Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

Abstract

For n 2, we consider V R n the class of germs of real analytic vector fields on R 2 , 0 with zero (n − 1)-jet and nonzero n-jet. We prove, for generic germs of V R n , that the real-formal orbital equivalence implies the real-analytic orbital equivalence, that is, the real-formal orbital rigidity takes place. This is the real analytic version of Voronin's formal orbital rigidity theorem.Keywords Nondicritic vector fields · Real-formal orbital equivalence · Real-analytic orbital equivalence · Real-formal orbital rigidity · Desingularization of singularities · Holonomy group Mathematics Subject Classification (2010) 32S65 · 37F75 · 32M25 · 34A25 · 34C20 · 57R30

Regularisation for Planar vector fields

“…There is a classification theory for singularities in complex systems, and first steps towards a related real classification have recently been made [14]. An interesting consequence of our work are the invariants found of the real dynamics, in particular the degree of smoothness of the block-map.…”

“…The question is addressed more directly in section 6. It is addressed in the literature on holomorphic vector fields on C 2 , see for instance [14]. In this literature r(X) is often referred to as the Camacho-Sad index.…”

“…[19,23], also see [6]. There is a classification theory in the complex, and first steps towards a related real classification have recently been made [14]. Our approach is different and less ambitious from both of these traditional roots, in that we only consider a particular sub-class of singularities for which the notion of regularisability makes sense.…”