Orbital linearization of smooth completely integrable vector fields (2012)

 

24/Apr/2012 15h20:

The first version of this paper is now available:

Orbital linearization of smooth completely integrable vector fields

(Click on the above link for the PDF file)

Abstract: The main purpose of this note is to prove the smooth local orbital linearization theorem for smooth  vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg–Chen isomorphism theorem for  formally-equivalent smooth hyperbolic vector fields. 11 pages. MSC: 37G05, 58K50,37J35

This paper will probabaly be submitted to Nonlinearity.

24/April/2012 at 10:35am

10 pages now. Nearly finished. This small paper will be circa 12 pages.

Completed the proof of the main thm. Have to write a refined thm in the 2-dim case, and then edit the introduction.

24/Apr/2012 at 1:00 am

9 pages. Slow like a snail. TIRED.

Completed the proof in the case without eigenvalue 0. Need to write the reduction of the case with eigenvalue 0 to the case without 0.

22/Apr/2012

6 pages. Tired of this, but have to finish.

Voted this morning in the presidential election of France.

Wrote a simple analog of Ziglin’s lemma.

20/Apr/2012

This is a  (hopefully short) paper that i’m writing up tight now. The main result is that if X is a smooth vector field in n dimension which admits n-1 smooth first integrals, and O is a non-degenerate singular point, then X is smoothly orbitally linearizable near O.

I’m a bit tired now (and still have lots of other things behind schedule) and would rather go swimming or gardening than writing this stuff. (Someone who is underpaid like me is not supposed to write many papers after all ?!). But I have to finish it because it’s results have been already used in another paper. After this one I’ll probably take a loooooong break.

Let’s see if I can finish this one before May. Maybe just 10-12 pages. Will just sketch the proofs instead of writing detailed ones.

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