Smooth normal forms for integrable hamiltonian systems near a focusfocus singularity
Abstract
We prove that completely integrable systems are normalisable in the category near focusfocus singularities.
1 Introduction and exposition of the result
In his PhD Thesis [4], Eliasson proved some very important results about symplectic linearisation of completely integrable systems near nondegenerate singularities, in the category. However, at that time the socalled elliptic singularities were considered the most important case, and the case of focusfocus singularities was never published. It turned out that focusfocus singularities became crucially important in the last 15 years in the topological, symplectic, and even quantum study of Liouville integrable systems. The aim of this article is to fill in some nontrivial gaps in the original treatment, in order to provide the reader with a complete and robust proof of the fact that a completely integrable system is linearisable near a focusfocus singularity.
Note that in the holomorphic category, the result is already well established (see Vey [7]).
Let us first recall the result.
Thoughout the paper, we shall denote by the quadratic focusfocus model system on equipped with the canonical symplectic form :
(1) 
Let be functions on a 4dimensional symplectic manifold , such that . Here the bracket is the Poisson bracket induced by the symplectic structure. We assume that the differentials and are independent almost everywhere on . Thus is a completely (or “Liouville”) integrable system.
If is a critical point for a function , we denote by the Hessian of at , which we view as a quadratic form on the tangent space .
Definition 1.1.
For an integrable system on a symplectic 4manifold , is a critical point of focusfocus type if

;

the Hessians and are linearly independent;

there exist canonical symplectic coordinates on such that these hessians are linear combinations of the focusfocus quadratic forms and .
Concretely, this definition amounts to requiring the existence of a linear symplectomorphism such that :
From a dynamical viewpoint, this implies that there exists , such that the linearization at of the hamiltonian vector field associated to the hamiltonian has four distinct complex eigenvalues. Thus the Lie algebra spanned by the Hessians of and is generic (open and dense) within 2dimensional abelian Lie algebras of quadratic forms on . This is the nondegeneracy condition as defined by Williamson [9].
The purpose of this paper is to give a complete proof of the following theorem, which was stated in [4].
Theorem 1.2.
Let be a symplectic 4manifold, and an integrable system on ( ie. ). Let be a nondegenerate critical point of of focusfocus type.
Then there exist a local symplectomorphism , defined near the origin, and sending the origin to the point , and a local diffeomorphism , defined near , and sending to , such that
The geometric content of this normal form theorem becomes clear if, given any completely integrable system , one considers the singular foliation defined by the level sets of . Thus, the theorem says that the foliation defined by may, in suitable symplectic coordinates, be made equal to the foliation given by the quadratic part of . With this in mind, the theorem can be viewed as a “symplectic Morse lemma for singular lagrangian foliations”.
The normal form and the normalization are not unique. However, the degrees of liberty are well understood. We cite the following results for the reader’s interest, but they are not used any further in this article.
Theorem 1.3 ([8]).
If is a local symplectomorphism of preserving the focusfocus foliation near the origin, then there exists a unique germ of diffeomorphism such that
(2) 
and is of the form , where and is flat at the origin, with .
Theorem 1.4 ([6]).
If is a local symplectomorphism of preserving the map , then is the composition where is a linear symplectomorphism preserving and is the time1 flow of a smooth function of .
1.1 The formulation in complex variables
Since Theorem 1.2 is a local theorem, we can always invoke the Darboux theorem and formulate it in local coordinates . Throughout the whole paper, we will switch whenever necessary to complex coordinates. But here, the complex coordinates are not defined in the usual way but instead we set and . We set then :
Introducing such notation is justified by the next properties :
Proposition 1.5.

.

The hamiltonian flows of and in these variables are
(3) 
In complex coordinates, the Poisson bracket for realvalued functions is :
2 Birkhoff normal form for focusfocus singularities
In this section we show 1.2 in a formal context (i.e. : with formal series instead of functions), and use the formal result to solve the problem modulo a flat function. For people familiar with Birkhoff normal forms, we compute here simultaneously the Birkhoff normal forms of 2 commuting hamiltonians.
2.1 Formal series
We consider the space of formal power expansions in . We recall that this is a topological space for the adic topology, where is the maximal ideal generated by the variables.
If , we write
For , designates the Taylor expansion of at 0. We have the following definitions
Definition 2.1.
Definition 2.2.
is (note the difference with the previous definition) if one of the 3 equivalent conditions is fulfilled :

and all its derivatives of order at 0 are 0.

There exists a constant such that, in a neighbourhood of the origin,
(4) 
.
The equivalence of the above conditions is a consequence of the Taylor expansion of . Recall, however, that if were not supposed to be smooth at the origin, then the estimates (4) alone would not be sufficient for implying the smoothness of .
Definition 2.3.
is flat at the origin or if for all , it is . Its Taylor expansion is equally zero as a formal series.
Smooth functions can be flat and yet nonzero in a neighbourhood of 0. The most classical example is the function , which is in fact used in some proofs of the following Borel lemma.
Lemma 2.4 ([1]).
Let . Then there exists a function whose Taylor expansion in the variables is . This function is unique modulo the addition of a function that is flat in the variables.
We define the Poisson bracket for formal series the same way we do in the smooth context : for ,
The same notation will designate the smooth and the formal bracket, depending on the context. We have that Poisson bracket commutes with taking Taylor expansions : for formal series ,
(5) 
From this we deduce, if we denote the subspace of homogeneous polynomials of degree in the variables :
We also define . We still have two preliminary lemmas needed before starting the actual proof of the Birkhoff normal form.
Lemma 2.5.
For and a smooth function, we have
for each for which the flow on the lefthand side is defined.
Notice that, since , the righthand side
is always convergent in . In order to prove this lemma, we shall also use the following result which we prove immediately :
Lemma 2.6.
For and , if and , then . Moreover if and depend on a parameter in such a way that their respective estimates (4) are uniform with respect to that parameter, then the corresponding estimates for are uniform as well.
Proof.
Let denote the Euclidean norm in . In view of the estimates (4), given any two neighbourhoods of the origin and , there exist, by assumption, some constants and such that
for and . Since , we may choose such that . So we may write
which proves the result.
∎
Proof of Lemma 2.5.
We write the transport equation for
When integrated, it comes as
(6) 
We now show by induction that for all :
uniformly for .
Initial step :
Induction step :
We suppose, for a given that
uniformly for .
Composing by on the left and by on the right, and then integrating, we get for any ,
The last equality uses that is symplectic and . The induction hypothesis gives
uniformly for , which concludes the induction. ∎
The next lemma will be needed to propagate through the induction the commutation relations.
Lemma 2.7.
For formal series and
Proof.
The Borel lemma gives us whose Taylor expansion is . Since a hamiltonian flow is symplectic, we have
(7) 
We know from Lemma 2.5 that the Taylor expansion of is . Because the Poisson bracket commutes with the Taylor expansion (see equation (5)), we can simply conclude by taking the Taylor expansion in the above equality (7). ∎
2.2 Birkhoff normal form
We prove here a formal Birkhoff normal form for commuting Hamiltonians near a focusfocus singularity. Recall that the focusfocus quadratic forms and were defined in (1).
Theorem 2.8.
Let , , such that
Then there exists , with , and there exist , such that :
(8) 
Proof.
Let’s first start by leftcomposing by and . Let and . , to reduce to the case where
Of course, an element in can be written as a formal series in the variables . We consider a generic monomial . Using (3), it is now easy to compute :
(9)  
For the same reasons, we have :
(10) 
Hence, the action of and is diagonal on this basis. A first consequence of this is the following remark: any formal series such that has the form with (and the reverse statement is obvious). Indeed, let
If , then all must vanish, except when and . For these monomials, and , that is, they are of the form (remember that ). Thus we can write
Specializing to we have . Since , we see that , which establishes our claim.
We are now going to prove the theorem by constructing and by induction.
Initial step :
Taking , the equation (8) is already satisfied modulo , by assumption.
Induction step :
Let and suppose now that the equation is solved modulo : we have then constructed polynomials , such that
With the lemma 2.7, we have that
Hence
Since and are polynomials in , they
must commute :
. On the other hand, so
which implies .
One then looks for , and , with for , such that
(11) 
We have
Here, one needs to expand a noncommutative binomial. We set
for the summand of all possible words formed with occurrences of and occurences of . We have then the formula
What are the terms that we have to keep modulo ? As far
as we only look here to modifications of the total valuation, the
order of the composition between ’s doesn’t matter here. Let’s
examinate closely :
, and
so for any and any ,
Thus
Therefore, equation (11) becomes
and hence
(12) 
Using the notation , , and , we have
We can then solve the first equation by setting,

when :
and (these choices are necessary); 
when :
(necessarily), and (this one is arbitrary).
Similarly, we solve the second equation by setting,

when :
and (necessarily); 
when :
(necessarily), and (arbitrarily).
Of course we need to check that the choices for in (a) and (c) are compatible with each other. This is ensured by the “cross commuting relation” , which reads
∎
Remark 2.9 The relation is a cocycle condition if we look at (12) as a cohomological equation. The relevant complex for this cohomological theory is a deformation complex “à la ChevalleyEilenberg”, described, for instance, in [5].
As a corollary of the Birkhoff normal form, we get a statement concerning smooth functions, up to a flat term.
Lemma 2.10.
Let , where and satisfy the same hypothesis as in the Birkhoff theorem 2.8. Then there exist a symplectomorphism of , tangent to the identity, and a smooth local diffeomorphism , tangent to the matrix such that :
Proof.
We use the notation of (8). Let , be Borel summations of the formal series , and let be a Borel summation of . Let and . Applying Lemma 2.5, we see that the Taylor series of
is flat at the origin.
∎
We can see that Lemma 2.10 gives us the main theorem modulo a flat function. The rest of the paper is devoted to absorbing this flat function.
3 A Morse lemma in the focusfocus case
One of the key ingredients of the proof is a (smooth, but non symplectic) equivariant Morse lemma for commuting functions. In view of the Birkhoff normal form, it is enough to state it for flat perturbations of quadratic forms, as follows.
Theorem 3.1.
Let be functions in such that
Assume for the canonical symplectic form on .
Then there exists a local diffeomorphism of of the form such that
Moreover, we can choose such that the symplectic gradient of for and for are equal, which we can write as
That is to say, preserves the action generated by .
3.1 Proof of the classical flat Morse lemma
In a first step, we will establish the flat Morse lemma without the equivariance property. This result, that we call the “classical“ flat Morse lemma, will be used in the next section to show the equivariant result.
Proof.
Using Moser’s path method, we shall look for as the time1 flow of a timedependent vector field , which should be uniformly flat for .
We define : to satisfy . We want
Differentiating this equation with respect to , we get
So it is enough to find a neighbourood of the origin where one can solve, for , the equation
(13) 
Notice that is flat and .
Let be an open neighborhood of the origin in . Let’s consider as a linear operator from , the space of smooth vector fields, to , the space of pairs of smooth functions. This operator sends flat vector fields to flat functions.
Before going on, we wish to add here a few words concerning flat functions. Assume is contained in the euclidean unit ball. Let denote the vector space of flat functions defined on . For each integer , and each , the quantity
is finite due to (4), and thus the family is an increasing^{(1)}^{(1)}(1) family of norms on . We call the corresponding topology the “local topology at the origin”, as opposed to the usual topology defined by suprema on compact subsets of . Thus, a linear operator from to itself is continuous in the local topology if and only if