m1 +m2

foc

3) Pi = pi qi +pi+1 qi+1 (focus-focus type) i=m1 +m2 +1, m1 +m2 +3, foc Pi+1 = pi qi+1 −pi+1 qi . . . , m1 +m2 +2m3 −1

where m1 + m2 + 2m3 = n.

Thus, the type of a Cartan subalgebra in sp(2n, R) is completely determined by the triple (m1 , m2 , m3 ), where m1 is the number of basic elements of elliptic type, m2 is the number of basic elements of hyperbolic type, and m3 is the number of pairs of basic elements of focus-focus type. Therefore, for a given singular point x, one can call the triple (m1 , m2 , m3 ) determined by the corresponding Cartan subalgebra hx the type of the singular point x. Now let us describe the notion of non-degeneracy in the general case. Let x ∈ M be a singular point of rank r. Then the orbit Ox of the corresponding Hamiltonian action passing through the point x has dimension r. Let us consider the following two linear subspaces in the tangent space Tx M : the subspace Lx = Tx Ox (spanned by sgrad f1 , . . . , sgrad fn ) and its skew-orthogonal complement L#x . Since the functions f1 , . . . , fn commute, Lx is the kernel of the restriction of the symplectic structure ω to the space L#x . This implies that ω induces a well-defined symplectic form ω " on the quotient L#x /Lx . The dimension of the stabilizer Stx of the point x (under the Hamiltonian action generated by the functions f1 , . . . , fn ) is equal to n−r. The connected component of the identity of the group Stx is isomorphic to Rn−r and its action on M generates an action on Tx M by linear symplectic transformations. Since both the subspaces Lx and L#x are invariant under this action, we have a symplectic action (with respect to the form ω " ) of the group Rn−r # on the space Lx /Lx of dimension 2(n − r).


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Thus, after this reduction, the situation becomes similar to the case of singularities of rank 0 (see above). In particular one obtains a commutative subalgebra " hx ⊂ sp(L#x /Lx , ω " ).

Definition 8. A singular point x ∈ M 2n (of rank r) is called non-degenerate if " hx is a Cartan subalgebra in sp(L#x /Lx , ω " ). It turns out that the (local) structure of the Liouville foliation in a small neighborhood of a non-degenerate singularity is completely determined by its rank and type. This can be explained as follows. Consider the space R2n with coordinates q1 , . . . , qn , p1 , . . . , pn and n ! ell hyp foc the standard symplectic structure ω = dpi ∧ dqi . Let Pj , Pk , Pl be i=1

functions as in the Williamson theorem and Ps = qs , where j = 1, . . . , m1 , k = m1 + 1, . . . , m1 + m2 , l = m1 + m2 + 1, . . . , m1 + m2 + 2m3 , ell hyp foc s = m1 + m2 + 2m3 + 1, . . . , n. Then the functions Pj , Pk , Pl , Ps are functionally independent and pairwise commute. Therefore, they define a canonical Liouville foliation L can in R2n with singular point 0. It is clear that this point is non-degenerate, its rank is equal to r = n−m1 −m2 −2m3 , and its type is (m1 , m2 , m3 ). Theorem 3 (Eliasson theorem [14]). The Liouville foliation in a neighborhood of a non-degenerate singular point of rank r and of type (m1 , m2 , m3 ) is locally symplectomorphic to the canonical Liouville foliation L can with the same parameters m1 , m2 , m3 , r. In particular, in the case of rank 0, the Liouville foliation in a neighborhood of a non-degenerate singular point is locally symplectically equivalent to the Liouville foliation defined by quadratic parts of integrals. In the realanalytic case this theorem was proved by J. Vey in [51]. A complete proof of Theorem 3 in C ∞ -case can be found in [35]. Consider the integrable Hamiltonian system in R2n defined by the funcell hyp foc tions Pj , Pk , Pl , Ps . The bifurcation diagram of the corresponding momentum mapping is called the canonical bifurcation diagram for the singularity of type (m1 , m2 , m3 ) and rank r = n − m1 − m2 − 2m3 . It is the union of m1 + m2 pieces of codimension 1 and m3 pieces of codimension 2 (in Rn ) each of which is the product of coordinate axes and (closed) semi-axes. The Eliasson theorem, in particular, states that the functions defining the Liouville foliation in a small neighborhood of a non-degenerate singular point x of type (m1 , m2 , m3 ) and rank r can be chosen in such a way that the local bifurcation diagram Σx (i. e., the bifurcation diagram of the momentum mapping restricted to the neighborhood of x) becomes canonical with the same parameters m1 , m2 , m3 , r. (See also Remark 3.) Nguyen Tien Zung and E. Miranda have recently shown (see [36,45] that the statement of Theorem 3 remains true if one replaces the non-degenerate singular point in the formulation of the theorem by a compact non-degenerate singular orbit of the Hamiltonian action associated with the system (see Definition 2).

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2.3. S TABILITY

and

a.a oshemkov

OF SINGULARITIES

There are many examples of mechanical and physical systems which depend on some parameters. A small variation of parameters can be considered as a perturbation of the initial system. A natural question is how singularities of systems (or invariants of singularities) behave under such perturbations. The stability means that a singularity does not change its structure under such perturbations. Let us give the formal definition. Given an integrable Hamiltonian system (M, ω, H, f1 , . . . , fn ) one can consider its arbitrary perturbation (M, ωε , Hε , f1,ε , . . . , fn,ε ), where ωε (x) = ω(x, ε), Hε (x) = H(x, ε), fi,ε (x) = fi (x, ε) smoothly depend on x and ε, such that the initial system corresponds to ε = 0. Definition 9. The system (M, ω, H, f1 , . . . , fn ) is called stable if for any integrable perturbation (M, ωε , Hε , f1,ε , . . . , fn,ε ) there exists δ such that the initial system is equivalent to the perturbed one for all ε ∈ [0, δ]. The two types of equivalence defined above (Definitions 4 and 5) give two types of stability. Also, we can speak of local and semi-local stability (in the sense of Definition 6). For example, the Liouville theorem implies the semi-local symplectic stability (and, of course, the Liouville stability) of an integrable Hamiltonian system in a neighborhood of a Liouville torus.

3. B ASIC

SINGULARITIES

As will be shown in Section 4, singularities of an integrable Hamiltonian system can be obtained from some simple ones by the operations of direct product and factorization. Here we describe such “basic” (in the semi-local sense) singularities, which are “atoms” and “focus-focus” singularities. More detailed description can be found in book [6]. Also, various properties of focus-focus singularities were investigated in [9, 10, 29, 32, 41–44].

3.1. E LLIPTIC

AND HYPERBOLIC SINGULARITIES .

A TOMS

A Hamiltonian system with one degree of freedom is simply a symplectic two-dimensional manifold (M 2 , ω) (phase space) with a smooth function H on it (Hamiltonian). We will assume that levels of the function H are compact. Then the system (M, ω, H, f1 = H) is integrable by definition. In this case the non-degeneracy of singularities simply means that H is a Morse function. Hence, the local structure of the Liouville foliation is described by classical Morse lemma, which has the following symplectic analog. Theorem 4 (Morse–Darboux lemma [8]). If a point x is a non-degenerate critical point of a function H on a symplectic manifold (M 2 , ω), then there exist symplectic coordinates q, p in a neighborhood of the point x such that the function H has one of the following forms: H = h(q 2 + p2 ) H = h(qp) where h# (0) '= 0.

(elliptic case) (hyperbolic case)


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Thus, from the local point of view, the Liouville classification and even the symplectic classification of singularities for integrable systems with one degree of freedom are trivial: there are exactly two equivalence classes. From the semi-local point of view the situation is more complicated. If a leaf of a Liouville foliation contains exactly one singular point, then there are only two possibilities shown in Figure 1.

A

B

Figure 1. Elliptic and hyperbolic singularities with one singular point (one degree of freedom)

Figure 2. Typical (semi-local) situation for Liouville foliation with several singular points

If a leaf contains several singular points, then they should be hyperbolic. A typical situation is presented in Figure 2. Definition 10. The class of the semi-local Liouville equivalence of a singular non-degenerate leaf (in the case of one degree of freedom) is called an atom. The complexity of an atom is the number of singular points on the singular leaf (which are also called vertices of an atom). Other formal definitions of atoms see in [6, 48]. Remark 2. We will use the same word “atom” for representatives of equivalence classes, i. e., for an integrable system (V, ω, H) with a singularity, where (V, ω) is a compact symplectic two-dimensional surface with boundary and H is a Morse function with one critical value 0 such that V = H −1 [−ε, ε] and ∂V = H −1 (±ε). Note that the atom defined by such a pair (V, H) is completely determined (as a class of the semi-local Liouville equivalence) by the singular level Γ = H −1 (0) ⊂ V . This allows us to think of an atom as a pair (V, Γ), and we will depict atoms in this way. The number of topologically different (i. e., non-equivalent in the sense of the semi-local Liouville equivalence) cases is obviously infinite. However, there is an explicit algorithm allowing us to enumerate all atoms of a given complexity (see [6, 48]). Using computer, one can obtain the list of atoms (represented, for example, as a sequence of some symbols allowing to reconstruct a neighborhood of the singular leaf) and, in particular, the number of them. For atoms of complexity ≤ 7 this was done by A. A. Oshemkov. Theorem 5. For k = 1, 2, 3, 4, 5, 6, 7 the number of atoms of complexity k is equal to 2, 4, 10, 58, 322, 3044, 33917 respectively.

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There are only two atoms of complexity 1 (see Figure 1): “elliptic” atom A and “hyperbolic” atom B. All atoms of complexity 2 and 3 are shown in Table 1. In what follows, we will use standard names for these atoms (indicated in the table). More detailed description of atoms of complexity 4 and 5 can be found in [4]. Table 1. The list of atoms of small complexity complexity 2

C1

C2

D1

D2

complexity 3

E1

G1

E2

G2

E3

G3

F1

H1

F2

H2

Theorem 5 gives the semi-local Liouville classification of non-degenerate singularities of small complexity for the case of one degree of freedom. The symplectic classification of such singularities is more complicated. In particular, for each hyperbolic atom there are infinitely many symplectically non-equivalent singularities [12] (see also Section 6.2). Note that all atoms, except for the atoms A and B, are not stable. More precisely, each hyperbolic atom can be decomposed into atoms B by a small perturbation. This statement is equivalent to the fact that simple Morse functions are dense in the space of all Morse functions.


3.2. F OCUS - FOCUS

13

SINGULARITIES

Let us now discuss the semi-local structure of focus-focus singularities. We start with a description of a model singularity of this type. According to the Eliasson theorem, the Liouville foliation near a focusfocus point is given by two functions f1 = p1 q1 + p2 q2 and f2 = p1 q2 − p2 q1 in symplectic coordinates q1 , q2 , p1 , p2 . To understand the local structure of the foliation in a neighborhood of a focus-focus point, it is convenient to pass to complex variables z = q1 + iq2 ,

w = p1 − ip2 .

Then the functions f1 and f2 can be represented as the real and imaginary parts of one complex function F = zw, and the symplectic structure takes the form ω = Re(dw∧dz). In particular, this implies that locally the singular leaf L is represented as a pair of Lagrangian discs transversally intersecting at the focus-focus points and given by the equations z = 0 and w = 0. One can use this local representation of a focus-focus singularity to construct a Liouville foliation with compact leaves. Let U be an open domain in C2 (z, w) given by the following inequalities: |zw| < ε,

|z| < 1 + δ,

|w| < 1 + δ,

where ε, δ > 0 and ε(1 + δ)2 < 1. This domain is a regular neighborhood of two two-dimensional discs {|z| ≤ 1, w = 0}

and {z = 0, |w| ≤ 1}

intersecting transversally at the point (0, 0). Consider the following regular neighborhoods of the boundary circles of these discs in U : Uz = U ∩ {(1 + δ)−1 < |z| < 1 + δ},

Uw = U ∩ {(1 + δ)−1 < |w| < 1 + δ}.

Topologically, Uz and Uw are homeomorphic to the direct product S 1 ×D3 (and the condition ε(1 + δ)2 < 1 implies that they are disjoint). Let us glue the neighborhoods Uz and Uw by the mapping ξ : Uw → Uz , where ξ : (z, w) %→ (w−1 , zw2 ). Clearly, ξ is a complex mapping. Moreover, it is easily seen that ξ is symplectic (since dw ∧ dz = d(zw2 ) ∧ d(w−1 )) and preserves the function zw. Hence, as a result, from the domain U one obtains a complex symplectic 4-manifold U1 with a holomorphic function F on it, which has the form zw in terms of local coordinates z, w. This function has exactly one critical point (0, 0). The singular level {F = 0} is obtained as a result of gluing two transversally intersecting discs along their boundaries, i. e., is homeomorphic to the sphere with one self-intersection point. The manifold U1 is a regular neighborhood of this level (i. e., U1 = {|F | < ε}) foliated into compact non-singular levels of F , each of which is diffeomorphic to a 2-torus.

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From the real point of view, we have obtained two commuting functions f1 = Re F and f2 = Im F , which determine the structure of a Liouville foliation on U1 with the only singular leaf of focus-focus type, which contains one singular point. Thus, U1 can be considered as the simplest model example of a focusfocus singularity of complexity 1. In a similar way, we can construct a model example of a focus-focus singularity of complexity k, i. e., a singularity whose singular leaf contains k focus-focus points (a definition of complexity for arbitrary singularities is given in Section 4.1). To this end, we only need to glue successively k copies of the domain U as was described above. As a result, we obtain a manifold Uk with the desired properties. The same manifold can be obtained in another way. Namely, we can just consider a k-sheeted covering over U1 . Here we use the fact that the fundamental group of a singular leaf in U1 (as well as that of U1 itself) is isomorphic to Z. So, we can produce such a covering in the standard way by taking the subgroup kZ ⊂ Z.

(a)

(b)

Figure 3. Singular leaf for a focus-focus singularity and its universal covering

Notice that the universal covering U∞ over U1 coincides with the universal covering over Uk . Moreover, the structure of the (non-compact) Liouville foliation on U∞ lifted from the base of the covering will be the same for all Uk , k ∈ N (see Figure 3). On this universal covering, there is a natural action of Z, and every focus-focus singularity (considered up to the Liouville equivalence) can be obtained from this universal model by taking a quotient with respect to the subgroup in Z of index k. Let us emphasize that this statement holds only in the topological sense. From the symplectic point of view, it is not the case. The point is that there exist non-trivial symplectic invariants which distinguish focus-focus singularities of the same complexity [52] (see also Section 6.3). The following statement describes focus-focus singularities up to the Liouville equivalence in the case when the singular leaf contains only points of focus-focus type. Theorem 6 (V. S. Matveev [32], Nguyen Tien Zung [42]). Focus-focus singularities of the same complexity are semi-locally Liouville C 0 -equivalent.


15

For focus-focus singularities of complexity 1, the above statement holds in the smooth case as well. For focus-focus singularities of complexity k > 1, it seems that there exist non-trivial smooth invariants. Some arguments for this conjecture can be found in book [6], but the nature of those invariants is not well understood yet. Problem 1. To classify focus-focus singularities up to the smooth Liouville equivalence. Theorem 6 says that all focus-focus singularities of a given complexity k are topologically the same. Let us discuss some propertiers of such singularity. The topology of the singular leaf. The singular leaf of a focus-focus singularity of complexity k is homeomorphic to the torus with k pinching points (see Figure 3(b)). It can be also considered as either an immersed sphere with one point of self-intersection (for k = 1) or the chain of embedded spheres transversally intersecting at focus-focus points (for k > 1). Note that these spheres are Lagrangian submanifolds (immersed for k = 1) in the phase space. It is well-known that not every symplectic manifold can contain an embedded Lagrangian sphere. For, example, R4 with standard symplectic structure ω does not contain an embedded Lagrangian sphere (in particular, if an integrable Hamiltonian system on an arbitrary 4-dimensional phase space has a focus-focus singularity of complexity k > 1, then its singular leaf is not “small”, i. e., not contained in a neighborhood symplectomorphic to a domain in (R4 , ω)). This follows from general M. Gromov’s theorems [23], but for the case S 2 ⊂ R4 can be easily explained as follows: on the one hand, the normal bundle to an embedded sphere S 2 in R4 is trivial, on the other hand, the normal bundle to an arbitrary Lagrangian submanifold is isomorphic to its cotangent bundle, but the cotangent bundle of the sphere S 2 is non-trivial. Thus, the topology of the phase space can give obstructions to existence of a focus-focus singularity of complexity k. Problem 2. To describe obstructions to existence of a focus-focus singularity of complexity k on a given symplectic manifold. Bifurcation diagram and monodromy. All leaves of the Liouville foliation close to the singular leaf L containing focus-focus points are Liouville tori, i. e., non-singular. Thus, the bifurcation diagram for the momentum mapping F locally consists of an isolated point y = F(L) on the plane. In such situation a phenomenon called Hamiltonian monodromy naturally appears. To describe it, one takes a circle γε of small radius ε centered at the point y and its preimage Qγε = F −1 (γε ) (Figure 4(a)). It is clear that the 3-manifold Qγε is a fiber bundle over the circle γε whose fibers are Liouville tori T 2 . This fiber bundle is completely determined by its monodromy group, i. e., the group of the automorphisms of the fundamental group of a fiber π1 (T 2 ) corresponding to closed loops on the base. Since π1 (T 2 ) = Z ⊕ Z and the base is the circle γε , the monodromy group is just a cyclic subgroup in SL(2, Z).

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Clearly, Qγε can be represented as the result of identification of the boundary tori T0 and T1 of the 3-cylinder T 2 × [0, 1] by some diffeomorphism ψ : T0 → T1 (Figure 4(b)). The gluing diffeomorphism ψ induces

(a)

(b)

Figure 4. Local bifurcation diagram for a focus-focus singularity

an automorphism ψ∗ of the fundamental group of the torus Z ⊕ Z. The automorphism ψ∗ is uniquely defined by an integer unimodular matrix. This matrix depends, of course, on the choice of a basis on the torus, but its conjuT2

gacy class is a well-defined complete invariant of the fiber bundle Qγε −→ γε . This matrix is called the monodromy matrix. Theorem 7 (V. S. Matveev [32], Nguyen Tien Zung [42]). The monodro# $ my matrix for a focus-focus singularity of complexity k has the form 10 k1 .

Circle action. Let us consider again a domain U ⊂ C2 with coordinates z, w, which was used above for constructing a model focus-focus singularity. The function f2 = Im(zw) defines a Hamiltonian S 1 -action on U preserving the foliation, since the integral curves of the vector field sgrad f2 are all closed with period 2π. Moreover, this action is free everywhere except for the only fixed point (0, 0). This follows from the explicit form of sgrad f2 in terms of the complex coordinates z, w: Hence the

S 1 -action

z˙ = −iz,

w˙ = iw.

is given by the simple formula: (z, w) %→ (e−iϕ z, eiϕ w).

The model focus-focus singularity of complexity k is the result of gluing k copies of U . Since the gluing is symplectic and preserves the function f2 , one can define a smooth S 1 -action on the neighborhood of the singular leaf. A similar result holds for general focus-focus singularity although such a singularity is not necessary diffeomorphic (but only homeomorphic) to the model singularity of focus-focus type. Theorem 8 (Nguyen Tien Zung [42], see also [45]). On the neighborhood of the singular leaf of a focus-focus singularity, there is a smooth Hamiltonian S 1 -action which is free everywhere except for the singular points. Each leaf of the Liouville foliation is invariant under this action. Such an action is uniquely defined up to reversing orientation on the acting circle S 1 .


4. S INGULARITIES

17

AS ALMOST DIRECT PRODUCTS

Now we turn to the semi-local study of arbitrary (non-degenerate) singularities. First of all, one needs to generalize some notions to the semi-local case.

4.1. B ASIC

DEFINITIONS IN THE SEMI - LOCAL CASE

We will say that a singular leaf of a Liouville foliation (or, simply, singularity) is non-degenerate if the singular points of the momentum mapping lying on the singular leaf are all non-degenerate in the sense of Definition 8. Let L be a non-degenerate singular leaf. Consider those points x ∈ L where the rank of the momentum mapping F is minimal: % & Lr = x ∈ L | rank dF(x) = r = min rank dF(y) . y∈L

For each point x ∈ Lr , we can determine its type that is the triple of integers (m1 , m2 , m3 ) described in Section 2.2. It can be shown that the type (m1 , m2 , m3 ) is the same for all x ∈ Lr , i. e., for all singular points of minimal rank on L. Therefore, one can speak of the type (m1 , m2 , m3 ) of the singular leaf L itself as well as of its rank r. From the topological point of view, Lr is a disjoint union of r-dimensional orbits of the Hamiltonian action associated with the system (each of which is diffeomorphic to a torus T r ). The number of connected components of Lr is called the complexity of the singularity. (For atoms and focus-focus singularities, this notion of course coincides with ones introduced in Section 3.) Clearly, the number of non-equivalent non-degenerate singularities is infinite. In the case of the Liouville classification, the notions of type, rank, and complexity allows one to divide this infinite list into finite parts. (As is shown in Section 6, this is not the case for the symplectic classification.) Thus, the classification problem for the semi-local Liouville equivalence can be formulated as follows: to make the complete list of non-equivalent singularities of given type (m1 , m2 , m3 ), rank r, and complexity k. This problem is completely solved for some values of the parameters m1 , m2 , m3 , r, k (see examples in Sections 5.1, 5.2). In particular, its solution for basic singularities, i. e., singularities of rank 0 and types (1, 0, 0), (0, 1, 0), (0, 0, 1), was described in Section 3. We will denote representatives of the corresponding equivalence classes (of arbitrary complexity) by V ell , V hyp , V foc respectively. (Note that V ell can be of complexity 1 only, i. e., it is just the atom A.) Also, denote by V reg the trivial foliation D 1 × S 1 without singularities (it can be considered as the class of the semi-local Liouville equivalence of a non-singular leaf S 1 for a system with one degree of freedom).

4.2. D IRECT

AND ALMOST DIRECT PRODUCTS

Now we describe one obvious method for constructing “complicated” singularities from “simple” ones.

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'2l , ω Let (M 2k , ω, f1 , . . . , fk ) and (M " , f"1 , . . . , f"l ) be integrable Hamil'2l . The 2-forms ω, ω tonian systems. Consider the direct product M 2k × M " " '2l and the integrals fi , fj can be naturally lifted to the manifold M 2k × M (we will leave the same notation for lifted objects). As a result, we obtain a new integrable Hamiltonian system (with k + l degrees of freedom) '2l , ω + ω (M 2k × M " , f1 , . . . , fk , f"1 , . . . , f"l ).

'2l is a singular point of this system if and only if A point (x, x ") ∈ M 2k × M at least one of the points x, x " is singular. Moreover, a singular point (x, x ") is non-degenerate if and only if x and x " are non-degenerate (or non-singular), and the rank of a singular point (x, x ") is the sum of the ranks of x and x ". The system so obtained is called of direct product type, and this operation can be naturally extended to an arbitrary number of factors. When considering systems from the local or semi-local point of view, one can also speak of singularities of direct product type. This means that a singularity is locally (or semi-locally) equivalent to the direct product of small neighborhoods V and V" of singular points (or singular leaves). Remark 3. The Liouville theorem says that an integrable Hamiltonian system with n degrees of freedom in a neighborhood of a Liouville torus can be represented as the direct product of n trivial systems with one degree of freedom. Analogously, the Eliasson theorem says that locally each non-degenerate singularity can be decomposed into the direct product of basic non-degenerate singularities. Singularities of direct product type can naturally be generalized to the class of singularities which are called “almost direct products”. To this end, we consider a singularity U = V1 × · · · × Vm of direct product type and actions ψ1 , . . . , ψm of a finite group G on the factors which satisfy the following conditions: 1) each mapping ψi (g) : Vi → Vi is a symplectomorphism preserving the Liouville foliation (moreover, we will assume that ψi preserves the corresponding functions that define the Liouville foliation on Vi ); 2) the action ψ of the group G on U = V1 ×· · ·×Vm defined by the formula ψ(g)(x1 , . . . , xm ) = (ψ1 (g)(x1 ), . . . , ψm (g)(xm )) is free. Consider now the quotient space of U with respect to the action ψ of G. Since the action ψ is free, this quotient U/G is a smooth manifold. Moreover, the symplectic structure and Liouville foliation can also be transferred to U/G from U . As a result, we obtain a symplectic manifold U/G with the Liouville foliation defined by commuting independent functions. Clearly, U/G can be considered as a neighborhood of a singular leaf all of whose singular points are non-degenerate. In other words, U/G represents a model non-degenerate singularity of an integrable Hamiltonian system. Definition 11. Singularities of type U/G (as well as those which are Liouville equivalent to them) are called singularities of almost direct product type (or, simply, almost direct products).


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In the next section we discuss an important result (due to Nguyen Tien Zung) which says that, in essense, all non-degenerate singularities are of almost direct product type.

4.3. D ECOMPOSITION

THEOREM

Before formulating Zung’s decomposition theorem, we need one more additional assumption to the singularities of Liouville foliations. This assumption (the so-called “non-splitting condition”) distinguishes a wide and natural class of singularities. For all specific examples of integrable systems which we know this condition is fulfilled. We begin with a simple example. Consider an integrable system with two degrees of freedom with Hamiltonian H and integral f on a compact phase space M . Let L ∈ M be a non-degenerate singular leaf of corank 1 and hyperbolic type (this case has been studied in detail by A. T. Fomenko; see, for example, [15–17] and Sections 5.1, 7.2). Assume that the level Qh = {H = h} of the function H containing the leaf L is not critical, i. e., sgrad H '= 0 on Qh . Then each singular point x ∈ L belongs to a critical one-dimensional submanifold (homeomorphic to a circle S 1 ) of the function f |Qh , which coincides with the orbit of x with respect to the Hamiltonian action of R2 generated by H and f . It is easy to check that such a submanifold is always included in a one-parameter family of non-degenerate critical submanifolds. The image of this family under the (local) momentum mapping F : U (L) → R2 is a smooth curve. Let us assume that L contains two critical submanifolds S1 , S2 each of which is included in its own one-parameter family Si (τ ), i = 1, 2, where as the parameter τ ∈ (h − ε, h + ε) we take the value of the function H , i. e., Si (h) = Si .

Figure 5. Splitting of the bifurcation diagram

Each family determines a certain curve Σi (τ ) = F(Si (τ )) of the bifurcation diagram. There are two posiibilities: these curves either coincide or are different (but intersect at the point F(L)!). The first possibility means that under varying τ the critical circles S1 (τ ), S2 (τ ) still remain on the same singular leaf L(τ ), whereas in the second case they pass to different leaves. In other words, the second possibility means that L splits into two singular leaves (see Figure 5). Thus to exclude such a possibility we must assume that Σ1 = Σ2 . In a similar way, the non-splitting condition can be formulated in the case of multi-dimensional non-degenerate singularities.

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Consider a non-degenerate singularity of a Liouville foliation on M 2n . Let L be the corresponding singular leaf, and ΣL denote its local bifurcation diagram, i. e., the bifurcation diagram of the momentum mapping F restricted to a sufficiently small neighborhood of L in M 2n . Now consider the points of minimal rank r on L. Such points form one or several critical tori of dimension r. For each of them, choose a point-representative xj and construct the local bifurcation diagrams Σxj ⊂ Rn for small neighborhood of xj . It is required that, for all the points-representatives {xj }, these local bifurcation diagrams {Σxj } are the same. Moreover, we must also require that ΣL does not contain anything else, i. e., ΣL = Σxj for each xj . Recall that local bifurcation diagrams Σxj have been already described in Section 2.2 as canonical bifurcation diagrams. Definition 12. We will say that a non-degenerate singularity of the Liouville foliation in M 2n satisfies the non-splitting condition if its bifurcation diagram Σ in Rn can be reduced by some diffeomorphism to the canonical diagram corresponding to the type and rank of the given singularity (see Section 2.2). Remark 4. We have partially changed the terminology used in the original work [41] by Nguyen Tien Zung. He called the above singularities stable. We say instead that they satisfy the non-splitting condition. The point is that, speaking of stability, one usually means the stability under certain perturbations (see Definition 9). In fact, the stability in this sense and the non-splitting condition are closely connected (in the above example the non-splitting condition is equivalent to the stability of the leaf L under small variation of the level {H = h} → {H = h + ε}). Nevertheless, these two conditions are, in general, not equivalent. The following “decomposition theorem” states that a finite covering of a non-degenerate singularity satisfying the non-splitting condition can be represented as the direct product of basic singularities (described in Section 3). Theorem 9 (Nguyen Tien Zung [41]). Each non-degenerate singularity that satisfies the non-splitting condition is Liouville equivalent to a singularity of almost direct product type whose factors have one of the following four types: 1) an elliptic singularity V ell with one degree of freedom (i. e., atom A); 2) a hyperbolic singularity V hyp with one degree of freedom; 3) a focus-focus singularity V foc with two degrees of freedom; 4) a trivial Liouville foliation V reg without singularities with one degree of freedom (i. e., D 1 × S 1 ).

Recall that, unlike elliptic and regular factors (V ell and V reg ), hyperbolic and focus factors (V hyp and V foc ) can be of different topological types described in Sections 3.1 and 3.2 respectively. It should be emphasized that Theorem 9 has topological, but not symplectic character. More precisely, it states that every non-degenerate singu-


21

larity satisfying the non-splitting condition is (smoothly) Liouville equivalent to a certain model singularity of almost direct product type. But the corresponding diffeomorphism does not have to be a symplectomorphism. That is, the symplectic structure on the direct product is not necessarily the product of symplectic forms of the factors. Note that a representation of a singularity as an almost direct product is, in general, not unique (one of simplest examples of such situation is described after Theorem 10; see also Problem 5 and the discussion preceding it).

5. E XAMPLES The decomposition theorem is a powerful tool for studying the topological structure of non-degenerate singularities. However, to get the list of all such singularities (of a given complexity) is still a non-trivial problem. Some special cases for which the classification problem is completely solved (in the sense of the Liouville equivalence) are described in Sections 5.1 and 5.2, where we assume the non-splitting condition to be fulfilled. In Section 5.3 we give examples of another kind. Namely, we discuss different situations when the non-splitting condition is not fulfilled (and therefore, the decomposition theorem is not applicable).

5.1. H YPERBOLIC

SINGULARITIES OF CORANK

1

In the case of one degree of freedom, corank 1 hyperbolic singularities (i. e., singularities of type (0, 1, 0)) are just atoms considered in Section 3.1. For two degrees of freedom, the classification was obtained by A. T. Fomenko and H. Zieschang [20]. Here is a reformulation of that result in terms of almost direct products. Theorem 10. In dimension 4, the hyperbolic corank 1 singularities can be of the following two types: 1) direct products V hyp × V reg ; 2) almost direct products (V hyp ×V reg )/Z2 with the action of the group Z2 defined by (x, s, ϕ) %→ (τ (x), s, ϕ + π),

where x ∈ V hyp , (s, ϕ) are action-angle variables on V reg , and τ is an involution V hyp → V hyp whose fixed points are some vertices of the hyperbolic atom V hyp .

As is seen from this theorem, a singularity of the second type (i. e. a 4-dimensional corank 1 hyperbolic singularity which is not a direct product) is completely determined by a pair (V hyp , τ ). But it should be noted that the almost direct product representation (described in the theorem) is not uniquely defined: two different pairs (V hyp , τ ) and (V"hyp , τ") may define Liouville equivalent 4-dimensional singularities. This happens if and only if the quotient spaces V hyp /τ and V"hyp /" τ are the same (see [6] for details).

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Topologically we represent these quotients as surfaces with embedded graphs on which we indicate (by stars) the projections of the fixed points of the involution. For example, quotients of the atoms G1 and G3 (see Table 1) by the central symmetry are obviously the same (they define a 4-dimensional singularity denoted by B ∗ in Table 2). Table 2. Four-dimensional corank 1 hyperbolic singularities of small complexity

The list of corank 1 hyperbolic singularities in dimension 4 is presented in Table 2 for complexity 1, 2, 3. Each singularity is presented as a graph Γ (immersed into the plane) some of whose vertices are marked by stars (star-vertices). To reconstruct the singularity from this graph, we first take a regular tubular neighborhood U (Γ) of Γ immersed into the plane. If Γ has no star-vertices, this pair (U (Γ), Γ) is considered as a certain hyperbolic atom V hyp , and then we take the direct product V hyp × V reg . If Γ has some star-vertices, we need first to construct a two-sheeted covering over U (Γ) whose branched points are exactly these star-vertices. Speaking more precisely, we must find a hyperbolic atom V hyp with involution τ such that V hyp /τ = (U (Γ), Γ). After this, we apply the procedure indicated in the second item of Theorem 10.


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Each singularity in Table 2 is given with its notation represented as a vertex of the “Reeb graph” describing the base of the Liouville foliation in a neighborhood of the singularity (see Sections 7.1, 7.2.) The following generalization of Theorem 10 to the multi-dimensional case has been recently obtained by A. S. Lermontova [30]. Theorem 11. Corank 1 hyperbolic singularities in the case of n degrees of freedom can be represented in the form (V hyp × V reg × · · · × V reg )/(Z2 )n−1 , where each Z2 -factor acts on V hyp and on exactly one regular factor V reg (i. e., it acts trivially on all other n − 2 regular factors). In other words, a 2n-dimensional corank 1 hyperbolic singularity is completely determined by a hyperbolic atom V hyp with n − 1 commuting involutions τ1 , . . . , τn−1 . As in the 4-dimensional case, this representation is not unique, and one can consider the quotient space V hyp /(Z2 )n−1 as a surface with embedded graph whose star-vertices are the projections of fixed points of involutions from (Z2 )n−1 . The difference with the case of two degrees of freedom is that now star-vertices can correspond to different involutions from (Z2 )n−1 and, therefore, each star-vertex should be marked by a non-trivial element of the group (Z2 )n−1 . Clearly, if two collections of marks can be transformed to one another by an automorphism of the group (Z2 )n−1 , then they define the same singularities. As is shown in [30], the converse is always true. This allows one to make the list of corank 1 hyperbolic singularities for given dimension 2n and complexity k. For example, the complete list of such singularities for n = 3 and k ≤ 3 can be described as follows. The first part of this list consists of direct products of the singularities from Table 2 by V reg . The second part consists of the following five additional singularities: A∗∗ , B1∗∗ , B2∗∗ with two different star-vertices, A∗∗∗ with three different star-vertices, and A∗∗∗ with star-vertices of two types.

5.2. H YPERBOLIC

SINGULARITIES OF RANK

0

The next example is hyperbolic singularities of rank 0, i. e. 2n-dimensional singularities of type (0, n, 0). The classification of such singularities is non-trivial starting from two degrees of freedom. The first result in this direction is due to L. M. Lerman and Ya. L. Umanskiˇı [28, 29]. This is the semi-local Liouville classification of hyperbolic singularities of rank 0 and complexity 1. Thus, we consider singularities of type (0, 2, 0) in dimension 4, which are also sometimes called saddle-saddle singularities. It is quite natural to start with the topological structure of the singular leaf L. It follows immediately from the decomposition theorem that L can be represented as a 2-dimensional polyhedron glued from squares. In what follows, we denote (oriented) edges of these squares by letters. To reconstruct L we need just to glue together the edges denoted by the same letter (of course, taking into account their orientation).

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Theorem 12 (L. M. Lerman, Ya. L. Umanskiˇı [28]). Suppose that the singular leaf L contains exactly one singular point of type (0, 2, 0) and rank 0. Then L is homeomorphic to one of the four following complexes:

Two saddle-saddle singularities of complexity 1 are semi-locally Liouville equivalent if and only if their singular leaves are homeomorphic. To each saddle-saddle singularity one can naturally assign another topological invariant, the so-called loop molecule, which is rather natural for applications (see, for instance, [3, 7]). We now give its brief description (for details, see [6] and Section 7.3). Consider the image of a neighborhood U (L) of the singular leaf L under the momentum mapping F . The local bifurcation diagram Σ of the mapping F is homeomorphic to a cross (two lines transversally intersecting at the point F(L), see Figure 6). Consider a small loop γ around

Figure 6. Local bifurcation diagram for a saddle-saddle singularity

the point F(L) and its preimage F −1 (γ). It is easy to see that F −1 (γ) is a closed three-dimensional manifold foliated into two-dimensional tori everywhere except those points where γ intersects the bifurcation diagram. These points correspond to hyperbolic singularities of corank 1, which were described above (see Section 5.1). The loop molecule is a graph W whose edges correspond to one-parameter families of Liouville tori and vertices correspond to the bifurcations of these tori or, which is the same, to hyperbolic corank 1 singularities related to four segments of the cross Σ. Taking into account the classification of such singularities (see Theorem 10 and Table 2), we just replace each vertex of W by a certain letter associated with the corresponding bifurcation (or several letters, if Liouville tori can be divided into groups which bifurcate independently).


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The next theorem describes the loop molecules corresponding to the saddle-saddle singularities of complexity 1 listed in Theorem 12. Theorem 13 (A. V. Bolsinov [3]). The loop molecules for saddle-saddle singularities of complexity 1 corresponding to cases 1–4 indicated in Theorem 12 have the following form:

Here, rational marks assigned to edges of the loop molecules contains necessary information on gluing of neighboring Liouville tori of the bifurcations corresponding to vertices (the formal definition of these marks is given in [6], see also Section 7.2). We now turn to the classification of saddle-saddle singularities of complexity 2. In this case the singular leaf L contains exactly two saddle-saddle points. Theorem 14 (A. V. Bolsinov [3]). There exist exactly 39 different (up to the Liouville equivalence) saddle-saddle singularities of complexity 2. The structure of the corresponding singular leaves L is presented in Table 3. Singular leaves are represented in the table in the same way as above, i. e., as a collection of squares which should be glued together along edges marked by the same letters, where all horizontal and all vertical edges are assumed to be oriented from left to right and from bottom to top respectively. Note that some complexes L from Table 3 are homeomorphic. This means that the topology of the singular leaf L does not determine, in general, the topology of its neighborhood U (L). Theorem 15 (V. S. Matveev [5]). The loop molecules of the saddlesaddle singularities of complexity 2 have the form presented in Table 3. Corollary. In the case of saddle-saddle singularities of complexity 2, the loop molecule is a complete invariant of the Liouville foliation in a neighborhood of a given singularity. Note that this is not the case for saddle-saddle singularities of arbitrary complexity (see Problem 12 and an example in Section 7.3).

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Table 3. Saddle-saddle singularities of complexity 2 No

SINGULAR LEAF b

1

α

γ a

β

2

β

α c

δ

β d

b

c

a

d

α

γ a

β d

γ

δ b

β

α c

δ

α

b

c

a

d

a

c

b

d

α

γ a

α c

δ

γ b

β

α d

δ

β

b

d

a

c

a

c

b

d

α

γ a

α c

δ

γ b

β

α d

δ

β

a

c

b

d

a

c

b

d

α

γ a

α c

δ

δ b

β

α d

γ

β

a

c

b

d

a

c

b

d

α

γ a

β

7

γ b a

β

6

d

c

β

5

α d

δ

β

4

a

b

α

3

c

α c

δ

γ b

β

β d

δ

α

a

c

b

d

a

c

b

d

α

γ a

β

β c

δ a

δ b

α c

α d

γ b

LOOP MOLECULE

β d

singularities of integrable hamiltonian systems Table 3. Saddle-saddle singularities of complexity 2 (continued) No

SINGULAR LEAF a

8

α

γ b

β

9

β

α c

δ

β d

b

c

b

c

α

γ a

β d

γ

δ a

β

α d

δ

α

b

c

b

c

a

c

b

d

α

γ a

α c

δ

γ b

β

α d

δ

β

a

c

b

d

a

b

d

c

α

γ a

α b

δ

γ d

β

α c

δ

β

a

b

d

c

a

b

d

c

α

γ a

β b

δ

γ d

α

α c

δ

β

a

b

d

c

a

b

d

c

α

γ a

β

14

γ a b

β

13

d

c

β

12

α d

δ

β

11

b

a

α

10

c

β b

δ

δ d

α

β c

γ

α

a

b

d

c

a

b

d

c

α

γ a

β

β b

δ a

δ d

α b

α c

γ d

LOOP MOLECULE

β c

27

28

a.v. bolsinov

and

a.a oshemkov

Table 3. Saddle-saddle singularities of complexity 2 (continued) No

SINGULAR LEAF a

15

δ

δ c

γ

16

α

δ b

α

γ d

a

b

c

d

δ

α d

β a

α

γ b

δ

δ c

γ

β

a

b

c

d

c

b

d

a

α

γ a

α d

β

γ b

δ

α c

β

δ

c

b

d

a

c

b

c

b

α

γ a

β d

α

δ a

δ

α d

β

γ

c

b

c

b

b

c

a

d

α

γ a

β d

δ

γ b

β

α c

δ

α

b

c

a

d

b

c

b

c

α

γ a

α

21

β a c

α

20

d

b

γ

19

β d

γ

δ

18

c

a

β

17

b

β d

δ

δ a

β

α d

γ

α

b

c

b

c

a

b

c

d

β

β a

α

γ b

α a

δ c

δ b

α d

γ c

LOOP MOLECULE

β d

singularities of integrable hamiltonian systems Table 3. Saddle-saddle singularities of complexity 2 (continued) No

SINGULAR LEAF a

22

β

β a

α

23

δ

β d

δ

α d

a

b

c

d

β

α a

δ b

β

γ c

γ

β d

δ

α

a

b

c

d

a

b

c

d

β

β a

γ b

α

γ c

δ

α d

δ

β

a

b

c

d

a

b

c

d

β

α a

δ b

β

γ c

γ

α d

δ

β

a

b

c

d

a

b

c

d

β

α a

δ b

β

δ c

γ

β d

γ

α

a

b

c

d

a

b

c

d

α

γ b

β

28

γ c c

α

27

d

b

α

26

γ b

α

α

25

c

a

α

24

b

α c

δ

γ d

β

α a

δ

β

b

c

d

a

a

b

c

d

α

γ b

β

α c

δ d

γ d

β a

α a

δ b

LOOP MOLECULE

β c

29

30

a.v. bolsinov

and

a.a oshemkov


SINGULAR LEAF a

29

β

β a

α

30

δ

α d

γ

β d

a

b

c

d

β

β a

γ b

α

γ c

δ

β d

δ

α

a

b

c

d

a

b

c

d

β

α a

δ b

β

γ c

γ

β d

δ

α

a

b

c

d

a

b

c

d

β

β a

γ b

α

γ c

δ

α d

δ

β

a

b

c

d

a

b

c

d

β

α a

δ b

β

γ c

γ

α d

δ

β

a

b

c

d

a

b

c

d

β

α a

α

35

δ c c

α

34

d

b

α

33

γ b

α

α

32

c

a

α

31

b

δ b

β

δ c

γ

β d

γ

α

a

b

c

d

a

b

c

d

α

γ b

β

α c

δ c

γ d

β d

α a

δ a

LOOP MOLECULE

β b


31


SINGULAR LEAF a

36

α

γ d

β

37

α a

δ

d γ

b β

α c

δ

β

b

c

d

c

d

a

b

α

α a

γ b

δ

γ c

β

α d

β

δ

c

d

a

b

c

d

a

b

α

β a

γ

39

c

a

δ

38

b

δ b

δ

γ c

β

α d

α

γ

c

d

a

b

c

a

d

b

α

γ a

δ

β d

α d

δ c

γ b

α b

β c

LOOP MOLECULE

δ a

Some atoms appearing in the loop molecules from Table 3 have complexity 4 (and hence were not presented in Table 1 or Table 2). All of them are described in Figure 8. Remark 5. Saddle-saddle singularities of complexity 3 can be classified up to the Liouville equivalence in a similar way. This classification was carried out by N. A. Maksimova by computer analysis. It turns out that the total number of such singularities is 256. In principle, there is an enumeration algorithm for saddle-saddle singularities of arbitrary complexity (see, for example, [33]). But the number of singularities grows very fast as complexity increases. It is interesting that the classification Theorems 12 and 14 were proved several years before Zung’s decomposition theorem. We now give the description of the same singularities in terms of almost direct products. In the case of complexity 1 we have the following four representations corresponding to the four singularities described in Theorem 12 (here, in the first three cases we use the standard representations and notation for

32

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atoms from Table 1, and in the fourth case we consider the representation of the atom C2 shown in Figure 7): 1) the direct product B × B; 2) (B × C2 )/Z2 , where Z2 acts on each factor as the central symmetry; 3) (B × D1 )/Z2 , where Z2 acts on each factor as the central symmetry; 4) (C2 × C2 )/(Z2 × Z2 ), where one generator of the group Z2 × Z2 acts on the first factor as the symmetry relative to the x-axis and acts on the second one as the symmetry relative to the z-axis, and the second generator, on the contrary, acts on the first factor as the symmetry relative to the z-axis and acts on the second one as the symmetry relative to the x-axis.

Figure 7. Symmetric representation of the atom C2

It is an interesting fact that the first three singularities from the above list occur in rigid body dynamics. The first and third occur in the Kovalevskaya case, the second occurs in the Goryachev–Chaplygin–Sretenskiˇı case. In order to formulate the result about almost direct product representations for saddle-saddle singularities of complexity 2, one needs to describe some atoms of complexity 4 and their “symmetry groups”. Let us explain what we mean by symmetries of an atom. Since an atom (V, ω, H) (see Remark 2) is defined as a class of the semi-local Liouville equivalence, it could be natural to take all homeomorphisms V → V preserving the Liouville foliation. But we want to consider only those symmetries which can be used for almost direct product representations of singularities. So, by the definition of almost direct products, they should be of finite order and preserve the function H : V → R and the orientation of V . This leads to the following definition: the symmetry group of an atom (V, ω, H) is the quotient of the group of all homeomorphisms V → V preserving H and the orientation by the subgroup of homeomorphisms isotopic to the identity. Considering an atom as a pair (V, Γ), one can also describe the same notion in terms of the pair itself. The point is that each Morse function H : V → R with the only critical level Γ = H −1 (0) defines the partition of the set of annuli V \ Γ into positive and negative ones according to the sign of H , and this partition does not depend on the choice of such a function.

33


I2

J2

L1

K1

M1

L2

M2

V4!

V4

(a) Atoms with the symmetry group Z2 = !α"

C1

C1

I1

J1

(b) Atoms with the symmetry group Z4 = !γ"

K2 Z2 × Z2 = !α" × !β" β(1, 2, 3, 4, 5, 6, 7, 8) = (6, 5, 8, 7, 2, 1, 4, 3)

K3 Z4 × Z2 = !γ" × !β" β(1, 2, 3, 4, 5, 6, 7, 8) = (7, 8, 5, 6, 3, 4, 1, 2)

P4 D4 = Z4 ! Z2 = !γ" ! !β" β(1, 2, 3, 4, 5, 6, 7, 8) = (7, 6, 5, 8, 3, 2, 1, 4)

(c) Atoms with other symmetry groups Figure 8. Atoms from Tables 3 and 4 (with their symmetry groups)

34

a.v. bolsinov

and

a.a oshemkov

Thus, one can say that symmetries of an atom (V, Γ) are homeomorphisms preserving the orientation and this partition considered up to isotopy. All necessary atoms are presented in Figure 8(a,b,c) (or in Table 1). We also indicate there the symmetry groups of the atoms and their generators. Moreover, the atoms themselves are presented in a symmetric form in order to make their symmetry groups more visual. In all the cases, α denotes the central symmetry of an atom, and for each atom in Figure 8(a) it is the only non-trivial symmetry (as well as for the atoms B and D1 from Table 1). Similarly, in all the cases, γ denotes the rotation of an atom through the angle π/2, and for each atom in Figure 8(b) it is the generator of the symmetry group Z4 . Finally, β denotes an additional symmetry of an atom which has a different meaning for different atoms. Three such atoms are presented in Figure 8(c), where we indicate the symmetry β explicitly by showing how it acts on the edges of the graph Γ (from this information, the action of β on the whole atom is uniquely reconstructed). For the atom C2 , the symmetry group is Z2 × Z2 , whose generators α and β are the symmetries relative to the x-axis and y-axis respectively (here we use the representation of the atom C2 shown in Figure 7). Now we can describe the almost direct product representations for all saddle-saddle singularities of complexity 2 (i. e., for the 39 cases listed in Table 3). This result obtained by V. V. Korneev [26] is presented in Table 4. In the second column of Table 4, we indicate the numbers that are assigned to the same singularities in Table 3. The numbers in Table 3 and Table 4 turn out to be different because of using two essentially different approaches: in Table 3 the singularities are ordered according to their “l-types” (the definition of this invariant can be found in [3, 5]; roughly speaking, it describes the Liouville foliation on the set of critical points of the momentum mapping), whereas in Table 4 they are ordered according to the types of the groups G acting on direct products. In the third column of Table 4, we indicate the atoms that are the factors in the almost direct products. The groups G acting on the direct products of these atoms are indicated in the last column. All of these groups are commutative, with the only exception in the last case when G is the dihedral group D4 . In each case, the group G has at most two generators. The component-wise action of these generators is described in the fourth column. For example, in case 32, the first generator e1 of the group G = Z2 × Z2 acts on the direct product C2 × P4 as follows: e1 (C2 × P4 ) = α(C2 ) × γ 2 (P4 ).

This means that, on the first component (i. e., on C2 ), the generator e1 acts as the symmetry α, and, on the second component (i. e., on P4 ), it acts as the symmetry γ 2 . In this case, α is the central symmetry of C2 , and γ is the rotation of P4 through the angle π/2 (in particular, γ 2 is also the central symmetry). Similarly, the second generator e2 of the group G = Z2 × Z2 acts according to the rule e1 (C2 × P4 ) = β(C2 ) × β(P4 ).

35

singularities of integrable hamiltonian systems Table 4. Saddle-saddle singularities of complexity 2 as almost direct products No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

NUMBER FROM TABLE 3 4 11 22 30 3 15 24 32 31 34 26 25 33 13 6 21 29 1 17 8 37 10 5 14 7 12 23 27 35 36 28 38 18 20 19 9 2 39 16

FACTORS

ACTION

GROUP

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

trivial trivial trivial trivial ( γ2 , α ) ( α , α ) ( α , γ2 ) ( α , α ) ( α , α ) ( α , α ) ( α , α ) ( α , γ2 ) ( α , α ) ( α , α ) ( α , α ) ( α , α ) ( α , α ) ( γ2 , α ) ( α , α ) ( α , αβ ) ( α , α ) ( α , αβ ) ( α , α ) ( α , γ2 ) ( α , β ) ( α , γ2 ) ( α , β ) ( γ , γ ) ( γ , γ ) ( γ , γ ) ( γ , γ ) γ2 ) , ( β , β ), ( β , γ3β ) , ( β , β ), ( β , βγ 2 ) , ( β , βγ 2 ) , ( β , γ ), ( β , γ ) , (γ 3 β,

{e} {e} {e} {e} Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z4 Z4 Z4 Z4 Z2 × Z2 Z2 × Z2 Z2 × Z2 Z2 × Z2 Z2 × Z2 Z2 × Z2 Z4 × Z2 D4

B B B B C1 D1 B B B B B B B B B B B C1 C2 C2 D1 D1 B B B B B C1 C1 C1 C1 C2 C2 C2 C2 C2 C2 K3 P4

C1 C2 D1 D2 D1 D1 I1 I2 K1 L1 L2 J1 J2 M1 M2 V4 V4! C2 C2 C2 C2 C2 K2 K3 K3 P4 P4 I1 K3 P4 J1 P4 P4 P4 K2 K3 K3 K3 P4

( ( ( ( ( ( ( (

α α α α α α γ γ

, , , , , , , ,

β ) βγ 2 ) βγ 3 ) αβ ) β ) γ2 ) βγ 2 ) β )

36

a.v. bolsinov

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a.a oshemkov

Table 5. Saddle-saddle-saddle singularities of complexity 1 No

FACTORS

1

B×B×B

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

C2 × B × C2 B × B × D1 B × B × C2

ACTION

GROUP {e}

trivial

Z2 × Z2

(α, id, αβ), (αβ, id, α) (α, id, α)

Z2

(α, id, α)

Z2

C2 × C2 × C2

(α, id, αβ), (id, α, αβ), (αβ, αβ, α)

B × C2 × C2

(id, α, αβ), (α, αβ, α)

D1 × C2 × C2 C2 × D1 × C2 D1 × D1 × B B × D1 × C2

(α, id, αβ), (id, α, αβ), (id, αβ, α)

(α, id, α), (id, α, α)

B × C2 × C2 C2 × C2 × B

D1 × B × C2

C2 × C2 × C2

Z2 × Z2

(id, α, αβ), (α, id, α)

D1 × C2 × B

C2 × C2 × C2

Z2 × Z2

(α, id, αβ), (id, α, αβ), (αβ, id, α)

Z2 × Z2

(id, α, αβ), (α, id, α)

Z2 × Z2

(α, id, α), (id, α, α)

Z2 × Z2

(α, id, αβ), (id, α, α) (α, αβ, αβ), (αβ, α, αβ), (αβ, αβ, α) (α, αβ, id), (αβ, α, αβ), (αβ, αβ, α)

19

P4 × P4 × B

(β, γβ, α), (γ β, γβ, id), (γβ, β, α), (γβ, γ 2 β, id)

20

C2 × C2 × P4

(α, αβ, id), (αβ, α, id), (αβ, αβ, β), (αβ, id, γ 2 β)

21

C2 × C2 × C2

(α, αβ, id), (αβ, α, αβ), (αβ, id, α)

2

C2 × C2 × B

(α, αβ, α), (αβ, α, α)

23

C2 × C2 × P4

(α, αβ, id), (αβ, α, id), (αβ, αβ, β), (id, αβ, γ 2 β)

24

D1 × C2 × B

(α, αβ, α), (id, α, α)

26 27 28 29 30 31 32

(β, αβ, id), (γ β, id, α), (id, α, α)

D1 × B × C2

(α, α, αβ), (id, α, α)

P4 × B × C2 P4 × B × C2 C2 × P4 × B

C2 × C2 × C2 C2 × C2 × B

2

(β, αβ, α), (γ β, id, α), (id, α, α)

G16 Z2 × Z2 × Z2 × Z2 Z2 × Z2 × Z2 Z2 × Z2 × Z2 × Z2 Z2 × Z2 × Z2 Z2 × Z2 × Z2 Z2 × Z2

2

(β, α, id), (γ β, id, αβ), (id, α, α) (β, α, αβ), (γ 2 β, id, αβ), (id, α, α) 2

(α, id, α), (αβ, β, α), (id, γ β, α) (α, αβ, id), (id, α, αβ), (αβ, id, α) (α, αβ, α), (id, α, α)

Z2 × Z2 × Z2

Z2 × Z2

2

P4 × C2 × B P4 × C2 × B

Z2 × Z2 × Z2

Z2 × Z2

22

25

Z2 × Z2 × Z2 Z2 × Z2

C2 × C2 × C2

18

Z2 × Z2 × Z2 Z2 × Z2

(α, id, α), (id, α, α) (α, id, α), (id, α, α)

Z2 × Z2 × Z2 Z2 × Z2

(α, id, αβ), (id, α, αβ), (αβ, id, α)

C2 × D1 × B

Z2 × Z2 × Z2

Z2 × Z2 × Z2 Z2 × Z2 × Z2 Z2 × Z2 × Z2

Z2 × Z2 × Z2 Z2 × Z2


37

The next result is the classification of six-dimensional hyperbolic rank 0 singularities of complexity 1. We will call such singularities (i. e. singularities of type (0, 3, 0) and rank 0) saddle-saddle-saddle singularities. Theorem 16 (V. V. Kalashnikov [25]). In dimension 6, there exist exactly 32 saddle-saddle-saddle singularities of complexity 1 (up to the Liouville equivalence). Their almost direct product represenations are listed in Table 5. It is worth noticing that all these singularities are produced from the following four atoms only: B, D1 , C2 , P4 . These atoms and their symmetries are described above (see Figures 7 and 8 and comments to them). The above examples show that to get a complete list of hyperbolic rank 0 singularities is a quite non-trivial combinatorial problem, and it is hard to expect its reasonable solution for arbitrary complexity and dimension. However, we think that some particular cases are worth discussing. Problem 3. To classify hyperbolic rank 0 singularities of complexity 1 in dimension 2n > 6. It is probable that the number l(n) of such singularities increases as n → ∞ (for n = 1, 2, 3, this number is 1, 4, 32 respectively). Is it possible to find explicit formulas for l(n) or at least to obtain both estimates from below and from above? For applications it would be also interesting to find a complete list of hyperbolic singularities of small complexity for low dimensions. We distinguish here the case of hyperbolic singularities because of its specific combinatorial nature. However the other types of singularities in dimension higher than 2 are also quite interesting. Here is, for example, one of the natural questions: how many topologically different (i. e., non-equivalent in the sense of the Liouville equivalence) singularities of type (0, 0, 2) are there for complexity 1? According to the decomposition theorem, such a singularity must be a quotient of the direct product of two focus-focus singularities by an action of a finite group. It is not hard to construct a desired singularity in the form V foc (p) × V foc (q)/Zpq , where V foc (p) and V foc (q) are “basic” focus-focus singularities of complexities p and q respectively and the action of Zpq is effective on each factor. But it is not clear whether all of them are different or, on the contrary, Liouville equivalent. A more general problem is as follows. Problem 4. To classify non-degenerate rank 0 singularities of complexity 1 for dimension 2n > 4. For n = 1 the answer is trivial; for n = 2 the classification was first obtained by L. M. Lerman and Ya. L. Umanskiˇı [28,29], but now easily follows from the decomposition theorem. It is clear that the classification problem is closely connected with the uniqueness problem for the almost direct product representation of a given singlarity. In [41] Nguyen Tien Zung introduced a notion of a minimal model (or canonical model). This is an almost direct product (V1 × · · · × Vk )/G such that each element g ∈ G acts non-trivially at least

38

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on two components of the product V1 × · · · × Vk . The point is that any almost direct product is naturally reduced to a minimal model. Indeed, if we take the normal subgroup Gi ⊂ G formed by the elements acting trivially on all the factors except Vi , then Gi acts on Vi freely, and we can consider the quotient V"i = Vi /Gi . This quotient can be considered as a new factor, and the original almost direct product is reduced to a simpler one, namely: (V1 × · · · × Vi × · · · × Vk )/G = (V1 × · · · × V"i × · · · × Vk )/(G/Gi ).

After such reductions each model becomes, finally, minimal. Unfortunately even minimal models are not uniquely defined for some non-degenerate singularities. It seems, however, that singularities of rank 0 admit a unique minimal representation. But the strong proof of this result has not yet appeared. Problem 5. Which non-degenerate singularities admit a unique minimal model? Does the uniqueness of the minimal model take place for singularities of rank 0? If a certain singularity has several minimal models, how many and of what kind can they be? Another question is about topological stability of non-degenerate singularities. Notice that non-degenerate singularities of corank 1 are stable only in the case of complexity 1. More precisely, according to [40], any non-degenenerate corank 1 singularity of complexity k can be decomposed (by a small integrable perturbation of the system) into k singularities of complexity 1. For singularities of higher corank this statement is not true. Problem 6. To describe all stable non-degenerate singularities.

5.3. S PLITTABLE

SINGULARITIES

We now give several examples of non-degenerate singularities which do not decompose into an almost direct product. Example 1. Singular leaf with points of different rank. Let us describe a non-degenerate, but splittable singularity. In Figure 9 we show a singular leaf L (of the Liouville foliation for a system with two degrees of freedom) which contains one focus-focus point and a hyperbolic circle. The circle is the curve along which the 2-torus touches the sphere with two points identified. These points just give a focus-focus singularity. It is clear that the bifurcation diagram of this singularity is a smooth curve passing through the point that is the projection both of the focus-focus point and the hyperbolic circle. In the sense of Definition 12, this singularity is splittable. It would have satisfied the non-splitting condition if the local bifurcation diagram had consisted of the only point. But, in the case under consideration, there is an additional piece, the curve passing through this point. Here we see one of the mechanisms generating splittable singularities. The appearance of the additional curve is caused by the fact that, together with a focus-focus point, the singular leaf contains a closed orbit of


39

the Hamiltonian R2 -action. In general, the situation is similar: our assumption forbids the existence of closed orbits of rank greater than r on a singular leaf L of rank r. Note that, in the above example, the focus-focus point and one-dimensional closed orbit can be moved onto different leaves by a small perturbation of the Hamiltonian action of R2 . As a result, the singularity splits into two simpler singularities each of which satisfies the non-splitting condition.

Figure 9. Splittable singularity with focus-focus point and hyperbolic circle

Figure 10. Hyperbolic singularity of type D2

Example 2. Splittable singularity with two elliptic-hyperbolic points. To describe another example of a non-degenerate splittable singularity, let us consider the two-dimensional surface P in R3 (x, y, z) shown in Figure 10. The height function f (x, y, z) = z has the only critical value z = 0 on this surface, and the corresponding singular level contains two saddle critical points a = (x1 , y1 , 0) and b = (x2 , y2 , 0). Let P be a regular level surface of some smooth function H , i. e., P = {H(x, y, z) = 0}. Consider the four-dimensional Euclidean space as a symplectic manifold M 4 = R4 (x, y, u, v) with the symplectic structure ω = dx ∧ dy + du ∧ dv. Take two more functions " and f" on M 4 by setting H " = H(x, y, u2 + v 2 ), H

f" = f (x, y, u2 + v 2 ).

" and f" commute on M 4 and define the momentum mapping The functions H 4 2 " f"). The points " F : M → R (H, a = (x1 , y1 , 0, 0) and "b = (x2 , y2 , 0, 0) are non-degenerate critical points of type (1, 1, 0) for the Liouville foliation " f". Both points belong to the same singular leaf L of on M 4 defined by H, the Liouville foliation. The bifurcation diagram of F is presented in Figure 11(a). The set of critical points of the momentum mapping consists of three components. The first one is the 2-plane that consists of the points (x, y, 0, 0). All such points are critical for the function f". The second and third components are also two-dimensional and are generated by the points a and b. Under the momentum

40

a.v. bolsinov

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mapping F , the first component is mapped onto the horizontal straight line (Figure 11), the boundary of the upper half-plane. The second and third components are mapped onto the left and right rays respectively. Local bifurcation diagrams for each of the points " a and " b are shown in Figure 11(b). It is seen that none of them coincides with the whole bifurcation diagram, which consists of the horizontal straight line with both rays. Therefore, this singularity does not satisfy the non-splitting condition.

Figure 11. Bifurcation diagrams

Figure 12. Perturbation splitting the singularity

As in the preceding example, the described singularity of the Liouville foliation can be split into two non-splittable singularities by small perturbation of the Hamiltonian action of R2 . Such a perturbation is illustrated in Figure 12. Example 3. Non-analytic case. Another mechanism for generating splittable singularities occurs in the smooth case. This is splitting of the bifurcation diagram at a singular point. Here we can see the difference from the analytic case, where such a mechanism does not work. It turns out that there may be situations when, under an analytic perturbation, a non-degenerate singularity does not change its topological type, whereas, under a suitable smooth perturbation, the topological type changes. This is connected with the fact that non-critical levels of the-function f (x, y) = xy are not connected, but consist of two components, that allows one to perturb the momentum mapping on these components independently. For example, in the case of two degrees of freedom, such a perturbation can be described as follows. Consider two functions f1 = p1 q1 and f2 = p2 q2 + λ(p1 , q1 ) in R4 with standard symplectic structure ω = dp1 ∧ dq1 + dp2 ∧ dq2 , where λ(p1 , q1 ) is a function commuting with f1 . Note that in the analytic case this implies that λ(p1 , q1 ) is a function of f1 , i. e., λ(p1 , q1 ) = g(p1 q1 ). The critical points of the momentum mapping defined by the functions f1 , f2 fill two planes {q1 = p1 = 0} and {q2 = p2 = 0}, and the corresponding bifurcation diagram consists of the straight line {f1 = 0} and the curve {f2 = g(f1 )}. For example, if λ(p1 , q1 ) ≡ 0, then the bifurcation diagram consists of two straight lines {f1 = 0} and {f2 = 0} (Figure 13(a)).


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Now consider (C ∞ -smooth, but non-analytic) function λ given by the formula ( h(p1 q1 ) for p1 > 0, q1 > 0, λ(p1 , q1 ) = 0 in the other cases, where the function h(x) has zero of infinite order at x = 0. Here the bifurcation diagram consists of two straight lines {f1 = 0}, {f2 = 0} and the curve {f2 = h(f1 )} (Figure 13(b)).

Figure 13. Non-analytic splittable singularity

It is seen that this smooth perturbation splits one of the curves of the bifurcation diagram Σ into two curves which touch each other at a singular point of Σ (with infinite tangency order). Analytic perturbations do not allow us to produce such an effect. Example 4. Splittable singularity with saddle-saddle points. Consider the simplest non-degenerate hyperbolic singularity of type B (Figure 1). We assume B to be a subset of a plane with symplectic coordinates q, p such that ω = dp ∧ dq. Let f (p, q) be the corresponding Hamiltonian with the singular leaf (eight-figure) given by f = 0. Let us introduce a closed 1-form α = Q(p, q)dq + P (p, q)dp, which will be used as an analog of angle-variable, with the following properties: ∂f 1) Q ∂f ∂p − P ∂q = α(sgrad f ) > 0 everywhere except for the singular point S ); ) 2) Γ α = Γ α = 1, where Γ1 and Γ2 are the edges of the singular 1

2

leaf (i. e., the halves of the eight-figure); in particular, α is integer-valued in cohomologies. Equivalently, instead of α we can consider a multi-valued function g(q, p) on B such that dg = α and, in addition, the values of g at the singular point S are integer. It is clear that such a function g exists. Now we take three identical copies of such singularity B1 , B2 , B3 with singular points S1 , S2 , S3 , Hamiltonians f1 (q1 , p1 ), f2 (q2 , p2 ), f3 (q3 , p3 ), and 1-forms α1 (q1 , p1 ), α2 (q2 , p2 ), α3 (q3 , p3 ) (or multi-valued functions gi (qi , pi )). Let us construct from these components the direct product singularity M 6 = B1 × B2 × B3 with the symplectic form dp1 ∧ dq1 + dp2 ∧ dq2 + dp3 ∧ dq3 and three commuting functions f1 (q1 , p1 ), f2 (q2 , p2 ), f3 (q3 , p3 ).

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To describe the desired singularity with two degrees of freedom we, in fact, make the reduction of this three degrees of freedom system with respect to the integral f1 + f2 + f3 . More precisely, the construction is as follows. Consider a subset X 4 in M 6 given by the following two equations: f1 + f2 + f3 = 0,

g1 + g2 + g3 =

1 mod Z. 2

Since the forms dgi = αi are integer, we obtain a four-dimensional closed subset in M 6 . It is easy to see that X 4 ⊂ M 6 is a four-dimensional submanifold. The smoothness of X 4 follows from the fact that the forms df1 + df2 + df3 and α1 + α2 + α3 are linearly independent on X 4 . Indeed, the condition αi (sgrad fi ) > 0 on Bi \ Si implies that these forms are dependent only at the point S1 × S2 × S3 , but this point is outside of X4 . Furthermore, X 4 is symplectic because the Poisson bracket of g1 +g2 +g3 and f1 + f2 + f3 is non-zero on X 4 . To see this, we notice that {g1 +g2 +g3 , f1 +f2 +f3 } = {g1 , f1 } + {g2 , f2 } + {g3 , f3 } = α1 (sgrad f1 ) + α2 (sgrad f2 ) + α3 (sgrad f3 ) > 0. One can show that the restrictions of f1 , f2 , f3 to X 4 commute with respect to the symplectic structure restricted to X 4 . Thus we have an integrable system on X 4 given by any two of these three functions. We claim that this new reduced system has a non-degenerate singularity of type (0, 2, 0) (saddle-saddle singularity), but it is not of almost direct product type. To prove the last statement it suffices to look at the topological structure of the singular leaf L red ⊂ X 4 . As in the standard situation, this leaf is a two-dimensianal polyhedron and the dimensions of its cells are stratified by the rank of the momentum mapping. But in the case of almost direct product singularities, all 2-cells are squares (since they also must be direct products!). In our example it is not the case. Indeed, the singular leaf L red can be considered as a subset in the three-dimensional singular leaf L of the original system on M 6 defined by the condition g1 + g2 + g3 = 12 mod Z. This singular leaf L is the direct product of three copies of the eight-figure. This means that L is glued from eight cubes each of which can be considered as the direct product of three edges of the eight-figures (one from each of three copies). Take one of these cubes and denote the corresponding edges by Γ1 , Γ2 , Γ3 . By construction, gi increases monotonically on Γi taking all the values from 0 to 1. So, we may think of gi as a coordinate on Γi and, therefore, the triple (g1 , g2 , g3 ) can be taken as a standard coordinate system on a unit cube. The two-dimensional cells of L red are just the intersections of the cubes with the “hypersurface” g1 + g2 + g3 = 12 mod Z. Inside the cube Γ1 × Γ2 × Γ3 = [0, 1] × [0, 1] × [0, 1] this equation defines two triangles and one hexagon, but no squares!

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Thus, the singular leaf L red is glued from 8 hexagons and 16 triangles and, therefore, is not of almost direct product type. In Figure 14(a) we show the singular leaf L red in the same manner as in Section 5.2. Identifying edges marked by the same letters, one obtains a two-dimensional complex homeomorphic to the singular leaf L red . The same complex can also be described as follows (see Fifure 14(b)). Its 1-skeleton has 6 vertices (0-cells) and 24 edges (1-cells), which may be viewed as vertices and edges of the octahedron (where two oppositely oriented 1-cells correspond to each edge of the octahedron). All 16 triangular 2-cells are glued to this 1-skeleton along 16 oriented cycles of length 3 (i. e., each such 2-cell corresponds to an oriented face of the octahedron). Each of 8 hexagonal 2-cells also corresponds to a face of the octahedron and is glued to the 6 oriented edges of that face in such a way that each two heighboring edges have different orientations with respect to orientaion on the boundary of the hexagon. a!4 a3 a!2 a1 c!1 b!1

c1

a!1

b2 a2 b!2

c1

c!1 a!2

b1 a1 b!4

c!4

c4

a!1

a2

c!4

b3 b!3

c4

b!2 c2 b1 a4 b!1 c2 b2 a3 b!3 c3 b4 a4 ! b4 c3 b3 a3

b!1 c!2 b2 a!3 b!2

b!1

c1

c!2

b1

b1 a!4 b!4 c!3 b3 a!3

b!2

a4

a!1

a3 a2

b!4 b4

a!3

a!4 a!2

a1

b2

c!2

c!1

b!3 c!3

c2

c4 c!4

c!3

b4

c3

b3 b!3

a!4

a!2

b4 a1

(a)

(b)

Figure 14. Singular leaf for the reduced system

The bifurcation diagram of the momentum mapping (f1 , f2 ) : X 4 → R2 is shown in Figure 15(a). It consists of three straight lines intersecting at one point. Since the non-splitting condition (for saddle-saddle points) requires the bifurcation diagram to be locally a cross (i. e., two intersecting lines), it fails for our example.

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The bifurcations of Liouville tori corresponding to the three lines of the bifurcation diagram are described by the atoms B (Figure 1). The loop molecule for the point of intersection of this lines is shown in Figure 15(b).

Figure 15. Bifurcation diagram and loop molecule for the reduced system

It should be noted that the obtained system is not stable in the sense of Definition 9. The perturbation which decomposes the singularity into simpler ones is quite natural: in our construction we should just replace the condition f1 + f2 + f3 = 0 by f1 + f2 + f3 = ε. For ε = 0 we obtain our example, but if ε '= 0, then the straight lines of the bifurcation diagram pass to a “general position”: they intersect one another, but have no common point. As a result, the singular leaf L red splits into three direct product type singularities. Problem 7. Do there exist topologically stable non-degenerate saddlesaddle singularities similar to the example described in this section? Let us note that all the above examples of splittable singularities do not seem to be typical for integrable systems in geometry, physics, and mechanics. The non-degenerate singularities which occure in concrete examples are usually non-splittable and, therefore, of almost direct product type.

6. S YMPLECTIC 6.1. A CTIONS

INVARIANTS

AND AFFINE STRUCTURE

Let us be given two integrable Hamiltonian systems (M, ω, f1 , . . . , fn ) ', ω and (M " , f"1 , . . . , f"n ). How one can recognize whether they are symplectically equivalent or not? First of all, they must be Liouville equivalent. So, we will assume from the very beginning that this condition is satisfied, i. e., ' of the corresponding Liouville there exists a diffeomorphism Ψ : M → M foliations. What are additional invariants for the symplectic equivalence? When is it possible to modify the diffeomorphism Ψ so that it becomes a symplectomorphism of the Liouville foliations?


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First of all, one should notice that any symplectomorphism of Liouville foliations must “preserve” action variables. Recall that to each regular leaf (i. e., Liouville torus T n ) we can assign a set of actions s1 (T n ), . . . , sn (T n ), which depend smoothly on the torus T n and, therefore, can be considered as smooth functions on the base of the Liouville foliation. Notice that these functions are defined not uniquely but modulo the natural action of the group GL(n, Z) " Rn (the semidirect product of the group of linear transformations preserving a lattice Zn in Rn and the commutative group of all translations in Rn ). This means exactly that on the base of the Liouville foliation (more precisely, on its smooth part) we obtain a natural integer affine structure. ' is a diffeomorphism of the Liouville foliations, then we If Ψ : M → M can naturally reduce it to the level of the bases Y and Y" : ψ : Y → Y" ,

' → Y" are natural so that ψ ◦ π = π " ◦ Ψ, where π : M → Y and π ": M projections. Obviously, if Ψ is a symplectomorphism, then ψ “preserves” the action variables (or, what is the same, ψ transforms the affine structures on Y and Y" one to the other). Thus, the affine structure on the base Y is a symplectic invariant of the Liouville foliation. For the first time, this was observed by J. J. Duistermaat in his famous paper [13]. Notice that in a neighborhood of a regular leaf this invariant is obviously trival. Indeed, the actions give a regular coordinate system on the base, and two regular coordinate systems can be transformed one to the other by an appropriate diffeomorphism. However, this invariant becomes non-trivial if we consider the global structure of the Liouville foliation (see Section 7) or if we study it in a neighborhood of a singular leaf L. Indeed, in a neighborhood of the point y = π(L) ∈ Y for a singular leaf L, the situation is more complicated, since the action variables s1 , . . . , sn are not smooth any more. Thus, the question on the existence of a smooth mapping ψ : Y → Y" with the property si = s"i ◦ ψ becomes non-trivial. Suppose, for a moment, that we can answer this question. If the answer is “no”, then the given systems are not (symplectically) equivalent; if the answer is “yes”, then we obtain the following natural question. " (L) " be a diffeomorphism of the Liouville Problem 8. Let Ψ : U (L) → U foliations in neighborhoods of singular leaves which “preserves” the actions, i. e., si = s"i ◦ ψ. Is it possible to “deform” Ψ into a symplectomorphism of the Liouville foliations? More precisely, does there exist a smooth homotopy " (L) " such that Ψ1 is a symplectomorphism, Ψ0 = Ψ, and Ψt : U (L) → U t ψ = ψ for all t ∈ [0, 1]?

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It turns out that in the case of the simplest singularities the answer to this question is positive. If L is a singular leaf of elliptic type, then this fact is just a reformulation of the Eliasson theorem. We now briefly discuss the hyperbolic and focus-focus cases completely studied by J.-P. Dufour, P. Molino, A. Toulet [12] and S. V˜ u Ngo.c [52].

6.2. H YPERBOLIC

CASE

( ONE

DEGREE OF FREEDOM )

Consider the simplest hyperbolic one degree of freedom singularity, i. e., the atom B in our notation (see Figure 1). In this case, the Liouville foliation is defined by a Morse function H : V → R with an eight-figure singular level L = {H = 0}. Its neighborhood V = H −1 [−ε, ε], after removing the singular leaf L, is decomposed into three famillies of one-dimensional Liouville tori T1 (h), T2 (h), T3 (h). Without loss of generality, we assume that for h ∈ (−ε, 0), the level {H = h} is connected and T1 (h) = {H = h}. Then for h ∈ (0, ε), the level {H = h} consists of two components, which we denote by T2 (h) and T3 (h). The base of the Liouville foliation is a graph with one vertex and three edges incident to it. The parameter h can be considered as a coordinate on the edges of this graph. For each family of Liouville tori Ti (h), we can naturally define the action variable by the formula * α, si (h) = Ti (h)

where α is the action 1-form such that dα = ω. The action form is not uniquely defined. To make the choice of α unambigiuos we will assume that si (h) → 0 as h → 0. As we already noticed, a symplectomorphism of Liouville foliations preserves action variables: if Ψ : V → V" is such a symplectomorphism, then for each regular torus T and its image T" = Ψ(T ) we have s(T ) =

*

T

α=

*

T

α " = s"(T"),

where α = Ψ∗ (" α). So, the three functions si (h) are, in essence, symplectic invariants of the singularity. The only problem is that the parameter h in our construction is determined by the choice of the Hamiltonian H which is not unique (or, which is the same, by the parametrization of the edges of the base of the Liouville foliation). For example, we can replace H by any function of the form f (H). However, by using the Morse–Darboux lemma, one can choose H in an “almost” canonical way.

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Lemma. The Hamiltonian H can be chosen in such a way that the asymptotics of the action variables si (h) are as follows: s1 (h) = 2h ln |h| + c1 (h),

s2 (h) = h ln |h| + c2 (h),

s3 (h) = h ln |h| + c3 (h),

where c1 (h) : R− → R, c2 (h) : R+ → R, c3 (h) : R+ → R are smooth functions such that the function ( c1 (h), if h ≤ 0, c(h) = if h > 0 c2 (h) + c3 (h),

is smooth in a neignborhood of zero. The functions ci in these decompositions are defined up to a flat function. In particular, the Taylor expansions of ci are uniquely defined. Remark 6. If H is just an arbitrary Morse function which defines the foliation in question, then the formulas for si are quite similar. Namely, s1 (h) = 2a(h) ln |h| + b1 (h),

si (h) = a(h) ln |h| + bi (h),

i = 2, 3,

where a(h) is a positive smooth function such that a(0) = 0 and a# (0) '= 0. Since the action variables are preserved under symplectomorphisms, the Taylor expansions of ci I2 =

∞ + k=1

(k)

c2 hk ,

I3 =

∞ +

(k)

c3 hk

k=1

are symplectic invariants of the singular foliation in question. Moreover, the following classification theorem holds. Theorem 17 (J.-P. Dufour, P. Molino, A. Toulet [12]). Two hyperbolic " of type B are semi-locally singularities L = (V, ω, H) and L" = (V" , ω " , H) " and I3 (L) = I3 (L). " symplectically equivalent if and only if I2 (L) = I2 (L) Note that the Taylor expansion of c1 is a symplectic invariant as well, but we can omit it because this series is just the sum of I2 and I3 . As we see, the above theorem says that, in essence, the only semi-local symplectic invariant of the singularity B is the action. In particular, the following statement holds. Let us consider two Liouville foliations (V, ω, H) " of type B. They are symplectomorphic if and only if there and (V" , ω " , H) exists a diffeomorphism Ψ : V → V" such that si = s"i ◦ ψ (for an appropriate choice of the action variables si and s"i ). If the singularity contains several critical points on the singular leaf, then the classification theorem becomes more complicated. The asymptotics of the action function give us, of course, invariants similar to I2 and I3 . But they are not enough to solve the classification problem. Roughly speaking, in addition, we need to take into account how the action is “distributed” between singular points and separatrices. This work is completely done by A. Toulet in her PhD thesis [54].

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6.3. F OCUS - FOCUS

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a.a oshemkov

CASE

The focus-focus case was studied by S. V˜ u Ngo.c in [52]. The construction of symplectic invariants in this situation is quite similar to the hyperbolic case. Consider the simplest focus-focus singularity (described in Section 3.2). Let L be the singular leaf and S ∈ L the only singular point on it. As we have already seen, the base of the Liouville foliation in this case is a 2-disc D whose center corresponds to the singular leaf L. As above, consider the actions s1 and s2 as two functions on D. Notice, first of all, that only one of them can give us non-trivial symplectic information. Indeed, in a neighborhood of L there is a canonical Hamiltonian action of the circle (see Section 3.2), and one of the actions, say s1 , is just the Hamiltonian of this action. As a function on D, the action s1 is smooth and has no singularity. Thus, for all focus-focus singularities, s1 has absolutely the same properties and, therefore, gives no symplectic invariant. The other action s2 , in contrast, has a logarithmic singularity. The situation is similar to the hyperbolic case, but the logarithm becomes a complex function. To formulate the exact statement we first need to choose local coordinates (x, y) on the disc. For instance, we can just put x = f1 , y = f2 , where f1 and f2 are the functions that give us the local canonical form of a focus-focus singularity (see Section 3.2). Then s1 (x, y) = y, and the other action is given by s2 (x, y) = Re z ln z + c(x, y), where z = x + iy and c(x, y) is a smooth function on D (see [52]). As is seen, the second action variable is a multi-valued function on D. This is just the reflection of the fact that a focus-focus singularity has a non-trivial monodromy. Since the action s2 is defined modulo a constant, we can assume without loss of generality that c(0, 0) = 0. The choice of x = f1 and y = f2 in this construction is almost unique (namely, the coordinate x is defined modulo a flat function). In particular, the Taylor expansion of c(x, y) is uniquely defined. Thus, to each focus-focus singularity we can naturally assign a power series in two variables with zero first term + ck,l xk y l , I= (k,l)&=(0,0)

which will be a symplectic invariant of a focus-focus singularity. Theorem 18 (S. V˜ u Ngo.c [52]). The invariant I classifies focus-focus singularities of complexity 1 up to the symplectic equivalence. In other words, two focus-focus singularities L and L" are symplectically equivalent if and " only if I(L) = I(L).


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This theorem, in particular, shows that the action variables give us the complete symplectic description of focus-focus singularities (in the case of complexity 1). Problem 9. To classify focus-focus singularities with several singular points up to the symplectic equivalence. After solving this problem we will have the symplectic classification of all “basic” non-degenerate singularities from which all the other singularities can be constructed by the “almost direct product” procedure. Thus, one can formulate the next natural question. Problem 10. To classify non-degenerate singularities (of arbitrary dimension) up to the symplectic equivalence. In particular, it would be interesting to verify the following conjecture. Problem 11. Is it true that for non-degenerate singularities of complexity 1 the answer to the question from Problem 8 is positive? If it is so, then the symplectic invariants for these singularities can be obtained just in the same way as above. The Eliasson theorem gives us a possibility to choose the “coordinates” on the base of the Liouville foliation almost uniquely (i. e., up to flat terms). The actions being written in those coordinates can be naturally decomposed into two parts, singular and regular. The singular part is standard and can be omitted, the regular part is a smooth function defined modulo a flat function. So, the Taylor expansions of the “regularized” actions would give us a complete set of symplectic invariants.

7. G LOBAL

ASPECTS

In this section we discuss some global results concerning the symplectic topology of Liouville foliations.

7.1. O NE

DEGREE OF FREEDOM SYSTEMS

We start with the global classification of Liouville foliations in the case of one degree of freedom. A complete invariant for this problem can be described as follows. Consider a two-dimensional surface M with a Morse function H . The first ingredient of the invariant is the so-called Reeb graph of the function H . It is simply the base of the Liouville foliation, i. e., the quotient of the surface M , where points are equivalent if and only if they belong to the same leaf. Evidently, such a base is a finite graph whose vertices correspond to singular leaves. It is clear that the Reeb graph is not a complete invariant of a Liouville foliation on M , because singular leaves (or, more precisely, atoms) can be of different types and, therefore, the information about their structure should be added to the Reeb graph. Roughly speaking, we should assign an atom to

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each vertex of the Reeb graph and fix some correspondence between boundary circles of the atom and edges of the Reeb graph incident to the corresponding vertex. An example is shown in Figure 16.

Figure 16. One degree of freedom: Liouville foliation and corresponding molecule

The graph with labelled vertices so obtained is called the molecule corresponding to the Liouville foliation. Let us emphasize that the correspondence between boundary circles of atoms and edges of the Reeb graph is very important to understand which molecules are considered as “identical”. To explain this, it is convenient to assume that each vertex of the Reeb graph is assigned with a certain standard model of the corresponding atom. ' are said to be identical if two conditions Then two molecules W and W are satisfied. First one is natural: there exists an isomorphism between them as labelled graphs. This isomorphism naturally induces a permutation of boundary circles of the standard model (for each pair of the corresponding vertices). The second condition is that this permutation must be generated by a symmetry of the standard model onto itself (see Section 5.2). It should be emphasized that not every permutation of boundary circles of a surface V st can be extended up to a homeomorphism of the atom (V st , Γ st ) onto itself. This just means that, generally speaking, the boundary circles of an atom are not equivalent.

Figure 17. Identity of molecules

An example is shown in Figure 17: the boundary circles 1 and 3 are equivalent, but 1 and 2 are not; therefore, the molecules W1 and W2 are identical, whereas W2 and W3 differ.


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Theorem 19 [6]. Two (non-degenerate) integrable systems with one degree of freedom are Liouville equivalent if and only if their molecules are identical. By using this invariant (molecule), we can get the list of equivalence classes for global Liouville classification of integrable systems of small complexity. Theorem 20 [6]. The Liouville equivalence classes for non-degenerate integrable systems on the sphere and torus with at most 6 singular points are described by the molecules shown in Table 6. Table 6. Molecules on sphere and torus

The global symplectic classification is also known. In the case of the simplest singularities (atoms A and B; see Figure 1), the classification theorem is obtained in [12]. The complete invariant is the Reeb graph W of the Hamiltonian endowed with additional information which characterizes symplectic invariants of singular leaves and one-parameter families of Liouville 1-tori.

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Since in the case of the simplest singularities these invariants can be expressed in terms of action variables, the additional structure on the graph is just an affine structure on its edges (see Section 6.1). One only needs to define in a correct way an equivalence relation between “affine graphs” by introducing the notion of a “smooth mapping” for graphs. The meaning of “smoothness” is very natural. Since each graph W is the base of a one-dimensional Liouville foliation M 2 → W , a mapping between two such graphs W1 → W2 is defined to be smooth if and only if it can be lifted up to a diffeomorphism of Liouville foliations M1 → M2 . It is not hard to see that such a smooth structure on W can be defined in an intrinsic way. After this, the classification theorem takes the following formulation. Theorem 21 (J.-P. Dufour, P. Molino, A. Toulet [12]). The graph W endowed with an integer affine structure is a complete symplectic invariant of a one-dimensional Liouville foliation with singularities of types A and B only. In other words, two Liouville foliations L1 and L2 are symplectomorphic if and only if there exists a smooth isomorphism ψ : W1 → W2 which preserves the integer affine structure on the edges. If the bifurcations are not simplest, then the action variables are not enough for the classification. Besides the affine structure, the graph W must be endowed with some additional semi-local invariants of singular leaves. These invariants are completely described by A. Toulet in [54].

7.2. F OMENKO –Z IESCHANG

INVARIANT

In fact, the above construction first appeared in the study of the topology of three-dimensinal isoenergy surfaces of integrable Hamiltonian systems with two degrees of freedom (A. T. Fomenko [15, 17]). Consider an integrable Hamiltonian system with two degrees of freedom on a symplectic manifold (M 4 , ω) with a Hamiltonian H and an additional integral f . Take the restriction of the system to a regular isoenergy surface Q3 = {H = const}. Then the Liouville foliation on Q3 is given by the function f : Q3 → R. Under the assumption of non-degeneracy of singularities, the structure of singular leaves is described by Theorem 10. Since dH '= 0 on Q3 , only hyperbolic and elliptic singularities of rank 1 can appear. The global topological structure of the Liouville foliation on Q3 is described by the Fomenko–Zieschang invariant. This invariant is a graph endowed with additional information (marked molecule W ∗ ). The graph is the base of the Liouville foliation, its vertices correspond to singular leaves, and to each vertex one assigns a label that determines the type of the corresponding singular leaf in the sense of (semi-local) Liouville equivalence. This gives us the complete information about three-dimensional bricks which Q3 is glued from. However, unlike the case of one degree of freedom, gluing should be made along two-dimensional tori and, consequently, is not unique. The rules of gluing are coded by a collection of numerical marks. The formal description of the Fomenko–Zieschang invariant (or, what is the same, marked molecule) can be found in [6]. The classification theorem for integrable systems on isoenergy 3-manifolds is as follows.


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Theorem 22 (A. T. Fomenko, H. Zieschang [20]). Marked molecule W ∗ classifies non-degenerate integrable Hamiltonian systems on isoenergy 3-surfaces up to the Liouville equivalence. In other words, two Liouville foliations (Q31 , L1 ) and (Q32 , L2 ) are homeomorphic if and only if the corresponding marked molecules W1∗ and W2∗ are identical. Notice that Fomenko–Zieschang’s theory also describes completely the class of admissible invariants. Roughly speaking, it asserts that any abstract marked molecule can be realized as the Fomenko–Zieschang invariant of an appropriate Hamiltonian system. It is worth noticing that the problem of isoenergy classification is quite natural, since the isoenergy reduction is often used in the study of specific integrable systems in geometry, physics, and mechanics. For example, due to homogeneity of the Hamiltonian in the case of geodesic flows on 2-manifolds, the topology of such flows is uniquely defined by a single isoenergy surface. For many concrete integrable systems the Fomenko–Zieschang invariants are explicitly found (see examples in [6] and references therein). The above construction gives, in principle, the complete list of Liouville foliations on 3-manifolds and, in particular, allows one to describe the class of 3-manifolds that admit Liouville foliations with non-degenerate singularities (i. e., can be represented as isoenergy surfaces of integrable systems). Theorem 23 (A. T. Fomenko, H. Zieschang [18]). An orientable 3-manifold M admits a Liouville foliation with non-degenerate singularities if and only if M is a graph-manifold. The class of graph-manifolds is well-known in the 3-topology. Such manifolds were introduced and studied by F. Waldhausen [55]. They can, in fact, be defined just in the same way as they appear in the theory of integrable systems. These are manifolds which can be glued from pieces of two kinds: solid tori D 2 × S 1 and direct products N 2 × S 1 , where N 2 is a disk with two holes. Notice that these pieces can be regarded as regular neighborhoods of elliptic and simplest hyperbolic leaves of corank 1. A non-trivial fact is that not every 3-manifold admits the above representation. For example, a hyperbolic manifold is not a graph-manifold. It is interesting to notice that, if we omit the integrability condition, then any 3-manifold can be an isoenergy surface of a certain Hamiltonian system (in general, non-integrable).

7.3. L OOP

MOLECULES

In [41] Nguyen Tien Zung conjectured that the topological type of a non-degenerate singularity of corank k (i. e., its Liouville equivalence class) can be completely reconstructed from the information on neighboring singularities of corank 1. A similar conjecture was discussed in [6]. To formulate it more precisely, let us consider the following example. Assume we have an integrable Hamiltonian system with two degrees of freedom, we know the bifurcation diagram, and we want to describe the topological types of all the singularities. In a typical situation, the bifurcation

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diagram Σ consists of several smooth curves and “exceptional points” (isolated points, cusps, points of intersection of smooth parts of Σ, and so on). Smooth parts correspond to corank 1 singularities (their classification is given by Theorem 10). Usually it is not so hard to describe (or to guess) their structure. The structure of singularities corresponding to exceptional points is much harder to understand. But each of them has the following very natural topological invariant. Let us consider a small loop γ around an “exceptional” point y ∈ Σ (for instance, circle of a small radius). Assume that inside this loop there is no more exceptional points, and if we shrink γ into y, it remains always transversal to the smooth parts of the bifurcation diagram (in most examples we know this condition holds). The preimage of γ is a three-dimensional submanifold in M 4 with the structure of a Liouville foliation. This structure is completely described by the Fomenko–Zieschang invariant W ∗ (γ) (i. e., marked molecule), which does not depend on the choice of γ. This invariant is called the loop molecule of the singularity corresponding to the “exceptional” point y (we have already used this invariant in the case of saddle-saddle singularities; see Section 5.2). The question is: can we reconstruct the topology of the singularity, if we know the corresponding loop molecule? It is easy to see that the preimage F −1 (γ) is nothing else but the boundary of a neighborhood of the singular leaf. Thus, a general version of this question can be formulated as follows. Problem 12. Consider two non-degenerate singularities. Let L1 , L2 be the singular leaves, and U1 (L1 ), U2 (L2 ) their regular neighborhoods. Suppose that the Liouville foliations on U1 (L1 ) \ L1 and U2 (L2 ) \ L2 (slightly different version: on the boundaries of Ui (Li )) are homeomorphic. Is it true that the Liouville foliations on U1 (L1 ) and U2 (L2 ) are homeomorphic (i. e., the corresponding systems are semi-locally Liouville equivalent)? The positive answer to this question would simplify the topological analysis of integrable systems in geometry, mechanics and physics. As is seen from Table 3, all 39 loop molecules for singularities of saddle-saddle type (of complexity 2) are different. Thus, in this case, the answer to the question is positive, and this fact was used for topological analysis of some concrete integrable systems (see, for example, [7, 27, 38, 39, 46]). However, in general, the answer to the above question is negative as is seen from the following example constructed by A. V. Grabezhnoˇı [22]. Let us describe two different singularities of saddle-saddle type with the same loop molecule. The first one is just the direct product C1 × C1 of two copies of the atom C1 (see Table 1 and Figure 8). It is easy to see that the loop molecule of this singularity has the form presented in Figure 18. If we choose natural basic cycles on the boundary tori of the atoms of this molecule induced by the direct product structure, # $ then the gluing matrices on the edges of the molecule will be 01 10 .


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The second singularity is the almost direct product (C1 × K2 )/Z2 , where K2 is the atom presented in Figure 8, which is a two-sheeted covering of C1 . The action of Z2 on the factors C1 and K2 is the central symmetry (here we use the “symmetric” representations of C1 and K2 shown in Figure 8).

Figure 18. Loop molecule for singularities C1 × C1 and (C1 × K2 )/Z2

It is not hard to verify that the loop molecule for this singularity will be just the same as for the above direct product C1 × C1 . The described singularities C1 × C1 and (C1 × K2 )/Z2 are not Liouville equivalent. This can be seen, for example, from the fact that their singular leaves are not homeomorphic.

7.4. N ON - SINGULAR L IOUVILLE

FOLIATIONS

If we want to describe the global structure and global invariants of Liouville foliations, then we have to solve the following two problems. First we need to be able to describe the semi-local structure of singularities. Then we need to understand global characteristics of foliations which are not related directly with the structure of singularities. As the first step, it would be interesting to describe Liouville foliations without any singularities at all. The question about a complete set of invariants for such foliations is solved by J. J. Duistermaat [13]. Consider a locally trivial fibration M 2n → Y n whose total space is a symplectic manifold M 2n and fibers are Lagrangian tori. The first symplectic invariant of this fibration is the integer affine structure on the base Y n . The second invariant is the topological structure of the fibration. Finally, there is another invariant which is purely symplectic. This is the so-called “Lagrangian Chern class”, which determines, roughly speaking, obstruction to the existence of a global Lagrangian section of the fibration. Although all necessary invariants are known, it is still an open problem to describe the complete list of Liouville foliations without singularities. This problem is solved only for two degrees of freedom. Classification of Lagrangian bundles with compact fibers over surfaces. The base of a Lagrangian T 2 -bundle carries an integer affine structure. Only two closed surfaces possess this property: the torus and Klein bottle. Here we describe all Lagrangian T 2 -bundles over the torus following the paper [37] by K. N. Mishachev (a similar construction can be applied in the case of the Klein bottle; see [31, 37]).

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There is the trivial Lagrangian bundle, which is simply the direct product T 2 × T 2 with coordinates (x, y, ϕ1 , ϕ2 ), where x, y are angle coordinates on the base T 2 and ϕ1 , ϕ2 are angle coordinates on the fiber T 2 , with the symplectic structure ω = dx ∧ dϕ1 + dy ∧ dϕ2 . Starting from this trivial bundle, it is convenient to describe all possible Lagrangian T 2 -bundles over the torus with the help of some operations. 1. The first operation, which changes the affine structure on the base of the bundle, can be described as follows. Let us consider the cotangent bundle T ∗ T 2 with the standard symplectic structure dx ∧ dpx + dy ∧ dpy , where px , py are the coordinates on fibers R2 corresponding to coordinates x, y on the base T 2 . We want to obtain a Lagrangian bundle with compact fibers (tori) from the cotangent bundle factorizing each fiber R2 with respect to a lattice. Locally such a family of lattices can be defined by a pair of 1-forms α1 , α2 on the base T 2 . The factorization should preserve the standard symplectic structure on the cotangent bundle. This is equivalent to the fact that the forms α1 , α2 are closed. It turns out that it suffices to consider the following two series of lattices: 1) α1 = a dx+b dy, α2 = c dx+d dy where a, b, c, d ∈ R and ad−bc '= 0; 2) α1 = b dy, α2 = a dx + bky dy where a, b ∈ R and k ∈ N. For the lattice generated by α1 = dx and α2 = dy, one obtains the trivial T 2 -bundle described above. A choice of another lattice can be codsidered as a choice of another affine structure on the base. Note that all Lagrangian bundles obtained from the trivial one by this operation can be characterized as the set of bundles admitting a global Lagrangian section. 2. The second operation is some surgery of T 2 -bundle. We simply cut a small disk from the base together with fibers over it, and then glue the removed piece D 2 × T 2 back by some fiberwise diffeomorphism. The difference from usual topological surgery here is that after the gluing we should obtain a symplectic manifold with Lagrangian foliation. This can be done in the following way. Without loss of generality, one can assume that the symplectic structure on D 2 ×T 2 has the form ω = dx∧dϕ1 +dy∧dϕ2 , where x, y are coordinates on the base D2 and ϕ1 , ϕ2 are angle coordinates on the fiber T 2 . Let the boundary circle S 1 of the disk D2 is parametrized by ϕ. Consider the form β on D 2 , which is equal to dϕ on the neighborhood of the boundary circle S 1 . Let us change the form ω as follows: ω " = dx∧dϕ1 +dy∧dϕ2 +d((mx+ny)β). " ) foliated into Lagrangian tori. One obtains a symplectic manifold (D 2 ×T 2 , ω Now let us glue back the manifold (D 2 × T 2 , ω " ) by the following diffeomorphism: (ϕ, ϕ1 , ϕ2 ) → (ϕ, ϕ1 + mϕ, ϕ2 + nϕ). Evidently, one can assume that this operation glues symplectic structures on the pieces. As a result, one obtains a new Lagrangian bundle. Let us note that the operation described above does not change the affine structure on the base.


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3. The last operation over Lagrangian bundles changes the symplectic structure, but preserves both the topology of the foliation and the affine structure on the base. We simply replace the original symplectic structure ω by a new one of the form ω " = ω + π ∗ σ, where σ is a 2-form on the base, and π is the projection to the base. Theorem 24 (K. N. Mishachev [37]). All Lagrangian T 2 -bundles over T 2 can be obtained from Lagrangian bundles admitting a global Lagrangian section (described in item 1) by applying operations 2 and 3.

7.5. T ORUS

ACTION

In the previous section we have described Liouville foliations without singularities. Now let us consider the case when there can be singularities, but of special type, namely elliptic ones, which are the simplest non-degenerate singularities of integrable Hamiltonian systems. It is easy to see that the Liouville foliation in a neighborhood of an elliptic 2n-dimensional singularity always admits a Hamiltonian n-torus action. Indeed, the existence of a Hamiltonian torus action means that one can choose periodic first integrals, but for elliptic singularities such integrals exist by virtue of the Eliasson theorem. It turns out that for the case when the torus action is defined on the whole phase space M 2n , it is possible to classify all systems up to symplectic equivalence (T. Delzant [11]). The first results about Hamiltonian torus actions were obtained by M. F. Atiyah [2], V. Guillemin and S. Sternberg [24]. In particular, they describe all possible singularities of an integrable Hamiltonian system admitting a Hamiltonian torus action. Theorem 25 (M. F. Atiyah [2], V. Guillemin, S. Sternberg [24]). Let integrals f1 , . . . , fn of an integrable Hamiltonian system defined on a closed symplectic manifold (M 2n , ω) be periodic. Then the image of the momentum mapping F : M → Rn is a convex polytop. The next achievement in this direction is due to T. Delzant. He gave a complete description of all possible polytops and corresponding symplectic manifolds with torus actions.

Theorem 26 (T. Delzant [11]). Let M1 and M2 be 2n-dimensional symplectic manifolds with Hamiltonian effective n-torus actions. If the images of the corresponding momentum mappings F1 (M1 ) and F2 (M2 ) coincide, then there exists a T n -equivariant symplectomorphism Ψ : M1 → M2 such that F2 ◦ Ψ = F1 . In particular, Ψ is a symplectomorphism of Liouville foliations, i. e., establishes the symplectic equivalence between the corresponding integrable Hamiltonian systems on M1 and M2 . All possible polytops are described in the following theorem.

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Theorem 27 (T. Delzant [11]). A polytop P can be the image of the momentum mapping F : M 2n → Rn corresponding to a Hamiltonian effective n-torus actions if and only if each vertex O of the polytop P is incident to exactly n edges v1 , . . . , vn and there exist λ1 , . . . , λn ∈ R such that λ1 v1 , . . . , λn vn is a basis of the integer lattice Zn ⊂ Rn . As we already mentioned, an integrable Hamiltonian systems with a Hamiltonian torus action has only non-degenerate elliptic singularities. The converse is not true: there are integrable systems with elliptic singularities which do not admit any global torus action (see Theorem 28 below, case (ii)).

7.6. A LMOST

TORIC MANIFOLDS

Another class of Liouville foliations has been introduced and described in papers by M. Symington [50], N. C. Leung and M. Symington [31], S. V˜ u Ngo.c [53]. These are the so-called “almost toric” manifolds. Speaking non-formally, a symplectic manifold is called almost toric, if it admits an integrable Hamiltonian system with non-degenerate singularities of elliptic or focus-focus type. In the 4-dimensional case, such manifolds and the corresponding Liouville foliations are studied in detail in [31, 50]. Similarly to the case of a torus action, in this situation the topology of the manifold together with the Liouville foliation on it can be described in terms of the base of the foliation. Notice that this property distinguishes elliptic and focus-focus singularities from hyperbolic ones: in the hyperbolic case there are many singularities which are topologically different, but this difference disappears after projection onto the base (see examples in Section 5). Note that a slightly more general notion than the Liouville foliation is considered in [31, 50]. It is the singular Lagrangian fibration π : M 2n → Y , and the difference is that one allows the base Y to be a space not immersed into Rn . We restrict ourselves with formulating one of the main results of [31,50]. Definition 13. An almost toric fibration of a closed symplectic manifold (M 2n , ω) is a singular Lagrangian fibration π : M 2n → Y such that any critical point of π is non-degenerate and has no hyperbolic components in the sense of Section 2.2. An almost toric manifold is a symplectic manifold equipped with an almost toric fibration. In the case of two degrees of freedom, the possible singularities can be of three types only: elliptic of rank 0 or 1 and focus-focus. The base of the fibration π : M 4 → Y is a two-dimensional surface Y perhaps with boundary. Non-singular points of the boundary ∂Y correspond to elliptic singularities of rank 1, singular points of ∂Y correspond to elliptic singularities of rank 0 (they are called vertices). Singular leaves of focus-focus type are projected onto some interior isolated points of the base (following [31, 50], we will call them nodes). For simplicity, let us assume that each singular focus-focus leaf


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contains exactly one singular point. Notice that we can always obtain this condition by a small perturbation in the class of almost toric fibrations. The next theorem describes the topological types of almost toric manifolds and the structure of the corresponding singular Lagrangian fibrations. Theorem 28 (N. C. Leung, M. Symington [31]). If π : (M, ω) → Y is an almost toric fibration of a closed four-manifold, then the total space M must be diffeomorphic to one of the following manifolds: ( i ) S2 × S2 , ( ii ) S 2 × T 2 , " 2 and n ≥ 0, (iii) N # nCP 2 with N = CP 2 or S 2 ×T (iv) the K3-surface, ( v ) the Enriques surface, ,# 1 0 $ # $1k , (vi) a torus bundle over the torus with monodromy 01 , 0 1 ,# 1 0 $ # $(vii) a torus bundle over the Klein bottle with monodromy 0 −1 , 10 k1 . Furthermore, the classification of all such fibrations is given in Table 7 according to the homeomorphism type of the base Y , the number of nodes, and the number of vertices on the boundary ∂Y . Table 7. Closed almost toric four-manifolds Base

Number of nodes

Number of vertices

Total space 2

D2

n≥0

k ≥ max(0, 3−n)

S 1 × D1

n≥0

0

S 1 ×D1

n≥0

0

S2

24

0

12

0

RP

2

CP # (n + k − 3)CP 2

or S 2 × S 2 (if n + k = 4) (S 2 × T 2 ) # nCP 2

or (S 2 ×T 2 ) # nCP 2

(S 2 × T 2 ) # nCP 2

or (S 2 ×T 2 ) # nCP 2 K3 surface Enriques surface 2

T2 1

S ×S

0 1

7.7. Z UNG ’ S

0

0

T -bundle with monodromy 1 0 1λ 0 1 , 0 1

0

T 2 -bundle with monodromy 1 0 1λ 0 −1 , 0 1

CLASSIFICATION THEOREMS

The most general approach to the classification of Liouville foliations has been suggested by Nguyen Tien Zung in [44]. He succeded in combining together the semi-local theory of Liouville foliation singularities and the theory of global invariants like monodromy and Lagrangian Chern class. His construction can be briefly presented as follows.

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Suppose that we want to compare two given integrable systems (with n degrees of freedom) from the point of view of the Liouville or symplectic equivalence. The first natural (topological) invariant of a foliation is its base. It is natural to assume that the base of a Liouville foliation has the structure of a stratified manifold, where points of strata of maximal dimension n correspond to regular leaves (i. e., Liouville tori), and points of other strata correspond to certain singularities of the Liouville foliation. All types of singularities for our two systems must coincide. In other words, the topological types of singularities (i. e. their Liouville equivalence classes) are invariants. Finally, there is another topological invariant called “homological monodromy”, which establishes the relationship between the fundamental (homological) groups of different leaves. If all the above invariants coincide, then the systems are said to be “roughly Liouville equivalent” (in [44] such systems are called “roughly topologically equivalent”). More precisely, this notion can be described as follows. As above, for a Liouville foliation L on (M, ω), we denote by π the natural projection M → Y onto the base.

Definition 14. Let us be given two Liouville foliations (two integrable systems) L1 and L2 . They are called roughly Liouville equivalent if there exists a homeomorphism between the bases ψ : Y1 → Y2 which is locally in a neighborhood of each point can be lifted up to a homeomorphism of the Liouville foliations. In other words, there exists a covering of Y1 by open subsets Uα and homeomorphisms Ψα : π1−1 (Uα ) → π2−1 (ψ(Uα )) such that π2 ◦ Ψα = ψ ◦ π1 . Besides, it is required that on intersections the lifted homeomorphisms are trivial in the homotopic sense. This means that Ψ−1 α Ψβ induces the identity mapping on the fundamental groups of the strata of π1−1 (x) and the identity mapping on H1 (π1−1 (x), Z) for each point x ∈ Uα ∩ Uβ . The systems are called roughly symplectically equivalent if, in addition, Ψi are (semi-local) symplectomorphisms. It is clear that the rough equivalence is a necessary condition for the global equivalence. Now the question is what additional invariants are sufficient for the complete solution of the classification problem. Such invariants have been completely described in [44]. These are some characteristic classes of a Liouville foliation, which are called the Chern class (for the case of the Liouville equivalence) and the Lagrangian class (for the case of the symplectic equivalence). Note that these classes generalize the corresponding notions introduced by J. J. Duistermaat [13] to the case when the base of Liouville foliation is not smooth but stratified manifold. The classification theorems (for the Liouville and symplectic equivalences) can be formulated as follows. Theorem 29 (Nguyen Tien Zung [44]). Two roughly Liouville (resp., symplectically) equivalent systems are equivalent in the sense of Liouville (resp., symplectic) equivalence if and only if the corresponding characteristic Chern (resp., Lagrangian) classes coincide.


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Notice that these theorems do not assume non-degeneracy of singularities. One only requires that the structure of Liouville foliations satisfies some additional, but very natural conditions. It is worth saying that the possibility to include degenerate singularities to the general theory is very important because degenerate singularities appear in integrable systems in a generic way and cannot be avoided by a small perturbation. Theorem 29 is a powerful tool for comparing systems up to the global Liouville or symplectic equivalence. However, it does not allow us to describe possible types of systems. The above mentioned results by N. C. Leung and M. Symington [31,50], T. Delzant [11], A. T. Fomenko and H. Zieschang [20], K. N. Mishachev [37] solve this problem for important particular cases, but in the general situation the question about the list of integrable Hamiltonian systems still remains open. In particular, the following problem seems to be quite interesting. Problem 13. Which symplectic manifolds admit integrable Hamiltonian systems all of whose singularities are non-degenerate? Are there any topological (or might be symplectic) obstructions to the existence of integrable Hamiltonian systems on a given compact symplectic manifold? It is worth mentioning that if we do not put any additional conditions to the structure of singularities, then the answer to the above question becomes trivial. Each symplectic manifold M admits a smooth integrable systems with compact regular leaves. Such a system is easily constructed by using the partition of unity: symplectic manifold is covered almost everywhere by a disjoint union of symplectic balls, then inside each ball we construct an integrable Hamiltonian system f1 , . . . , fn , such that each fi vanishes on the boundary of the ball together with all derivatives. This gives us a possibility to sew all these local systems together in one global integrable system on M .

7.8. T OPOLOGY

OF THE SET OF SINGULARITIES

Here we consider some topological properties of the set of singular points of an integrable Hamiltonian system. Note that this set is not the union of singular leaves of the Liouville foliation, but the union of singular orbits of the corresponding Hamiltonian action. Also note that speaking of the topology of the set of singularities, we mean the following two aspects: the topology of this set itself, and the “imbedding” of it to the phase space of the system. Let, as above, (M 2n , ω, f1 , . . . , fn ) be an integrable Hamiltonian system, ρ the Hamiltonian action of Rn on M associated with this system, and K the union of all orbits of the action ρ of dimension less than n (i. e., the set of singular points of the system). Denote by Kr ⊂ K the set of singular points of rank ≤ r (where r < n). The Hamiltonian vector fields v1 = sgrad f1 , . . . , vn = sgrad fn are dependent at points x ∈ K , and the dimension of the subspace 1v1 (x), . . . , vn (x)2 spanned by these vector fields at a point x ∈ K is equal to the rank of this singular point.

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For generic vector fields v1 , . . . , vk on a compact manifold, sets of type Kr realize some specific homology classes of that manifold. This allows one to use methods of algebraic topology for describing topological (or homological) properties of sets Kr . However, vector fields vi = sgrad fi corresponding to an integrable Hamiltonian system are not generic, because they commute and preserve the symplectic structure. Nevertheless, it turns out that one can use homological methods for investigating the sets Kr in the case of integrable Hamiltonian system as well. The idea is to define a complex structure on the tangent bundle T M (compatible with the symplectic structure) and then to consider the fields vi = sgrad fi as sections of that complex bundle. First, let us describe a necessary complex structure on the tangent bundle of M or, what is the same, an almost complex structure on M . Recall that an almost complex structure J on a symplectic manifold (M 2n , ω) is said to be compatible with the symplectic structure ω if the bilinear form defined by 1u, v2 = ω(u, Jv) is symmetric and positive definite (i. e., is a Riemannian metric on M ). It is well-known (see, for example, [34]) that any symplectic manifold possesses an almost complex structure compatible with the symplectic structure, and it is uniquely defined up to homotopy (i. e., the set of all such almost complex structures is contractible). Thus, considering the tangent bundle of a symplectic manifold as a complex bundle, one can speak of Chern classes of this complex vector bundle, which are independent of the choice of a compatible almost complex structure. Further, if we consider the tangent bundle of a symplectic manifold as a complex bundle, then for its arbitrary sections (i. e., vector fields) the dependence over C and the dependence over R are different things. But it turns out that for a set of vectors generating isotropic subspaces, this notions are the same. More precisely, if a collection of vectors u1 , . . . , uk in a linear symplectic space (V, ω) equipped with a compatible complex structure J generates (over R) an isotropic subspace L, then dimR 1u1 , . . . , uk 2 = dimC 1u1 , . . . , uk 2. This statement easily follows from the fact that L ∩ JL = {0} for any isotropic subspace L. Since Hamiltonian vector fields v1 = sgrad f1 , . . . , vn = sgrad fn defining an integrable Hamiltonian system (M 2n , ω, f1 , . . . , fn ) commute, they generate isotropic subspaces in all tangent spaces Tx M . Therefore, one can consider the sets Kr of singular points of the system as degeneracy sets for the collection of sections v1 , . . . , vn of the complex vector bundle (T M, J). There is the classical construction relating such degeneracy sets of a collections of sections with the Chern classes of the complex vector bundle (see, for example, book [21], where this construction is called the Gauss–Bonnet formula). Roughly speaking, if the phase space M is compact, then the sets Kr realize the homology classes which are Poincaré dual to the Chern classes of the bundle (T M, J). Now let us describe some properties of the set K of singular points of an integrable Hamiltonian system in the case of two degrees of freedom (see [49] for details).


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We will suppose that the phase space M 4 is a closed manifold and that the Hamiltonian action of the group R2 on M 4 is non-degenerate (i. e. all singularities of the system are non-degenerate). In this case the set K of all singular points of the system (i. e., the union of all 0-dimensional and 1-dimensional orbits of the corresponding Hamiltonian action) can be represented in the form K = (P1 ∪ · · · ∪ Pl ) ∪ (Q1 ∪ · · · ∪ Qm ) ∪ (T1 ∪ · · · ∪ Ts ), where T1 , . . . , Ts are isolated singular points of type focus-focus, and P1 , . . . , Pl , Q1 , . . . , Qm are closed immersed 2-dimensional submanifolds in M 4 satisfying the following properties: l . (1) all elliptic singular orbits are contained in Pi , all hyperbolic sinm i=1 . gular orbits are contained in Qj ; j=1

(2) all intersections of the submanifolds Pi and Qj (including selfintersections) are transverse and occur only at singular points, the intersections of the form Pi ∩ Pj contain only singular points of type (2, 0, 0), the intersections of the form Pi ∩ Qj contain only singular points of type (1, 1, 0), and the intersections of the form Qi ∩ Qj contain only singular points of type (0, 0, 2). (3) the restriction of the symplectic form ω to the submanifolds Pi and Qj is non-degenerate. For an oriented two-dimensional (immersed) submanifold N ⊂ M 4 , denote by [N ] the corresponding integral homology class in H2 (M 4 , Z). Also denote by [K] the sum of all classes realizing by the submanifolds Pi , Qj : [K] = [P1 ] + · · · + [Pl ] + [Q1 ] + · · · + [Qm ] ∈ H2 (M 4 , Z), where the orientation on the submanifolds Pi filled by elliptic orbits is defined by the form ω, and the orientation on the submanifolds Qj filled by hyperbolic orbits is defined by −ω. We define the orientation on M 4 by the form ω ∧ ω. The following statement is a special case of the general construction described above. Theorem 30 (A. A. Oshemkov [49]). Let K be the set of singularities of a non-degenerate Hamiltonian action of the group R2 on a compact symplectic manifold (M 4 , ω). The class [K] ∈ H2 (M 4 , Z) is Poincaré dual to the first Chern class c1 (M 4 ) ∈ H 2 (M 4 ). The next theorems describe some other topological properties of the set of singularities. Theorem 31 (A. A. Oshemkov [49]). Let K be the set of singularities of a non-degenerate Hamiltonian action of the group R2 on a symplectic manifold (M 4 , ω). Any submanifold Qj (which belongs to the complex K and which is filled by hyperbolic orbits) has trivial normal bundle in the manifold M 4 .

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Theorem 32 (A. A. Oshemkov [49]). Let K = (P1 ∪· · ·∪Pl )∪(Q1 ∪· · ·∪ Qm ) ∪ (T1 ∪ · · · ∪ Ts ) be the set of singularities of a non-degenerate Hamiltonian action of the group R2 on a compact symplectic manifold (M 4 , ω). For the intersection indices of the homology classes [Pi ], [Qj ] the following conditions are satisfied: (1) [Pi ] · [Pj ] ≥ 0, [Pi ] · [Qj ] ≤ 0, [Qi ] · [Qj ] ≥ 0, (2) [K] · [Pi ] = χ(Pi ) + [Pi ] · [Pi ] − 2nPi , (3) [K] · [Qj ] = −χ(Qj ), (4) χ(P1 ) + · · · + χ(Pl ) − χ(Q1 ) − · · · − χ(Qm ) = 2(χ(M 4 ) − s). where nPi and nQj are the numbers of selfintersection points of the (immersed) submanifolds Pi and Qj respectively, and the Euler characteristic of a manifold X is denoted by χ(X). These theorems give some conditions on the intersection form for a 4-dimensional symplectic manifold admitting a non-degenerate integrable Hamiltonian system. Thus, if there are symplectic manifolds not admitting integrable Hamiltonian systems with non-degenerate singularities (see Problem 13), one can try to find such an example among those manifolds which do not satisfy the “homological” conditions from Theorems 30–32. Notice that the above conditions on the submanifolds Pi , Qj are sufficiently strong that allows us to describe all possibilities for symplectic 4-manifolds with “small” 2-dimensional homology group. For example, in the simplest case M 4 = CP 2 , these conditions imply that there are only four variants: they correspond to almost toric fibrations from Table 7 with the base D 2 (where n + k = 3); in particular, any integrable Hamiltonian system on CP 2 with non-degenerate singularities has no hyperbolic singularities.

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