# Nondegenerate singularities of integrable dynamical systems (2012)

Last updated: 14/March/2012

I have now updated my preprint “Nondegenerate singularities of integrable dynamical systems”. I’ve also changed the title a bit, from “non-Hamiltonian” to “dynamical”.

PDF file: Nondegenerate_V2_2012.pdf

Preprint on Arxiv: http://arxiv.org/abs/1108.3551

ABSTRACT: We give a natural notion of nondegeneracy for singular points of integrable non-Hamiltonian systems, and show that such nondegenerate singularities are locally geometrically linearizable and deformation rigid in the analytic case. We conjecture that the same result also holds in the smooth case,  and prove this conjecture for systems of type $(n,0)$, i.e. $n$ commuting smooth vector fields on a $n$-manifold. 15 pages

Comment: At first I wanted to prove the smooth linearization theorem in the general case (for any type (p,q)). But it seems to be a bit too hard at the moment, so in the new version of the paper I left it as a conjecture, and only proved it for the case of systems of type (n,0). This is the case which is needed in our most recent work on the geometry of Rn actions on n-manifolds.

I just submitted this paper to Ergodic Th Dyn Syst

For the record, from the Journal: 14/March/2012

Your manuscript entitled “Nondegenerate singularities of integrable dynamical systems” has been successfully submitted online for consideration for publication in Ergodic Theory and Dynamical Systems.

Please mention the above manuscript ID in all future correspondence. If there are any changes in your contact details, please log in to ScholarOne Manuscripts at http://mc.manuscriptcentral.com/etds and edit your user information as appropriate. etds@warwick.ac.uk

_____

Old notes (Jan/2012):

I’m currently revising this article, changing its format. I’ll send it to Acta Mathematica Vietnamica (special issue devoted to an international conference held in Vietnam, of which Eva Miranda, Viktor Ginzburg, Do Duc Thai and I will be guest editors). On second thoughts, I’ll send another paper to Acta Vietnamica instead of this one. I’ll send this one to a more “standard” journal, e.g. Ergodic Th Dyn Syst or Crelle or J Diff Eq.

I’m extending the papers to include some results on the smooth case. These smooth results are based on formal normal forms + Chaperon:

Chaperon, Marc Géométrie différentielle et singularités de systèmes dynamiques.  Astérisque No. 138-139 (1986), 440 pp.

http://zung.zetamu.net/2011/04/nondegenerate-singularities-of-non-hamiltonian-integrable-systems/

Chaperon’s result that we use (copied from MathScinet, review by Robbin): he allows $G$ to be an elementary group, which means that it is isomorphic to a direct product $F\times T^l\times {\bf Z}^k\times {\bf R}^m$, where $F$ is finite abelian and $T^l$ is an $l$-dimensional torus. For a linear action of such a group he defines the notion of weak hyperbolicity and proves that a linear action with weakly hyperbolic fixed point can be smoothly linearized if and only if it can be formally linearized. He shows that this result is best possible in that given a linear action which is not weakly hyperbolic there is a nonlinear action with a fixed point having the given linear action as its linear part which can be formally linearized but not smoothly linearized. He also gives a $C^r$ version of the linearization theorem (generalizing the $C^r$ Sternberg theorem) and a very careful explanation of formal linearization. His results do not require that the linear part be semisimple, but this more general result is relegated to an appendix. Special cases of his linearization theorem were known earlier: in addition to Sternberg’s original theorem there is the work of Dumortier and Roussarie who treated the case $G={\bf R}^2$.

### 2 comments to Nondegenerate singularities of integrable dynamical systems (2012)

An old reference which may be related to our stuff:

Cerveau & Lins Neto, Formes tangentes à des actions commutatives, Annales Toulouse, 1984.

Section: The smooth case

Definition of nondegeneracy in the smooth case:
a) The linear part of vector fields forms a nondegenerate family
b) The first integrals are formally independent, i.e. their Taylor series are independent.

(Condition b) is tronger than the condition that the functions are locally independent)

Steps of the Proof of linearizability in the smooth case:
1) The system is formally linearizable (the same proof as in the anlytic case)
2) Arrange so that the elliptic part (real toric degree) is given by smooth vector fields in the system. Make things in a Te-equivariant way.
3) Each vector field can be formally written as
X_i = \sum a_ij Y_j
where a_ij are formal first integrals, and Y_ij are formal vector fields which are simultaneously linearizable nondegenerate
4) Replace a_ij and Y_j by smooth first integrals and vector fields with these Taylor series
5) Y_ij is then generating a hyperbolic Rh X Te action in the sense of Chaperon, and it is formally linearizable. Invoke Chaperon’s theorem –> it is smoothly linearizable.

Remark: the matrix (a_ij) is invertible, i.e.
can write
Y_ij = \sum b_ij X_j
where b_ij are formal first integrals.

In Step 4, work with b_ij instead of working with a_ij. Make b_ij smooth, then Y_ij will be automatically smooth.