On the first integral conjecture of Rene Thom

Jacky Cresson (1, 2) , Aris Daniilidis (3) , Masahiro Shiota (4)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 IMCCE - Institut de Mécanique Céleste et de Calcul des Ephémérides
3 UAB - Departament de Matemàtiques [Barcelona]
4 MATH-NAGOYA - Deapartment of Mathematics

More that half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals. Though somehow unprecise for what concerns the interpretation of the word \textquotedblleft generically\textquotedblright, this statement is ostensibly true and is nowadays commonly accepted. On the other hand, the (few) known formal proofs of Thom's conjecture are all relying to the classical Sard theorem and are thus requiring the technical assumption that first integrals should be of class $C^{k}$ with $k\geq d,$ where $d$ is the dimension of the manifold. In this work, using a recent nonsmooth extension of Sard theorem we establish the validity of Thom's conjecture for locally Lipschitz first integrals, interpreting genericity in the $C^{1}$ sense.

Mots clés: structural stability; first integral; o-minimal structure; Sard theorem

Domaines: Systèmes dynamiques [math.DS]

Jacky Cresson, Aris Daniilidis, Masahiro Shiota. On the first integral conjecture of Rene Thom. 2007. ⟨hal-00182463⟩