Hidden toric symmetry and structural stability of singularities in integrable systems
aa r X i v : . [ m a t h . S G ] A ug Hidden toric symmetry and structural stabilityof singularities in integrable systems
E. Kudryavtseva ∗ Abstract
The goal of the paper is to develop a systematic approach to the study of (per-haps degenerate) singularities of integrable systems and their structural stability.As the main tool, we use “hidden” system-preserving torus actions near singularorbits. We give sufficient conditions for the existence of such actions and showthat they are persistent under integrable perturbations. We find toric symmetriesfor several infinite series of singularities and prove, as an application, structuralstability of Kalashnikov’s parabolic orbits with resonances in the real-analyticcase. We also classify all Hamiltonian k -torus actions near a singular orbit on asymplectic manifold M n (or on its complexification) and prove that the normalforms of these actions are persistent under small perturbations. As a by-product,we prove an equivariant version of the Vey theorem (1978) about local symplecticnormal form of nondegenerate singularities. Key words: integrable system, Hamiltonian torus action, degenerate singu-larity of integrable system, structural stability
MSC:
Let ( M n , Ω , f , . . . , f n ) be an integrable Hamiltonian system with n degrees of freedom,where the momentum map F = ( f , . . . , f n ) is proper. Consider the Hamiltonian R n -action on M generated by the momentum map F . We will call orbits of this actionsimply orbits . Consider the singular fibration (called the Liouville fibration ), whosefibers are connected components of the level sets F − ( a ), a ∈ R n . ∗ Moscow State University, Moscow, Russia; Moscow Center for Fundamental and Applied Mathe-matics, Moscow, Russia. E-mail address: [email protected]
1y a local (respectively semilocal ) singularity of such a singular fibration we will meana singular orbit (respectively fiber). We recall that a point m ∈ M is called a singular (or critical ) point of this fibration if rank d F ( m ) < n . An orbit (or a fiber) is called singular if it contains a singular point. The minimal rank of singular points belongingto a fiber is called rank of the fiber.It is known that a compact rank- r orbit of an n -degrees of freedom integrable systemalways admits a locally-free F -preserving Hamiltonian ( S ) r -action on some neighbour-hood of this orbit, provided that the orbit is either regular (Liouville theorem), orsingular and has one of the following types:– nondegenerate (Ito [20], Zung [38] for the general case, Fomenko [15, Proposition4], [3, Theorem 3.2] for the twisted hyperbolic case with n = 2 and r = 1),– not too degenerate (Bao and Zung [1, Theorem 2.1]),– having finite type (Zung [40], [41, Theorem 3.7], see § r = n − R L ≤ n (Zung [39, Theorem 1.2]) where L is the fibercontaining the given orbit.Furthermore, if the singular orbit is nondegenerate with Williamson type ( k e , k h , k f ),then this action extends to an effective (not locally-free) F -preserving Hamiltonian( S ) r + k e + k f -action (Zung [38, Theorem 6.1]), moreover the system is fiberwise symplec-tomorphic to a linear model ([20] for the real-analytic case, [30] for the C ∞ case).The purpose of this paper is to develop a systematic approach for study and classi-fication of singularities (especially, structurally stable singularities, maybe degenerateones) of singular Lagrangian fibrations associated with integrable systems. In this pa-per, three natural circles of questions are addressed. The first circle of questions is asfollows:(1a) Does there exist an effective Hamiltonian ( S ) r + κ e -action on a neighbourhood of a degenerate orbit O , that preserves the momentum map, where the ( S ) r -subactionis locally-free, and the ( S ) κ e -subaction leaves the orbit O fixed?(1b) Can this torus action be extended to an effective Hamiltonian ( S ) r + κ e + κ h -actionon a small open complexification U C of a degenerate (respectively, nondegenerate )orbit O , that preserves the holomorphic extension F C of the momentum map to U C , where the ( S ) κ h -subaction leaves the orbit O fixed?(1c) Is this torus action persistent under small real-analytic integrable perturbationsof the system? 2he second circle of questions is about symplectic normalization of a torus action:(2a) Can a Hamiltonian torus action be written in a “simple” normal form (so-calledcanonical model) w.r.t. some symplectic coordinates on a neighbourhood of anorbit O ?(2b) Can a (more general) Hamiltonian torus action be written in a “simple” symplecticnormal form, when some of S -subactions fixing the orbit O are allowed to begenerated by real-analytic functions multiplied with √− r + κ e + κ h = n . For illustration, we answer the question(3a) for several infinite series of local singularities (Examples 3.6, 3.12), and conjecturestructural stability of all these singularities (Example 4.2 (B)). As an application, wesolve the questions (3b)–(3d) for parabolic orbits with resonances, which are degeneratelocal singularities with r = n − κ e = κ h = 0 (Proposition 4.3). As a by-product,we prove an equivariant version of the Vey theorem [35] about symplectic local normalform for nondegenerate singularities (Lemmata 6.1 and 6.2). Our proofs are analogousto the proofs of theorems about torus actions in [38, 1, 3, 39, 40].When using the term “hidden” for toric symmetries, we mean the following: • This symmetry is a Hamiltonian ( S ) r + κ e + κ h -action generated by some smoothfunctions (“actions”) depending on the first integrals of the system, but these“action functions” are often not given or not known in advance.3 The ( S ) r + κ e -subaction is defined only on a small neighbourhood U of the givensingular orbit, so this subaction does not necessarily extend globally to the wholephase space. • The ( S ) κ h -subaction is generated by imaginary-valued “action functions”, so thissubaction is defined only on a small open complexification U C of the neighbour-hood U of the singular orbit.The above properties of being “hidden” for the ( S ) r + κ e + κ h -action and its subactionsare observed in many integrable mechanical systems, e.g. for regular orbits (so-calledLiouville tori, r = n , κ e = κ h = 0) and nondegenerate singular orbits ( r < n , r + κ e + κ h = n ).Remarks.1) In the case of a locally-free ( S ) r -action, the question (1a) was solved for nondegen-erate singularities by Ito [20] (for real-analyltic case), Fomenko [15, Proposition 4], [3,Theorem 3.2] (for the twisted hyperbolic case when n = 2 and r = 1), Zung [38]; for nottoo degenerate singularities by Bao and Zung [1, Theorem 2.1]; for finite type singu-larities by Zung [40], [41, Theorem 3.7] (see § S ) r + κ e -subaction were solved by Ito [20] (for real-analytic case),Miranda and Zung [30] (for equivariant C ∞ case). For nondegenerate singularities, thequestions (3b) and (3d) w.r.t. the ( S ) r + κ e -subaction were partially solved by Mirandaand Zung [30] (for equivariant C ∞ case), for a weaker notion of structural stability(resp. persistence), namely for structural stability (resp. persistence) under parametricfamilies of integrable perturbations.3) For (degenerate) rank-0 singularities, a solution to the questions (1a), and partially(1b), (2a), (2b) was described by Zung [41] in terms of the Poincar´e-Birkhoff normalform, by proving its convergence.4) For an arbitrary smooth symplectic action of a compact Lie group on a neighbour-hood of its fixed point, the question (2a) was partially solved by Weinstein [37, Lecture5], [9] by linearizing the action in some symplectic coordinates. For an arbitrary smoothsymplectic action of a Lie group on a neighbourhood of its orbit, a solution to the ques-tions (2a), (2b) was given under some natural assumptions by Marle [28, Propositions1.9 and 1.10], Guillemin and Sternberg [18] in terms of a linear model of the action.5) For parabolic orbits (without resonance), the question (3b) was solved by Lermanand Umanskii [25]. For parabolic orbits with resonances, the question (3b) was par-tially solved by Kalashnikov [21], for a weaker notion of structural stability, namely for4tructural stability under S -symmetry-preserving integrable perturbations. Infinites-imal stability (i.e. stability under infinitesimal integrable deformations of the system[16, Definition 8]) was studied for 2-degrees of freedom integrable systems, namely:nondegenerate rank-0 and rank-1 singular points and a rank-1 parabolic singular pointare infinitesimally stable [16, Definition 9, Theorems 2 and 3]. This partially solves thequestion (3b) for these singularities, for a weaker notion of structural stability, namelyfor infinitesimal stability.6) For a corank-2 singularity “integrable Hamiltonian Hopf bifurcation” of integrableHamiltoninan systems with 3 degrees of freedom, the question (3c) was partially solvedby van der Meer [34] (without studying the symplectic structure).7) For saddle-saddle fibers satisfying a “non-splitting” condition, structural stabilityunder “component-wise” C ∞ integrable perturbations was proved by Oshemkov andTuzhilin [32]. This partially solves the question (3b) for such semilocal singularities,for a weaker notion of structural stability.Our solutions to the questions (1a)–(3d) have the following advantages: • In our solution to the questions (1a)–(1c) (in Theorems 2.1, 2.2), we do not assumethat the orbit is nondegenerate, or has corank 1, or the torus action is locally-free. • In our solution to the questions (2a)–(2c) (in Theorems 3.4, 3.10), we do notassume that the torus action preserves an integrable system with nondegeneratesingularities; furthermore we show that our canonical model is not only linear butalso has a “diagonal” form (in contrast to [37, 9, 28, 18]). • In our solution to the question (3a) (in Examples 3.6, 3.12), local singularitieshave resonances of different qualitative nature (so-called elliptic, hyperbolic andtwisting resonances), and these resonances cannot be reduced or simplified. Inour solution to the questions (3b)–(3d) for parabolic orbits with resonances (inProposition 4.3), we study structural stability of a singularity (resp., persistence ofa preliminary normal form) under arbitrary small integrable perturbations (Defi-nition 4.1). In particular, we do not assume that the S -action is preserved underthe perturbation. We also do not assume that the perturbation is parametric, soour “perturbed” system is not necessarily included into a parametric family of in-tegrable systems containing the “unperturbed” one. We also do not assume thatthe “perturbed” system has a singular orbit close to the “unperturbed” orbit.We expect that our solutions to the questions (1a)–(2c), and (3a) for several infiniteseries of local singularities, as well as (3b)–(3d) for parabolic orbits with resonances,will be helpful for solving the questions (3b)–(3d) for other singularities, including thosefrom Examples 3.6 and 3.12, as we conjectured in Example 4.2 (B).5he author is grateful to A. Bolsinov for helpful comments on Cartan subalgebras of theLie algebra sp (2 n, R ) and valuable suggestions on a preliminary version of the paper,to A. Oshemkov for useful discussion on proving extendability of homomorphisms toa circle from a finite subgroup of a torus (cf. (10)), and to S. Nemirovski for helpfulcomments on topologies on the spaces of analytic functions. This work was supportedby the Russian Science Foundation (project 17-11-01303). This section is devoted to solving the questions (1a)–(1c) from Introduction.The following theorem tells us how to partially solve the questions (1a) and (1c) w.r.t.some of the S -subactions of the desired torus action. In detail: we should apply thistheorem several times, in order to find several S -actions (some of them will be locallyfree, and the others will leave our orbit fixed). Such S -actions automatically pairwisecommute, so all together they form a single ( S ) r ′ + κ ′ e -action, that will be a subactionof the desired ( S ) r + κ e -action, where 0 ≤ r ′ ≤ r , 0 ≤ κ ′ e ≤ κ e . This theorem givessufficient conditions for • the existence of a (not necessarily locally-free) Hamiltonian S -action that pre-serves the momentum map, and • persistence of such an action under small integrable perturbations.The case of a nondegenerate orbit (the existence part only) was treated in [20, 15, 3, 38].The case of a locally-free action (also the existence part only) was treated in [1, Theorem2.1], [39, Theorem 1.2] (the corank-1 case), [40]. Actually our proof is analogous to theproof of theorems about torus actions in [38, 1, 3, 39, 40].Denote by O m the orbit of a point m ∈ M under the (local) Hamiltonian action of R n on ( M, Ω) generated by the functions f , . . . , f n . Denote by X f the Hamiltonian vectorfield with the Hamilton function f .In the following two theorems, the momentum map F is not necessarily proper. Theorem 2.1.
Let ( M n , Ω , F ) be a real-analytic integrable Hamiltonian system, m ∈ M a singular point of the momentum map F = ( f , . . . , f n ) , F ( m ) = (0 , . . . , , and L = F − (0 , . . . , the singular fiber containing the point m .Suppose there exists a point m ∈ L satisfying the following conditions: F ( m ) = n , i.e. m is a regular point of the momentum map F , (ii) there exist a compact trajectory γ (i.e. a closed trajectory or an equilibrium) and acontinuous one-parameter family of π -periodic trajectories γ u ⊂ O m , < u ≤ ,of the vector field X f such that m ∈ γ =: γ and m ∈ γ ⊂ S such that, for any (“perturbed”) real-analytic integrable Hamiltonian system ( M n , ˜Ω , ˜ F ) that is ε − close to the initial systemin C k -norm, the following properties hold. There exist a bigger neighbourhood ˜ U ⊃ U ′ and a unique ˜ F -preserving Hamiltonian (w.r.t. the “perturbed” symplectic structure) S -action on ˜ U generated by a function ˜ I ( ˜ f , . . . , ˜ f n ) , where ˜ I ( z , . . . , z n ) is a real-analytic function on some bigger neighbourhood ˜ V = ˜ F ( ˜ U ) ⊃ V ′ that is O ( ε ) − close tothe function I ( z , . . . , z n ) in C k -norm. The “perturbed” action function ˜ I ( z , . . . , z n ) can be computed by the Mineur-Arnold integral formula ˜ I ( z , . . . , z n ) = 12 π I ˜ γ ( z ,...,zn ) ˜ α + const , ( z , . . . , z n ) ∈ ˜ V , (2) where ˜ γ ( z ,...,z n ) ⊂ ˜ F − ( z , . . . , z n ) denotes a closed curve close to γ , and ˜ α is any analytic1-form on a neighbourhood of γ such that ˜Ω = d ˜ α (such a 1-form always exists). In the following theorem, we show how to solve the questions (1b) and (1c) fromIntroduction, as well as the remaining part of the questions (1a) and (1c). Similarly7o the previous theorem, we formulate our solution w.r.t. some of the S -subactionsof the desired torus action. By using this theorem, one can obtain the remaining( S ) r − r ′ + κ e − κ ′ e + κ h -subaction of the desired ( S ) r + κ e + κ h –action.Denote by M C a small open complexification of M , on which Ω C and F C are defined.Denote by O C m the orbit of a point m ∈ M C under the (local) Hamiltonian action of C n on M C generated by the functions f C , . . . , f C n . For a holomorphic function f on M C , denote by X f the Hamiltonian vector field on M C with the Hamilton function f .Denote by Sing( F C ) the set of singular points of F C . Theorem 2.2.
Let ( M n , Ω , F ) be a real-analytic integrable Hamiltonian system, m ∈ M a singular point of the momentum map F = ( f , . . . , f n ) , F ( m ) = (0 , . . . , .Suppose there exists a point m ∈ L C = ( F C ) − (0 , . . . , and λ ∈ C \ { } such that (i) rank d F C ( m ) = n , i.e. m is a regular point of the map F C , (ii) there exist a compact trajectory γ (i.e. a closed trajectory or an equilibrium) and acontinuous one-parameter family of π -periodic trajectories γ u ⊂ O C m , < u ≤ ,of the vector field X λf such that m ∈ γ =: γ and m ∈ γ ⊂ S , there exists a ∈ R n , | a | < ε , such that the closed path γ a from (3) is homological in the fiber ( F C ) − ( a ) \ Sing( F C ) to its C -conjugated path γ a (respectively, to the closed path obtained from γ a by reversing orientation). hen λ ∈ R (respectively λ ∈ i R ) and the “normalized” action function λ I ( z , . . . , z n ) is real-valued (and, hence, real-analytic) on the domain V ∩ R n . (b) This S -action is persistent under real-analytic integrable perturbations in the fol-lowing sense. Suppose we are given k ∈ Z + , a neighbourhood U ′ of the set C in M C anda neighbourhood V ′ of the origin in C n having compact closures U ′ ⊂ U and V ′ ⊂ V .Then there exists ε > such that, for any (“perturbed”) real-analytic integrable Hamilto-nian system ( M, ˜Ω , ˜ F ) whose holomorphic extension to M C is ε − close to ( M C , Ω C , F C ) in C − norm, the following properties hold. On some neighbourhood ˜ U ⊃ U ′ , there existsa unique ˜ F C -preserving Hamiltonian (w.r.t. the “perturbed” symplectic structure ˜Ω ) S -action generated by a function ˜ I ( ˜ f C , . . . , ˜ f C n ) , where ˜ I ( z , . . . , z n ) is a holomorphic func-tion on some neighbourhood ˜ V ⊃ V ′ that is O ( ε ) − close to I ( z , . . . , z n ) in C k − norm.The “perturbed” action function ˜ I ( z , . . . , z n ) can be computed by the Mineur-Arnoldintegral formula ˜ I ( z , . . . , z n ) = 12 π I ˜ γ ( z ,...,zn ) ˜ α C + const , ( z , . . . , z n ) ∈ ˜ V , (4) where ˜ γ ( z ,...,z n ) ⊂ ( ˜ F C ) − ( z , . . . , z n ) is a closed curve close to γ , ˜ α C is a holomorphic1-form on a neighbourhood of γ such that ˜Ω C = d ˜ α C (such a 1-form always exists).Let, in addition, the curve γ in (ii) be homologically symmetric , i.e. satisfy (iii) from (a) . Then the “perturbed” “normalized” action function λ ˜ I ( z , . . . , z n ) is real-valued(and, hence, real-analytic) on the domain ˜ V ∩ R n . Proofs of Theorems 2.1 and 2.2 are given in § A singular point m of rank 0 is called nondegenerate (cf. e.g. [10]) if the linearizations A j of the Hamiltonian vector felds X f j at the point m span a Cartan subalgebra ofthe Lie algebra of the Lie group Symp( T m M, Ω | m ) ≃ Symp(2 n, R ), i.e. the operators A , . . . , A n span an n -dimensional commutative subalgebra and there exists a linearcombination A = n P j =1 c j A j , c j ∈ R , having a simple spectrum: | Spec A | = 2 n . Asingular point m of rank r is called nondegenerate (cf. e.g. [10]) if the correspondingrank-0 singular point of the corresponding reduced integrable Hamiltonian system with n − r degrees of freedom (obtained by local symplectic reduction under the action of f , . . . , f r such that df ∧ · · · ∧ df r | m = 0) is nondegenerate.A singular orbit (respectively, fiber) is called nondegenerate if each singular point con-tained in this orbit (fiber) is nondegenerate.9 singular orbit O m is called of finite type [41, Definition 3.6] if there is only a fi-nite number of orbits of the infinitesimal action of C n ≈ R n on the fiber L C ⊂ M C containing m , and L C contains a regular point of the map F C .Due to the Vey theorem [35], each nondegenerate orbit is of finite type. Remark 2.3.
Let us explain the meaning of periodicity condition (ii) in Theorem 2.1(and its analogue in Theorem 2.2). Consider the commuting vector fields X f , . . . , X f n on the regular fiber L \ Sing( F ). Since they are linearly independent, they form a basisof T m L at each point m ∈ L \ Sing( F ). Since they pairwise commute, they define aflat affine connection on L \ Sing( F ). Moreover, this flat affine connection is integer .The condition (ii) in Theorem 2.1 simply means that there exists a closed geodesic γ on L \ Sing( F ) w.r.t. this integer flat affine connection, moreover the following conditionshold: the velocity vector of γ equals X f | γ and there exists a compact trajectory γ of X f passing through m . The following condition is sufficient for the above condition:there exist a closed curve ˆ γ (not necessarily a geodesic) on U ∩ L \ Sing( F ) such that R ˆ γ θ j = 2 πδ j , 1 ≤ j ≤ n , and there exist a sequence ε k ց γ k ⊂ U ∩ L \ Sing( F ) homotopic to ˆ γ in U ∩ L \ Sing( F ), k ∈ N , such that γ k liesin the ε k -neighbourhood of m (w.r.t. a fixed local coordinates on U ) and R γ k d s = o (1 /ε k )(i.e., γ k is “not too long”). Here θ , . . . , θ n are 1-forms on L \ Sing( F ) forming a dualbasis of T ∗ m L to the basis X f | m , . . . , X f n | m of T m L at each point m ∈ L \ Sing( F ),and d s := n P j =1 ( θ j ) is the flat Riemannian metric on L \ Sing( F ).The periodicity condition (ii) in Theorem 2.2 means that there exists a closed geodesic γ on L C \ Sing( F C ) w.r.t. the similar flat affine connection on the complexified fiber L C \ Sing( F C ), moreover the following conditions hold: the velocity vector of γ equals X λf | γ and there exists a compact trajectory γ ⊂ M of X λf passing through m . Asufficient condition is that there exist a closed curve ˆ γ (not necessarily a geodesic) on L C \ Sing( F C ) such that R ˆ γ θ j = 2 πλδ j , 1 ≤ j ≤ n , and there exist a sequence ε k ց γ k ⊂ L C \ Sing( F C ) homotopic to ˆ γ in L C \ Sing( F C ), k ∈ N , such that γ k lies in the ε k -neighbourhood of m (w.r.t. a fixed local coordinateson M C ) and R γ k d s = o (1 /ε k ) (i.e., γ k is “not too long”). Here d s := n P j =1 θ j θ j is the flatRiemannian metric on L C \ Sing( F C ). Example 2.4.
Let us verify the condition (iii) from Theorem 2.2 (a) on homologicalsymmetry for basic nondegenerate singularities: elliptic, hyperbolic and focus-focus.10e) Consider an elliptic nondegenerate rank 0 singularity, given by f = ( p + q )and Ω = d p ∧ d q on R with coordinates ( p, q ). Take a small ε > m = ( ε, iε ). Since f C ( m ) = 0, we have m ∈ L C . Then the Hamiltoniansystem has the form d p d t = − ∂f ∂q = − q , d q d t = ∂f ∂p = p . Its solutions are γ a,b ( t ) =( ae − it + be it , iae − it − ibe it ) with arbitrary constants a, b ∈ C . So we have a solution γ ( t ) = γ ε, ( t ) = εe − it (1 , i ) with γ (0) =: m ∈ L C . Since this solution is 2 π -periodic,we have λ = 1 ∈ R . Take a real regular value a = 4 ε ∈ R close to 0. Considerthe closed path γ ′ ( t ) = γ ε,ε ( t ) = εe − it (1 + e it ε, i − ie it ε ) in the (Milnor’s) fiber( F C ) − ( a ) = { p + q = 4 ε } . This path is a closed path obtained from γ by a smalldeformation. The C -conjugated path is γ ′ ( t ) = εe it (1 + e − it ε, − i + ie − it ε ) = γ ε ,ε ( t ).Then γ ′ ( t ) and γ ′ ( t ) are homological in the (Milnor’s) fiber ( F C ) − ( a ). Indeed, theyare orbits of the Hamiltonian S -action generated by f C , thus they can be connectedwith each other by a 1-parameter family of such orbits in the regular (Milnor’s) fiber( F C ) − ( a ). We have a real λ = 1 ∈ R and a real-analytic 2 π -periodic first integral I ( f ) = f , as Theorem 2.2 (a) asserts.(h) Consider a hyperbolic nondegenerate rank 0 singularity, given by f = pq andΩ = d p ∧ d q on R with coordinates ( p, q ). Then the Hamiltonian system with theHamilton function if C has the form d p d t = − i ∂f ∂q = − ip , d q d t = i ∂f ∂p = iq . Its solutionsare γ a,b ( t ) = ( ae − it , be it ) with arbitrary constants a, b ∈ C . Take a small ε > m = ( ε, ∈ L . So we have a solution γ ( t ) = γ ε, ( t ) = ( εe − it , γ (0) = m ∈ L . Since this solution is 2 π -periodic, we have λ = i ∈ i R . The C -conjugated path is γ ( t ) = ( εe it ,
0) = γ ( − t ). Thus γ ( t ) and γ ( − t ) are homological in theregular part of the fiber L C , since they just coincide. We have an imaginary λ = i ∈ i R and a holomorphic 2 π -periodic first integral I ( f ) = if , moreover iI ( f ) = − f isreal-valued, as Theorem 2.2 (a) asserts.(f) Consider a focus-focus nondegenerate rank 0 singularity, given by f = p q − p q , f = p q + p q and Ω = d p ∧ d q + d p ∧ d q on R with coordinates ( p, q ) =( p , p , q , q ). We have two commuting Hamiltonian S -actions on a small open com-plexification of the origin, namely those generated by f C and if C .The Hamiltonian S -action generated by f is given by the Hamiltonian system d p d t = − ∂f ∂q = p , d p d t = − ∂f ∂q = − p , d q d t = ∂f ∂p = q , d q d t = ∂f ∂p = − q . Its trajectory γ ( t ) = ε (cos t, − sin t, ,
0) with γ (0) = ( ε, , ,
0) =: m lies on a regular part of L ,and Theorem 2.1 can be applied to it. We have a real λ = 1 ∈ R and a real-analytic2 π -periodic first integral I ( f , f ) = f , as Theorem 2.1 (a) asserts.The Hamiltonian S -action generated by if C is given by the Hamiltonian system d p d t = − i ∂f ∂q = − ip , d p d t = − i ∂f ∂q = − ip , d q d t = i ∂f ∂p = iq , d q d t = i ∂f ∂p = iq . Its orbit γ ( t ) = ε ( e − it , , ,
0) with γ (0) = ( ε, , ,
0) =: m lies on a regular part of L C . Sincethis solution is 2 π -periodic, we have λ = i ∈ i R . The C -conjugated path is γ ( t ) =( εe it , , ,
0) = γ ( − t ). Thus γ ( t ) and γ ( − t ) are homological in the regular part of the11ber L C , since they just coincide. We have an imaginary λ = i ∈ i R and a holomorphic2 π -periodic first integral I ( f , f ) = if , moreover iI ( f , f ) = − f is real-valued, asTheorem 2.2 (a) asserts. In this section, we solve the questions (2a)–(2c) and (3a) from Introduction.In particular, we describe any Hamiltonian torus action on a neighborhood of its orbit,and prove persistence of its canonical model under peturbations.Such a torus action can be obtained e.g. from a Hamiltonian R n -action generated bythe momentum map F = ( f , . . . , f n ) of an integrable Hamiltonian system, via eitherresults of [38, 1, 15, 3, 39, 40] or our results of the previous section. If we do so,we will obtain a Hamiltonian action of a torus generated by some functions of theform I j = I j ( f , . . . , f n ), 1 ≤ j ≤ r + κ . As we are mostly interested in the singularLagrangian fibration (rather than specific commuting functions f , . . . , f n ), we allowourselves to replace f , . . . , f n with I , . . . , I r + κ , f r + κ +1 , . . . , f n where ∂ ( I ,...,I r + κ ) ∂ ( f ,...,f r + κ ) = 0.So, we can assume that some of the components of the momentum map generate atorus action. In this section (except for Examples 3.6, 3.12), we forget about othercomponents of the momentum map (even about their existence). After that, one canstudy normal form of the system and its perturbations, see Examples 3.6, 3.12 below.It is well known that a smooth action of a compact Lie group G is linearizable ona neighbourhood of its fixed point ([2], [7, Sec 3.1.4]). According to the Darboux-Weinstein theorem (which is an equivariant Darboux theorem), a smooth symplecticaction of a compact Lie group G is symplectically linearizable on a neighbourhood of itsfixed point ([37, Lecture 5], [9]). This extends to arbitrary Lie groups and their arbitraryorbits under natural assumptions [28, 18]. Moreover, an integrable Hamiltonian systemadmitting a symplectic action of a compact Lie group G preserving the momentummap of the system, is equivariantly fiberwise symplectomorphic to a linear model on aneighbourhood of an invariant compact nondegenerate singular orbit O [30].In this section, we formulate Theorems 3.4 and 3.10 about • reduction to a symplectic normal form, so-called canonical model, for a Hamil-tonian torus action generated by smooth (real-analytic, respectively) functions(some of which are multiplied by √−
1, respectively) on a neighbourhood of a(may be degenerate) singular orbit O in M (in M C , resp.);12 persistence of the canonical model of a torus action under small perturbations ofthe action in the class of Hamiltonian torus actions.In particular, we will define discrete parameters that completely determine the canonicalmodel up to symplectomorphism: so-called Williamson type of the orbit, κ tuples ofintegers and r tuples of ratios called elliptic, hyperbolic and twisting resonances of theorbit, respectively. Here r and r + κ denote dimensions of the orbit and the torus,respectively. Proofs of Theorems 3.4 and 3.10 will be given in Sections 6, 7. In this subsection, we solve the questions (2a), (2c) and partially (3a) from Introduction.Denote by λ = ( λ , . . . , λ r ) a linear coordinate system on a small ball D r of dimension r centred at the origin, ϕ = ( ϕ , . . . , ϕ r ) a standard periodic coordinate system of thetorus ( S ) r , and ( x, y ) = ( x , y , . . . , x n − r , y n − r ) a linear coordinate system on a smallpolydisc ( D ) n − r of dimension 2( n − r ) centred at the origin. Consider the manifold V = D r × ( S ) r × ( D ) n − r , (5)with the standard symplectic form r P s =1 d λ s ∧ d ϕ s + n − r P j =1 d x j ∧ d y j , and the following map:( λ, H ) = ( λ , . . . , λ r , h , . . . , h k e ) : V → R r + k e (6)where r, k e ∈ Z + , r + k e ≤ n , h j = x j + y j ≤ j ≤ k e . (7)Let Γ be a group acting on the product D r × ( S ) r × ( D ) n − r by symplectomorphismspreserving the map ( λ, H ). We will say that the action of Γ is linear (compare [30, § acts on the product V = D r × ( S ) r × ( D ) n − r componentwise; the action of Γ on D r is trivial, its action on ( S ) r is by translations (w.r.t. the coordinate system ϕ ),and its action on ( D ) n − r is linear w.r.t. the coordinate system ( x, y ) . Suppose now that Γ is a finite group with a free symplectic action on V that is linear (see(L) above) and preserves the map ( λ, H ). Then we can form the quotient symplecticmanifold V /
Γ, with a ( S ) r + κ e -action on it generated by the following momentum map ,all whose components are linear or quadratic functions:( λ, Q ) = ( λ , . . . , λ r , Q . . . , Q κ e ) : V / Γ → R r + κ e , where Q ℓ := k e X j =1 p jℓ h j , (8)13 ≤ ℓ ≤ κ e , for some integers κ e ∈ Z + and p jℓ ∈ Z . Suppose also that(i) rank k p jℓ k = κ e , thus κ e ≤ k e ,(ii) Γ acts on ( D ) n − r = ( D ) k e × ( D ) n − r − k e componentwise, and the induced actionon ( D ) n − r − k e is by involutions.The set O := { λ s = x j = y j = 0 } / Γ ⊂ V /
Γis a rank- r orbit of the above ( S ) r + κ e -action on V / Γ. Definition 3.1.
Consider the above Hamiltonian ( S ) r + κ e -action on V /
Γ generated bythe momentum map (8), satisfying the assumptions (i) and (ii) from above. We willcall this action the linear ( S ) r + κ e -action (or linear model ) of rank r , Williamson type ( k e , , elliptic resonances ( p ℓ : · · · : p k e ,ℓ ) ∈ Q P k e − , ≤ ℓ ≤ κ e , (9)and twisting group Γ (or, more precisely, twisting linear action of Γ on V ), providedthat the integer k e cannot be made smaller via a linear change of coordinates ( x, y ) on( D ) n − r (which is equivalent to the fact that, for each j ∈ { , . . . , k e } , either κ e P ℓ =1 | p jℓ | > a ∈ { , . . . , r } such that 2 q ψ a ,j Z , see (11) below). Remark and Definition 3.2. (A) Since Γ freely acts on V componentwise and theinduced action on ( S ) r is by translations, we can regard Γ as a subgroup of ( S ) r . Onecan show that a (free) twisting linear action of Γ on V has the form( λ, ϕ, z , . . . , z n − r ) ( λ, ϕ + ψ, e i h m ,ψ i z , . . . , e i h m n − r ,ψ i z n − r ) , ψ ∈ Γ ⊂ ( S ) r , (10)for some m , . . . , m n − r ∈ Z r , for an appropriate choice of symplectic coordinates ( x, y )on ( D ) n − r satisfying (8) (corresponding to a root decomposition of R n − r ) w.r.t. thecommuting ( S ) κ e -action and Γ-action). Here we used the notation m j =: ( m j , . . . , m jr ), ψ := ( ψ , . . . , ψ r ), h m j , ψ i := r P s =1 m js ψ s , z j := x j + iy j , 1 ≤ j ≤ n − r . In other words, Indeed, it follows from Theorem 3.10 that the action of generators ψ a := p ( γ a ) of Γ on V has theform ( λ, ϕ, z , . . . , z n − r ) ( λ, ϕ + ψ a , e πiq γa, z , . . . , e πiq γa,n − r z n − r ) , ≤ a ≤ r, for some q γ a ,j ∈ Q . Here γ , . . . , γ r is a generating set of the lattice p − (Γ) ⊂ R r , p : R r → ( S ) r denotes the projection. Since p − (Γ) is a lattice in R r , there exists a unique linear map R r → R n − r sending γ a π ( q γ a , , . . . , q γ a ,n − r ), 1 ≤ a ≤ r . Clearly, this linear map has the form γ ( h m , γ i , . . . , h m n − r , γ i ), γ ∈ R r , for some m , . . . , m n − r ∈ R r . From the short exact sequence 0 → π Z r → p − (Γ) → Γ →
0, we conclude that h m j , γ i ∈ π Z , provided that γ ∈ π Z r . Therefore m j ∈ Z r . This proves (10).
14 (free) twisting linear action of Γ on V can be extended to a (free) linear Hamiltonianaction of ( S ) r ⊃ Γ preserving the momentum map (8).(B) We will call such a twisting linear action of Γ on V the twisting linear action with twisting resonances ( q ψ a , , . . . , q ψ a ,n − r ) ∈ ( Q / Z ) n − r , where q ψ a ,j := h m j , ψ a π i mod 1 ∈ Q / Z , (11)1 ≤ a ≤ r , 1 ≤ j ≤ n − r , and 2 q ψ a ,j ⊂ Z for k e + 1 ≤ j ≤ n − r (due to assumption (ii)from above). Here γ , . . . , γ r denote a basis of the homology group H ( O ) ≃ p − (Γ) ⊂ π Q r ⊂ R r , thus ψ a := p ( γ a ), 1 ≤ a ≤ r , is a generating set of the group Γ, where p : R r → ( S ) r is the projection.(C) We always may assume that each basic cycle γ a has coordinates γ as = πN a δ as , 1 ≤ a, s ≤ r , where N a is a positive integer. In this case, we have q ψ a ,j = m ja N a mod 1. Definition 3.3.
In notations of Definition 3.1, Remark and Definition 3.2, we will callthe linear ( S ) r + κ e -action on V /
Γ from Definition 3.1 the linear ( S ) r + κ e -action with • dimension 2 n , rank r , Williamson type ( k e , , • elliptic resonances (9) assigned to the basic cycles of the subtorus ( S ) κ e of( S ) r + κ e (which is the isotropy subgroup of some and, hence, any point of O ), • twisting resonances (11) assigned to the basic cycles of the orbit O = { λ s = x j = y j = 0 } / Γ ≈ ( S ) r / Γ relatively the action of the subtorus ( S ) r of ( S ) r + κ e .Clearly, dimension, rank, the Williamson type together with elliptic and twisting res-onances completely determine the symplectic manifold V /
Γ with the linear ( S ) r + κ e -action on it, up to symplectomorphism. See also Remark 3.5 about uniqueness of theWilliamson type and the resonances.Now we can formulate our result in the elliptic case, which is the symplectic normal-ization theorem for singular orbits of Hamiltonian torus actions: Theorem 3.4.
Suppose we are given an effective Hamiltonian action of the ( r + κ e ) -torus ( S ) r + κ e generated by C ∞ -smooth functions I , . . . , I r , J , . . . , J κ e on a C ∞ -smoothsymplectic manifold ( M, Ω) . Suppose a point m ∈ M is fixed under the ( S ) κ e -subaction, and its orbit O m is r -dimensional. Then: (a) There exist an invariant neighbourhood U of O m , a finite group Γ , a linear ( S ) r + κ e -action of rank r on the symplectic manifold V / Γ given by (5)–(8), and a smooth action-preserving symplectomorphism φ from U to V / Γ , that sends the orbit O m to the torus O = { λ s = x j = y j = 0 } / Γ . If the system is real-analytic, φ is real-analytic too. n particular, dimensions of the torus ( S ) r + κ e and its subtorus ( S ) κ e satisfy the esti-mates r + κ e ≤ r + k e ≤ n . If r + κ e = n then the orbit O m is nondegenerate. (b) The symplectic normalization (5)–(8) for a torus action is persistent under C ∞ -smooth perturbations in the following sense. For any integer k ≥ and any neighbour-hood U ′ of the orbit O m having a compact closure U ′ ⊂ U , there exists ε > satisfyingthe following. Suppose we are given a C ∞ -smooth Hamiltonian ( S ) r + κ e -action on ˜ M (w.r.t. the “perturbed” symplectic structure ˜Ω that is ε -close to Ω in C k + n − r − − norm)generated by a (“perturbed”) momentum map ˜ F = ( ˜ I , . . . , ˜ I r , ˜ J , . . . , ˜ J κ e ) that is ε -closeto F = ( I , . . . , I r , J , . . . , J κ e ) in C k − norm, where U ⊂ ˜ M ⊂ M . Then there exist aninvariant neighbourhood ˜ U ⊃ U ′ , and a smooth action-preserving symplectomorphism ˜ φ from ˜ U to V / Γ that is O ( ε ) − close to φ in C k − − norm. If the systems are real-analytic, ˜ φ is real-analytic too. If the system depends smoothly (resp., analytically) on a localparameter (i.e. we have a local family of systems), φ can also be chosen to dependsmoothly (resp., analytically) on that parameter. Remark 3.5. (A) Consider a linear ( S ) r + κ e -action on V /
Γ (see Definition 3.1, Re-mark and Definition 3.2, Definition 3.3). What is a dynamical meaning of the ellipticresonances (9) and the twisting resonances (11)? The eigenvalues of the linearized ℓ -th infinitesimal generator of the κ e -subtorus action on ( D ) k e are pure imaginary andare in (9) resonance, which is exactly the elliptic resonance assigned to this generator.Further, the multipliers of a cycle γ ∈ H ( O ) ≈ p − (Γ) ⊂ π Q r ⊂ R r of the torus O = { λ s = x j = y j = 0 } / Γ under the r -subtorus action on ( D ) k e equal e ± πiq ψ,j ,where ψ := p ( γ ) ∈ Γ and q ψ,j mod 1 := h m j , ψ π i ∈ Q / Z , 1 ≤ j ≤ n − r . So, they arein ( q ψ, , . . . , q ψ,n − r ) resonance, which is exactly the twisting resonance assigned to thecycle γ . Here p : R r → ( S ) r denotes the projection.(B) Clearly, Williamson type is well defined, i.e. completely determined by the (“hid-den”) torus-symmetry of the orbit O . Are the elliptic resonances also well defined(perhaps, up to some natural transformations)? Clearly, the subtorus ( S ) κ e of thetorus ( S ) r + κ e trivially acts on O , moreover this subtorus is the isotropy subgroup ofany point of O , thus this subtorus is well defined, i.e. does not depend on the choice ofgenerators of the ( S ) r + κ e -action. It follows that the elliptic resonances (9), 1 ≤ ℓ ≤ κ e ,are well defined up to replacing the vectors ( p ℓ , . . . , p k e ,ℓ ), 1 ≤ ℓ ≤ κ e , by their linearcombinations with integer coefficients forming a nondegenerate κ e × κ e -matrix.(C) Are the twisting resonances also well defined (perhaps, up to some natural trans-formations)? Suppose we allow ourselves to change the generators of the subtorus ( S ) r that acts locally-freely on our manifold V /
Γ. At the same time, suppose that we fixedbasic cycles γ , . . . , γ r ∈ H ( O ) ⊂ Q r ⊂ R r of the orbit O , and we want that the cycle γ a has coordinates γ as = πN a δ as w.r.t. to the generators of the subtorus ( S ) r , both beforeand after the change, may be with different integers N a , 1 ≤ a, s ≤ r (see Remark andDefinition 3.2 (C)). Thus, the above change is equivalent to replacing the coordinates16 s , ϕ s , z j withˆ λ s := ν s λ s + κ e X ℓ =1 n ℓs Q ℓ , ˆ ϕ s := 1 ν s ϕ s , ˆ z j := e − i h ( pn ) j ,ϕ i z j for 1 ≤ s ≤ r and 1 ≤ j ≤ n − r . Here ν s , n s , . . . , n κ e s are coprime integers such that ν s = 0, and we denoted ( pn ) js := κ e P ℓ =1 p jℓ n ℓs if 1 ≤ j ≤ k e , ( pn ) js := 0 if k e + 1 ≤ j ≤ n − r , ( pn ) j := (( pn ) j , . . . , ( pn ) jr ), ϕ := ( ϕ , . . . , ϕ r ), h ( pn ) j , ψ i := r P s =1 ( pn ) js ψ s . Thisreplacement will lead to the following replacements: ψ aj with ˆ ψ aj such that ν j ˆ ψ aj = ψ aj ,and m j ∈ Z r with ˆ m js := ν s ( m js − ( pn ) js ). Thus, the twisting resonances (11) will bereplaced with(ˆ q ψ a , , . . . , ˆ q ψ a ,n − r ) ∈ ( Q / Z ) n − r , where ˆ q ψ a ,j = q ψ a ,j + h ( pn ) j , ψ a π i mod 1 ∈ Q / Z , ≤ a ≤ r , 1 ≤ j ≤ n − r . If we recall that γ as = πN a δ as , 1 ≤ a, s ≤ r , then ˆ q ψ a ,j = q ψ a ,j + ( pn ) ja /N a . In other words, the twisting resonances (11) are well defined upto adding any linear combinations of “extended” elliptic resonances (9) with rationalcoefficients forming a r × κ e -matrix. Here, by the ℓ -th extended elliptic resonance ,1 ≤ ℓ ≤ κ e , we mean the vector ( p ,ℓ , . . . , p k e ,ℓ , , . . . , ∈ Z n − r . Example 3.6.
Suppose O is a compact r -dimensional orbit of the momentum mapof a real-analytic integrable Hamiltonian system with n degrees of freedom. In otherwords, O is a rank- r local singularity. Suppose this singularity has one of the followingtypes:(a) a parabolic orbit with resonance ℓ/s (see [21] or § s ≥
5, given by a mo-mentum map F = ( H, I ) : ( D λ ) × S ϕ ) × D x,y ) ) / Z s → R where I = λ , H = Re ( z s ) + | z | + λ | z | , we denote z = x + iy ; a generator of Z s acts on D × S × D by the transformation ( λ, ϕ, z ) ( λ, ϕ + 2 π/s, e πiℓ/s z ), s, ℓ ∈ Z ,0 ≤ ℓ < s , ( s, ℓ ) = 1;(b) an integrable Hamiltonian Hopf bifurcation with resonance p : q [11, 19], p, q ∈ Z ,0 < p < q , ( p, q ) = 1, pq = , given by a momentum map F = ( H, I, J ) : D λ ) × S ϕ ) × D z ,z ) → R where I = λ , J = p | z | + q | z | , H = Re ( z p ¯ z q )+ a | z | + λ | z | ,one denotes z j = x j + iy j , a is a real parameter;(c) a normally-elliptic parabolic orbit with resonance ℓ/s (compare [6]), s ≥
5, givenby a momentum map F = ( H, I, J ) : ( D λ ) × S ϕ ) × D z ,z ) ) / Z s → R where I = λ , J = | z | , H = Re ( z s ) + | z | + ( I ± J ) | z | , one denotes z j = x j + iy j ,a generator of Z s acts on D × S × D by the transformation ( λ, ϕ, z , z ) ( λ, ϕ + 2 π/s, z , e πiℓ/s z ), s, ℓ ∈ Z , 0 ≤ ℓ < s , ( s, ℓ ) = 1;17d) a Hamiltonian swallow-tail bifurcation with resonance ℓ/
5, given by a momentummap F = ( H, I , I ) : ( D λ ,λ ) × ( S ) ϕ ,ϕ ) × D x,y ) ) / Z → R where I = λ , I = λ , H = Re ( z ) + | z | + λ | z | + λ | z | , one denotes z = x + iy , a generator of Z acts on D × ( S ) × D by the transformation ( λ , λ , ϕ , ϕ , z ) ( λ , λ , ϕ +2 π/ , ϕ , e πiℓ/ z ), ℓ ∈ Z , 1 ≤ ℓ < § F -preserving Hamiltonian ( S ) r -action on someneighbourhood of O . It can be shown from Theorem 2.1 or 2.2 that this action extendsto a “hidden” ( S ) r + κ e -symmetry, i.e. to an effective (not locally-free) F -preservingHamiltonian ( S ) r + κ e -action, with r + κ e = n −
1. Moreover, this action is persistentunder real-analytic integrable perturbations. By Theorem 3.4, the latter action is sym-plectomorphic to a linear ( S ) r + κ e -action, that is also persistent under real-analyticintegrable perturbations. In Table, we show Williamson type ( k e , , S ) r + κ e -action, for each type of thesingularity O from above.Case n Subtori dim’s Williamson Resonances r κ e type ( k e , ,
0) elliptic twisting(a) 2 1 0 (1 , ,
0) no ℓ/s mod 1 ∈ Q / Z (b) 3 1 1 (2 , ,
0) ( p : q ) ∈ Q P (0 , ∈ ( Q / Z ) (c) 3 1 1 (2 , ,
0) (0 : 1) ∈ Q P ( ℓ/s mod 1 , ∈ ( Q / Z ) (d) 3 2 0 (1 , ,
0) no ℓ/s mod 1 , ∈ Q / Z In this subsection, we solve the questions (2b), (2c) and (3a) from Introduction.As above, consider the manifold V = D r × ( S ) r × ( D ) n − r as in (5), with the standardsymplectic form r P s =1 d λ s ∧ d ϕ s + n − r P j =1 d x j ∧ d y j , and the following map:( λ, H ) = ( λ , . . . , λ r , h , . . . , h k f + k e + k h ) : V → R r +2 k f + k e + k h (12)where r, k e , k h , k f ∈ Z + , r + k e + k h + 2 k f ≤ n , h j − = x j − + y j − − x j + y j and h j = x j − y j + x j y j − for 1 ≤ j ≤ k f ,h j = x j + y j for 2 k f + 1 ≤ j ≤ k f + k e ,h j = x j y j for 2 k f + k e + 1 ≤ j ≤ k f + k e + k h . (13)18et Γ be a group acting on the product D r × ( S ) r × ( D ) n − r by symplectomorphismspreserving the map ( λ, H ) given by (12), (13). Suppose the group Γ is finite, and itsaction on V is free and linear (see Property (L) in § V /
Γ, with a real-analytic momentum map ( λ, Q ) = ( λ , . . . , λ r , Q . . . , Q κ ) : V / Γ → R r + κ , where Q ℓ := k e + k h +2 k f X j =1 p jℓ h j , (14)1 ≤ ℓ ≤ κ = κ e + κ h , and a ( S ) r + κ -action on its small open complexification ( V / Γ) C generated by the map( λ , . . . , λ r , Q . . . , Q κ e , iQ κ e +1 , . . . , iQ κ e + κ h ) : V / Γ → R r + κ e × ( i R ) κ h , (15)for some integers κ e , κ h ∈ Z + and p jℓ ∈ Z such that each component Q ℓ (respectively iQ ℓ ) of the map (15) is a linear combination of elliptic (respectively hyperbolic) h j , i.e. p jℓ = 0 if at least one of the following conditions (i) and (ii) holds:(i) 1 ≤ ℓ ≤ κ e and j ∈ { , , , . . . , k f } ∪ { k f + k e + 1 , . . . , k f + k e + k h } ,(ii) κ e < ℓ ≤ κ e + κ h and j ∈ { , , , . . . , k f − } ∪ { k f + 1 , . . . , k f + k e } .Suppose also that(iii) rank k p jℓ k = κ e + κ h , thus κ e ≤ k e + k f , κ h ≤ k h + k f , and κ e + κ h ≤ k e + k h +2 k f ,(iv) Γ acts on ( D ) n − r = ( D ) k e + k h +2 k f × ( D ) n − r − k e − k h − k f componentwise, and theinduced action on ( D ) n − r − k e − k h − k f is by involutions.The set O := { λ s = x j = y j = 0 } / Γ ⊂ V /
Γis a rank- r orbit of the above ( S ) r + κ e + κ h -action on ( V / Γ) C . Definition 3.7.
Consider the above Hamiltonian ( S ) r + κ e + κ h -action on ( V / Γ) C gen-erated by the map (15), satisfying the assumptions (i)–(iv) from above. We will callthis action the linear ( S ) r + κ e + κ h -action (or linear model ) of rank r , Williamson type ( k e , k h , k f ), elliptic resonances ( p ,ℓ : − p ,ℓ : p ,ℓ : − p ,ℓ : · · · : p k f − ,ℓ : − p k f − ,ℓ : p k f +1 ,ℓ : · · · : p k f + k e ,ℓ ) ∈ Q P k f + k e − , (16)1 ≤ ℓ ≤ κ e , hyperbolic resonances ( p ,ℓ : p ,ℓ : p ,ℓ : p ,ℓ : · · · : p k f ,ℓ : p k f ,ℓ : p k f + k e +1 ,ℓ : · · · : p k f + k e + k h ,ℓ ) ∈ Q P k f + k h − , (17)19 e + 1 ≤ ℓ ≤ κ e + κ h , and twisting group Γ (or, more precisely, twisting linear action of Γ on V ), provided that the integer k e + k h + 2 k f cannot be made smaller via alinear change of coordinates ( x, y ) on ( D ) n − r (which is equivalent to the fact that,for each j ∈ { , , , . . . , k f − } ∪ { k f + 1 , . . . , k f + k e } , either κ e P ℓ =1 | p jℓ | > a ∈ { , . . . , r } such that 2 q ψ a ,j Z , see (18) below; furthermore for each j ∈ { , , , . . . , k f } ∪ { k f + k e + 1 , . . . , k f + k e + k h } , we have κ e + κ h P ℓ = κ e +1 | p jℓ | > Remark and Definition 3.8. (A) Since Γ freely acts on V componentwise and theinduced action on ( S ) r is by translations, we can regard Γ as a subgroup of ( S ) r . Onecan show similarly to (10) that a (free) twisting linear action of Γ on V has the form( λ, ϕ, z , . . . , z n − r ) ( λ, ϕ + ψ, e i h m ,ψ i z , e − i h m ,ψ i z , . . . , e i h m kf ,ψ i z k f − , e − i h m kf ,ψ i z k f ,e i h m kf +1 ,ψ i z k f +1 , . . . , e i h m kf + ke ,ψ i z k f + k e , χ ( ψ ) z k f + k e +1 , . . . , χ k h ( ψ ) z k f + k e + k h ,e i h m kf + ke + kh +1 ,ψ i z k f + k e + k h +1 , . . . , e i h m n − r − kf ,ψ i z n − r ) , ψ ∈ Γ ⊂ ( S ) r , for some integer vectors m , . . . , m k f + k e , m k f + k e + k h +1 , . . . , m n − r − k f ∈ Z r and characters χ , . . . , χ k h : Γ → { , − } , for an appropriate choice of coordinates ( x, y ) on ( D ) n − r satisfying (14) (corresponding to a root decomposition of R n − r ) w.r.t. the commuting( S ) κ e + κ h -action and Γ-action). Here we used the notation m j =: ( m j , . . . , m jr ), ψ =( ψ , . . . , ψ r ) ∈ ( R / π Z ) r , h m j , ψ i := r P a =1 m ja ψ a , z j := x j + iy j .In other words, the twisting group Γ freely acts on the product V = D r × ( S ) r × ( D ) k f + k e × ( D ) k h × ( D ) n − r − k f − k e − k h componentwise, where its action on the “hy-perbolic” component ( D ) k h has the form( z k f + k e +1 , . . . , z k f + k e + k h ) ( χ ( ψ ) z k f + k e +1 , . . . , χ k h ( ψ ) z k f + k e + k h ) , ψ ∈ Γ , while its action on the product D r × ( S ) r × ( D ) k f + k e × ( D ) n − r − k f − k e − k h of theremaining components can be extended to a (free) linear Hamiltonian action of ( S ) r ⊃ Γ on V preserving the momentum map (14).(B) We will call such a twisting linear action of Γ on V the twisting linear action with twisting resonances ( q ψ a , , − q ψ a , , . . . , q ψ a ,k f , − q ψ a ,k f , q ψ a ,k f +1 , . . . , q ψ a ,n − r − k f ) ∈ ( Q / Z ) n − r , (18)1 ≤ a ≤ r , where q ψ a ,j := h m j , ψ a π i mod 1 ∈ Q / Z for j ∈ { , . . . , k f + k e } ∪ { k f + k e + k h + 1 , . . . , n − r − k f } , q ψ a ,j := − χ j ( ψ a )4 for k f + k e + 1 ≤ j ≤ k f + k e + k h , and2 q ψ a ,j ⊂ Z for k f + k e + k h + 1 ≤ j ≤ n − r − k f (due to assumption (iv) from above).Here γ , . . . , γ r denote a basis of the homology group H ( O m ) ≃ p − (Γ) ⊂ π Q r ⊂ R r ,thus ψ a := p ( γ a ), 1 ≤ a ≤ r , is a generating set of Γ, p : R r → ( S ) r is the projection.20C) Similarly to the elliptic case (see Remark and Definition 3.2 (C)), we always mayassume that γ as = πN a δ as , 1 ≤ a, s ≤ r , where N a is a positive integer. In this case, wehave q ψ a ,j = m ja N a mod 1. Definition 3.9.
We will call the linear ( S ) r + κ e + κ h -action on V /
Γ from Definition 3.7the linear ( S ) r + κ e + κ h -action with • dimension 2 n , rank r , Williamson type ( k e , k h , k f ), • elliptic resonances (16) and hyperbolic resonances (17) assigned to the basic cyclesof the subtori ( S ) κ e and ( S ) κ h of ( S ) r + κ e + κ h , respectively, • twisting resonances (18) assigned to the basic cycles of the orbit O = { λ s = x j = y j = 0 } / Γ ≈ ( S ) r / Γ relatively the action of the subtorus ( S ) r of ( S ) r + κ e + κ h .Clearly, dimension, rank, the Williamson type, the elliptic, hyperbolic and twisting res-onances completely determine the symplectic manifold V /
Γ with the linear ( S ) r + κ e + κ h -action on ( V / Γ) C , up to symplectomorphism. See also Remark 3.11 about uniquenessof the Williamson type and the resonances.Now we can formulate our result in the general real-analytic case, which is the sym-plectic normalization theorem for singular orbits of Hamiltonian torus actions: Theorem 3.10.
Suppose we are given real-analytic functions I , . . . , I r , J , . . . , J κ e + κ h on a real-analytic symplectic manifold ( M, Ω) . Suppose the functions I C , . . . , I C r , J C , . . . , J Cκ e , iJ Cκ e +1 , . . . , iJ Cκ e + κ h generate an effective Hamiltonian ( S ) r + κ e + κ h -action on ( M C , Ω C ) .Suppose a point m ∈ M is fixed under the ( S ) κ e + κ h -subaction, and its orbit O m is r -dimensional. Then: (a) There exist an ( S ) r + κ e + κ h -invariant neighbourhood U of O m in M C , a finite group Γ , a linear ( S ) r + κ e + κ h -action of rank r on the symplectic manifold ( V / Γ) C given by(5), (12)–(14), and a real-analytic symplectomorphism φ from U ∩ M to V / Γ having anaction-preserving holomorphic extension to U and sending the orbit O m to the torus O = { λ s = x j = y j = 0 } / Γ .In particular, dimensions of the torus ( S ) r + κ e + κ h and its subtori ( S ) κ e and ( S ) κ h satisfy the estimates κ e ≤ k e + k f , κ h ≤ k h + k f , and r + κ e + κ h ≤ r + k e + k h + 2 k f ≤ n .If r + κ e + κ h = n then the orbit O m is nondegenerate. (b) The symplectic normalization (5), (12)–(14) for a torus action is persistent underanalytic perturbations in the following sense. For any k ∈ Z + and any neighbourhood U ′ of the orbit O m in M C having a compact closure U ′ ⊂ U , there exists ε > satisfyingthe following. Suppose ˜ F = ( ˜ I , . . . , ˜ I r , ˜ J , . . . , ˜ J κ e + κ h ) and ˜Ω are analytic (“perturbed”)momentum map and symplectic structure, whose holomorphic extensions to M C are -close to F C = ( I C , . . . , I C r , J C , . . . , J Cκ e + κ h ) and Ω C , respectively, in C − norm. Sup-pose the functions ˜ I C , . . . , ˜ I C r , ˜ J C , . . . , ˜ J Cκ e , i ˜ J Cκ e +1 , . . . , i ˜ J Cκ e + κ h generate a Hamiltonian ( S ) r + κ e + κ h -action on ( ˜ M C , ˜Ω C ) , where U ⊂ ˜ M C ⊂ M C . Then there exist an invariantneighbourhood ˜ U ⊃ U ′ , and an analytic symplectomorphism ˜ φ from ˜ U ∩ M to V / Γ , whoseholomorphic extension to ˜ U is action-preserving and O ( ε ) − close to φ C in C k − norm.If the system depends on a local parameter (i.e. we have a local family of systems),moreover its holomorphic extension to M C depends smoothly (resp., analytically) onthat parameter, then φ can also be chosen to depend smoothly (resp., analytically) onthat parameter. Remark 3.11. (A) Consider a linear ( S ) r + κ e + κ h -action on ( V / Γ) C (see Definition 3.7,Remark and Definition 3.8, Definition 3.9). What is a dynamical meaning of the ellipticresonances (16), hyperbolic resonances (17) and the twisting resonances (18)? Theeigenvalues of the linearized ℓ -th infinitesimal generator of the κ e -dimensional subtorusaction on ( D ) k e + k h +2 k f are pure imaginary and in (16) resonance, which is exactly theelliptic resonance assigned to this generator, 1 ≤ ℓ ≤ κ e . Further, the eigenvaluesof the linearized ℓ -th infinitesimal generator of the κ h -dimensional subtorus actionon (( D ) k e + k h +2 k f ) C are real and in (17) resonance, which is exactly the hyperbolicresonance assigned to this generator, κ e +1 ≤ ℓ ≤ κ e + κ h . Furthermore, the multipliersof a cycle γ ∈ H ( O ) ≈ p − (Γ) ⊂ π Q r ⊂ R r of the torus O = { λ s = x j = y j =0 } / Γ under the r -subtorus action on ( D ) n − r equal e ± πiq ψ,j , e ± πiq ψ,j for 1 ≤ j ≤ k f , e ± πiq ψ,j for k f + 1 ≤ j ≤ n − r − k f . So, they are in ( q ψ, , − q ψ, , . . . , q ψ,k f , − q ψ,k f , q ψ,k f +1 , . . . , q ψ,n − r − k f ) ∈ ( Q / Z ) n − r resonance, which is exactly the twisting resonanceassigned to the cycle γ . Here p : R r → ( S ) r denotes the projection, ψ := p ( γ ) ∈ Γ.(B) Clearly, the Williamson type is well defined, i.e. completely determined by the(“hidden”) torus-symmetry of the orbit O . Are the elliptic and hyperbolic resonancesalso well defined (perhaps, up to some natural transformations)? At first, we recallthat the Hamiltonian action of the subtorus ( S ) r + κ e of the torus ( S ) r + κ e + κ h is gen-erated by real-valued functions, while the Hamiltonian action of the subtorus ( S ) κ h is generated by imaginary-valued functions. Thus, the subtori ( S ) r + κ e and ( S ) κ h of the torus ( S ) r + κ e + κ h are well defined, i.e. they do not depend on the choice ofgenerators of the ( S ) r + κ e + κ h -action. At second, the subtorus ( S ) κ e of the subtorus( S ) r + κ e trivially acts on O , moreover this subtorus is the isotropy subgroup of anypoint of O under the ( S ) r + κ e -action, thus the subtorus ( S ) κ e is also well defined.It follows that the elliptic resonances (16) are well defined up to replacing the vec-tors ( p ℓ , p ℓ , . . . , p k f − ,ℓ , p k f +1 ,ℓ , . . . , p k f + k e ,ℓ ), 1 ≤ ℓ ≤ κ e , by their linear com-binations with integer coefficients forming a nondegenerate κ e × κ e -matrix. Sim-ilarly, the hyperbolic resonances (17) are well defined up to replacing the vectors( p ℓ , p ℓ , . . . , p k f ,ℓ , p k f + k e +1 ,ℓ , . . . , p k f + k e + k h ,ℓ ), κ e + 1 ≤ ℓ ≤ κ e + κ h , by their lin-ear combinations with integer coefficients forming a nondegenerate κ h × κ h -matrix.(C) Are the twisting resonances also well defined (perhaps, up to some natural trans-22ormations)? Suppose we allow ourselves to change the generators of the subtorus ( S ) r that acts locally-freely on our manifold ( V / Γ) C . At the same time, suppose that wefixed basic cycles γ , . . . , γ r ∈ H ( O ) of the orbit O , and we want that each cycle γ a has coordinates γ as = πN a δ as , 1 ≤ a, s ≤ r (see Remark and Definition 3.8 (C)). Thus,the above change is equivalent to replacing the variables λ s , ϕ s , z j withˆ λ s := ν s λ s + κ e X ℓ =1 n ℓs Q ℓ , ˆ ϕ s := 1 ν s ϕ s , ˆ z j := e − i h ( pn ) j ,ψ i z j for 1 ≤ s ≤ r and 1 ≤ j ≤ n − r . Here ν s , n s , . . . , n κ e s are coprime integers suchthat ν s = 0, and we denoted ( pn ) j − ,s = − ( pn ) j,s := κ e P ℓ =1 p j − ,ℓ n ℓs if 1 ≤ j ≤ k f ,( pn ) js := κ e P ℓ =1 p k f + j,ℓ n ℓs if 2 k f + 1 ≤ j ≤ k f + k e + k h , ( pn ) js := 0 if 2 k f + k e + k h + 1 ≤ j ≤ n − r , ( pn ) j := (( pn ) j , . . . , ( pn ) jr ), ψ := ( ψ , . . . , ψ r ), h ( pn ) j , ϕ i := r P s =1 ( pn ) js ϕ s .This replacement will lead to the following replacements: ψ a = ( ψ a , . . . , ψ ra ) → ˆ ψ a =( ˆ ψ a , . . . , ˆ ψ ra ) := ( ψ a ν , . . . , ψ ra ν r ), and m j = ( m j , . . . , m jr ) → ˆ m js where ˆ m js := ν s ( m js − ( pn ) j − ,s ) if 1 ≤ j ≤ k f , ˆ m js := ν s ( m js − ( pn ) k f + j,s ) if k f + 1 ≤ j ≤ n − r − k f . Thus,the twisting resonances (18) will be replaced with(ˆ q ψ a , , − ˆ q ψ a , , . . . , ˆ q ψ a ,k f , − ˆ q ψ a ,k f , ˆ q ψ a ,k f +1 , . . . , ˆ q ψ a ,n − r − k f ) ∈ ( Q / Z ) n − r , ≤ a ≤ r , where ˆ q ψ a ,j = q ψ a ,j + h ( pn ) j − , ψ a π i mod 1 ∈ Q / Z if 1 ≤ j ≤ k f ,ˆ q ψ a ,j = q ψ a ,j + h ( pn ) k f + j , ψ a π i mod 1 ∈ Q / Z if k f + 1 ≤ j ≤ n − r − k f . If we re-call that ψ as = πN a δ as , 1 ≤ a, s ≤ r , then ˆ q ψ a ,j = q ψ a ,j + ( pn ) j − ,a /N a if 1 ≤ j ≤ k f ,ˆ q ψ a ,j = q ψ a ,j + ( pn ) k f + j,a /N a if k f + 1 ≤ j ≤ n − r − k f . In other words, thetwisting resonances (18) are well defined up to adding any linear combinations ofthe “extended” elliptic resonances (16) with rational coefficients forming a r × κ e -matrix. Here, by the ℓ -th extended elliptic resonance , 1 ≤ ℓ ≤ κ e , we mean the vector( p ,ℓ , − p ,ℓ , p ,ℓ , − p ,ℓ , . . . , p k f − ,ℓ , − p k f − ,ℓ , p k f +1 ,ℓ , . . . , p k f + k e ,ℓ , , . . . , ∈ Z n − r . Example 3.12.
In Table below, we give hyperbolic analogues of the singularities (b)and (c) from Example 3.6. In detail, suppose that O is a compact r -dimensional orbitof the momentum map of a real-analytic integrable Hamiltonian system with n degreesof freedom. Suppose this singularity has one of the following types:(b) a hyperbolic integrable Hamiltonian Hopf bifurcation [27, § F = ( H, I, J ) : D λ ) × S ϕ ) × D x ,y ,x ,y ) → R where I = λ , J = x y + x y , H = x y + ( x y ) + λx y ;(c) a normally-hyperbolic parabolic orbit with resonance ℓ/s ( s ≥ F = ( H, I, J ) : ( D λ ) × S ϕ ) × D z ,z ) ) / Z s → R where I = λ ,23 = x y , H = Re ( z s ) + | z | + ( I ± J ) | z | , one denotes z j = x j + iy j , agenerator of Z s acts on D × S × D by the transformation ( λ, ϕ, z , z ) ( λ, ϕ + 2 π/s, z , e πiℓ/s z ), s, ℓ ∈ Z , 0 ≤ ℓ < s , ( s, ℓ ) = 1.One shows from Theorem 2.2 that this singularity possesses a “hidden” ( S ) r + κ e + κ h -symmetry, i.e. an effective (not locally-free) F C -preserving Hamiltonian ( S ) r + κ e + κ h -action, with r + κ e + κ h = n −
1. Moreover, this action is persistent under real-analyticintegrable perturbations. By Theorem 3.10, the latter action is symplectomorphic to alinear ( S ) r + κ e + κ h -action, that is also persistent under real-analytic integrable pertur-bations. In Table, we show the Williamson type ( k e , k h , k f ), the elliptic, hyperbolic andtwisting resonances (Definition 3.9) of this linear ( S ) r + κ e + κ h -action, for each type ofthe singularity O from above.Case n Subtori dim’s Williamson Resonances r κ e κ h type ( k e , k h , k f ) elliptic hyperbolic twisting(b) 3 1 0 1 (0 , ,
0) no (1 : 1) ∈ Q P (0 , ∈ ( Q / Z ) (c) 3 1 0 1 (1 , ,
0) no 1 ∈ Q P ( ℓ/s mod 1 , ∈ ( Q / Z ) In this section, we solve the questions (3b)–(3d) from Introduction, for parabolic orbitswith resonances.Two singularities will be called equivalent if there exists a fiberwise homeomorphism offibration germs at these singularities.Our central object will be structurally stable singularities (Definition 4.1 below), whichare those singularities whose equivalence classes are open in the topology described be-low. Such singularities are met in typical integrable systems, so they cannot disappearafter small integrable perturbations. We will assume that the manifold M , the sym-plectic structure Ω and the momentum map F are real-analytic . Notice that, due to theCauchy theorem or the Weierstrass theorem [33, Ch. I, §
2, Theorem 8], all compact-open C k − topologies on the space of holomorphic pairs (Ω C , F C ) on M C are pairwiseequivalent for all k ∈ Z + . Here M C denotes a small open complexification of M , whileΩ C , F C are holomorphic extensions of Ω , F to M C . Thus, below we can take k = 0. Definition 4.1.
A singularity of a singular Liouville fibration ( M, Ω , F ) will be called structurally stable under real-analytic integrable perturbations , or simply structurallystable if it has a neighbourhood U such that, for any smaller neighbourhood U witha compact closure U ⊂ U , there exist ε > U C of U satisfying the following. For any real-analytic integrable perturbation ( U , ˜Ω , ˜ F )24f ( U , Ω | U , F | U ) such that k ˜Ω C − Ω C k C k + k ˜ F C − F C k C k < ε , the singular Liouvillefibrations ( U, Ω | U , F | U ) and ( ˜ U , ˜Ω | ˜ U , ˜ F | ˜ U ) are equivalent (i.e., fiberwise homeomorphic)for some neighbourhoods U, ˜ U ⊂ U each of which contains U . In a similar way, onedefines structural stability under integrable perturbations of some class , e.g. the classesof C ∞ perturbations , symmetry-preserving perturbations etc. Example 4.2. (A) Consider a nondegenerate singular orbit O of a real-analytic inte-grable system. One can show, using the Vey-Eliasson theorem [35] (cf. [14] for C ∞ case), that every point m ∈ O is structurally stable under integrable perturbations(both in real-analytic and C ∞ cases). Let us show that the orbit O is structurally sta-ble (under real-analytic integrable perturbations), provided that O is compact. Due tothe result by Ito [20], our singular Lagrangian fibration on a neighbourhood U of O canbe defined by a momentum map F = ( I , . . . , I r , J , . . . , J n − r ) having a standard form(called a symplectic normalization, see § r + κ e + κ h = n ) w.r.t. some real-analyticcoordinate system. In particular, on some small open complexification U C of U , we havea F -preserving linear ( S ) n -action generated by I , . . . , I r , J , . . . , J κ e , iJ κ e +1 , . . . , iJ n − r (Definitions 3.1, 3.7). One directly checks (see e.g. Example 2.4) that, for each S -subaction, there exists a point m ∈ ( F C ) − ( F ( O )) satisfying the conditions (i)–(iii)of Theorem 2.2 (a). Hence, by Theorem 2.2 (b), for any “perturbed” real-analyticintegrable system ( U, e Ω , e F ), there exists a e F -preserving “perturbed” ( S ) n -action gen-erated by e I , . . . , e I r , e J , . . . , e J κ e and i e J κ e +1 , . . . , i e J n − r , for some real-analytic functions e I s , e J j close to I s , J j . By Theorem 3.10 (b), the latter “perturbed” ( S ) n -action is alsolinear, moreover there exists an ( S ) n -action-preserving symplectomorphism (close tothe identity) between the “unperturbed” and the “perturbed” fibrations.(B) We conjecture that all degenerate local singularities from Examples 3.6 and 3.12are structurally stable. We expect that, similarly to (A), one can derive this fromTheorems 2.2 and 3.10. In Proposition 4.3 below, we do this for parabolic orbits withresonances, that were briefly described in Example 3.6 (a). Consider integrable systems with 2 degrees of freedom. Such a system is defined by apair F = ( H, K ) of Poisson-commuting functions on a symplectic 4-manifold ( M , Ω).An important property of parabolic orbits is their structural stability under small inte-grable perturbations (see Lerman and Umanskii [25]). This is one of the reasons whysuch orbits can be observed in many examples of integrable Hamiltonian systems: Ko-valevskaya top [4], other integrable cases in rigid body dynamics including Steklov case,Clebsch case, Goryachev–Chaplygin–Sretenskii case, Zhukovskii case, Rubanovskii caseand Manakov top on so (4) [3], as well as systems invariant w.r.t. rotations [22, 23, 24],25ee also examples discussed in [13], [12]. Unlike nondegenerate singularities, however,in the literature on topology and singularities of integrable systems there are only fewpapers devoted to degenerate singularites including parabolic ones. We refer, first ofall, to the following six — L. Lerman, Ya. Umanskii [26], V. Kalashnikov [21], N. T.Zung [39], H. Dullin, A. Ivanov [12], K. Efstathiou, A. Giacobbe [13] and Y. Colinde Verdi`ere [8] — which we consider to be very important in the context of generalclassification programme for bifurcations occurring in integrable systems.Parabolic orbits with resonances were discovered by Kalashnikov [21], who proved thatthey form a complete list of typical degenerate rank-1 singularities, and are structurallystable under S -symmetry-preserving integrable perturbations (in C ∞ case). As we willshow in Proposition 4.3 below, they are structurally stable (in real-analytic case) evenin the following stronger sense: structurally stable under all integrable perturbations,not necessarily preserving the S -action. Parabolic singularity with resonance and aplus sign (known as “elliptic period-doubling bifurcation”, cf. (20) with s = 2 and a plussign) can be observed in the Sretenskii system; its Z -symmetric 2-fold cover (knownas “elliptic pitchfork”) can be observed in the problems by Kovalevskaya, Steklov,Neumann, Clebsch [4]. Parabolic singularity with resonance and a minus sign (knownas “hyperbolic period-doubling bifurcation”, cf. (20) with s = 2 and a minus sign) is alocal singularity that corresponds to two topologically different semilocal singularities ofcomplexity one. Both these semilocal singularities can be observed in the problems byKovalevskaya and Sretenskii; their Z -symmetric 2-fold covers (known as “hyperbolicpitchforks”) are observed in the Kovalevskaya problem [4].It would be interesting to find examples of mechanical integrable systems having aparabolic orbit with “higher order” resonances (i.e., resonances different from 0 and1 / r local singularity with “higher order” twisting resonances (18), i.e. resonances( q a, , . . . , q a,n − r ) ∈ ( Q / Z ) n − r , 1 ≤ a ≤ r , where at least some q a,j is different from 0 and1 / S ) r -subaction, see Remark 3.11 (C)).It is well known that from the smooth point of view, all parabolic orbits withoutresonances are equivalent, i.e., any two parabolic orbits admit fiberwise diffeomorphicneighborhoods (Lerman-Umanskii [25, 26], Kalashnikov [21]). The same is true forcuspidal tori [13]. A symplectic classification of parabolic orbits is studied in [5].Below we describe the structure of the singular Lagrangian fibration in a neighborhoodof a parabolic orbit with resonance. As we are mostly interested in this fibration (ratherthan specific commuting functions H and K ), we allow ourselves to replace H and K with H = H ( H, K ) and K = K ( H, K ) where ∂ ( H ,K ) ∂ ( H,K ) = 0.A model (called “preliminary normal form”) for a parabolic singularity with resonanceis as follows. Denote by D a small interval centred at 0 with coordinate λ , by S a26ircle with a standard periodic angle coordinate ϕ mod 2 π , and by D a small opendisk centred at the origin with coordinates x, y . Consider the manifold V = D × S × D . Let s be a positive integer (called the resonance order ). Consider the free action ρ ofthe group Γ = Z s on V generated by the map( λ, ϕ, z ) ( λ, ϕ + 2 πs , e πi ℓs z ) , where z = x + iy, (19) ℓ ∈ Z is an integer coprime with s (we may assume that 0 ≤ ℓ < s ). Consider thefollowing 1-parameter family of Z s -invariant functions f a,λ,s ( z ) on D with parameter λ ∈ R (where a ∈ R \ { , − } is an additional parameter appearing for s = 4 only): f λ, ( x, y ) = x + y + λy,f λ, ( x, y ) = x ± y + λy ,f λ, ( z ) = Re ( z ) + λ | z | ,f a,λ, ( z ) = Re ( z ) + a | z | + λ | z | , a = 1 ,f λ,s ( z ) = Re ( z s ) + | z | + λ | z | , s ≥ . (20)We endow V with the Z s -invariant symplectic 2-formΩ = dα ( λ ) ∧ dϕ + π ∗ ω, (21)where π : R × S → R is the projection; α ( λ ) is any real-analytic function such that α ′ ( λ ) >
0, moreover α ( λ ) ≡ λ if s < ω is an arbitrary Z s -invariant closed 2-form on D × D such that ω ∧ d λ nowhere vanish. The latter condition on ω is equivalent tothe following: ω = R ( x, y, λ )d x ∧ d y + d λ ∧ ( Q ( x, y, λ )d x − P ( x, y, λ )d y ) , (22)for some Z s -invariant real-analytic divergence-free vector field ( P, Q, R ) on D × D having a nowhere vanishing “vertical component” R ( x, y, λ ). In a simplest case, wehave R ( x, y, λ ) ≡ ω = d x ∧ d y . However, in general, we cannot assume that thecoordinates α ( λ ) , ϕ, x, y are canonical, since ω can depend on λ in an essential way.Now we can form the quotient symplectic manifold V / Z s , with an integrable system onit given by the following two functions: H = f a ( λ ) ,λ,s ( x, y ) and K = λ, (23)where a ( λ ) is any real-analytic function such that a ( λ ) = ± s = 4 only). The functions (23) Poisson-commute w.r.t. the symplectic 2-form (21).Due to a result by V. Kalashnikov [21], the curve γ ( t ) = (0 , , , t ) is a parabolic orbitwith ℓ/s resonance of an integrable Hamiltonian system defined by commuting functions27 and K on ( V / Z s , Ω). A formal definition of a parabolic orbit with ℓ/s resonance isgiven in [21] (see also [5] for s = 1).In according to the definitions 3.1 and 3.3, γ ( t ) is a rank-1 orbit admitting a linear S -action (see Example 3.6 (a) for properties of this S -action). Proposition 4.3.
Consider the rank-1 orbit O = { γ ( t ) } (called a parabolic orbit with ℓ/s resonance) of the real-analytic singular Lagrangian fibration ( V / Z s , Ω , F ) , whosesymplectic 2-form Ω and momentum map F = ( H, K ) are in a “preliminary normalform” (20)–(23) on V / Z s , and the Z s -action on V has the form (19). Then: (a) The orbit O is structurally stable under real-analytic integrable perturbations. (b) Moreover, the preliminary normal form (19)–(23) is persistent under real-analyticintegrable perturbations (up to some left-right change of variables) in the followingsense. For any k ∈ Z + and any neighbourhood U ′ of O having a compact closure U ′ ⊂ V / Z s , there exists ε > satisfying the following. For any (“perturbed”) real-analytic integrable system ( V / Z s , e Ω , e F ) whose holomorphic extension to ( V / Z s ) C is ε -close to (( V / Z s ) C , Ω C , F C ) in C − norm, with e F = ( e H, e K ) , one can choose a neigh-bourhood e U ⊃ U ′ in V / Z s , real-analytic coordinate changes e φ : e U → V / Z s and e χ : ( ˜ H, ˜ K ) ( ˜ H ( ˜ H, ˜ K ) , ˜ K ( ˜ H, ˜ K )) that are O ( ε ) − close to the identities in C k − normand bring the “perturbed” singular Lagrangian fibration on e U to a preliminary normalform ( e φ ( e U ) , ( e φ − ) ∗ e Ω = d e α ( λ ) ∧ d ϕ + π ∗ e ω, e χ ◦ e F ◦ e φ − = ( f λ, ˜ a ( λ ) ,s ( z ) , λ )) , (24) for some analytic (“perturbed”) Z s -invariant closed 2-form e ω and functions ˜ a ( λ ) , e α ( λ ) that are O ( ε ) − close to the 2-form ω and the functions a ( λ ) , α ( λ ) in C k − norm, e α ( λ ) ≡ λ if s < (the function ˜ a ( λ ) appears only for resonance order s = 4 ).In particular, the “unperturbed” and the “perturbed” singular Lagrangian fibrations arelocally (i.e., on some neighbourhoods U, e U ⊃ U ′ of O ) fiberwise homeomorphic. More-over, they are fiberwise diffeomorphic if s = 4 . Comment 4.4.
We stress that the 2-forms ω and e ω , that appear in the preliminarynormal form for the “unperturbed” and the “perturbed” fibrations, may be different.Similarly, the functions α ( λ ) and e α ( λ ) may be different (if s ≥ a ( λ ) and e a ( λ ) (if s = 4) may be different.However they affect only the smooth structure (rather than the topology) of our singularLagrangian fibrations, provided that | a ( λ ) | − | e a ( λ ) | − Proof. Step 1.
Consider the linear S -action on V by shifts along the ϕ -axis. Observethat it is a Hamiltonian S -action generated by the function I ( H, K ) = α ( λ ), w.r.t.28he symplectic structure Ω in (21). Therefore, a function on V Poisson commutes with K = λ if and only if it is S -invariant, i.e. does not depend on ϕ . Step 2.
Suppose we are given a “perturbed” real-analytic integrable system ( U, e Ω , e F ),where e F = ( e H, e K ). Let us show that it admits a real-analytic 2 π -periodic first integral˜ I ( e H, e K ) on ˜ U ⊃ U ′ , where ˜ I ( z , z ) is close to I ( z , z ) = α ( z ). One directly checksthat there exists a point m ∈ ( F C ) − ( F ( O )) satisfying either the conditions (i) and(ii) of Theorem 2.1 or the conditions (i)–(iii) of Theorem 2.2, moreover the paths γ a and γ a in (iii) are homological to each other in the fiber. Hence, by Theorem 2.1 (b)or Theorem 2.2 (b), the “perturbed” system has a 2 π -periodic first integral ˜ I ( ˜ H, ˜ K ),for some real-analytic function ˜ I ( z , z ) close to I ( z , z ) = α ( z ). Define the followingchange of first integrals: χ : ( ˜ H, ˜ K ) ( ˜ H, ˜ I ( ˜ H, ˜ K )) . Step 3.
Consider the 1-parameter family of (“unperturbed”) symplectic 2-forms ω λ := R ( x, y, λ )d x ∧ d y with real parameter λ . By assumption, this 2-form is invariant underthe action ρ ′ of the group Γ := Z s on D , where ρ ′ (1 mod s )( z ) = e πi/s z .Due to Theorem 3.4 (b), this Γ-action ρ ′ is symplectically normalizable, i.e. there existsa real-analytic Γ-equivariant change of variables φ λ : ( x, y ) ( x ( x, y, λ ) , y ( x, y, λ ))that analytically depends on λ and brings the symplectic form ω λ to the standard formd x ∧ d y , so ω λ = φ ∗ λ d x ∧ d y .Define the (Γ-equivariant and S -equivariant) change φ : ( λ, ϕ, x, y ) ( λ = α ( λ ) , ϕ, x , y ).Then Ω = φ ∗ (d λ ∧ (d ϕ + β )+d x ∧ d y ) for some 1-form β . Since dΩ = 0, it follows thatd λ ∧ β = d λ ∧ d g for some real-analytic function g = g ( x, y, λ ), that is Γ-invariant and S -invariant. Define ϕ = ϕ + g ( x, y, λ ). Then, with respect to the coordinate system( λ , ϕ , x , y ), the symplectic 2-form Ω has the standard form, the S -action is linear,and the free action ρ of Γ is also linear.Clearly, the change φ : ( λ, ϕ, x, y ) ( λ , ϕ , x , y ) is (Γ × S )-equivariant. Step 4.
By Step 3, the (“unperturbed”) S -action is linear with respect to the coordi-nate system ( λ , ϕ , x , y ) on V . By Step 2, there exists a “perturbed” Hamiltonian S -action generated by a function ˜ I ( ˜ H, ˜ K ) close to I ( H, K ) = α ( K ). Hence, by The-orem 3.4 (b), the latter “perturbed” S -action is linear too, moreover there exists a“perturbed” coordinate change e φ : ( λ, ϕ, x, y ) (˜ λ , ˜ ϕ , ˜ x , ˜ y ) such that, with re-spect to the “perturbed” coordinate system (˜ λ , ˜ ϕ , ˜ x , ˜ y ), the “perturbed” symplectic2-form e Ω has the standard form, moreover the “perturbed” Hamiltonian S -action gen-erated by ˜ I ( ˜ H, ˜ K ) and the “perturbed” free Γ-action e ρ are linear. This implies that e I ◦ ( e H, e K ) ◦ e φ − = ˜ λ + const. Without loss of generality, the latter constant is 0.How do φ and e φ transform the Hamilton functions H and e H (respectively)?29he “unperturbed” coordinate change φ brings the “unperturbed” function H = f a ( λ ) ,λ,s ( z ) to a function H ◦ φ − =: G = g λ ( z ). The “perturbed” coordinate change e φ brings the “perturbed” function e H to a function e H ◦ e φ − =: e G = e g ˜ λ (˜ z ). Indeed, bothfunctions G and e G are S -invariant (since H and I ( H, K ) Poisson commute, and e H and e I ( e H, e K ) Poisson commute), hence they don’t dependent on ϕ and ˜ ϕ , respectively.The functions G and e G are invariant under the free Γ-action ρ , since φ and e φ are Γ-equivariant, and H and e H are invariant under the free Γ-actions ρ and e ρ , respectively. Step 5.
Let us compose these two functions G = g λ ( z ) and e G = e g ˜ λ (˜ z ) with the(Γ-equivariant and S -equivariant, see Step 3) coordinate change φ . We will obtaintwo functions on V : G ◦ φ = H = f a ( λ ) ,λ,s ( z ) and e G ◦ φ = e H ◦ ( e φ − ◦ φ ) =: e f λ ( z ) , that both are also Γ-invariant and S -invariant (since G and e G are, by Step 4). We canand will regard the “perturbed” function e f λ ( z ) as a 1-parameter family of functions invariables z = ( x, y ) with parameter λ .The diffeomorphism e φ − ◦ φ is Γ-equivariant, since both φ and e φ are. Since ( φ − ) ∗ Ωis standard (by Step 3) and ( e φ − ) ∗ e Ω is standard (by Step 4), they coincide. Hence( e φ − ◦ φ ) ∗ e Ω = Ω . (25)By Step 4 and Step 3, e I ◦ ( e H, e K ) ◦ e φ − ◦ φ = ˜ λ ◦ φ = α ( λ ) . (26) Step 6.
Define the 1-parameter family of functions on D with parameter λ : b f a,λ,s ( z ) := f λ,s ( z ) , ≤ s ≤ ,f a,λ, ( z ) , a = 1 , s = 4 , Re ( z s ) + a | z | + λ | z | , a > , s ≥ , (27)cf. (20), where a is an additional real parameter which is inessential if s ≤ f λ,s ( z ) = b f ,λ,s ( z ) for s ≥ λ, z ) → ( λ, b z ( z, λ )) close to the identity and smooth (real-analytic in ourcase) functions e a ( λ ) , e b ( λ ) , e c ( λ ) close to a ( λ ) , λ,
0, respectively, such that e f λ ( z ) = b f e a ( λ ) , e b ( λ ) ,s ( b z ( z, λ )) + e c ( λ ) . Consider the change of variables b φ : ( λ, ϕ, z ) ( λ, ϕ, b z ) .
30y above, it is close to the identity and transforms our “perturbed” functions e f λ ( z ) = e H ◦ ( e φ − ◦ φ ) and (26) to e H ◦ e φ − ◦ φ ◦ b φ − = b f e a ( λ ) , e b ( λ ) ,s ( b z ) + e c ( λ ) and e I ◦ ( e H, e K ) ◦ e φ − ◦ φ ◦ b φ − = α ( λ ) . (28)Clearly, b φ is Γ-equivariant and brings the symplectic structure (25) to a form( e φ − ◦ φ ◦ b φ − ) ∗ e Ω = ( b φ − ) ∗ Ω = dα ( λ ) ∧ dϕ + π ∗ b ω (29)similar to (21), with some Γ-invariant 2-form b ω close to ω . Step 7.
Recall that we are mostly interested in our singular Lagrangian fibration (ratherthan specific commuting functions e H and e K ), so we allow ourselves to replace the“perturbed” momentum map e F = ( e H, e K ) with a composition of it from the left withsome diffeomorphism. Our goal is to show that some of the functions e a ( λ ) , e b ( λ ) , e c ( λ ) in(28) can be simplified by an appropriate Γ-equivariant change of variables close to theidentity, both in the sourse and in the target (so-called left-right change of variables),perhaps at the expense of spoiling the function α ( λ ) in (29) if s ≥ b χ : ( e H, e I ) ( e H − e c ◦ α − ( e I ) , α − ( e I )), we can killthe function e c ( λ ) and reduce e I to λ .Secondly, the function e b ( λ ) can be reduced to λ (at the expense of replacing e a ( λ )with b a ( λ ) = e a ( λ ) b ( λ ) (4 − s ) / ( s − if s ≥
5, but preserving λ and e I ) via the followingΓ-equivariant change of variables and first integrals: φ × χ : ( λ, ϕ, b x, b y ; e H, e I ) ( λ, ϕ, u ( λ ) b x, v ( λ ) b y ; w ( e I ) e H, e I ) , where • u ( λ ) = b ( λ ) − / − s/ , v ( λ ) = b ( λ ) − / , w = b ( λ ) − − s/ if s = 1 , • u ( λ ) = v ( λ ) = b ( λ ) − / ( s − , w = b ( λ ) − s/ ( s − if s ≥ b ( λ ) > e b ( λ ) = λb ( λ ).Finally, if s ≥
5, we can reduce the function b a ( λ ) > e I with a composition of e I from the left with some diffeomorphism) by thefollowing Γ-equivariant change of variables and first integrals: φ × χ : ( λ, ϕ, b z ; e H, e I ) ( t ( λ ) s − λ, ϕ, t ( λ ) b z ; t ( I ) s e H, t ( e I ) s − e I ) , t ( λ ) := b a ( λ ) / (4 − s ) > . After applying the changes e φ = φ ◦ φ ◦ b φ ◦ φ − ◦ e φ and e χ = χ ◦ χ ◦ b χ ◦ χ ,the “perturbed” symplectic structure e Ω and the momentum map e F = ( e H, e K ) will betransformed to the desired form (24). 31 Proof of Theorems 2.1 and 2.2 on a “hidden” toricsymmetry near a singular orbit (a) On a small neighbourhood U ( m ) of the point m , we can extend the functions f , . . . , f n to canonical coordinates f i , g i such that Ω | U ( m ) = n P i =1 d f i ∧ d g i (Darbouxcoordinates). This is possible because the point m is regular by (i). Without lossof generality, g i ( m ) = 0. Consider the time-2 π map h = φ πf : U ( m ) → M of theHamiltonian flow generated by the function f . Here t φ tH ( m ) denotes the trajectoryof the vector field X H with initial condition φ H ( m ) = m . Since h ( m ) = m , h ∗ Ω = Ωand F ◦ h = F , it follows that the map h w.r.t. the local coordinates f i , g i on U ′ ( m ) := U ( m ) ∩ h − ( U ( m ))has the form h ( f , . . . , f n , g , . . . , g n ) = ( f , . . . , f n , g + h ( f , . . . , f n ) , . . . , g n + h n ( f , . . . , f n ))for some functions h i ( f , . . . , f n ) such that h i (0 , . . . ,
0) = (0 , . . . , U ′ ( m ) ∩ F − ( a ) is connected (this can be achievedby choosing a smaller neighbourhood U ( m ) of the point m in M ).Since the map h preserves the symplectic structure Ω | U ′ ( m ) = n P i =1 d f i ∧ d g i , we con-clude that the functions g i + h i ( f , . . . , f n ) pairwise Poisson commute. Hence { g i + h i ( f , . . . , f n ) , g j + h j ( f , . . . , f n ) } = ∂h j /∂f i − ∂h i /∂f j equals 0. It follows from theTheory of PDE’s that h i ( z , . . . , z n ) = ∂S ( z , . . . , z n ) /∂z i , for some function S = S ( z , . . . , z n ) on the neighbourhood F ( U ′ ( m )) of the origin, i.e. h = φ S ◦ F on U ′ ( m ).Here we used the connectedness of the sets U ′ ( m ) ∩ F − ( a ). Due to h i (0 , . . . ,
0) =(0 , . . . , S (0 , . . . ,
0) = 0. We can and will assume that S (0 , . . . ,
0) = 0. Thefunction S is real-analytic, since the functions h i are. Let us define a function I ( z , . . . , z n ) := z − π S ( z , . . . , z n )on F ( U ′ ( m )). Let us show that it has the required properties.From the properties of the function S , we have I ( z , . . . , z n ) = z + O ( n P j =1 | z j | ). Fromthe equalities φ πf = h = φ S ◦ F = φ π π S ◦ F , we obtain φ πf − π S ◦ F = Id on U ′ ( m ). That is,the function f − π S ◦ F = I ◦ F is a 2 π -periodic first integral on a neighbourhood of γ containing U ′ ( m ).The function I ◦ F is defined and is real-analytic on the neighbourhood U ( L ) = F − ( F ( U ′ ( m ))) of the singular fiber L := F − (0 , . . . , φ πI ◦ F
32f the function I ◦ F in time 2 π is defined on some neighbourhood U ⊆ U ( L ) of theset C , due to the condition (ii) and the equality X I ◦ F = X f on F − (0 , . . . , U ′ := U ∩ ( φ πI ◦ F ) − ( U ) is connected, since C is. Let U bethe union of all trajectories through points of U ′ of the Hamiltonian vector field X I ◦ F ,which all are 2 π -periodic by construction. Let V := F ( U ).Thus, the map φ πI ◦ F is real-analytic, is defined on the connected open domain U ′ , andits restriction to the sub-domain U ′ ( m ) ∩ U = ∅ is the identity. Therefore, by theuniqueness of analytic continuation, this map is the identity on the whole U ′ , therefore U is filled by 2 π -periodic trajectories of X I ◦ F .Let us prove the integral formula (1) for the action function I . Similarly to the beginningof the proof, we can extend the functions I ◦ F, f , . . . , f n to canonical coordinates I ◦ F, f , . . . , f n , u , . . . , u n on a smaller neighbourhood of m (we can and will denotethis neighbourhood again by U ′ ) such that Ω | U ′ = d( I ◦ F ) ∧ d u + n P i =2 d f i ∧ d u i (Darbouxcoordinates). Recall that U denotes the union of all trajectories through points of U ′ of the Hamiltonian vector field X I ◦ F . Observe that the functions u , . . . , u n and the 1-form d u on U ′ are preserved by the Hamiltonian flow generated by the action function I ◦ F . We extend these functions and 1-form to U by making them invariant under theHamiltonian S -action generated by I ◦ F . Define the 1-form α := ( I ◦ F )d u + n X i =2 f i d u i (30)on U , then d α = Ω | U , thus the symplectic structure is exact on U . By the Stokesformula, in order to prove (1), it suffices to prove the integral formula I ( z , . . . , z n ) = 12 π Z C γ,γ ( z ,...,zn ) Ω , ( z , . . . , z n ) ∈ V, (31)where C γ,γ ( z ,...,zn ) denotes the cylinder (with boundary ∂C γ,γ ( z ,...,zn ) = γ ( z ,...,z n ) − γ )formed by the closed curves γ ( tz ,...,tz n ) , 0 ≤ t ≤
1. The integral formula (31) followsfrom the following facts: • the Hamiltonian flow generated by I ◦ F is 2 π -periodic on a neighbourhood U ( γ ) ⊂ U of γ = γ , and • its closed orbits on each fiber U ( γ ) ∩ F − ( z , . . . , z n ) are homological in this fiberto the closed curve γ ( z ,...,z n ) , ( z , . . . , z n ) ∈ V ∩ F ( U ( γ )).Indeed, the right-hand side of (31) will not change if we replace the closed curves γ ( z ,...,z n ) by 2 π -periodic trajectories ˆ γ ( z ,...,z n ) ⊂ F − ( z , . . . , z n ) of the Hamiltonian flow33enerated by I ◦ F . If we compute the right-hand side of (31) in this way, via theFubini formula and taking into account that Ω( · , dd t ˆ γ ( z ,...,z n ) ) = d( I ◦ F ) | γ ( z ,...,zn ) , thenthe resulting value equals I ( z , . . . , z n ) − I (0 , . . . ,
0) = I ( z , . . . , z n ), which is the left-hand side of (31), as required.(b) Let us fix a smaller neighbourhood U ′ of the set C and a smaller neighbourhood V ′ of the origin in R n having compact closures U ′ ⊂ U and V ′ ⊂ V .We can extend the functions ˜ f , . . . , ˜ f n on U ( m ) to canonical coordinates ˜ f i , ˜ g i suchthat ˜Ω | U ( m ) = n P i =1 d ˜ f i ∧ d˜ g i (“perturbed” Darboux coordinates). If one follows anexplicit construction of such coordinates, one can manage to have k ˜ g i − g i k C k − = O ( ε ).Let us consider the “perturbed” map ˜ h = φ π ˜ f : U ( m ) → M , where t φ t ˜ H ( m ) isthe trajectory of the Hamiltonian vector field X ˜ H with the Hamilton function ˜ H , thesymplectic structure ˜Ω and initial condition φ H ( m ) = m . Since ˜ h ∗ ˜Ω = ˜Ω and ˜ F ◦ ˜ h = ˜ F ,it follows that the map ˜ h w.r.t. the local coordinates ˜ f i , ˜ g i on˜ U ( m ) := U ′ ( m ) ∩ ˜ h − ( U ′ ( m ))has the form ˜ h ( ˜ f , . . . , ˜ f n , ˜ g , . . . , ˜ g n ) = ( ˜ f , . . . , ˜ f n , ˜ g +˜ h ( ˜ f , . . . , ˜ f n ) , . . . , ˜ g n +˜ h n ( ˜ f , . . . , ˜ f n )),for some functions ˜ h i ( z , . . . , z n ) on ˜ V := ˜ F ( ˜ U ( m )) O ( ε ) − close to h i ( z , . . . , z n ) in C k − − norm.Since the map ˜ h preserves the symplectic structure ˜Ω | ˜ U ( m ) , we have similarly to (a) that˜ h i ( z , . . . , z n ) = ∂ ˜ S ( z , . . . , z n ) /∂z i for some real-analytic function ˜ S = ˜ S ( z , . . . , z n ) O ( ε ) − close to the function S in C k − − norm on some (a bit perturbed) neighbourhood˜ V of the origin. We have ˜ V ⊃ V ′ if the perturbation is small enough. Let us define on˜ V the real-analytic function˜ I ( z , . . . , z n ) := z − π ˜ S ( z , . . . , z n ) . Observe that the “perturbed” function ˜ I ◦ ˜ F is close to the “unperturbed” function I ◦ F ,moreover the “unperturbed” map φ πI ◦ F is defined and is real-analytic on the connecteddomain U ⊃ U ′ (cf. (a)). This implies that the “perturbed” map φ π ˜ I ◦ ˜ F is defined and isreal-analytic on (a bit smaller) connected domain ˜ U ⊃ U ′ . But the “perturbed” map φ π ˜ I ◦ ˜ F is the identity on the sub-domain ˜ U ( m ) ∩ ˜ U = ∅ (by construction of the function˜ I ). Therefore, by the uniquess of analytic continuation, it equals the identity on thewhole of ˜ U . Thus, ˜ I ◦ ˜ F is a 2 π -periodic first integral of the perturbed system on ˜ U .The integral formula (2) for the “perturbed” action function ˜ I follows by the samearguments as for the “unperturbed” case. The estimate k ˜ α − α k C k − = O ( ε ) followsfrom construction. The estimate k ˜ I − I k C k = O ( ε ) follows from (31) and its analoguefor the “perturbed” system. 34 .2 Proof of Theorem 2.2 Similarly to the proof of Theorem 2.1 (a) (respectively, (b)), we can construct a complex-valued 2 π -periodic first integral I ◦ F C (respectively, ˜ I ◦ ˜ F C ). Such a construction can beperformed on a small open complexification U (respectively, ˜ U ⊃ U ′ ) of a small neigh-bourhood of the set C , since, by assumption, C is connected and invariant under theHamiltonian flow generated by the function λf C . As a result, we obtain a holomorphicfunction I = I ( z , . . . , z n ) on a neighbourhood V (respectively, a holomorphic function˜ I = ˜ I ( z , . . . , z n ) on a smaller neighbourhood ˜ V ⊃ V ′ ) of the origin in C n such that • I ( z , . . . , z n ) = λz + O ( n P j =1 | z j | ) (respectively, ˜ I is O ( ε ) − close to I and ˜ I (0 , . . . ,
0) =0), • the set U (respectively, ˜ U ) is invariant under the Hamiltonian flow generated by I ◦ F C (respectively, by ˜ I ◦ ˜ F C ), and this flow is 2 π -periodic on U (respectively,on ˜ U ), • the integral formula (3) for I ( z , . . . , z n ) (respectively, (4) for ˜ I ( z , . . . , z n )) holds.Due to the Cauchy integral formula for holomorphic functions and the first propertyfrom above, on a smaller neighbourhood, the function ˜ I is O ( ε ) − close to I in C k − norm,for any k ∈ Z + . We want to prove that λ I (resp., λ ˜ I ) is real-valued on V ∩ R n (resp.,on ˜ V ∩ R n ), moreover λ ∈ R ∪ i R , provided that the corresponding condition (iii) forthe circle γ of being homologically symmetric holds. Consider two cases. Case 1: the trajectory γ ∋ m of the Hamiltonian flow generated by the function λf C (and, hence, by I ◦ F C ) is homological to its conjugate in the following sense. Thereexists m ′ ∈ U ( m ) such that a := F C ( m ′ ) ∈ R n and the closed path γ a in the fiber L C a := ( F C ) − ( a ) is homological in the fiber L C a \ Sing( F C ) to the closed path γ a (obtained from γ a by C -conjugation).(a) Choose a small neighbourhood U ( m ′ ) of m ′ in U ( m ). Take any point m in U R ( m ′ ) := U ( m ′ ) ∩ ( F C ) − ( R n ), thus a := F C ( m ) ∈ R n . Consider the closed path γ a in the fiber L C a := ( F C ) − ( a ). Since the momentum map F = ( f , . . . , f n ) isreal-analytic and a ∈ R n , it follows that the closed path γ a is contained in the samefiber L C a , moreover the paths γ a and γ a are homological to each other in the fiber L C a \ Sing( F C ), due to the homological symmetry condition. On the other hand, wehave the integral formula (3) for I ( z , . . . , z n ), which by Stokes’ formula reads: I ◦ F C ( m ) − I ◦ F C ( m ′ ) = I ( a ) − I ( a ) = 12 π Z C γa ,γa Ω C , m ∈ U R ( m ′ ) , C γ a ,γ a is a cylinder in U with boundary ∂C γ a ,γ a = γ a − γ a . Since (the realand imaginary parts of) Ω C is closed, the integral does not depend on the choice of thecylinder in the given homotopy class relatively boundary. We want to show that thelatter integral is in fact real. By changing coordinates under the integral, we obtain Z C γa ,γa Ω C = Z C γa ,γa Ω C = Z C γa ,γa Ω C . (32)Since the symplectic structure vanishes on each fiber, the resulting integral in (32) doesnot depend on the choice of closed paths in given homology classes, so this integral willnot change if we replace γ a , γ a by their conjugates. Thus Z C γa ,γa Ω C = Z C γa ,γa Ω C = Z C γa ,γa Ω C , which shows that the integral is in fact real. We conclude that I ◦ F C ( m ) − I ◦ F C ( m ′ ) ∈ R , m ∈ U R ( m ′ ) , thus Im ( I ◦ F C ) is constant on U R ( m ′ ). Since m ′ ∈ U ( m ) and U ( m ) consists of regularpoints of F C , the set V ( a ) := F ( U R ( m ′ )) is open in R n , so it is a neighbourhoodof a in R n . By above, Im I ( z , . . . , z n ) is constant on this neighbourhood. Hence I ( z , . . . , z n ) up to an additive constant is real-analytic on V ( a ) ⊂ V ∩ R n . Thisimplies, by the uniqueness of analytic continuation, that I ( z , . . . , z n ) is real-analyticon the entire neighbourhood V ∩ R n of (0 , . . . , I (0 , . . . ,
0) = 0 by properties of the function I = I ( z , . . . , z n ).It remains to show that λ ∈ R . On one hand, λ = ∂I∂z (0 , . . . , I ( z , . . . , z n ) ∈ R for any ( z , . . . , z n ) ∈ V ∩ R n . Hence λ ∈ R .(b) For a perturbed system, the proof follows the same arguments. In more detail, wetake any two points ˜ m , ˜ m in ˜ U R ( m ′ ) := U ( m ′ ) ∩ ( ˜ F C ) − ( R n ), thus ˜ a j := ˜ F C ( ˜ m j ) ∈ R n ( j = 1 , γ ˜ a j in the fiber ˜ L ˜ a j := ( ˜ F C ) − (˜ a j ) ( j = 1 , F = ( ˜ f , . . . , ˜ f n ) is real-analytic and ˜ a j ∈ R n , itfollows that the closed path ˜ γ ˜ a j is contained in the same fiber ˜ L ˜ a j , moreover ˜ γ ˜ a j and˜ γ ˜ a j are homological to each other in the fiber ˜ L ˜ a j \ Sing( ˜ F C ), due to the homologicalsymmetry condition ( j = 1 , I ( z , . . . , z n ), which by Stokes’ formula reads:˜ I ◦ ˜ F C ( ˜ m ) − ˜ I ◦ ˜ F C ( ˜ m ) = ˜ I (˜ a ) − ˜ I (˜ a ) = 12 π Z C ˜ γ ˜ a , ˜ γ ˜ a ˜Ω C , ˜ m , ˜ m ∈ ˜ U R ( m ′ ) , C ˜ γ ˜ a , ˜ γ ˜ a is a cylinder with boundary ∂C ˜ γ ˜ a , ˜ γ ˜ a = ˜ γ ˜ a − ˜ γ ˜ a . Similarly to (32), wecompute Z C ˜ γ ˜ a , ˜ γ ˜ a ˜Ω C = Z C ˜ γ ˜ a , ˜ γ ˜ a ˜Ω C = Z C ˜ γ ˜ a , ˜ γ ˜ a ˜Ω C . (33)Similarly to the “unperturbed” system case, the resulting integral in (33) will notchange if we replace the closed curves ˜ γ , ˜ γ by their conjugates, due to the homologicalsymmetry condition. Thus Z C ˜ γ ˜ a , ˜ γ ˜ a ˜Ω C = Z C ˜ γ ˜ a , ˜ γ ˜ a ˜Ω C = Z C ˜ γ ˜ a , ˜ γ ˜ a ˜Ω C , thus the integral is in fact real, thus˜ I ◦ ˜ F C ( ˜ m ) − ˜ I ◦ ˜ F C ( ˜ m ) ∈ R , ˜ m , ˜ m ∈ ˜ U R ( m ′ ) . Similarly to the “unperturbed” system case, we conclude that Im ˜ I ( z , . . . , z n ) is con-stant on the neighbourhood ˜ V ( a ) := ˜ F ( ˜ U R ( m ′ )) of a in R n . Hence ˜ I ( z , . . . , z n ) up toan additive constant is real-analytic on ˜ V ( a ) ⊂ ˜ V ∩ R n . This implies that ˜ I ( z , . . . , z n )is real-analytic on the entire neighbourhood ˜ V ∩ R n of (0 , . . . , I (0 , . . . ,
0) = 0 by construction(see properties of the function ˜ I ). Case 2: the trajectory γ ∋ m of the Hamiltonian flow generated by the function λf C (and, hence, by I ◦ F C ) is reverse-homological to its conjugate in the following sense.There exists m ′ ∈ U ( m ) such that a := F C ( m ′ ) ∈ R n and the closed path γ a in thefiber L C a := ( F C ) − ( a ) is homological in the fiber L C a \ Sing( F C ) to the closed pathobtained from the ( C -conjugated to γ a ) path γ a by reversing orientation.Similarly to the proof in the case λ ∈ R , we conclude that each resulting integral in(32) and (33) will change to the opposite if we replace the closed curves γ a , γ a and˜ γ ˜ a , ˜ γ ˜ a by their conjugates. This immediately shows that each of these integrals is infact purely imaginary. We conclude that I ◦ F C ( m ) − I ◦ F C ( m ′ ) ∈ i R , ˜ I ◦ ˜ F C ( ˜ m ) − ˜ I ◦ ˜ F C ( ˜ m ) ∈ i R . Thus Re I ( z , . . . , z n ) is constant on V ( a ), and Re ˜ I ( z , . . . , z n ) is constant on ˜ V ( a ).Hence I ( z , . . . , z n ) and ˜ I ( z , . . . , z n ) up to additive constants are imaginary-valued onopen subsets V ( a ) ⊂ V ∩ R n and ˜ V ( a ) ⊂ ˜ V ∩ R n , respectively. Therefore iI ( z , . . . , z n )and i ˜ I ( z , . . . , z n ) are real-analytic on the entire neighbourhoods V ∩ R n and ˜ V ∩ R n of the origin in R n , up to additive constants. The latter additive constants are in factreal, since I (0 , . . . ,
0) = ˜ I (0 , . . . ,
0) = 0 by construction.It remains to show that λ ∈ i R . On one hand, λ = ∂I∂z (0 , . . . , I ( z , . . . , z n ) ∈ i R for any ( z , . . . , z n ) ∈ V ∩ R n . Hence λ ∈ i R .37 Equivariant symplectic normalization of a torusaction near a fixed point
In this section, we study equivariant version of the theorems 3.4 and 3.10 in the casewhen the orbit O m is a point, and we prove that the torus action can be normalizedsymplectically in an equivariant way.For proving Theorem 3.4, we will need the following lemma about an equivariant normalform of a Hamiltonian torus action near a fixed point. Lemma 6.1.
Suppose we are given a Hamiltonian ( S ) κ -action generated by C ∞ -smooth functions J , . . . , J κ on a C ∞ -smooth symplectic manifold ( M, Ω) , where themomentum map F = ( J , . . . , J κ ) is not necessarily proper. Suppose a point m ∈ M isfixed under this ( S ) κ -action. Suppose a finite Abelian group Γ acts on a neighbourhoodof m by F -preserving symplectomorphisms ρ ( ψ ) , ψ ∈ Γ , fixing the point m . Here ρ : Γ → Symp( M, Ω) is a homomorphism from Γ to the group of (local) symplectomor-phisms. Then: (a) There exist a Γ × ( S ) κ -invariant neighbourhood U of m and smooth local coordi-nates x , y , . . . , x n , y n on U such that x j ( m ) = y j ( m ) = 0 , ≤ j ≤ n , and Ω | U = n X j =1 d x j ∧ d y j , J ℓ | U = c ℓ + k e X j =1 p jℓ x j + y j , ≤ ℓ ≤ κ , (34) for some constants c ℓ ∈ R and integers k e ∈ Z + , p jℓ ∈ Z . Moreover, the ρ (Γ) -action on U has the form ρ ( ψ ) : ( z , . . . , z n ) ( e πiq ψ, z , . . . , e πiq ψ,n z n ) , ψ ∈ Γ , (35) for some q ψ, , . . . , q ψ,n ∈ Q depending on ψ such that q ψ,j ∈ Z for k e + 1 ≤ j ≤ n .Here we used the notation z j := x j + iy j , ≤ j ≤ n . If the system is analytic, thecoordinates x j , y j are analytic too. (b) The equivariant symplectic normalization (34)–(35) for a torus action is persistentunder C ∞ -smooth perturbations in the following sense. For any integer k ≥ andany neighbourhood U ′ of m having a compact closure U ′ ⊂ U , there exists ε > satisfying the following. Suppose ˜ F = ( ˜ J , . . . , ˜ J κ ) is a (“perturbed”) momentum map ε -close to F in C k − norm, and ˜Ω is a (“perturbed”) symplectic structure ε -close to Ω in C k + n − − norm. Suppose the functions ˜ J , . . . , ˜ J κ generate a Hamiltonian ( S ) κ -actionon ( ˜ M , ˜Ω) , where U ⊂ ˜ M ⊂ M . Suppose we are given a “perturbed” action of the group Γ on ˜ M by ˜ F -preserving symplectomorphisms ˜ ρ ( ψ ) , ψ ∈ Γ , where ˜ ρ : Γ → Symp( ˜
M , ˜Ω) is a homomorphism ε − close to ρ in C k − − norm. Then there exist a point ˜ m ∈ U ′ fixed under the “perturbed” Γ × ( S ) κ -action, an invariant (under the “perturbed” Γ × S ) κ -action) neighbourhood ˜ U ⊃ U ′ , and smooth local coordinates ˜ x , ˜ y , . . . , ˜ x n , ˜ y n on ˜ U O ( ε ) − close to x , y , . . . , x n , y n in C k − − norm such that ˜ x j ( ˜ m ) = ˜ y j ( ˜ m ) = 0 , ≤ j ≤ n , and an analogue of (34) holds for ˜Ω | ˜ U and ˜ J ℓ | ˜ U w.r.t. the “perturbed”coordinates, with the same integers p jℓ as in (a) and with some constants ˜ c ℓ ∈ R closeto c ℓ . Moreover, an analogue of (35) holds for the “perturbed” action ˜ ρ ( ψ ) of eachelement ψ ∈ Γ on ˜ U , with the same rational numbers q ψ,j as in (a) . If the systems areanalytic, the coordinates ˜ x j , ˜ y j are analytic too. If the system ( M, Ω , F ) and the action ρ (Γ) depend smoothly (resp., analytically) on a local parameter, i.e. we have a localfamily of systems with actions, then the local coordinates can also be chosen to dependsmoothly (resp., analytically) on that parameter. The following lemma generalizes the real-analytic part of the previous lemma to the casewhen the torus acts holomorphically on a small open complexification of the manifold,and the generating functions of this action are real-analytic functions some of whichare multiplied by √−
1. This lemma will be used for proving Theorem 3.10.
Lemma 6.2.
Suppose we are given real-analytic functions J , . . . , J κ e + κ h on a real-analytic symplectic manifold ( M, Ω) , where the map F = ( J , . . . , J κ e + κ h ) is not neces-sarily proper. Suppose the functions J C , . . . , J Cκ e , iJ Cκ e +1 , . . . , iJ Cκ e + κ h generate a Hamil-tonian ( S ) κ e + κ h -action on ( M C , Ω C ) , where κ e , κ h ∈ Z + . Suppose a point m ∈ M is fixed under this ( S ) κ e + κ h -action. Suppose a finite Abelian group Γ acts on a neigh-bourhood of m by F -preserving real-analytic symplectomorphisms ρ ( ψ ) , ψ ∈ Γ , fixingthe point m . Here ρ : Γ → Symp( M, Ω) is a homomorphism from Γ to the group of(local) symplectomorphisms. Then: (a) There exist a Γ × ( S ) κ e -invariant neighbourhood U of m in M and real-analyticlocal coordinates x , y , . . . , x n , y n on U such that x j ( m ) = y j ( m ) = 0 , ≤ j ≤ n , and Ω | U = n X j =1 d x j ∧ d y j , (36) J ℓ | U = c ℓ + k f X j =1 p jℓ h j − + k e X j =1 p k f + j,ℓ h k f + j , ≤ ℓ ≤ κ e , (37) J ℓ | U = c ℓ + k f X j =1 p j,ℓ h j + k h X j =1 p k f + j,ℓ h k f + k e + j , κ e + 1 ≤ ℓ ≤ κ e + κ h (38) (cf. (13)), for some real constants c ℓ ∈ R and integers p jℓ ∈ Z and k f , k e , k h ∈ Z + suchthat k f + k e + k h ≤ n . Moreover, the ρ (Γ) -action on U has the form ρ ( ψ ) : ( z , . . . , z n ) ( e πiq ψ, z , . . . , e πiq ψ,n z n ) , ψ ∈ Γ , (39) for some q ψ, , . . . , q ψ,n ∈ Q depending on ψ such that q ψ, j − + q ψ, j ∈ Z for ≤ j ≤ k f ,and q ψ,j ∈ Z for k f + k e + 1 ≤ j ≤ n . The equivariant symplectic normalization (36)–(39) for a torus action is persis-tent under analytic perturbations in the following sense. For any k ∈ Z + and anyneighbourhood U ′ of the point m in M having a compact closure U ′ ⊂ U , there ex-ists ε > satisfying the following. Suppose ˜ F = ( ˜ J , . . . , ˜ J κ e + κ h ) and ˜Ω are ana-lytic (“perturbed”) momentum map and symplectic structure whose holomorphic ex-tensions to M C are ε -close to F C and Ω C , respectively, in C − norm. Suppose thefunctions ˜ J C , . . . , ˜ J Cκ e , i ˜ J Cκ e +1 , . . . , i ˜ J Cκ e + κ h generate a Hamiltonian ( S ) κ e + κ h -action on ( ˜ M C , ˜Ω C ) , where U ⊂ ˜ M C ⊂ M C . Suppose we are given a “perturbed” action of thegroup Γ on ˜ M = ˜ M C ∩ M by ˜ F -preserving analytic symplectomorphisms ˜ ρ ( ψ ) , ψ ∈ Γ ,where ˜ ρ : Γ → Symp( ˜
M , ˜Ω) is a homomorphism whose holomorphic extension to ˜ M C is ε − close to ρ C in C − norm. Then there exist a point ˜ m ∈ U ′ fixed under the“perturbed” Γ × ( S ) κ e -action, an invariant (under the “perturbed” Γ × ( S ) κ e -action)neighbourhood ˜ U ⊃ U ′ , and analytic (“perturbed”) local coordinates ˜ x , ˜ y , . . . , ˜ x n , ˜ y n on ˜ U O ( ε ) − close to x , y , . . . , x n , y n in C k − norm, such that ˜ x j ( ˜ m ) = ˜ y j ( ˜ m ) = 0 , ≤ j ≤ n , and analogues of (36)–(38) hold for ˜Ω | ˜ U and ˜ J ℓ | ˜ U w.r.t. the “perturbed”coordinates, with the same integers p jℓ and k f , k e , k h as in (a) and with some constants ˜ c ℓ ∈ R . Moreover, an analogue of (39) holds for the “perturbed” action ˜ ρ ( ψ ) of each ele-ment ψ ∈ Γ on ˜ U , with the same rational numbers q ψ,j as in (a) . If the system ( M, Ω , F ) and the action ρ (Γ) depend on a local parameter, i.e. we have a local family of systemswith actions, moreover the holomorphic extensions of the system and the action to M C depend smoothly (resp., analytically) on that parameter, then the local coordinates canalso be chosen to depend smoothly (resp., analytically) on that parameter. We will give a proof of Lemma 6.2 (a) on the real-analytic case. If κ h = 0 (i.e., allfunctions J ℓ generate 2 π -periodic flows), all our arguments and constructions literallywork both in the real-analytic and C ∞ cases. This gives a proof of Lemma 6.1 (a) too. Step 1.
On the vector space V := T m M , consider the linear operators • A ℓ , 1 ≤ ℓ ≤ κ e + κ h , the linearizations of the vector fields X J ℓ at their commonequilibrium point m , • M a , 1 ≤ a ≤ r , the linearizations of the symplectomorphisms ρ ( ψ a ) at theircommon fixed point m , where ψ a ∈ Γ are generators of the group Γ.Observe that the operators A ℓ (1 ≤ ℓ ≤ κ e + κ h ) and M a (1 ≤ a ≤ r ) are • Hamiltonian and symplectic (respectively) w.r.t. the symplectic form Ω | m ,40 semisimple (i.e. diagonalizable over C ) and pairwise commute, • the operators A , . . . , A κ e and M a are elliptic, while the operators A κ e +1 , . . . , A κ e + κ h are hyperbolic. In other words, all eigenvalues of A , . . . , A κ e belong to the unitcircle in C , all eigenvalues of M a are purely imaginary (i.e. belong to i R ), alleigenvalues of A κ e +1 , . . . , A κ e + κ h are real.It follows from a standard result of Linear Algebra that the linear hull of the operators A ℓ is contained in a maximal Abelian subalgebra consisting of semisimple elements(called a Cartan subalgebra ) of the Lie algebra sp (2 n, R ) of the Lie group Sp (2 n, R ) ofreal linear symplectomorphisms. Moreover, each symplectic operator M a is the expo-nent of some element of this subalgebra.Furthermore, it is known that each Cartan subalgebra of sp (2 n, R ) is uniquely de-termined (up to conjugation) by a triple (ˆ k e , ˆ k h , ˆ k f ) of non-negative integers such thatˆ k e +ˆ k h +2ˆ k f = n . If we decompose the symplectic vector space ( R n ( x ,...,x n ,y ,...,y n ) , n P j =1 d x j ∧ d y j ) into the direct product of ˆ k e + ˆ k h symplectic subspaces R x j ,y j ) , 1 ≤ j ≤ ˆ k e + ˆ k h , andˆ k f symplectic subspaces R x j − ,x j ,y j − ,y j ) , 1 ≤ j ≤ ˆ k f , then the corresponding Cartansubalgebra (as a vector space) is the direct product of ˆ k e copies of the “elliptic” Cartansubalgebra of sp (2 , R ), ˆ k h copies of the “hyperbolic” Cartan subalgebra of sp (2 , R ),and ˆ k f copies of the “focus-focus” Cartan subalgebra of sp (4 , R ). Here the “elliptic”(respectively, “hyperbolic”) Cartan subalgebra of sp ( R x j ,y j ) ) is one-dimensional and isspanned by the Hamiltonian operator with the quadratic Hamilton function ( x j + y j ) / x j y j ), while the “focus-focus” Cartan subalgebra of sp ( R x j − ,x j ,y j − ,y j ) )is two-dimensional and is spanned by two Hamiltonian operators with the quadraticHamilton functions x j − y j − y j − x j and x j − y j − + x j y j (cf. (37), (38)).Using the above facts and taking into account that the flows of X J ℓ , 1 ≤ ℓ ≤ κ e , and iX J ℓ , κ e + 1 ≤ ℓ ≤ κ e + κ h , are 2 π -periodic, one can show the existence of non-negativeintegers k e ≤ ˆ k e , k h ≤ ˆ k h , k f ≤ ˆ k f and coordinates ˆ x j , ˆ y j on V , in which the followingformulae (40)–(44) corresponding to (36)–(39) hold:Ω | m = n X j =1 dˆ x j ∧ dˆ y j , (40)d J ℓ | m = k f X j =1 p jℓ (dˆ x j − +dˆ y j − − dˆ x j − dˆ y j )+ k f + k e X j =2 k f +1 p j − k f ,ℓ (dˆ x j +dˆ y j ) , ℓ ≤ κ e , (41)d J ℓ | m = k f X j =1 p jℓ (dˆ x j − dˆ y j + dˆ x j dˆ y j − ) + k f + k e + k h X j =2 k f + k e +1 p j − k f − k e ,ℓ dˆ x j dˆ y j , ℓ > κ e , (42)41 ≤ ℓ ≤ κ e + κ h , for some integers p jℓ , M a : (ˆ z , . . . , ˆ z n ) ( e πiq a, ˆ z , . . . , e πiq a,n ˆ z n ) , ≤ a ≤ r, (43)for some q a, , . . . , q a,n ∈ Q such that q a, j − + q a, j ∈ Z for 1 ≤ j ≤ k f , q a, k f + k e + j ∈ Z for 1 ≤ j ≤ k h . (44)Here we denoted ˆ z j := ˆ x j + ˆ y j .Let us explicitely describe (in Substeps 1a–1d below) a construction of such coordinatesˆ x j , ˆ y j on V satisfying (40)–(44). Substep 1a.
Since the vector fields X J ℓ pairwise commute and are ρ (Γ) ∗ -invariant,moreover Γ is commutative, we conclude that the operators A ℓ and M a pairwise com-mute too. From a standard assertion of Linear Algebra, there exists a unique (up to apermutation of terms) decomposition V = M s V s such that each V s is invariant under each A ℓ and M a , moreover Spec( A ℓ | V s ) = {± λ ℓs , ± λ ℓs } ,Spec( M a | V s ) = { µ ± as , µ ± as } , and for any s = s ′ there exists either ℓ ∈ { , . . . , κ e + κ h } such that λ ℓs ′
6∈ {± λ ℓs , ± λ ℓs } or a ∈ { , . . . , r } such that µ as ′
6∈ { µ ± as , µ ± as } .Consider the symplectic form Ω | m on V , and denote it by Ω. Since the operators A ℓ and M a are Hamiltonian and symplectic respectively, moreover they pairwise commute,it follows that the subspaces V s are symplectic and pairwise skew-orthogonal [17, Propo-sition 3.1]. On the other hand, these subspaces are invariant under each operator A ℓ .Therefore V s are pairwise “orthogonal” w.r.t. the second differential of each function J ℓ , that is given by the symmetric bilinear form d J ℓ | m ( ξ , ξ ) = Ω( ξ , A ℓ ξ ) on V . So,it remains to compute the restrictions of all d J ℓ | m and M a to each subspace V s .Let us fix ℓ ∈ { , . . . , κ e + κ h } . By assumption, either the flow of X J ℓ is 2 π -periodic (for1 ≤ ℓ ≤ κ e ), or the flow of X C iJ ℓ is 2 π -periodic (for κ e < ℓ ≤ κ e + κ h ). As we observedat the beginning of Step 1, in the former case, A ℓ is elliptic ; all its eigenvalues are pureimaginary and belong to i Z . In the latter case, A ℓ is hyperbolic ; all its eigenvalues areintegers. Without loss of generality, we can assume that λ ℓs ∈ i Z + if A ℓ is elliptic (i.e.,1 ≤ ℓ ≤ κ e ), and λ ℓs ∈ Z + if A ℓ is hyperbolic (i.e., κ e < ℓ ≤ κ e + κ h ).Let us fix a ∈ { , . . . , r } . By assumption, the operator M a is of finite order, hence allits eigenvalues µ ± as , µ ± as belong to the unit circle in C and belong to { e πiq | q ∈ Q } .Thus, without loss of generality, we can assume that µ as = e πiq as for some q as ∈ Q ∩ [0 ,
12 ] , ≤ a ≤ r. (45)42 ubstep 1b. Fix a subspace V s ⊆ V from the above decomposition.Consider the set M s := { a ∈ { , . . . , r } | µ as
6∈ { , − }} . Thus q as ∈ Q ∩ (0 , /
2) for each a ∈ M s (see (45)). Define the linear operators L as := 1sin(2 πq a,s ) M a | V s − (cot(2 πq a,s ))Id V s , a ∈ M s , thus we have M a | V s = (cos(2 πq a,s ))Id V s + (sin(2 πq a,s )) L as , ≤ a ≤ r. (46)One can easily show that L as are Hamiltonian (and symplectic), pairwise commute andsatisfy the equalities L as = − Id V s , a ∈ M s . (47)If M s = ∅ , let us choose a s ∈ M s and consider the (elliptic Hamiltonian) operator on V s : E s := L a s ,s . (48)Consider the sets I s := { ℓ ∈ { , . . . , κ e } | λ ℓs ∈ i Z \{ }} , H s := { ℓ ∈ { κ e +1 , . . . , κ e + κ h } | λ ℓs ∈ Z \{ }} . So, all A ℓ | V s with ℓ ∈ I s are elliptic, while all A ℓ | V s with ℓ ∈ H s are hyperbolic. Thus λ ℓs = p sℓ i for some integer p sℓ > ℓ ∈ I s , λ ℓ ′ s =: p sℓ ′ > ℓ ′ ∈ H s .Define the linear operators B ℓs := 1 p sℓ A ℓ | V s , ℓ ∈ I s ∪ H s . (49)Clearly they pairwise commute, are diagonalizable over C , and hence satisfy the equal-ities B ℓs = − Id V s for ℓ ∈ I s , B ℓ ′ s = Id V s for ℓ ′ ∈ H s . (50)If M s = ∅ and I = ∅ , let us choose ℓ s ∈ I s and consider the (elliptic Hamiltonian)operator E s := B ℓ s ,s (51)on V s . If H = ∅ , let us choose ℓ ′ s ∈ H s and consider the (hyperbolic Hamiltonian)operator H s := B ℓ ′ s ,s (52)on V s . Consider the set of pairwise commuting symplectic involutions (see below) { E s L as | a ∈ M s } ∪ { E s B ℓs | ℓ ∈ I s } ∪ { H s B ℓ ′ s | ℓ ′ ∈ H s } (53)43n V s . Due to (47) and (50), these operators are involutions: ( E s L as ) = Id V s ,( E s B ℓs ) = Id V s and ( H s B ℓ ′ s ) = Id V s , hence they are symplectic (indeed: since A ℓ | V s isHamiltonian, B ℓs is also Hamiltonian, hence Ω( B ℓs B ℓ ′ s u, B ℓs B ℓ ′ s v ) = − Ω( B ℓ ′ s u, B ℓs B ℓ ′ s v ) =Ω( u, B ℓs B ℓ ′ s v ) = Ω( u, v ) for any u, v ∈ V s and either ℓ, ℓ ′ ∈ I s or ℓ, ℓ ′ ∈ H s ; the sym-plecticity of L as L a ′ s and L as B ℓs is proved similarly).It follows from Linear Algebra that there exists a unique (up to a permutation of terms)decomposition V s = M t V st such that each V st is invariant under each symplectic involution from the set (53), E s L as | V st = − ε ast Id V st , a ∈ M s ,E s B ℓs | V st = − η ℓst Id V st , ℓ ∈ I s , H s B ℓ ′ s | V st = η ℓ ′ st Id V st , ℓ ′ ∈ H s , (54)where ε ast , η ℓst , η ℓ ′ st ∈ { , − } , η a s st = 1, η ℓ s st = 1, η ℓ ′ s st = 1, and for any t = t ′ thereexists either ℓ ∈ I s ∪ H s such that η ℓst = η ℓst ′ or a ∈ M s such that ε ast = ε ast ′ .Clearly each subspace V st is symplectic, E s -invariant (if I s = ∅ or M s = ∅ ) and H s -invariant (if H s = ∅ ). Substep 1c.
Fix a subspace V st ⊆ V s from the above decomposition.We have four possibilities for the subspace V s : it is either of elliptic, hyperbolic orfocus-focus type, or trivial. Case 1 (elliptic): M s = ∅ or I s = ∅ , moreover H s = ∅ . Recall that, in Substep 1b,we fixed the elliptic Hamiltonian operator E s on V s , see (48) and (51).We have the Hamiltonian (and symplectic) operator E s | V st such that ( E s | V st ) = − Id V st .It follows from Linear Algebra that there exists a basis e , . . . , e dim V st / , f , . . . , f dim V st / of V st such that E s e j = ˆ η stj f j , E s f j = − ˆ η stj e j , Ω( e j , f j ) = 1 , ≤ j ≤
12 dim V st , where ˆ η stj ∈ { , − } and the planes Span { e j , f j } are pairwise skew-orthogonal.In the above symplectic basis of the “elliptic” subspace V st , we have from (54) that L as e j = − ε ast ˆ η stj f j , L as f j = ε ast ˆ η stj e j ,B ℓs e j = − η ℓst ˆ η stj f j , B ℓs f j = η ℓst ˆ η stj e j , ≤ j ≤ dim V st , for any ℓ ∈ I s and a ∈ M s , therefore, we have from (46) and (49) that M a e j = (cos(2 πq a,s )) e j − ε ast ˆ η stj (sin(2 πq a,s )) f j , A ℓ e j = η ℓst ˆ η stj p sℓ f j ,M a f j = ε ast ˆ η stj (sin(2 πq a,s )) e j + (cos(2 πq a,s )) f j , A ℓ f j = − η ℓst ˆ η stj p sℓ e j , (55)44 ≤ j ≤ dim V st , for any ℓ ∈ I s and a ∈ { , . . . , r } , while A ℓ | V st = 0 for any ℓ ∈ { , . . . , κ e + κ h } \ I s .Finally, let us compute the restriction to V st of the second differential d J ℓ | m ( ξ , ξ ) =Ω( ξ , A ℓ ξ ) of each function J ℓ at m . It has the formd J ℓ | m ( e i , e j ) = d J ℓ | m ( f i , f j ) = η ℓst ˆ η stj p sℓ δ ij , d J ℓ | m ( e i , f j ) = 0 , (56)1 ≤ i, j ≤ dim V st , for any ℓ ∈ I s , while d J ℓ | V st = 0 for any ℓ ∈ { , . . . , κ e + κ h } \ I s . Case 2 (hyperbolic): H s = ∅ and I s = M s = ∅ . Thus µ as ∈ { , − } and M a | V s = µ as Id V s , ≤ a ≤ r, (57)due to (45) and (46). Recall that, in Substep 1b, we fixed the hyperbolic Hamiltonianoperator H s = B ℓ ′ s s on V s , see (52).Consider the Hamiltonian operator H st := H s | V st = B ℓ ′ s s | V st . Since H st = Id V st , wehave a decomposition V st = V ′ st ⊕ V ′′ st where V ′ st , V ′′ st are the eigenspaces of H st withthe eigenvalues 1 , − H st is Hamiltonian, we have that each of V ′ st and V ′′ st is isotropic (in fact: 0 = Ω( H st u, v ) + Ω( u, H st v ) = ( λ + µ )Ω( u, v ), whenever u, v ∈ V s are eigenvectors of H st with eigenvalues λ, µ , respectively). Hence V ′ st , V ′′ st areLagrangian subspaces of V st .Choose a basis e , e , . . . , e dim V st / of the subspace V ′ st . Since Ω | V st is nondegenerate andvanishes on each V ′ st and V ′′ st , it gives a nondegenerate pairing between the subspaces V ′ st and V ′′ st , so there exists a unique basis f , f , . . . , f dim V st / of the subspace V ′′ st suchthat Ω( e i , f j ) = δ ij .For any ℓ ∈ H s , in the above basis of the “hyperbolic” subspace V st , we have from (54)that B ℓs e j = η ℓst H st e j = η ℓst e j , B ℓs f j = η ℓst H st f j = − η ℓst f j , ≤ j ≤
12 dim V st , Ω( e j , f j ) = 1 and the planes Span { e j , f j } are pairwise skew-orthogonal. Thus we havefrom (49) and (57) that M a e j = µ as e j , M a f j = µ as f j , A ℓ e j = η ℓst p sℓ e j , A ℓ f j = − η ℓst p sℓ f j (58)for any ℓ ∈ H s and any a ∈ { , . . . , r } , while A ℓ | V st = 0 for any ℓ ∈ { , . . . , κ e + κ h } \ H s .Finally, let us compute the restriction to V st of the second differential d J ℓ | m ( ξ , ξ ) =Ω( ξ , A ℓ ξ ) of each function J ℓ at m . It has the formd J ℓ | m ( e i , f j ) = − η ℓst p sℓ δ ij , d J ℓ | m ( e i , e j ) = d J ℓ | m ( f i , f j ) = 0 , (59)1 ≤ i, j ≤ dim V st , for any ℓ ∈ H s , while d J ℓ | V st = 0 for any ℓ ∈ { , . . . , κ e + κ h } \ H s .45 ase 3 (focus-focus): I s = ∅ or M s = ∅ , moreover H s = ∅ . Recall that, in Substep1b, we fixed two Hamiltonian operators E s and H s on V s , see (48), (51) and (52).Consider two commuting Hamiltonian operators E st := E s | V st , H st := H s | V st = B ℓ ′ s s | V st , which are elliptic and hyperbolic, respectively.Since H st = Id V st , we have a decomposition V st = V ′ st ⊕ V ′′ st where V ′ st , V ′′ st are theeigenspaces of H st with the eigenvalues 1 , − H st is a Hamiltonianoperator, we have that the subspaces V ′ st , V ′′ st are isotropic (see Case 2), so V ′ st , V ′′ st areLagrangian subspaces of V st .Since the operator E st commutes with H st , it leaves invariant each Lagrangian subspace V ′ st , V ′′ st . Since E st = − Id V st , it follows from Linear Algebra that dim V ′ st = dim V st / e , e , . . . , e dim V ′ st of V ′ st such that E st e j − = e j , E st e j = − e j − , ≤ j ≤
12 dim V ′ st . Since Ω | V st is nondegenerate and vanishes on each V ′ st and V ′′ st , it gives a nondegeneratepairing between the subspaces V ′ st and V ′′ st , so there exists a unique basis f , f , . . . , f dim V ′ st of the subspace V ′′ st such that Ω( e i , f j ) = δ ij . In this basis of V ′′ st , we can compute theoperator E st | V ′′ st as follows. Since E st is a Hamiltonian operator, we haveΩ( e j , E st f i ) = − Ω( E st e j , f i ) = Ω( e j − , f i ) = δ j − ,i , Ω( e j − , E st f i ) = − Ω( E st e j − , f i ) = − Ω( e j , f i ) = − δ j,i , therefore E st f j − = f j and E st f j = − f j − , 1 ≤ j ≤ dim V st / V st , due to the formulae(54), each elliptic Hamiltonian operator L as , a ∈ M s , acts by the formulae L as e j − = − ε ast e j , L as e j = ε ast e j − , L as f j − = − ε ast f j , L as f j = ε ast f j − , each elliptic Hamiltonian operator B ℓ , ℓ ∈ I s , acts by the formulae B ℓs e j − = − η ℓst e j , B ℓs e j = η ℓst e j − , B ℓs f j − = − η ℓst f j , B ℓs f j = η ℓst f j − , ≤ j ≤ dim V st /
4, while each hyperbolic Hamiltonian operator B ℓ ′ , ℓ ′ ∈ H s , acts bythe formulae B ℓ ′ s e j = η ℓ ′ st e j , B ℓ ′ s f j = − η ℓ ′ st f j , Ω( e j , f j ) = 1 , ≤ j ≤
12 dim V st , { e j , f j } are pairwise skew-orthogonal. Thus, each elliptic symplecticoperator M a , 1 ≤ a ≤ r , acts (due to (46)) by the formulae M a e j − = (cos(2 πq a,s )) e j − − ε ast (sin(2 πq a,s )) e j ,M a e j = ε ast (sin(2 πq a,s )) e j − + (cos(2 πq a,s )) e j ,M a f j − = (cos(2 πq a,s )) f j − − ε ast (sin(2 πq a,s )) f j ,M a f j = ε ast (sin(2 πq a,s )) f j − + (cos(2 πq a,s )) f j , ≤ j ≤ dim V st , (60)each elliptic operator A ℓ , ℓ ∈ I s , acts (due to (49)) by the formulae A ℓ e j − = − η ℓst p sℓ e j , A ℓ e j = η ℓst p sℓ e j − ,A ℓ f j − = − η ℓst p sℓ f j , A ℓ f j = η ℓst p sℓ f j − , ≤ j ≤ dim V st , (61)each hyperbolic operator A ℓ ′ , ℓ ′ ∈ H s , acts (due to (49)) by the formulae A ℓ ′ e j = η ℓ ′ st p sℓ ′ e j , A ℓ ′ f j = − η ℓ ′ st p sℓ ′ f j , ≤ j ≤
12 dim V st , (62)while A ℓ | V st = 0 for all remaining ℓ ∈ { , . . . , κ e + κ h } \ ( I s ∪ H s ).Finally, let us compute the restriction to V st of the second differential d J ℓ | m ( ξ , ξ ) =Ω( ξ , A ℓ ξ ) of each function J ℓ at m . For each “elliptic” function J ℓ , ℓ ∈ I s , we haved J ℓ | m ( e i , f j − ) = η ℓst p sℓ δ i, j , d J ℓ | m ( e i , e j ) = d J ℓ | m ( f i , f j ) = 0 , d J ℓ | m ( e i , f j ) = − η ℓst p sℓ δ i, j − , ≤ i, j ≤ dim V st . (63)For each “hyperbolic” function J ℓ , ℓ ∈ H s , we haved J ℓ | m ( e i , f j ) = − η ℓst p sℓ δ ij , d J ℓ | m ( e i , e j ) = d J ℓ | m ( f i , f j ) = 0 , (64)1 ≤ i, j ≤ dim V st . For all remaining ℓ ∈ { , . . . , κ e + κ h } \ ( I s ∪ H s ), we haved J ℓ | V st = 0. Case 4 (trivial): M s = ∅ and I s = H s = ∅ . Thus V st = V s . Similarly to Case 2, wehave µ as ∈ { , − } and (57) (e.g. if µ as equal 1 for all a ∈ { , . . . , r } , then M a | V s = Id V s for any a ∈ { , . . . , r } ; in fact such s is unique if any, let us denote it by s ). Moreover A ℓ | V s = 0 and d J ℓ | V s = 0 for any ℓ ∈ { , . . . , κ e + κ h } . Choose a symplectic basis e , . . . , e dim V s / , f , . . . , f dim V s / of V s , i.e. a basis satisfyingΩ( e i , e j ) = Ω( f i , f j ) = 0 , Ω( e i , f j ) = δ ij , ≤ i, j ≤
12 dim V s . Substep 1d.
Now take a symplectic basis e , . . . , e n , f , . . . , f n of V = T m M formedby the above symplectic bases of the subspaces V st (see Substep 1c, Cases 1–4). In detail:the basis e , . . . , e dim V st / , f , . . . , f dim V st / of a subspace V st appears as e n st +1 , . . . , e n st +dim V st / , f n st +1 , . . . , f n st +dim V st / in the basis of V , for some “shifting” integers n st s.t. orderingof the subspaces V st with integers n st is consistent with the following partial order:“focus-focus” subspaces, “elliptic” ones, “hyperbolic” ones, and “trivial” ones.Let ˆ x , ˆ y , . . . , ˆ x n , ˆ y n be the linear coordinates on V = T m M such that47 the coordinates ˆ x j − , ˆ y j − , ˆ x j , ˆ y j , 1 ≤ j ≤ k f , correspond to the bases e j − − f j √ , e j + f j − √ , e j − + f j √ , − e j + f j − √ of the “focus-focus” subspaces V st (if any); they wereconstructed in Substep 1c, Case 3, • the coordinates ˆ x j , ˆ y j , 2 k f + 1 ≤ j ≤ k f + k e , correspond to the bases e j , f j ofthe “elliptic” subspaces V st (if any); they were constructed in Substep 1c, Case 1, • the coordinates ˆ x j , ˆ y j , 2 k f + k e + 1 ≤ j ≤ k f + k e + k h , correspond to the bases e j , f j of the “hyperbolic” subspaces V st (if any); they were constructed in Substep1c, Case 2, • the remaining coordinates ˆ x j , ˆ y j , 2 k f + k e + k h + 1 ≤ j ≤ n , correspond tosymplectic bases of the “trivial” subspaces V s (if any), see Substep 1c, Case 4.In these coordinates, Ω | m , d J ℓ | m and M a have the desired form (40)–(44), due to(55), (56), (58)–(64). Step 2.
In Step 1, we actually constructed local coordinates on a neighbourhood of m ,that bring Ω to the desired canonical form (36) at m , and bring the functions J ℓ to thedesired form (37) and (38) up to 3rd order terms. We want to deform these coordinatesa little bit, in order to achieve the equalities (36)–(38) exactly.Consider the group G := Γ × ( S ) κ e + κ h , where ( S ) κ e + κ h is the ( κ e + κ h )-torus with a usual group structure. Since G is acompact Lie group acting analytically on a neighbourhood U C ⊂ M C of m ∈ M witha fixed point m , it acts by linear transformations w.r.t. to some holomorphic localcoordinates on a G -invariant neighbourhood U C of m ([2], [7, Sec 3.1.4]).Let us explicitely construct (in Substeps 2a–2c below) real-analytic symplectic coor-dinates x , y , . . . , x n , y n on a neighbourhood U ⊂ U of m such that G acts by lin-ear transformations on a G -invariant neighbourhood U C of m w.r.t. the coordinates x C , y C , . . . , x C n , y C n . Substep 2a.
Let U be a small neighbourhood of m in M , and U C its small open G -invariant complexification.Take any real-analytic (pseudo-)Riemannian metric ds on U such thatd s | m = k f X j =1 a j (dˆ x j − + dˆ y j − − dˆ x j − dˆ y j ) + k e X j =1 a k f + j (dˆ x k f + j + dˆ y k f + j )++ k h X j =1 a k f + k e + j dˆ x k f + k e + j dˆ y k f + k e + j + n X j = k e + k h +2 k f +1 a j − k f (dˆ x j + dˆ y j )48compare (41), (42)). Here ˆ x j , ˆ y j are linear coordinates on T m M constructed in Step 1,and a , . . . , a n − k f ∈ R \ { } are fixed real numbers (e.g. a j = 1). If a j >
0, this pseudo-Riemannian metric has signature (2 n − k h − k f , k h + 2 k f ), so it is a Riemannian metricif k h = k f = 0 (elliptic case).Construct a G -invariant (pseudo-)Riemannian metric h (d s ) C i on U C by averaging(d s ) C over G : h (d s ) C i := 1(2 π ) κ | Γ | X ψ ∈ Γ 2 π Z · · · π Z ( φ t κ iJ Cκ ◦· · ·◦ φ t κ e +1 iJ Cκ e +1 ◦ φ t κ e J Cκ e ◦· · ·◦ φ t J C ◦ ρ ( ψ )) ∗ (d s ) C d t . . . d t κ , where κ := κ e + κ h , φ tf denotes the flow of the vector field X f . We claim that h (d s ) C i has the following properties:(i) h (d s ) C i is G -invariant;(ii) h (d s ) C i is real-analytic;(iii) h (d s ) C i| m = (d s ) C | m and, in particular, it is a nondegenerate quadratic form.The property (i) is obvious.For proving the property (ii), observe that h (d s ) C i can be obtained from (d s ) C in κ +1steps, where the first step performs averaging over Γ, and the ( ℓ + 1)-st step performsaveraging over the subtorus { } ℓ − × S × { } κ − ℓ of the torus ( S ) κ , 1 ≤ ℓ ≤ κ + 1.After the first κ e + 1 steps, the resulting average will be real-analytic, since d s is real-analytic and we averaged it over a real-analytic action. The ( ℓ + 1)-st step with ℓ > κ e performs an average over the 1-parameter family of diffeomorphisms φ tiJ C ℓ : U C → U C ,0 ≤ t ≤ π , such that φ iJ C ℓ = Id , dd t φ tiJ C ℓ ( m ) = iX C J ℓ | φ tiJ C ℓ ( m ) . (65)This implies that, at each real point m ∈ U , we have ddt φ π − tiJ C ℓ ( m ) = i X C J ℓ | φ π − tiJ C ℓ ( m ) = iX C J ℓ | φ π − tiJ C ℓ ( m ) , therefore φ tiJ C ℓ ( m ) = φ π − tiJ C ℓ ( m ). Hence, averaging over the family of diffeomorphisms φ tiJ C ℓ is the same as averaging over the family of diffeomorphisms φ tiJ C ℓ . But, if d s is a real-analytic (pseudo-)Riemannian metric, then ( φ tiJ C ℓ ) ∗ (d s ) C = ( φ tiJ C ℓ ) ∗ (d s ) C on U . Hence, the resulting average coincides with its C -conjugate, which shows that it isreal-valued on U . 49or proving the property (iii), it is enough to check that (d s ) C | m is invariant under thelinearized G -action at m . Observe that d s | m = k f P j =1 a j d h j − | m + n − k f P j =1 a k f + j d h k f + j | m .Here h , . . . , h k f + k e + k h denote the quadratic functions (13) w.r.t. the coordinates ˆ x j , ˆ y j , h j := (ˆ x j + ˆ y j ) / k f + k e + k h +1 ≤ j ≤ n . But the quadratic forms d h j | m on T m M pairwise Poisson commute with respect to the symplectic form Ω | m . Therefore, eachd h j | m is invariant under the Hamiltonian flows generated by the quadratic functions(41) and (42), which coincide with the second differentials d J ℓ by Step 1. Therefore,each d h C j | m is invariant under the linearized ( S ) κ -action at m . Furthermore, eachd h j | m is M a -invariant due to (43) and (44) proved in Step 1. Hence each d h C j | m (and therefore (d s ) C | m ) is invariant under the linearized G -action at m , as required.By the properties (ii) and (iii), h ( ds ) C i is the holomorphic extension to U C of somereal-analytic (pseudo-)Riemannian metric on U , which we denote by h d s i . We have h (d s ) C i = h d s i C , and it is G -invariant due to the property (i). Substep 2b.
Let us identify U with a small neighbourhood ˆ U of the origin in T m M viathe exponential map exp m | ˆ U : ˆ U ≈ −→ U (66)corresponding to the (pseudo-)Riemannian metric h d s i on U constructed in Substep2a.Let us transfer the linear coordinates ˆ x , ˆ y , . . . , ˆ x n , ˆ y n from ˆ U to U by means of theidentification U ≈ ˆ U in (66). We will obtain some coordinates on U , which we denoteby u , v , . . . , u n , v n .Due to Substep 2a, the holomorphic extension of the map (66) is G -equivariant. Substep 2c.
Now we want to “deform” the coordinates u , v , . . . , u n , v n on U (by some G -equivariant transformation), in order to make Ω being constant. We will achievethis by means of Moser’s path method [31, Theorem 2], more precisely its equivariantversion, as follows. Let ˆΩ be the constant symplectic 2-form on T m M coinciding withΩ | m at the origin. Denote by Ω the symplectic structure on U obtained from ˆΩ underthe identification (66), thus Ω | m = Ω | m . Since ˆΩ C is G -invariant and the exponentialmap (66) is G -equivariant (by Substep 2b), we conclude that Ω C is also G -invariant.Denote Ω t := (1 − t )Ω + t Ω , ≤ t ≤ . Choose a real-analytic 1-form α on U such that α C is G -invariant, α | m = 0 (moreover, α | m = O ( | m − m | ) w.r.t. some local coordinates on U ; this property will be achievedand used in Step 4 below), and d α = Ω − Ω ≡ dd t Ω t G -invariant 1-form can be obtained by averaging some 1-form with the aboveproperties over G , due to the G -invariance of Ω C and Ω C , then the resulting average willbe real-analytic, due to the same arguments as in Substep 2a). Define a 1-parameterfamily of vector fields ξ t on U , 0 ≤ t ≤
1, by Moser’s equation i ξ t Ω t = − α (such a vector field exists and is unique, perhaps in a smaller neighbourhood of m ,since Ω t | m = Ω | m is a nondegenerate 2-form). Define a 1-parameter family of diffeo-morphisms φ t : U t → U , 0 ≤ t ≤
1, such that φ = Id U anddd t φ t = ξ t ◦ φ t in U t . Observe that m is fixed under each φ t , since ξ t | m = 0 because α | m = 0. Clearly,Ω C t , ξ C t and U C t are G -invariant, thus each φ C t is G -equivariant. We havedd t ( φ ∗ t Ω t ) = φ ∗ t ( L ξ t Ω t + dΩ t d t ) = φ ∗ t (( i ξ t d+d i ξ t )Ω t +Ω − Ω ) = φ ∗ t (d i ξ t Ω t +Ω − Ω ) = φ ∗ t ( − d α +Ω − Ω ) = 0 . Since the above equalities hold for any t ∈ [0 , φ ∗ t Ω t ≡ φ ∗ Ω = Ω for any t , in particular for t = 1 we obtain φ ∗ Ω = Ω . Define on U := φ ( U ∩ U ) the coordinates x j := u j ◦ φ − | U , y j := v j ◦ φ − | U , ≤ j ≤ n, i.e. x , y , . . . , x n , y n are induced from the coordinates u , v , . . . , u n , v n on U ′ := U ∩ U by means of the identification U ′ ≈ −→ U via the diffeomorphism φ : U → U . Clearly, x j ( m ) = y j ( m ) = 0 for any j = 1 , . . . , n (since m is fixed under each φ t ). Step 3.
We claim thatΩ | U = n X j =1 d x j ∧ d y j , J ℓ | U = c ℓ + n X a,b =1 c ℓab w a w b , (67)for some real constants c ℓ , c ℓab . Here we denoted ( w , . . . , w n ) := ( x , y , . . . , x n , y n ).Recall (see Substep 2b and the beginning of Substep 2c) thatΩ = ((exp m | ˆ U ) − ) ∗ ˆΩ , u j := ˆ x j ◦ (exp m | ˆ U ) − , v j := ˆ y j ◦ (exp m | ˆ U ) − . ThusΩ | U = (( φ ◦ exp m | ˆ U ′ ) − ) ∗ ˆΩ , x j = ˆ x j ◦ ( φ ◦ exp m | ˆ U ′ ) − , y j = ˆ y j ◦ ( φ ◦ exp m | ˆ U ′ ) − , U ′ := (exp m | ˆ U ) − ( U ′ ). We conclude from (40) and from the beginning ofSubstep 2c that Ω | U has the desired form as in (67).Observe that the above diffeomorphism φ ◦ exp m | ˆ U ′ : ˆ U ′ ≈ −→ U (68)has a G -equivariant holomorphic extension, since φ C and exp C m are G -equivariant. Sincethe coordinates w , . . . , w n on U are induced from the linear coordinates ˆ x , ˆ y , . . . , ˆ x n , ˆ y n on T m M under the diffeomorphism U ≈ ˆ U ′ ⊂ T m M in (68), moreover the group G acts on T m M C by linear transformations w.r.t. ˆ x C , ˆ y C , . . . , ˆ x C n , ˆ y C n , we conclude that itacts on U C by linear transformations w.r.t. w C , . . . , w C n . Therefore the Hamiltonianvector fields X J ℓ on U (which generate an infinitesimal action of the Lie algebra of G )are also linear w.r.t. w , . . . , w n , 1 ≤ ℓ ≤ k . Since Ω | U has constant components w.r.t. w , . . . , w n , as in (67), we conclude that d J ℓ | U has linear components w.r.t. w , . . . , w n ,i.e. d J ℓ | U = 2 n P a,b =1 c ℓab w a d w b for some constants c ℓab = c ℓba ∈ R . This immediately givesus the desired quadratic form for J ℓ | U as in (67). Step 4.
Let us show that the quadratic forms in (67) have the special form as in (37),(38). For this, it is enough to make sure that d x j | m = dˆ x j | m and d y j | m = dˆ y j | m ,1 ≤ j ≤ n , due to (41) and (42). So, it is enough to achieve that d φ t | m = Id foreach t ∈ [0 , ξ t | m = 0, which in turn isequivalent to the fact that each component of α | m w.r.t. some coordinates on U is oforder O ( | m − m | ).So, it is enough to show that, in Substep 2c, we can choose a 1-form α on U satisfyingthe above condition. We have Ω | U − Ω = d( α − φ ∗ α ), where α is any 1-form on U such that d α = Ω, and φ is a diffeomorphism such that φ ( m ) = m and d φ | m = Id.In canonical coordinates ( p, q ) = ( p , q , . . . , p n , q n ) such that p j | m = q j | m = 0, wehave Ω | U = d p ∧ d q := n P j =1 d p j ∧ d q j . Put α = p d q := n P j =1 p j d q j , α := α − φ ∗ α .Since φ ( m ) = m and d φ | m = Id, it follows from Hadamard’s lemma that φ ∗ p i = p i + Q ′ i ( p, q ), φ ∗ q j = q j + Q ′′ j ( p, q ), where Q ′ i ( p, q ) and Q ′′ j ( p, q ) are some quadraticforms whose coefficients are real-analytic functions in ( p, q ). Thus α = α − φ ∗ α = p d q − φ ∗ ( p d q ) = p d q − ( φ ∗ p )d( φ ∗ q ) = n P j =1 p j d q j − ( p j + Q ′ j ( p, q ))d( q j + Q ′′ j ( p, q )) = − n P j =1 ( p j + Q ′ j ( p, q ))d Q ′′ j ( p, q ) − Q ′ j ( p, q )d q j . Each component of the latter 1-form is oforder O ( | p | + | q | ), as required.Lemma 6.2 (a), and hence Lemma 6.1 (a), is proved.52 .2 Proof of part (b) of Lemmata 6.1 and 6.2 Similarly to the previous subsection, we will give a proof of Lemma 6.2 (b). Due tothe Cauchy integral formula for holomorphic functions, we can and will assume that k ˜ F C − F C k C k + k ˜Ω C − Ω C k C k + n − + k ˜ ρ C − ρ C k C k − < ε for some k ≥ M in assumptions of Lemma 6.1 (b)). If κ h = 0 (i.e.,all functions J ℓ generate 2 π -periodic flows), all our arguments and constructions willliterally work both in the real-analytic and C ∞ cases. This will give a proof of Lemma6.1 (b) too.Consider the subgroup G := Γ × ( S ) κ e × { } κ h of the group G = Γ × ( S ) κ e + κ h . Recallthat U is G -invariant, and U C is G -invariant.We will identify U with its image under the coordinate map ( x , y , . . . , x n , y n ) : U → R n from (a). We will equip R n with cartesian coordinates ( x , y , . . . , x n , y n ) and withthe standard symplectic structure Ω as in (67).Consider the “perturbed” G -action generated by ˜ ρ (Γ) and by the “perturbed” func-tions ˜ J , . . . , ˜ J κ e , i ˜ J κ e +1 , . . . , i ˜ J κ e + κ h w.r.t. the “perturbed” symplectic structure ˜Ω. Byabusing language, we will call this action the ˜ G -action , in order to distinguish it fromthe “unperturbed” G -action.Divide the proof into several steps. Step 1.
In this step, we prove that the ˜ G -action has a fixed point ˜ m ∈ U ′ O ( ε ) − closeto m . We will use the same notations (e.g. A ℓ , M a , V s ) as in the proof of (a). Substep 1a.
Denote R := r \ a =1 ker( M a − Id) , K := R ∩ κ e + κ h \ ℓ =1 ker A ℓ = κ e + κ h \ ℓ =1 ker A ℓ | R where Id is the identity operator in R n (here we used that M a and A ℓ commute, hencethe set R of fixed points of M a is A ℓ -invariant). By (a), the group G acts on U linearlyw.r.t. our coordinates ( x , y , . . . , x n , y n ). We conclude that R ∩ U is the set of fixedpoints of the (“unperturbed”) Γ-action, and ( K ∩ U ) C is the set of fixed points of the(“unperturbed”) G -action.Clearly, there exists a linear combination A = κ e + κ h X ℓ =1 c ℓ A ℓ such that R ∩ ker A = K . 53e claim that K is a symplectic subspace of R n . Indeed, each ker A ℓ and ker( M a − Id)is the direct product of several subspaces V s . Hence K is also the direct product ofsome subspaces V s (actually, it is one of the “trivial” subspaces V s , see the proof of (a),Case 4 of Substep 1c), therefore K is symplectic. Substep 1b.
Since K is symplectic (by Substep 1a), we have R n = W × K where W := K ⊥ denotes the skew-orthogonal complement of K w.r.t. Ω. In fact, W and K are the coordinate subspaces of R n : R n = R k × R n − k ) = W × K where 2 k f + k e + k h ≤ k ≤ n , with coordinates σ := ( x , y , . . . , x k , y k ) on W , τ := ( x k +1 , y k +1 , . . . , x n , y n ) on K. Denote by
P r K : R n = W × K → K, ( σ, τ ) τ, the projection along W . Clearly, the subspace W C and the map P r C K are G -invariant,furthermore every point of K is fixed under the G -action.Choose G -invariant neighbourhoods U ′ := U ′ W × U ′ K ⊂ U W × U K ⊂ W × K of m = 0 in R n such that U ′ ⊂ U W × U K and U W × U K ⊂ U . Then we can choose G -invariant neighbourhoods( U ′ ) C := ( U ′ W ) C × ( U ′ K ) C ⊂ U C W × U C K ⊂ W C × K C of m = 0 in C n such that ( U ′ ) C ⊂ U C W × U C K and U C W × U C K ⊂ U C .Consider the (“perturbed”) ˜ G -invariant neighbourhood ˜ U := ˜ G ( U W × U K ) of m in R n and the (“perturbed”) ˜ G -invariant neighbourhood ˜ U C := ˜ G ( U C W × U C K ) of m in C n . Clearly, U ′ ⊂ ˜ U ⊂ U and U ′ C ⊂ ˜ U C ⊂ U C if the perturbation is small enough.Denote by g P r C K : ( U ′ W ) C × U C K → K C the map obtained by averaging the map P r C K | ˜ U C over the ˜ G -action: g P r C K ( m ) := X ψ ∈ Γ π ) κ | Γ | π Z · · · π Z P r C K ◦ φ t κ i ˜ J Cκ ◦· · ·◦ φ t κ e +1 i ˜ J Cκ e +1 ◦ φ t κ + e ˜ J Cκ e ◦· · ·◦ φ t ˜ J C ◦ ˜ ρ ( ψ )( m )d t . . . d t κ , ∈ ( U ′ W ) C × U C K ⊂ ( ˜ U ) C ⊂ C n , where κ := κ e + κ h , φ tf denotes the flow of thevector field X f ; we use the identification of U with its image under the coordinate map( x , y , . . . , x n , y n ) : U → R n from (a). Clearly, g P r C K is ˜ G -invariant and O ( ε )-close to P r C K | ( U ′ W ) C × U C K in C k − − norm (since the G -action on W × K is componentwise, andthe G -action on K is trivial), thus it is a ˜ G -invariant submersion. The map g P r C K isreal-valued on U ′ W × U K by the same arguments as in Step 1 of the proof of (a). Thus, g P r C K = ˜ P r C K for some real-analytic map (submersion)˜ P r K : U ′ W × U K → K. Due to the Inverse Functions Theorem, for each point τ ∈ U ′ K , its pre-image˜ W τ := ˜ P r − K ( τ ) ⊂ U ′ W × U K ⊂ W × K is a real-analytic submanifold O ( ε )-close to the submanifold W τ := U ′ W × { τ } in C k − − norm. Clearly ˜ W C τ is ˜ G -invariant (since ˜ P r C K is ˜ G -invariant). Hence the sub-manifold ˜ W τ is symplectic and has the form˜ W τ = { ( σ, ˜ T ( σ, τ )) | σ ∈ U ′ W } ⊂ W × K, τ ∈ U ′ K , for some real-analytic map ˜ T : U ′ → U K close to P r K | U ′ in C k − − norm.Denote ˜ U ′ := ˜ P r − K ( U ′ K ), so ˜ U ′ = S τ ∈ U ′ K ˜ W τ is close to U ′ . Substep 1c.
Choose a ∈ { , . . . , r } and consider the cyclic subgroup Γ a of Γ generatedby ψ a (recall that ψ a ∈ Γ are the generators of the group Γ, as in the proof of (a),Step 1). Denote R a := ker( M a − Id), so R a ∩ U is the set of all fixed points of the(“unperturbed”) Γ a -action. Clearly, R a is the direct product of several coordinatesubspaces Ox j y j , so it is symplectic and has the form R a = W a × K , where W a is theskew-orthogonal complement of K in R a . Denote the skew-orthogonal complement of R a by W a := R ⊥ a , so it is symplectic too. We have R n = W × K = W a × ( W a × K ) = W a × R a , with coordinates σ a on W a , σ a on W a . By applying the arguments from Substep 1b to the Γ a -action and the (“perturbed”)˜Γ a -action, we construct a real-analytic ˜Γ a -invariant map (submersion)˜ P r R a : U ′ W a × U R a → R a close in C k − − norm to the projection P r R a along R a restricted to U ′ W a × U R a , and afamily of real-analytic ˜Γ a -invariant submanifolds˜ W aσ a ,τ := ˜ P r − R a ( σ a , τ ) ⊂ U ′ W a × U R a ⊂ W a × R a W aσ a ,τ := U ′ W a × { ( σ a , τ ) } in C k − − norm, ( σ a , τ ) ∈ U ′ R a = U ′ W a × U ′ K . This submanifold has the form˜ W aσ a ,τ = { ( σ a , ˜ S a ( σ a , σ a , τ ) , ˜ T a ( σ a , σ a , τ )) | σ a ∈ U ′ W a } ⊂ W a × R a , ( σ a , τ ) ∈ U ′ R a = U ′ W a × U ′ K , for some real-analytic maps ˜ S a : U ′ → U W a and ˜ T a : U ′ → U K close in C k − − norm to P r W a | U ′ , ( σ a , σ a , τ ) σ a , and P r K | U ′ , respectively.Observe that ker(( M a − Id) | W a ) = R a ∩ W a = { } , so 0 is a unique fixed point of the(“unperturbed”) map M a | W a . It follows from the Inverse Functions Theorem that the(“perturbed”) map ˜ ρ ( ψ a ) | ˜ W aσa,τ has a unique fixed point˜ m aσ a ,τ = ( ˜ S a ( σ a , τ ) , ˜ S a ( ˜ S a ( σ a , τ ) , σ a , τ ) , ˜ T a ( ˜ S a ( σ a , τ ) , σ a , τ )) ∈ ˜ W aσ a ,τ , for some real-analytic function ˜ S a : U ′ R a → U ′ W a close to S a ≡ C k − − norm. Hence,˜ R a := { ˜ m aσ a ,τ | ( σ a , τ ) ∈ U ′ R a } is the set of all fixed points in ˜ P r − R a ( U ′ R a ) of the ˜Γ a -action. Clearly, ˜ R a is a real-analyticsubmanifold close to U ′ R a = R a ∩ U ′ in C k − − norm (and, hence, it is symplectic).Since the group Γ is Abelian, it follows that, for any nonempty subset M ⊆ { , . . . , r } ,the set T a ∈ M ˜ R a is a (dim( T a ∈ M R a ))-dimensional submanifold close to ( T a ∈ M R a ) ∩ U ′ . Thus˜ R := ˜ R ∩ · · · ∩ ˜ R r is a (dim R )-dimensional submanifold close to R ∩ U ′ = U ′ R in C k − − norm, where R isthe same as in Substep 1a. In particular, it is symplectic. Clearly, ˜ R is nothing elsethan the set of all fixed points in ˜ P r − R ( U ′ R ) of the ˜Γ-action. Substep 1d.
Due to Substep 1c, the symplectic submanifold ˜ R is the set of all fixedpoints in ˜ P r − R ( U ′ R ) of the ˜Γ-action. Hence, it is invariant under the (local) Hamiltonianflows generated by ˜ J , . . . , ˜ J κ (since the ˜Γ-action commutes with these flows).Since R is symplectic and contains a symplectic subspace K , it has the form R = R ′ × K, where R ′ is the skew-orthogonal complement of K in R (and, hence, symplectic too).Moreover, its skew-orthogonal complement W ′ := ( R ′ ) ⊥ in W is also the direct productof several coordinate subspaces Ox j y j , so it is symplectic too. We have R n = W × K = W ′ × ( R ′ × K ) = W ′ × R, with coordinates σ ′ on W ′ , σ ′′ on R ′ . R has the form˜ R = { ( ˜ S ′ ( σ ′′ , τ ) , σ ′′ , τ ) | ( σ ′′ , τ ) ∈ R ∩ U ′ } , for some real-analytic map ˜ S ′ : R ∩ U ′ → W ′ ∩ U close to S ′ ≡ C k − − norm.By applying the arguments from Substep 1b to the G -action on ( R ∩ U ′ ) C and to the(“perturbed”) ˜ G -action on ˜ R C , we can construct a real-analytic map (submersion)˜ P r ′ K : ˜ R → K close in C k − − norm to the projection P r ′ K := P r K | R ∩ U ′ along R ′ , such that ( ˜ P r ′ K ) C :˜ R C → K C is ˜ G -invariant. Further, we construct a family of real-analytic submanifolds˜ R ′ τ := ( ˜ P r ′ K ) − ( τ ) ⊂ ˜ R close to the submanifold R ′ τ := ( R ′ ∩ U ′ ) × { τ } ⊂ R ∩ U ′ , τ ∈ U ′ K . Clearly, thesubmanifolds ( ˜ R ′ τ ) C are ˜ G -invariant, τ ∈ U ′ K . In particular, each of these submanifoldsis symplectic and has the form˜ R ′ τ = { ( ˜ S ′ ( σ ′′ , ˜ T ′ ( σ ′′ , τ )) , σ ′′ , ˜ T ′ ( σ ′′ , τ )) | σ ′′ ∈ R ′ ∩ U ′ } ⊂ W ′ × R, τ ∈ U ′ K , for some real-analytic map ˜ T ′ : R ∩ U ′ → U K close to P r K | R ∩ U ′ in C k − − norm.Consider the “unperturbed” and the “perturbed” functions f := κ X ℓ =1 c ℓ J ℓ , ˜ f := κ X ℓ =1 c ℓ ˜ J ℓ , see Substep 1a. It follows from Substep 1a that the “unperturbed” function f | R ′ τ isquadratic and has a unique critical point, namely at the “origin” ( σ ′ , σ ′′ , τ ) = (0 , , τ ) ∈ R ′ τ , τ ∈ U ′ K . Since ˜ R ′ τ is close to R ′ τ = ( R ′ ∩ U ′ ) ×{ τ } , and (due to the Inverse FunctionsTheorem) after a small perturbation, nondegenerate critical points are preserved anddeformed a little bit, we conclude that the “perturbed” function ˜ f | ˜ R ′ τ has a uniquecritical point˜ m τ = ( ˜ S ′ ( ˜ S ′′ ( τ ) , ˜ T ′ ( ˜ S ′′ ( τ ) , τ )) , ˜ S ′′ ( τ ) , ˜ T ′ ( ˜ S ′′ ( τ ) , τ )) , τ ∈ U ′ K , for some real-analytic function ˜ S ′′ : U ′ K → R ′ close to S ′′ ≡ C k − − norm.We claim that the point ˜ m τ is fixed under the ˜ G -action, τ ∈ U ′ K . Indeed, the function( ˜ f | ˜ R ′ τ ) C is ˜ G -invariant, hence its (unique) critical point ˜ m τ is also ˜ G -invariant.Thus, the intersection of U ′ with the fixed points set of the ˜ G -action on ( U ′ ) C coincideswith˜ K := { ˜ m τ | τ ∈ U ′ K } = { ( ˜ S ′ ( ˜ S ′′ ( τ ) , ˜ T ′ ( ˜ S ′′ ( τ ) , τ )) , ˜ S ′′ ( τ ) , ˜ T ′ ( ˜ S ′′ ( τ ) , τ )) | τ ∈ U ′ K } ⊂ ˜ U ′ , K )-dimensional submanifold close to { } × U ′ K . Consider the point˜ m := ( ˜ S ′ ( ˜ S ′′ (0) , ˜ T ′ ( ˜ S ′′ (0) , , ˜ S ′′ (0) , ˜ T ′ ( ˜ S ′′ (0) , ∈ ˜ K of ˜ K corresponding to the origin τ = 0 of K . It is a desired fixed point of the ˜ G -action. Step 2.
Now we will apply (a) to the “perturbed” ˜ G -action and its fixed point ˜ m .It follows from the proof of (a), Step 1, that one can provide an algorithm for con-structing a symplectic basis e , f , . . . , e n , f n of V := T m M in which Ω | m , d J ℓ | m and M a have the form (40)–(44). Let us perform the same algorithm for constructing asymplectic basis of ˜ V := T ˜ m M . Similarly to the proof of (a), Step 1, denote by ˜ A ℓ the linearization of the vector field X ˜ J ℓ at the equilibrium point ˜ m , 1 ≤ ℓ ≤ κ e + κ h .Furthermore, denote by ˜ M a the linearization of the symplectomorphism ˜ ρ ( ψ a ) at thefixed point ˜ m , 1 ≤ a ≤ r (where ψ a ∈ Γ are the generators of the group Γ).Since the “perturbed” operators ˜ A ℓ and ˜ M a pairwise commute and are close to the“unperturbed” operators A ℓ and M a (respectively), it follows from Linear Algebra thatthere exists a unique decomposition R n = M s ˜ V s such that each ˜ V s is close to V s and is invariant under each ˜ A ℓ and each ˜ M a . Clearly,the subspaces ˜ V s are symplectic and pairwise skew-orthogonal.Since the “perturbed” flow of each “perturbed” vector field X ˜ J ℓ , 1 ≤ ℓ ≤ κ e , and X i ˜ J ℓ , κ e + 1 ≤ ℓ ≤ κ e + κ h , is 2 π -periodic, the operator ˜ A ℓ is diagonalizable over C and eachits eigenvalue belongs either to i Z for 1 ≤ ℓ ≤ κ e , or to Z for κ e + 1 ≤ ℓ ≤ κ e + κ h .Since the eigenvalues of ˜ A ℓ | ˜ V s are close to the eigenvalues of A ℓ | V s and belong to Z ∪ i Z ,we conclude that Spec( ˜ A ℓ | ˜ V s ) = Spec( A ℓ | V s ) = {± λ ℓs , ± λ ℓs } . Thus ˜ A ℓ and A ℓ havethe same spectrum. Similarly, since the “perturbed” operators ˜ M a , 1 ≤ a ≤ r , are offinite order, the operator ˜ M a is diagonalizable over C and each its eigenvalue belongsto the unit circle of C . Since the eigenvalues of ˜ M a | ˜ V s are close to the eigenvaluesof M a | V s and belong to the unit circle in C , moreover ˜ M | Γ | a = Id, we conclude thatSpec( ˜ M a | ˜ V s ) = Spec( M a | V s ) = { µ ± as , µ ± as } . Thus ˜ M a and M a have the same spectrum.For each subspace ˜ V s from the above decomposition, we define (similarly to the proofof (a), Substep 1b) commuting symplectic linear operators ˜ L as , a ∈ M s , and ˜ B ℓs , ℓ ∈ I s ∪ H s , on ˜ V s , and obtain a unique decomposition˜ V s = M t ˜ V st such that each ˜ V st is close to V st and is invariant under each ˜ A ℓ and each ˜ M a . Clearly,the subspaces ˜ V st are symplectic and pairwise skew-orthogonal.58ollowing the proof of (a), Substeps 1c, 1d, one can provide an algorithm for construct-ing a symplectic basis of each subspace ˜ V st close to the corresponding symplectic basisof V st and satisfying analogues of (55)–(64) w.r.t. ˜ M a , ˜ B ℓs , ˜Ω | ˜ m , d ˜ J ℓ | ˜ m . The resultingsymplectic basis on ˜ V = T ˜ m M brings the symplectic structure ˜Ω | ˜ m , each operator ˜ M a and each quadratic form d ˜ J ℓ | ˜ m to the desired diagonal forms as in (40)–(44). Step 3.
Similarly to the proof of (a), Step 2, one constructs “perturbed” coordinates(˜ x, ˜ y ) on ˜ U that are close to ( x, y ) in C k − − norm, and such that ˜ x j ( ˜ m ) = ˜ y j ( ˜ m ) = 0.Indeed, (˜ x, ˜ y ) = (˜ u, ˜ v ) ◦ ˜ φ − and k d˜ s − d s k C k − + k (˜ u, ˜ v ) − ( u, v ) k C k − + k ˜Ω − Ω k C k − + k (˜ p, ˜ q ) − ( p, q ) k C k − + k ˜ α − α k C k − + k ˜ φ t − φ t k C k − = O ( ε ). Similarly to the proof of(a), Step 3, one proves that these coordinates satisfy an analogue of (67), where theconstants c j , c jab are replaced by some real constants ˜ c j , ˜ c jab close to c j , c jab , respectively.Similarly to the proof of (a), Step 4, one has ˜ c jab = c jab .Lemma 6.2 (b) (and hence Lemma 6.1 (b)) is proved. (a) Similarly to the proof of Lemmata 6.1 and 6.2 (cf. § § κ h = 0 (i.e., all functions J ℓ generate 2 π -periodic flows),all our arguments and constructions literally work both in the real-analytic and C ∞ cases. This will give a proof of Theorem 3.4 (a) too. Step 1.
Under the hypotheses of Theorem 3.10 (a), consider the Hamiltonian actionof the subtorus ( S ) r ⊂ ( S ) r + κ e + κ h on a small neighbourhood U of the orbit O m generated by the functions I , . . . , I r . An element ψ = ( ψ , . . . , ψ r ) ∈ ( S ) r = ( R / π Z ) r acts by the transformation φ ψ I ◦ · · · ◦ φ ψ r I r , where φ tf denotes the Hamiltonian flowgenerated by the function f . DenoteΓ := n ψ ∈ ( S ) r | φ ψ I ◦ · · · ◦ φ ψ r I r ( m ) = m o , so Γ is the isotropy subgroup of the point m (and, hence, of each point of the orbit O m )w.r.t. the ( S ) r -action. It is a closed subgroup of the torus ( S ) r . It is commutative,since the torus ( S ) r is commutative. It is discrete, since the ( S ) r -action is locally freeby assumption. Therefore Γ is a finite commutative group.Since the ( S ) r -action is transitive on the orbit O m , this orbit is a torus O m ≈ ( S ) r / Γ ≈ R r /p − (Γ)where p : R r → ( R / π Z ) r = ( S ) r denotes the projection.59hoose a set of generators γ , . . . , γ r ∈ R r of the lattice p − (Γ) ⊂ R r , so they are basiccycles of the homology group H ( O m ). Then { ψ a = p ( γ a ) ∈ Γ | ≤ a ≤ r } is agenerating set (perhaps, not minimal) of the finite Abelian group Γ.Following the arguments of [30, § S ) r + κ e -invarianttubular neighbourhood U ( O m ) of O m and a normal finite covering \ U ( O m ) of U ( O m )such that the (locally-free) ( S ) r -action on U ( O m ) can be pulled back to a free ( S ) r -action on \ U ( O m ). The symplectic form Ω, the momentum map F and its correspondingsingular Lagrangian fibration, and the action functions I , . . . , I r can be pulled back to \ U ( O m ). We will use b to denote the pull-back: for example, the pull-back of O m is denoted by b O m , and the pull-back of I is denoted by b I . The free action of Γon \ U ( O m ) (this action will be denoted by ρ ) commutes with the free ( S ) r -action.By cancelling out the translations (symplectomorphisms given by the ( S ) r -action arecalled translations), we get another action of Γ on \ U ( O m ) that fixes b O m . We willdenote this latter action by ρ ′ .Take a point b m ∈ b O m (a pullback of m ), a local disk b P of dimension 2 n − r thatintersects b O m transversally at b m and that is preserved by ρ ′ . Denote by ˇ ϕ , . . . , ˇ ϕ r the uniquely defined functions modulo 2 π on \ U ( O m ) that vanish on b P and such that b X I i ( ϕ j ) = δ ij , 1 ≤ i, j ≤ r . Then each local disk { b I = const , . . . , b I r = const } ∩ b P near b m has an induced symplectic structure, induced functions b J , . . . , b J κ e + κ h thatpairwise Poisson commute, and an induced Hamiltonian ( S ) κ e + κ h -action generated by b J , . . . , b J κ e , i b J κ e +1 , . . . , i b J κ e + κ h . Step 2.
Applying Lemma 6.2 (b) in the case with a fixed point, finite symmetry groupΓ, and parameters I , . . . , I r , we can define local functions x , y , . . . , x n − r , y n − r on b P ,such that they form a local symplectic coordinate system on each local disk { b I =const , . . . , b I r = const } ∩ b P , with respect to which the induced Hamiltonian ( S ) κ e + κ h -action is linear and the action ρ ′ of Γ is linear. We extend x , y , . . . , x n − r , y n − r tofunctions on \ U ( O m ) by making them invariant under the action of ( S ) r .It follows from [30, Lemma 4.2] that the symplectic structure b Ω on b P has the form b Ω = r X s =1 d b I s ∧ d( ˇ ϕ s + g s ) + n − r X j =1 d x j ∧ d y j , for some real-analytic functions g s in a neighbourhood of b O m in \ U ( O m ), that areinvariant under the ( S ) r -subaction.Define ϕ s := ˇ ϕ s + g s . Then with respect to the coordinate system ( b I s , ϕ s , x j , y j ), thesymplectic form b Ω has the standard form, the Hamiltonian ( S ) r + κ e + κ h -action is linear,and the free action ρ of Γ is also linear. 60his yields Theorem 3.10 (a), and hence Theorem 3.4 (a).(b) For a “perturbed” system, we follow the same proof as for the “unperturbed” system(cf. Steps 1 and 2 from above), with the only difference that we apply Lemma 6.2 (b) inStep 2 to the “perturbed” system. We can and will assume that k ˜ F C − F C k C k + k ˜Ω C − Ω C k C k + n − < ε (one has similar inequalities for the real objects on M in assumptionsof Theorem 3.4 (b)). Step 3.
Let us use e for all “perturbed” objects. Denote by U ′ ( O m ) ⊂ U ( O m )a tubular neighbourhood invariant under the “unperturbed” ( S ) r + κ e -action, and by e U ( O m ) ⊃ U ′ ( O m ) a tubular neighbourhood invariant under the “perturbed” ( S ) r + κ e -action.Clearly, we can lift the “perturbed” ( S ) r -action to a covering \ e U ( O m ) ⊂ \ U ( O m ).Consider the “perturbed” ( S ) κ e + κ h -action and the “perturbed” Γ-actions ˜ ρ and ˜ ρ ′ on e U ( O m ); they are close to the “unperturbed” ones in C k − − norm. By the samearguments as in Step 2, it follows from Lemma 6.2 (b) that there exists a coordinatesystem ( be I s , ˜ ϕ s , ˜ x j , ˜ y j ) on \ e U ( O m ) close to ( b I s , ϕ s , x j , y j ) in C k − − norm (if k ≥ be Ω has the standard form, the “perturbed”Hamiltonian ( S ) r + κ e + κ h -action is linear, and the “perturbed” action ˜ ρ ′ of Γ is alsolinear, moreover the “perturbed” Γ × ( S ) r -action on the (˜ x, ˜ y )-component is the sameas the “unperturbed” Γ × ( S ) r -action on the ( x, y )-component. In particular, the“perturbed” free action ˜ ρ of Γ is also linear.By above, the coordinate system ( be I s , ˜ ϕ s , ˜ x j , ˜ y j ) is close to ( b I s , ϕ s , x j , y j ), hence the“perturbed” normal form is close to the “unperturbed” one. 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