# Moscow Mathematical Journal

Volume 20, Issue 2, April–June 2020  pp. 375–404.

Orbital Chen Theorem for Germs of $\mathcal{C}^{\infty}$ Vector Fields with Degenerate Singularity

Authors:  Jessica Jaurez Rosas (1) and Laura Ortiz-Bobadilla (1)
Author institution:(1) Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de la Investigación Científica, Circuito exterior, Ciudad Universitaria, 04510, Ciudad de México, México

Summary:

We consider germs of $\mathcal{C}^{\infty}$ vector fields in $(\mathbb{R}^2, 0)$ with degenerate non-dicritic singularity (having ($n-1$)-jet zero and non-zero $n$-jet) and their corresponding foliations. Under some natural hypothesis we prove that the orbital formal equivalence of any two such vector fields implies their orbital $\mathcal{C}^{\infty}$ equivalence (and thus the $\mathcal{C}^{\infty}$ equivalence of the corresponding foliations). This result generalizes Chen Theorem for foliations defined by generic $\mathcal{C}^{\infty}$ germs of vector fields in $(\mathbb{R}^2, 0)$ having hyperbolic singularity.

2010 Math. Subj. Class. 34C07, 34C05, 34C08.

Keywords: Formal normal forms, foliations, flat vector fields, rigidity.